Engineering Systems Introduction Systems Chapter 1 Engineering Systems Introduction Systems Electrical and Electronic Systems System Inputs and Outputs Physical Quantities and Electrical Signals System Block Diagrams

Introduction Engineering is inherently interdisciplinary 1.1 Introduction Engineering is inherently interdisciplinary Electrical and Electronic Systems represent a major enabling technology important to all engineers and scientist A systems approach to engineering combines: a systematic/top-down approach a systemic approach

Systems A system can be defined as Examples include: 1.2 Systems A system can be defined as Any closed volume for which all the inputs and output are known Examples include: an engine management system an automotive system a transportation system an ecosystem Inputs and outputs will reflect the nature of the system

Electrical and Electronic Systems 1.3 Electrical and Electronic Systems Basic functions include elements concerned with the manipulation of electrical energy Common functions are: generation transmission of communication control or processing utilisation storage

An example – a power distribution system

System examples electrical and electronic systems often fall within a range of categories, such as those responsible for: power generation and distribution monitoring of some equipment or process control of some equipment or process signal processing communications

System Inputs and Outputs 1.4 System Inputs and Outputs Systems may often be described simply by their inputs, their output and the relationship between them

Nature of inputs/outputs will depend on where we draw our system boundaries. For example:

By changing the system boundary we change the nature of the inputs and outputs

We will look at sensors and actuators in more detail in later lectures Components that interact with the outside world are termed sensors and actuators in the previous example the microphone represents a sensor in the previous example the loudspeaker represents an actuator We will look at sensors and actuators in more detail in later lectures

Physical Quantities & Electrical Signals 1.5 Physical Quantities & Electrical Signals The world about us is characterised by a number of physical properties or quantities e.g. temperature, pressure, humidity, etc. Physical quantities may be continuous or discrete. Continuous quantities change smoothly and can take an infinite number of values Discrete quantities change abruptly from one value to another. most real-world quantities are continuous many man-made quantities are discrete

It is often convenient to represent physical quantities by electrical signals. These can also be continuous or discrete Continuous signals are often described as analogue

Discrete signals are often described as digital signals Many digital signals take only two values and are referred to as binary signals

1.6 System Block Diagrams It is often convenient to represent complex arrangements by a simplified block diagram

In an electrical system a flow of energy requires a circuit - a system with a single input and a single output is shown below this shows the input circuit and the output circuit the sensor represents the source the actuator represents the load

We often divide complex circuits into subsystems or modules – as shown below the output of each module represents a source for the following section the input of each module represents a load to the previous section

Key Points Engineering is inherently interdisciplinary Engineers often adopt a ‘systems approach’ Systems may be defined by their inputs, their outputs and the relationship between them Systems interact with the world using sensors and actuators Physical quantities can be either continuous or discrete Physical quantities are often represented by signals Complex systems are often represented by block diagrams

Basic Electric Circuits & Components Chapter 2 Basic Electric Circuits & Components Introduction SI Units and Common Prefixes Electrical Circuits Direct Currents and Alternating Currents Resistors, Capacitors and Inductors Ohm’s and Kirchhoff’s Laws Power Dissipation in Resistors Resistors in Series and Parallel Resistive Potential Dividers Sinusoidal Quantities Circuit Symbols

Introduction This lecture outlines the basics of Electrical Circuits 2.1 Introduction This lecture outlines the basics of Electrical Circuits For most students much of this will be familiar this lecture can be seen as a revision session for this material If there are any topics that you are unsure of (or that are new to you) you should get to grips with this material before the next lecture the following lectures will assume a basic understanding of these topics We will return to look at several of these topics in more detail in later lectures

SI Units 2.2 Quantity Quantity symbol Unit Unit symbol Capacitance C Farad F Charge Q Coulomb Current I Ampere A Electromotive force E Volt V Frequency f Hertz Hz Inductance (self) L Henry H Period T Second s Potential difference Power P Watt W Resistance R Ohm Ω Temperature Kelvin K Time t

Common Prefixes 2.3 Prefix Name Meaning (multiply by) T tera 1012 G giga 109 M mega 106 k kilo 103 m milli 10-3  micro 10-6 n nano 10-9 p pico 10-12

Electrical Circuits Electric charge Electric current 2.4 Electrical Circuits Electric charge an amount of electrical energy can be positive or negative Electric current a flow of electrical charge, often a flow of electrons conventional current is in the opposite direction to a flow of electrons Current flow in a circuit a sustained current needs a complete circuit also requires a stimulus to cause the charge to flow

Electromotive force and potential difference the stimulus that causes a current to flow is an e.m.f. this represents the energy introduced into the circuit by a battery or generator this results in an electric potential at each point in the circuit between any two points in the circuit there may exist a potential difference both e.m.f. and potential difference are measured in volts

A simple circuit A water-based analogy

Voltage reference points all potentials within a circuit must be measured with respect to some other point we often measure voltages with respect to a zero volt reference called the ground or earth

Representing voltages in circuit diagrams conventions vary around the world we normally use an arrow, which is taken to represent the voltage on the head with respect to the tail labels represent voltages with respect to earth

Direct Current and Alternating Current 2.5 Direct Current and Alternating Current Currents in electrical circuits may be constant or may vary with time When currents vary with time they may be unidirectional or alternating When the current flowing in a conductor always flows in the same direction this is direct current (DC) When the direction of the current periodically changes this is alternating current (AC)

Resistors, Capacitors and Inductors 2.6 Resistors, Capacitors and Inductors Resistors provide resistance they oppose the flow of electricity measured in Ohms () Capacitors provide capacitance they store energy in an electric field measured in Farads (F) Inductors provide inductance they store energy in a magnetic field measured in Henry (H) We will look at each component in later lectures

2.7 Ohm’s Law The current flowing in a conductor is directly proportional to the applied voltage V and inversely proportional to its resistance R V = IR I = V/R R = V/I

Kirchhoff’s Current Law 2.8 Kirchhoff’s Current Law At any instant the algebraic sum of the currents flowing into any junction in a circuit is zero For example I1 – I2 – I3 = 0 I2 = I1 – I3 = 10 – 3 = 7 A

Kirchhoff’s Voltage Law 2.8 Kirchhoff’s Voltage Law At any instant the algebraic sum of the voltages around any loop in a circuit is zero For example E – V1 – V2 = 0 V1 = E – V2 = 12 – 7 = 5 V

Power Dissipation in Resistors 2.9 Power Dissipation in Resistors The instantaneous power dissipation P of a resistor is given by the product of the voltage across it and the current passing through it. Combining this result with Ohm’s law gives: P = VI P = I2R P = V2/R

Resistors in Series and Parallel 2.10 & 2.11 Resistors in Series and Parallel Series R = R1 + R2 + R3 Parallel

Resistive Potential Dividers 2.12 Resistive Potential Dividers General case

Example

Example

Sinusoidal Quantities 2.13 Sinusoidal Quantities Length of time between corresponding points in successive cycles is the period T Number of cycles per second is the frequency f f = 1/T

2.14 Circuit Symbols

Key Points Understanding the next few lectures of this course relies on understanding the various topics covered in this session A clear understanding of the concepts of voltage and current is essential Ohm’s Law and Kirchhoff’s Laws are used extensively in later lectures Experience shows that students have most problems with potential dividers – a topic that is used widely in the next few lectures You are advised to make sure you are happy with this material now

Sensors Introduction Describing Sensor Performance Temperature Sensors Chapter 3 Sensors Introduction Describing Sensor Performance Temperature Sensors Light Sensors Force Sensors Displacement Sensors Motion Sensors Sound Sensors Sensor Interfacing

3.1 Introduction To be useful, systems must interact with their environment. To do this they use sensors and actuators Sensors and actuators are examples of transducers A transducer is a device that converts one physical quantity into another examples include: a mercury-in-glass thermometer (converts temperature into displacement of a column of mercury) a microphone (converts sound into an electrical signal). We will look at sensors in this lecture and at actuators in the next lecture

Almost any physical property of a material that changes in response to some excitation can be used to produce a sensor widely used sensors include those that are: resistive inductive capacitive piezoelectric photoresistive elastic thermal. in this lecture we will look at several examples

Describing Sensor Performance 3.2 Describing Sensor Performance Range maximum and minimum values that can be measured Resolution or discrimination smallest discernible change in the measured value Error difference between the measured and actual values random errors systematic errors Accuracy, inaccuracy, uncertainty accuracy is a measure of the maximum expected error

Precision a measure of the lack of random errors (scatter)

Linearity Sensitivity maximum deviation from a ‘straight-line’ response normally expressed as a percentage of the full-scale value Sensitivity a measure of the change produced at the output for a given change in the quantity being measured

Temperature sensors Resistive thermometers 3.3 Temperature sensors Resistive thermometers typical devices use platinum wire (such a device is called a platinum resistance thermometers or PRT) linear but has poor sensitivity A typical PRT element A sheathed PRT

Thermistors use materials with a high thermal coefficient of resistance sensitive but highly non-linear A typical disc thermistor A threaded thermistor

pn junctions a semiconductor device with the properties of a diode (we will consider semiconductors and diodes later) inexpensive, linear and easy to use limited temperature range (perhaps -50C to 150 C) due to nature of semiconductor material pn-junction sensor

Light Sensors Photovoltaic 3.4 Light Sensors Photovoltaic light falling on a pn-junction can be used to generate electricity from light energy (as in a solar cell) small devices used as sensors are called photodiodes fast acting, but the voltage produced is not linearly related to light intensity A typical photodiode

Photoconductive such devices do not produce electricity, but simply change their resistance photodiode (as described earlier) can be used in this way to produce a linear device phototransistors act like photodiodes but with greater sensitivity light-dependent resistors (LDRs) are slow, but respond like the human eye A light-dependent resistor (LDR)

Force Sensors Strain gauge 3.5 Force Sensors Strain gauge stretching in one direction increases the resistance of the device, while stretching in the other direction has little effect can be bonded to a surface to measure strain used within load cells and pressure sensors Direction of sensitivity A strain gauge

Displacement Sensors Potentiometers 3.6 Displacement Sensors Potentiometers resistive potentiometers are one of the most widely used forms of position sensor can be angular or linear consists of a length of resistive material with a sliding contact onto the resistive track when used as a position transducer a potential is placed across the two end terminals, the voltage on the sliding contact is then proportional to its position an inexpensive and easy to use sensor

Inductive proximity sensors coil inductance is greatly affected by the presence of ferromagnetic materials here the proximity of a ferromagnetic plate is determined by measuring the inductance of a coil we will look at inductance in later lectures Inductive proximity sensors

Switches simplest form of digital displacement sensor many forms: lever or push-rod operated microswitches; float switches; pressure switches; etc. A limit switch A float switch

A reflective opto-switch Opto-switches consist of a light source and a light sensor within a single unit 2 common forms are the reflective and slotted types A reflective opto-switch A slotted opto-switch

Absolute position encoders a pattern of light and dark strips is printed on to a strip and is detected by a sensor that moves along it the pattern takes the form of a series of lines as shown below it is arranged so that the combination is unique at each point sensor is an array of photodiodes

Incremental position encoder uses a single line that alternates black/white two slightly offset sensors produce outputs as shown below detects motion in either direction, pulses are counted to determine absolute position (which must be initially reset)

Other counting techniques several methods use counting to determine position two examples are given below Opto-switch sensor Inductive sensor

3.7 Motion Sensors Motion sensors measure quantities such as velocity and acceleration can be obtained by differentiating displacement differentiation tends to amplify high-frequency noise Alternatively can be measured directly some sensors give velocity directly e.g. measuring frequency of pulses in the counting techniques described earlier gives speed rather than position some sensors give acceleration directly e.g. accelerometers usually measure the force on a mass

Sound Sensors Microphones a number of forms are available 3.8 Sound Sensors Microphones a number of forms are available e.g. carbon (resistive), capacitive, piezoelectric and moving-coil microphones moving-coil devices use a magnet and a coil attached to a diaphragm – we will discuss electromagnetism later

Sensor Interfacing Resistive devices can be very simple 3.9 Sensor Interfacing Resistive devices can be very simple e.g. in a potentiometer, with a fixed voltage across the outer terminals, the voltage on the third is directly related to position where the resistance of the device changes with the quantity being measured, this change can be converted into a voltage signal using a potential divider – as shown the output of this arrangement is not linearly related to the change in resistance

Switches switch interfacing is also simple can use a single resistor as below to produce a voltage output all mechanical switches suffer from switch bounce

Capacitive and inductive sensors sensors that change their capacitance or inductance in response to external influences normally require the use of alternating current (AC) circuitry such circuits need not be complicated we will consider AC circuits in later lectures

Key Points A wide range of sensors is available Some sensors produce an output voltage related to the measured quantity and therefore supply power Other devices simply change their physical properties Some sensors produce an output that is linearly related to the quantity being measured, others do not Interfacing may be required to produce signals in the correct form

Actuators Introduction Heat Actuators Light Actuators Chapter 4 Actuators Introduction Heat Actuators Light Actuators Force, Displacement and Motion Actuators Sound Actuators Actuator Interfacing

4.1 Introduction In order to be useful an electrical or electronic system must be able to affect its external environment. This is done through the use of one of more actuators As with sensors, actuators are a form of transducer which convert one physical quantity into another Here we are interested in actuators that take electrical signals from our system and from them vary some external physical quantity

Heat Actuators Most heat actuators are simple resistive heaters 4.2 Heat Actuators Most heat actuators are simple resistive heaters For applications requiring a few watts ordinary resistors of an appropriate power rating can be used For higher power applications there are a range of heating cables and heating elements available

4.3 Light Actuators For general illumination it is normal to use conventional incandescent light bulbs or fluorescent lamps power ratings range from a fraction of a watt to perhaps hundreds of watts easy to use but relatively slow in operation unsuitable for signalling and communication applications

Light-emitting diodes (LEDs) produce light when electricity is passed though them a range of semiconductor materials can be used to produce light of different colours can be used individually or in multiple-segment devices such as the seven-segment display shown here LED seven-segment displays

Liquid crystal displays consist of 2 sheets of polarised glass with a thin layer of oily liquid sandwiched between them an electric field rotates the polarization of the liquid making it opaque can be formed into multi- element displays (such as 7-segment displays) can also be formed into a matrix display to display any character or image A custom LCD display

Fibre-optic communication used for long-distance communication removes the effects of ambient light fibre-optic cables can be made of: optical polymer inexpensive and robust high attenuation, therefore short range (up to about 20 metres) glass much lower attenuation allowing use up to hundreds of kilometres more expensive than polymer fibres light source would often be a laser diode

Force, Displacement & Motion Actuators 4.4 Force, Displacement & Motion Actuators Solenoids basically a coil and a ferromagnetic ‘slug’ when energised the slug is attracted into the coil force is proportional to current can produce a force, a displacement or motion can be linear or angular often used in an ON/OFF mode Small linear solenoids

Meters moving-iron moving-coil effectively a rotary solenoid + spring can measure DC or AC moving-coil most common form deflection proportional to average value of current f.s.d. typically 50 A – 1 mA use in voltmeters and ammeters is discussed later Moving-coil meters

Motors three broad classes AC motors primarily used in high-power applications DC motors used in precision position-control applications Stepper motors a digital actuator used in position control applications we will look at AC and DC motors in later lectures

Stepper motors a central rotor surrounded by a number of coils (or windings) opposite pairs of coils are energised in turn this ‘drags’ the rotor round one ‘step’ at a time speed proportional to frequency typical motor might require 48-200 steps per revolution

Stepper-motor current waveforms A typical stepper-motor

Sound Actuators Speakers Ultrasonic transducers 4.5 Sound Actuators Speakers usually use a permanent magnet and a movable coil connected to a diaphragm input signals produce current in the coil causing it to move with respect to the magnet Ultrasonic transducers at high frequencies speakers are often replaced by piezoelectric actuators operate over a narrow frequency range

Actuator Interfacing Resistive devices 4.6 Actuator Interfacing Resistive devices interfacing involves controlling the power in the device in a resistive actuator, power is related to the voltage for high-power devices the problem is in delivering sufficient power to drive the actuator high-power electronic circuits will be considered later high-power actuators are often controlled in an ON/OFF manner these techniques use electrically operated switches discussed in later lectures

Capacitive and inductive devices many actuators are capacitive or inductive (such as motors and solenoids) these create particular problems – particularly when using switching techniques we will return to look at these problems when we have considered capacitor and inductors in more detail

Key Points Systems affect their environment using actuators Most actuators take power from their inputs in order to deliver power at their outputs Some devices consume only a fraction of a watt while others consume hundreds or perhaps thousands of watts In most cases the efficiency of the energy conversion is less than 100%, in many cases it is much less Some circuits resemble resistive loads while others have considerable capacitance or inductance. The ease or difficulty of driving actuators varies with their characteristics.

Signals and Data Transmission Chapter 5 Signals and Data Transmission Introduction Analogue Signals Digital Signals Signal Properties System Limitations Modulation Demodulation Multiplexing Distortion and Noise

Introduction Earlier we looked at physical quantities 5.1 Introduction Earlier we looked at physical quantities e.g. temperature, pressure, humidity It is often convenient to represent these by signals sensors produce signal from physical quantities actuators take signals and affect external quantities Signals can be analogue or digital in nature In this lecture we will look at examples of electrical signal of various forms

5.2 Analogue Signals Analogue signals are free from discontinuities and can take an infinite number of values They have been used since the 19th century e.g. in telephone and wireless communication Perhaps the simplest form is where a voltage represents the amplitude of some physical quantity

Digital Signals Most applications use two values (binary signals) 5.3 Digital Signals Most applications use two values (binary signals) The two states are often represented by 2 voltages

Digital signals have also been used since the 19th century e.g. in telegraphy using Morse code

A single digital signal can represent the state of a single binary quantity or device the information represented by such a signal can be represented by a single binary variable or binary digit a binary digit is usually referred to as a bit

Bits can be grouped together to form digital words these can represent several signals or many values

A digital variable of n bits can take 2n values an 8-bit word can take 28 = 256 values a 16-bit word can take 216 = 65,536 values a 32-bit word can take 232 = 4,294,967,296 values Therefore an 8-bit word gives a resolution of 1 part in 256 or 0.39% a 16-bit word gives a resolution of 1 part in 65,536 or 0.0015% a 32-bit word gives a resolution of 1 part in 4,294,967,296 or 0.000000023%

Digital words can be communicated in either parallel or serial form

5.4 Signal Properties Signals may be unipolar or bipolar

Signals vary in their frequency range e.g. the human voice has a range from 50 Hz – 7 kHz The range can be described by a frequency spectrum shows the magnitude of the frequency components A frequency spectrum for a speech waveform

The difference between the highest and lowest frequencies in a signal is termed its bandwidth For example: a typical human voice might have a frequency range from about 50 Hz to about 7 kHz. Therefore Bandwidth = 7 kHz – 50 Hz = 6.95 kHz  7 kHz

5.5 System Limitations All systems impose restrictions on the signals that can be used with them e.g. limits to the magnitudes of the inputs and outputs limits to the range of frequencies that can be used usable frequency range is determined by the frequency response of the system

Problems can occur if the frequency response of the system is not appropriate Can use modulation to overcome such problems

5.6 Modulation Modulation can be used to change the frequency range of a signal There are many forms Common forms include: amplitude modulation (AM) frequency modulation (FM both forms are used in radio broadcasting

Amplitude modulation the amplitude of a carrier wave is varied (or modulated) to represent the magnitude of the input signal many forms of modulation shown here is full amplitude modulation (as used in medium wave transmission) here the envelope of the waveform represents the input signal

Frequency modulation here the amplitude of the carrier is constant but the frequency is varied to represent the input signal

Both AM and FM change the frequency range and frequency spectrum of the signal the example shown here is of full-AM

Amplitude and frequency modulation can also be applied to digital signal used in computer modems

5.7 Demodulation The inverse process of recovering the original signal is called demodulation.

5.8 Multiplexing The ability to shift the frequency range of a signal using modulation allows us to make more effective use of the bandwidth of communication channels Many forms of multiplexing are used including: frequency-division multiplexing time-division multiplexing After transmission the process is reversed using demultiplexing to recover the original signals

Frequency-division multiplexing

5.9 Distortion and Noise All systems distort electrical signal to some extent examples include clipping, crossover distortion and harmonic distortion Distortion is systematic and is repeatable

All systems also add noise to the signals that pass through them Unlike distortion, noise is random and not repeatable Noise can often be removed from digital signals but this is often impossible with analogue signals

Key Points Electrical signals can take many forms and can be analogue or digital A simple analogue form is where a voltage is proportional to the amplitude of a quantity being represented A simple digital form is where the voltage takes one of two values to represent the two states of a quantity Modulation is often used to change the frequency range Multiplexing can be used to combine signals All electrical circuits add distortion and noise to signals

Amplification Introduction Electronic Amplifiers Sources and Loads Chapter 6 Amplification Introduction Electronic Amplifiers Sources and Loads Equivalent Circuits of Amplifiers Output Power Power Gain Frequency Response and Bandwidth Differential Amplifiers Simple Amplifiers

6.1 Introduction Amplification is one of the most common processing functions Amplification means making things bigger Attenuation means making things smaller There are many non-electronic forms of amplification

Non-electronic amplifiers Levers Example shown on the right is a force amplifier, but a displacement attenuator Reversing the input and output would produce a force attenuator but a displacement amplifier This is an example of a non-inverting amplifier (since the input and output are in the same direction) A lever arrangement

Non-electronic amplifiers Pulleys Example shown right is a force amplifier, but a displacement attenuator This is an example of an inverting amplifier (since the input and output are in opposite directions) but other pulley arrangements can be non-inverting A pulley arrangement

Passive and active amplifiers levers and pulleys are examples of passive amplifiers since they have no external energy source in such amplifiers the power delivered at the output must be less than (or equal to) that absorbed at the input some amplifiers are not passive but are active amplifiers in that they have an external source of power in such amplifiers the output can deliver more power than is absorbed at the input

Non-electronic active amplifiers an example is the torque amplifier shown here A torque amplifier

Electronic Amplifiers 6.2 Electronic Amplifiers Can be passive (e.g. a transformer) but most are active We will concentrate on active electronic amplifiers take power from a power supply amplification described by gain Circuit symbol

6.3 Sources and Loads An ideal voltage amplifier would produce an output determined only by the input voltage and its gain irrespective of the nature of the source and the load in real amplifiers this is not the case the output voltage is affected by loading

Modelling the input of an amplifier the input can often be adequately modelled by a simple resistor the input resistance

Modelling the output of a circuit all real voltage sources have an output resistance for example, a battery can be represented by an ideal voltage source and a series resistance representing its output resistance

Modelling the output of an amplifier similarly, the output of an amplifier can be modelled by an ideal voltage source and an output resistance this is an example of a Thévenin equivalent circuit (we will return to such circuits later)

Modelling the gain of an amplifier can be modelled by a controlled voltage source the voltage produced by the source is determined by the input voltage to the circuit

Equivalent Circuits of Amplifiers 6.4 Equivalent Circuits of Amplifiers Having modelled the input, the output and the gain, we can now model the entire amplifier

The use of an equivalent circuit (see Example 6.1 in the course text): Example: An amplifier has a voltage gain of 10, an input resistance of 1 k and an output resistance of 10 . The amplifier is connected to a sensor that produces a voltage of 2 V and has an output resistance of 100 , and to a load of 50 . What will be the output voltage of the amplifier (that is the voltage across the load resistance)?

We start by constructing an equivalent circuit of the amplifier, the source and the load

From this we can calculate the output voltage:

The voltage gain of the circuit in the previous example is given by: note that this is considerably less than the stated gain of the amplifier (which is 10) this is due to loading effects the gain of the amplifier in isolation is its unloaded voltage gain

An ideal voltage amplifier would not suffer from loading it would have Ri =  and Ro = 0 consider the effect on the previous example

If Ri = , then Therefore the effects of loading are removed (see Example 6.3)

6.5 Output Power The output power Po is that dissipated in the load resistor Power transfer is at a maximum when RL = Ro maximum power theorem choosing a load to maximize power transfer is called matching often voltage gain is more important than power transfer

6.6 Power Gain Power gain is the ratio of the power supplied to the load to that absorbed at the input For numerical example see Example 6.5 in set text Gain often given in decibels

Sample gains expressed in dBs Using dBs simplifies calculation in cascaded circuits Power gain Decibels (dBs) 100 20 10 1 Power gain Decibels (dBs) 0.5 -3 0.1 -10 0.01 -20

Power gain is related to voltage gain If R1 = R2 This expression is often used even when R1  R2 see Example 6.7 and Example 6.8 in the course text

Frequency Response and Bandwidth 6.7 Frequency Response and Bandwidth All real amplifiers have limits to the range of frequencies over which they can be used The gain of a circuit in its normal operating range is termed its mid-band gain The gain of all amplifiers falls at high frequencies characteristic defined by the half-power point gain falls to 1/2 = 0.707 times the mid-band gain this occurs at the cut-off frequency In some amplifiers gain also falls at low frequencies these are AC coupled amplifiers

(a) shows an AC coupled amplifier (b) shows the same amplifier – with gain in dBs (c) shows a DC coupled amplifier – the gain is constant down to DC

The bandwidth is the difference between the upper and lower cut-off frequencies … … or the difference between the upper-cut-off frequency and zero in a DC coupled amplifier

Differential Amplifiers 6.8 Differential Amplifiers Differential amplifiers have two inputs and amplify the voltage difference between them inputs are called the non-inverting input (labelled +) and the inverting input (labelled –)

An example of the use of a differential amplifier

Equivalent circuit of a differential amplifier one of the commonest forms of differential amplifier is the operational amplifier – discussed in later lectures

6.9 Simple Amplifiers Operational amplifiers are relatively complex circuits Amplifiers can also be formed using a ‘control device’ circuit is similar to a potential divider with one resistor replaced with a ‘control device’ typically a transistor A potential divider A simple amplifier

Key Points Amplification forms part of most electronic systems Amplifiers may be active or passive Equivalent circuits are useful when investigating the interaction between circuits Amplifier gains are often measured in decibels (dBs) The gain of all amplifiers falls at high frequencies The gain of some amplifiers falls at low frequencies Differential amplifiers take as their input the difference between two input signals Some amplifiers are very simple in construction

Control and Feedback Introduction Open-loop and Closed-loop Systems Chapter 7 Control and Feedback Introduction Open-loop and Closed-loop Systems Automatic Control Systems Feedback Systems Negative Feedback The Effects of Negative Feedback Negative Feedback – A Summary

7.1 Introduction Earlier we identified control as one of the basic functions performed by many systems often involves regulation or command Invariably, the goal is to determine the value or state of some physical quantity and often to maintain it at that value, despite variations in the system or the environment

Open-loop and Closed-loop Systems 7.2 Open-loop and Closed-loop Systems Simple control is often open-loop user has a goal and selects an input to a system to try to achieve this

More sophisticated arrangements are closed-loop user inputs the goal to the system

Automatic Control Systems 7.3 Automatic Control Systems Examples of automatic control systems: temperature control using a room heater

Examples of automatic control systems: Cruise control in a car

Examples of automatic control systems: Position control in a human limb

Examples of automatic control systems: Level control in a dam

7.4 Feedback Systems A generalised feedback system

By inspection of diagram we can add values or rearranging

This the transfer function of the arrangement Terminology: Thus This the transfer function of the arrangement Terminology: A is also known as the open-loop gain G is the overall or closed-loop gain

Effects of the product AB If AB is negative If AB is negative and less than 1, (1 + AB) < 1 In this situation G > A and we have positive feedback If AB is positive If AB is positive then (1 + AB) > 1 In this situation G < A and we have negative feedback If AB is positive and AB >>1 - gain is independent of the gain of the forward path A

Negative Feedback Negative feedback can be applied in many ways 7.5 Negative Feedback Negative feedback can be applied in many ways Xi and Xo could be temperatures, pressures, etc. here we are mainly interested in voltages and currents Particularly important in overcoming variability all active devices suffer from variability their gain and other characteristics vary with temperature and between devices we noted above that using negative feedback we can produce an arrangement where the gain is independent of the gain of the forward path this gives us a way of overcoming problems of variability

Consider the following example (Example 7.1 in text) We will base our design on our standard feedback arrangement Example: Design an arrangement with a stable voltage gain of 100 using a high-gain active amplifier. Determine the effect on the overall gain of the circuit if the voltage gain of the active amplifier varies from 100,000 to 200,000.

We will use our active amplifier for A and a stable feedback arrangement for B Since we require an overall gain of 100 so we will use B = 1/100 or 0.01

Now consider the gain of the circuit when the gain of the active amplifier A is 100,000

Now consider the gain of the circuit when the gain of the active amplifier A is 200,000

However, it does require that B is stable Note that a change in the gain of the active amplifier of 100% causes a change in the overall gain of just 0.05 % Thus the use of negative feedback makes the gain largely independent of the gain of the active amplifier However, it does require that B is stable fortunately, B can be based on stable passive components

Implementing the passive feedback path to get an overall gain of greater than 1 requires a feedback gain B of less than 1 in the previous example the value of B is 0.01 this can be achieved using a simple potential divider

Thus we can implement our feedback arrangement using an active amplifier and a passive feedback network to produce a stable amplifier The arrangement on the right has a gain of 100 … … but how do we implement the subtractor?

A differential amplifier is effectively an active amplifier combined with a subtractor. A common form is the operational amplifier or op-amp The arrangement on the right has a gain of 100.

In this circuit the gain is determined by the passive components and we do not need to know the gain of the op-amp however, earlier we assumed that AB >> 1 that is, that A >> 1/B that is, open-loop gain >> closed-loop gain therefore, the gain of the circuit must be much less than the gain of the op-amp see Example 7.2 in the course text

The Effects of Negative Feedback 7.6 The Effects of Negative Feedback Effects on Gain negative feedback produces a gain given by there, feedback reduces the gain by a factor of 1 + AB this is the price we pay for the beneficial effects of negative feedback

Effects on frequency response from earlier lectures we know that all amplifiers have a limited frequency response and bandwidth with feedback we make the overall gain largely independent of the gain of the active amplifier this has the effect of increasing the bandwidth, since the gain of the feedback amplifier remains constant as the gain of the active amplifier falls however, when the open-loop gain is no longer much greater than the closed-loop gain the overall gain falls

therefore the bandwidth increases as the gain is reduced with feedback in some cases the gain x bandwidth = constant

Effects on input and output resistance negative feedback can either increase or decrease the input or output resistance depending on how it is used. if the output voltage is fed back this tends to make the output voltage more stable by decreasing the output resistance if the output current is fed back this tends to make the output current more stable by increasing the output resistance if a voltage related to the output is subtracted from the input voltage this increases the input resistance if a current related to the output is subtracted from the input current this decreases the input resistance the factor by which the resistance changes is (1 + AB) we will apply this to op-amps in a later lecture

Effects on distortion and noise many forms of distortion are caused by a non-linear amplitude response that is, the gain varies with the amplitude of the signal since feedback tends to stabilise the gain it also tends to reduce distortion - often by a factor of (1 + AB) noise produced within an amplifier is also reduced by negative feedback – again by a factor of (1 + AB) note that noise already corrupting the input signal is not reduced in this way – this is amplified along with the signal

Negative Feedback – A Summary 7.7 Negative Feedback – A Summary All negative feedback systems share some properties They tend to maintain their output independent of variations in the forward path or in the environment They require a forward path gain that is greater than that which would be necessary to achieve the required output in the absence of feedback The overall behaviour of the system is determined by the nature of the feedback path Unfortunately, negative feedback does have implications for the stability of circuits – this is discussed in later lectures

Key Points Feedback is used in almost all automatic control systems Feedback can be either negative or positive If the gain of the forward path is A, the gain of the feedback path is B and the feedback is subtracted from the input then If AB is positive and much greater than 1, then G  1/B Negative feedback can be used to overcome problems of variability within active amplifiers Negative feedback can be used to increase bandwidth, and to improve other circuit characteristics.

Operational Amplifiers Chapter 8 Operational Amplifiers Introduction An Ideal Operational Amplifier Basic Operational Amplifier Circuits Other Useful Circuits Real Operational Amplifiers Selecting Component Values Effects of Feedback on Op-amp Circuits

8.1 Introduction Operational amplifiers (op-amps) are among the most widely used building blocks in electronics they are integrated circuits (ICs) often DIL or SMT

A single package will often contain several op-amps

An Ideal Operational Amplifier 8.2 An Ideal Operational Amplifier An ideal op-amp would be an ideal voltage amplifier and would have: Av = , Ri =  and Ro = 0 Equivalent circuit of an ideal op-amp

Basic Operational Amplifier Circuits 8.3 Basic Operational Amplifier Circuits Inverting and non-inverting amplifiers

In analysing these we normally assume the use of ideal op-amps When looking at feedback we derived the circuit of an amplifier from ‘first principles’ Normally we use standard ‘cookbook’ circuits and select component values to suit our needs In analysing these we normally assume the use of ideal op-amps in demanding applications we may need to investigate the appropriateness of this assumption the use of ideal components makes the analysis of these circuits very straightforward

A non-inverting amplifier Analysis Since the gain is assumed infinite, if Vo is finite then the input voltage must be zero. Hence Since the input resistance of the op-amp is  and hence, since V– = V+ = Vi and

Example (see Example 8.1 in the course text) Design a non-inverting amplifier with a gain of 25 From above If G = 25 then Therefore choose R2 = 1 k and R1 = 24 k (choice of values will be discussed later)

An inverting amplifier Analysis Since the gain is assumed infinite, if Vo is finite the input voltage must be zero. Hence Since the input resistance of the op-amp is  its input current must be zero, and hence Now

Analysis (continued) Therefore, since I1 = -I2 or, rearranging Here V– is held at zero volts by the operation of the circuit, hence the circuit is known as a virtual earth circuit

Example (see Example 8.2 in the course text) Design an inverting amplifier with a gain of -25 From above If G = -25 then Therefore choose R2 = 1 k and R1 = 25 k (we will consider the choice of values later)

8.4 Other Useful Circuits In addition to simple amplifiers op-amps can also be used in a range of other circuits The next few slides show a few examples of op-amp circuits for a range of purposes The analysis of these circuits is similar to that of the non-inverting and inverting amplifiers but (in most cases) this is not included here For more details of these circuits see the relevant section of the course text (as shown on the slide)

A unity gain buffer amplifier 8.4.1 A unity gain buffer amplifier Analysis This is a special case of the non-inverting amplifier with R1 = 0 and R2 =  Hence Thus the circuit has a gain of unity At first sight this might not seem like a very useful circuit, however it has a high input resistance and a low output resistance and is therefore useful as a buffer amplifier

A current to voltage converter 8.4.2 A current to voltage converter

A differential amplifier (or subtractor) 8.4.3 A differential amplifier (or subtractor)

An inverting summing amplifier 8.4.4 An inverting summing amplifier

Real Operational Amplifiers 8.5 Real Operational Amplifiers So far we have assumed the use of ideal op-amps these have Av = , Ri =  and Ro = 0 Real components do not have these ideal characteristics (though in many cases they approximate to them) In this section we will look at the characteristics of typical devices perhaps the most widely used general purpose op-amp is the 741

Voltage gain typical gain of an operational amplifier might be 100 – 140 dB (voltage gain of 105 – 106) 741 has a typical gain of 106 dB (2  105) high gain devices might have a gain of 160 dB (108) while not infinite the gain of most op-amps is ‘high-enough’ however, gain varies between devices and with temperature

Input resistance typical input resistance of a 741 is 2 M very variable, for a 741 can be as low as 300 k the above value is typical for devices based on bipolar transistors op-amps based on field-effect transistors generally have a much higher input resistance – perhaps 1012  we will discuss bipolar and field-effect transistors later

Output resistance typical output resistance of a 741 is 75  again very variable often of more importance is the maximum output current the 741 will supply 20 mA high-power devices may supply an amp or more

Supply voltage range a typical arrangement would use supply voltages of +15 V and – 15 V, but a wide range of supply voltages is usually possible the 741 can use voltages in the range 5 V to 18 V some devices allow voltages up to 30 V or more others, designed for low voltages, may use 1.5 V many op-amps permit single voltage supply operation, typically in the range 4 to 30 V

Output voltage range the output voltage range is generally determined by the type of op-amp and by the supply voltage being used most op-amps based on bipolar transistors (like the 741) produce a maximum output swing that is slightly less than the difference between the supply rails for example, when used with 15 V supplies, the maximum output voltage swing would be about 13 V op-amps based on field-effect transistors produce a maximum output swing that is very close to the supply voltage range (rail-to-rail operation)

Frequency response typical 741 frequency response is shown here upper cut-off frequency is a few hertz frequency range generally described by the unity-gain bandwidth high-speed devices may operate up to several gigahertz

Selecting Component Values 8.6 Selecting Component Values Our analysis assumed the use of an ideal op-amp When using real components we need to ensure that our assumptions are valid In general this will be true if we: limit the gain of our circuit to much less than the open-loop gain of our op-amp choose external resistors that are small compared with the input resistance of the op-amp choose external resistors that are large compared with the output resistance of the op-amp Generally we use resistors in the range 1 k – 100 k

Effects of Feedback on Op-amp Circuits 8.7 Effects of Feedback on Op-amp Circuits Effects of feedback on the Gain negative feedback reduces gain from A to A/(1 + AB) in return for this loss of gain we get consistency, provided that the open-loop gain is much greater than the closed-loop gain (that is, A >> 1/B) using negative feedback, standard cookbook circuits can be used – greatly simplifying design these can be analysed without a detailed knowledge of the op-amp itself

Effects of feedback on frequency response as the gain is reduced the bandwidth is increased gain  bandwidth  constant since gain is reduced by (1 + AB) bandwidth is increased by (1 + AB) for a 741 gain  bandwidth  106 if gain = 1,000 BW  1,000 Hz if gain = 100 BW  10,000 Hz

Effects of feedback on input and output resistance input/output resistance can be increased or decreased depending on how feedback is used. we looked at this in an earlier lecture in each case the resistance is changed by a factor of (1 + AB) Example if an op-amp with a gain of 2  105 is used to produce an amplifier with a gain of 100 then: A = 2  105 B = 1/G = 0.01 (1 + AB) = (1 + 2000)  2000

Example (see Example 8.4 in the course text) determine the input and output resistance of the following circuit assuming op-amp is a 741 Open-loop gain (A) of a 741 is 2  105 Closed-loop gain (1/B) is 20, B = 1/20 = 0.05 (1 + AB) = (1 + 2  105  0.05) = 104 Feedback senses output voltage therefore it reduces output resistance of op-amp (75 ) by 104 to give 7.5 m Feedback subtracts a voltage from the input, therefore it increases the input voltage of the op-amp (2 M) by 104 to give 20 G

Example (see Example 8.5 in the course text) determine the input and output resistance of the following circuit assuming op-amp is a 741 Open-loop gain (A) of a 741 is 2  105 Closed-loop gain (1/B) is 20, B = 1/20 = 0.05 (1 + AB) = (1 + 2  105  0.05) = 104 Feedback senses output voltage therefore it reduces output resistance of op-amp (75 ) by 104 to give 7.5 m Feedback subtracts a current from the input, therefore it decreases the input voltage. In this case the input sees R2 to a virtual earth, therefore the input resistance is 1 k

Key Points Operational amplifiers are among the most widely used building blocks in electronic circuits An ideal operational amplifier would have infinite voltage gain, infinite input resistance and zero output resistance Designers often make use of cookbook circuits Real op-amps have several non-ideal characteristics However, if we choose components appropriately this should not affect the operation of our circuits Feedback allows us to increase bandwidth by trading gain against bandwidth Feedback also allows us to alter other circuit characteristics

Digital Systems Introduction Binary Quantities and Variables Chapter 9 Digital Systems Introduction Binary Quantities and Variables Logic Gates Boolean Algebra Combinational Logic Number Systems and Binary Arithmetic Numeric and Alphabetic Codes

Introduction Digital systems are concerned with digital signals 9.1 Introduction Digital systems are concerned with digital signals Digital signals can take many forms Here we will concentrate on binary signals since these are the most common form of digital signals can be used individually perhaps to represent a single binary quantity or the state of a single switch can be used in combination to represent more complex quantities

Binary Quantities and Variables 9.2 Binary Quantities and Variables A binary quantity is one that can take only 2 states S L OPEN OFF CLOSED ON S L 1 A simple binary arrangement A truth table

A binary arrangement with two switches in series L = S1 AND S2

A binary arrangement with two switches in parallel L = S1 OR S2

Three switches in series L = S1 AND S2 AND S3

Three switches in parallel L = S1 OR S2 OR S3

A series/parallel arrangement L = S1 AND (S2 OR S3)

Representing an unknown network

9.3 Logic Gates The building blocks used to create digital circuits are logic gates There are three elementary logic gates and a range of other simple gates Each gate has its own logic symbol which allows complex functions to be represented by a logic diagram The function of each gate can be represented by a truth table or using Boolean notation

The AND gate

The OR gate

The NOT gate (or inverter)

A logic buffer gate

The NAND gate

The NOR gate

The Exclusive OR gate

The Exclusive NOR gate

Boolean Algebra Boolean Constants Boolean Variables Boolean Functions 9.4 Boolean Algebra Boolean Constants these are ‘0’ (false) and ‘1’ (true) Boolean Variables variables that can only take the vales ‘0’ or ‘1’ Boolean Functions each of the logic functions (such as AND, OR and NOT) are represented by symbols as described above Boolean Theorems a set of identities and laws – see text for details

Boolean identities AND Function OR Function NOT function 00=0 0+0=0 01=0 0+1=1 10=0 1+0=1 11=1 1+1=1 A0=0 A+0=A 0A=0 0+A=A A1=A A+1=1 1A=A 1+A=1 AA=A A+A=A

Boolean laws Commutative law Absorption law Distributive law De Morgan’s law Associative law Note also

9.5 Combinational Logic Digital systems may be divided into two broad categories: combinational logic where the outputs are determined solely by the current states of the inputs sequential logic where the outputs are determined not only by the current inputs but also by the sequence of inputs that led to the current state In this lecture we will look at combination logic

Implementing a function from a Boolean expression Example – see Example 9.1 in the course text Implement the function

Implementing a function from a Boolean expression Example – see Example 9.2 in the course text Implement the function

Generating a Boolean expression from a logic diagram Example – see Example 9.3 in the course text

Example (continued) – work progressively from the inputs to the output adding logic expressions to the output of each gate in turn

Implementing a logic function from a description Example – see Example 9.4 in the course text The operation of the Exclusive OR gate can be stated as: “The output should be true if either of its inputs are true, but not if both inputs are true.” This can be rephrased as: “The output is true if A OR B is true, AND if A AND B are NOT true.” We can write this in Boolean notation as

Example (continued) The logic function can then be implemented as before

Implementing a logic function from a truth table Example – see Example 9.6 in the course text Implement the function of the following truth table A B C X 1 first write down a Boolean expression for the output then implement as before in this case

Example (continued) The logic function can then be implemented as before

Example – see Example 9.7 in the course text In some cases it is possible to simplify logic expressions using the rules of Boolean algebra Example – see Example 9.7 in the course text can be simplified to hence the following circuits are equivalent

Number Systems and Binary Arithmetic 9.6 Number Systems and Binary Arithmetic Most number systems are order dependent Decimal 123410 = (1  103) + (2  102) + (3  101) + (4  100) Binary 11012 = (1  23) + (1  22) + (0  21) + (1  20) Octal 1238 = (1  83) + (2  82) + (3  81) Hexadecimal 12316 = (1  163) + (2  162) + (3  161) here we need 16 characters – 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

Example – see Example 9.8 in the course text Number conversion conversion to decimal add up decimal equivalent of individual digits Example – see Example 9.8 in the course text Convert 110102 to decimal 110102 = (1  24) + (1  23) + (0  22) + (1  21) + (0  20) = 16 + 8 + 0 + 2 + 0 = 2610

Example – see Example 9.9 in the course text Number conversion conversion from decimal repeatedly divide by the base and remember the remainder Example – see Example 9.9 in the course text Convert 2610 to binary Number Remainder Starting point 26  2 13 0  2 6 1  2 3 0  2 1 1  2 0 1 read number from this end =11010

Binary arithmetic much simpler than decimal arithmetic can be performed by simple circuits, e.g. half adder

More complex circuits can add digital words Similar circuits can be constructed to perform subtraction – see text More complex arithmetic (such as multiplication and division) can be done by dedicated hardware but is more often performed using a microcomputer or complex logic device

Numeric and Alphabetic Codes 9.7 Numeric and Alphabetic Codes Binary code by far the most common way of representing numeric information has advantages of simplicity and efficiency of storage Decimal Binary 1 2 3 4 5 6 7 8 9 10 11 12 etc. 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 etc.

Numeric and Alphabetic Codes 9.7 Numeric and Alphabetic Codes Binary-coded decimal code formed by converting each digit of a decimal number individually into binary requires more digits than conventional binary has advantage of very easy conversion to/from decimal used where input and output are in decimal form Decimal Binary 1 2 3 4 5 6 7 8 9 10 11 12 etc. 0 1 10 11 100 101 110 111 1000 1001 10000 10001 10010 etc.

Numeric and Alphabetic Codes 9.7 Numeric and Alphabetic Codes ASCII code American Standard Code for Information Interchange an alphanumeric code each character represented by a 7-bit code gives 128 possible characters codes defined for upper and lower-case alphabetic characters, digits 0 – 9, punctuation marks and various non-printing control characters (such as carriage-return and backspace)

Numeric and Alphabetic Codes 9.7 Numeric and Alphabetic Codes Error detecting and correcting codes adding redundant information into codes allows the detection of transmission errors examples include the use of parity bits and checksums adding additional redundancy allows errors to be not only detected but also corrected such techniques are used in CDs, mobile phones and computer disks

Key Points It is common to represent the two states of a binary variable by ‘0’ and ‘1’ Logic circuits are usually implemented using logic gates Circuits in which the output is determined solely by the current inputs are termed combinational logic circuits Logic functions can be described by truth tables or using Boolean algebraic notation Binary digits may be combined to form digital words Digital words can be processed using binary arithmetic Several codes can be used to represent different forms of information

Sequential Logic Introduction Bistables Memory Registers Chapter 10 Sequential Logic Introduction Bistables Memory Registers Shift Registers Counters Monostables or one-shots Astables Timers

10.1 Introduction Sequential logic elements combine the characteristics of combinational logic with memory When constructing sequential logic our building blocks are often some form of multivibrator a term used to describe a range of circuits these have two outputs that are the inverse of each other the output are labelled Q and three basic forms: bistables monstables astables

Bistables The S-R latch (SET-RESET Latch) when R = S = 0 10.2 Bistables The S-R latch (SET-RESET Latch) when R = S = 0 circuit stays in current state when S = 1, R = 0 Q is SET to 1 ( = 0) when S = 0, R = 1 Q is RESET to 0 ( = 1) when S = 1, R = 1 both outputs at 0 – not allowed

The S-R latch Circuit symbols

The S-R latch A waveform diagram

A design example - see Example 10.1 in course text A Burglar Alarm close all doors and window (closing switches) open reset switch to initialise system opening any of the door/window switches will activate alarm alarm will continue if switch is then closed alarm is silenced by opening reset switch

The D latch (Data Latch)

Edge-triggered devices it is often necessary to synchronise many devices this can be done using a clock input such devices respond on a particular transition of the clock these are called edge-triggered devices or flip-flops can have positive-edge or negative-edge triggered devices

The D flip-flop symbol as in previous slide behaviour of positive-edge triggered device as below Q becomes equal to D at the time of the trigger event

The J-K flip-flop similar to S-R flip-flop but toggles when J = K = 1

Asynchronous inputs some flip-flops have asynchronous inputs

Propagation delays and races real logic gates take a finite time to react some circuits (as below) can suffer from race hazards where the operation of the circuit is uncertain in this circuit the output depends on which devices is fastest

Pulse-triggered or master/slave bistables these overcome race hazards by responding to the state of the inputs shortly before the clock trigger

10.3 Memory Registers Combining a number of bistables we can construct a memory register several forms of bistable can be used, for example:

Often we are not concerned with the internal construction of the register they are a standard integrated component

10.4 Shift Registers A slightly different configuration of bistables can produce a shift register

Behaviour of a shift register

An application of a shift register in serial/parallel and parallel/serial conversion used in serial communication

Counters Ripple counters 10.5 Counters Ripple counters can be constructed using several forms of bistable consider the following arrangement with J = K = 1 each bistable toggles on the falling edge of its clock input

acts as a frequency divider Each stage toggles at half the frequency of the previous stage acts as a frequency divider divides frequency by 2n (n is the number of stages)

Application of a frequency divider – see Example 10.3 Clock generator for a digital watch 15-stage counter divides signal from a crystal oscillator by 32,768 to produce a 1 Hz signal to drive stepper motor or digital display

the outputs count in binary from 0 to 2n-1 and then repeat Consider the pattern on the outputs of the counter shown earlier – displayed on the right the outputs count in binary from 0 to 2n-1 and then repeat the circuit acts as a modulo-2n counter since the counting process propagates from one bistable to the next this is called a ripple counter circuit shown is a 4-bit or modulo-16 (or mod-16) ripple counter

Modulo-N counters by using an appropriate number of stages the earlier counter can count modulo any power of 2 to count to any other base we add reset circuitry e.g. the modulo-10 or decade counter shown here

Down and up/down Counters a slight modification to the earlier circuit will produce a counter that counts from 2n-1 to 0 and then restarts this is a down counter a further modification can produce an up/down counter which counts up or down depending on the state of a control line (usually labelled ) when this is 1 the counter counts up when this is 0 the counter counts down

Propagation delay in counters while ripple counters are very simple they suffer from problems at high speed since the output of one flip-flop is triggered by the change of the previous device, delays produced by each flip-flop are summed along the chain the time for a single device to respond is termed its propagation delay time tPD an n-bit counter will take n  tPD to respond if read before this time the result will be garbled

Synchronous counters these overcome the propagation delay in ripple counters by connecting all the flip-flops to the same clock signal thus each stage changes state at the same time additional circuitry is used to determine which stages change state on each clock pulse faster than ripple counters but more complex available in many forms including up, down, up/down and modulo-N counters

Monostables or one-shots 10.6 Monostables or one-shots Monostables are another form of multivibrator while bistables have two stable output states monostables have one stable & one metastable states when in its stable state Q = 0 when an appropriate signal is applied to the trigger input (T ) the circuit enters its metastable state with Q = 1 after a set period of time (determined by circuit components) it reverts to its stable state it is therefore a pulse generator Circuit symbol

Monostables can be retriggerable or non-retriggerable

Astables The last member of the multivibrator family is the astable 10.7 Astables The last member of the multivibrator family is the astable this has two metastable states has the function of a digital oscillator circuit spends a fixed period in each state (determined by circuit components) if the period in each state is set to be equal, this will produce a square waveform

Timers The integrated circuit timer can produce a range of functions 10.8 Timers The integrated circuit timer can produce a range of functions including those of a monostable or astable various devices one of the most popular is the 555 timer can be configured using just a couple of external passive components internal construction largely unimportant – all required information on using the device is in its data sheet

Key Points Sequential logic circuits have the characteristic of memory Among the most important groups of sequential components are the various forms of multivibrator bistables monostables astables The most widely used form is the bistable which includes latches, edge-triggered flip-flops and master/slave devices Registers form the basis of various memories Counters are widely used in a range of applications Monostables and astables perform a range of functions

Measurement of Voltages and Currents Chapter 11 Measurement of Voltages and Currents Introduction Sine waves Square waves Measuring Voltages and Currents Analogue Ammeters and Voltmeters Digital Multimeters Oscilloscopes

11.1 Introduction Alternating currents and voltages vary with time and periodically change their direction

11.2 Sine Waves Sine waves by far the most important form of alternating quantity important properties are shown below

Instantaneous value shape of the sine wave is defined by the sine function y = A sin  in a voltage waveform v = Vp sin 

Angular frequency frequency f (in hertz) is a measure of the number of cycles per second each cycle consists of 2 radians therefore there will be 2f radians per second this is the angular frequency  (units are rad/s)  = 2f

Equation of a sine wave the angular frequency  can be thought of as the rate at which the angle of the sine wave changes at any time  = t therefore v = Vp sin t or v = Vp sin 2ft similarly i = Ip sin t or i = Ip sin 2ft

Example – see Example 11.2 in the course text Determine the equation of the following voltage signal. From diagram: Period is 50 ms = 0.05 s Thus f = 1/T =1/0.05 = 20 Hz Peak voltage is 10 V Therefore

Phase angles the expressions given above assume the angle of the sine wave is zero at t = 0 if this is not the case the expression is modified by adding the angle at t = 0

Phase difference two waveforms of the same frequency may have a constant phase difference we say that one is phase-shifted with respect to the other

Average value of a sine wave average value over one (or more) cycles is clearly zero however, it is often useful to know the average magnitude of the waveform independent of its polarity we can think of this as the average value over half a cycle… … or as the average value of the rectified signal

Average value of a sine wave

r.m.s. value of a sine wave the instantaneous power (p) in a resistor is given by therefore the average power is given by where is the mean-square voltage

While the mean-square voltage is useful, more often we use the square root of this quantity, namely the root-mean-square voltage Vrms where Vrms = we can also define Irms = it is relatively easy to show that (see text for analysis)

r.m.s. values are useful because their relationship to average power is similar to the corresponding DC values

Form factor for any waveform the form factor is defined as for a sine wave this gives

Peak factor for any waveform the peak factor is defined as for a sine wave this gives

11.3 Square Waves Frequency, period, peak value and peak-to-peak value have the same meaning for all repetitive waveforms

Phase angle we can divide the period into 360 or 2 radians useful in defining phase relationship between signals in the waveforms shown here, B lags A by 90 we could alternatively give the time delay of one with respect to the other

Average and r.m.s. values the average value of a symmetrical waveform is its average value over the positive half-cycle thus the average value of a symmetrical square wave is equal to its peak value similarly, since the instantaneous value of a square wave is either its peak positive or peak negative value, the square of this is the peak value squared, and

Form factor and peak factor from the earlier definitions, for a square wave

Measuring Voltages and Currents 11.4 Measuring Voltages and Currents Measuring voltage and current in a circuit when measuring voltage we connect across the component when measuring current we connect in series with the component

Measuring Voltages and Currents 11.4 Measuring Voltages and Currents Loading effects – voltage measurement our measuring instrument will have an effective resistance (RM) when measuring voltage we connect a resistance in parallel with the component concerned which changes the resistance in the circuit and therefore changes the voltage we are trying to measure this effect is known as loading

Measuring Voltages and Currents 11.4 Measuring Voltages and Currents Loading effects – current measurement our measuring instrument will have an effective resistance (RM) when measuring current we connect a resistance in series with the component concerned which again changes the resistance in the circuit and therefore changes the current we are trying to measure this is again a loading effect

Analogue Ammeters and Voltmeters 11.5 Analogue Ammeters and Voltmeters Most modern analogue ammeters are based on moving-coil meters see Chapter 4 of textbook Meters are characterised by their full-scale deflection (f.s.d.) and their effective resistance (RM) typical meters produce a f.s.d. for a current of 50 A – 1 mA typical meters have an RM between a few ohms and a few kilohms

Measuring direct currents using a moving coil meter use a shunt resistor to adjust sensitivity see Example 11.5 in set text for numerical calculations

Measuring direct voltages using a moving coil meter use a series resistor to adjust sensitivity see Example 11.6 in set text for numerical calculations

Measuring alternating quantities moving coil meters respond to both positive and negative voltages, each producing deflections in opposite directions a symmetrical alternating waveform will produce zero deflection (the mean value of the waveform) therefore we use a rectifier to produce a unidirectional signal meter then displays the average value of the waveform meters are often calibrated to directly display r.m.s. of sine waves all readings are multiplied by 1.11 – the form factor for a sine wave as a result waveforms of other forms will give incorrect readings for example when measuring a square wave (for which the form factor is 1.0, the meter will read 11% too high)

Analogue multimeters general purpose instruments use a combination of switches and resistors to give a number of voltage and current ranges a rectifier allows the measurement of AC voltage and currents additional circuitry permits resistance measurement very versatile but relatively low input resistance on voltage ranges produces considerable loading in some situations A typical analogue multimeter

11.6 Digital Multimeters Digital multimeters (DMMs) are often (inaccurately) referred to as digital voltmeters or DVMs at their heart is an analogue-to-digital converter (ADC) A simplified block diagram

Measurement of voltage, current and resistance is achieved using appropriate circuits to produce a voltage proportional to the quantity to be measured in simple DMMs alternating signals are rectified as in analogue multimeters to give its average value which is multiplied by 1.11 to directly display the r.m.s. value of sine waves more sophisticated devices use a true r.m.s. converter which accurately produced a voltage proportional to the r.m.s. value of an input waveform A typical digital multimeter

Oscilloscopes An oscilloscope displays voltage waveforms 11.7 A simplified block diagram

A typical analogue oscilloscope

Measurement of phase difference

Key Points The magnitude of an alternating waveform can be described by its peak, peak-to-peak, average or r.m.s. value The root-mean-square value of a waveform is the value that will produce the same power as an equivalent direct quantity Simple analogue ammeter and voltmeters are based on moving coil meters Digital multimeters are easy to use and offer high accuracy Oscilloscopes display the waveform of a signal and allow quantities such as phase to be measured.

Resistance and DC Circuits Chapter 12 Resistance and DC Circuits Introduction Current and Charge Voltage Sources Current Sources Resistance and Ohm’s Law Resistors in Series and Parallel Kirchhoff’s Laws Thévenin’s and Norton’s Theorems Superposition Nodal Analysis Mesh Analysis Solving Simultaneous Circuit Equations Choice of Techniques

12.1 Introduction In earlier lectures we have seen that many circuits can be analysed, and in some cases designed, using little more than Ohm’s law However, in some cases we need some additional techniques and these are discussed in this lecture. We begin by reviewing some of the basic elements that we have used in earlier lectures to describe our circuits

Current and Charge An electric current is a flow of electric charge 12.2 Current and Charge An electric current is a flow of electric charge At an atomic level a current is a flow of electrons each electron has a charge of 1.6  10-19 coulombs conventional current flows in the opposite direction Rearranging above expression gives For constant current

12.3 Voltage Sources A voltage source produces an electromotive force (e.m.f.) which causes a current to flow within a circuit unit of e.m.f. is the volt a volt is the potential difference between two points when a joule of energy is used to move one coulomb of charge from one point to the other Real voltage sources, such as batteries have resistance associated with them in analysing circuits we use ideal voltage sources we also use controlled or dependent voltage sources

Voltage sources

12.4 Current Sources We also sometimes use the concept of an ideal current source unrealisable, but useful in circuit analysis can be a fixed current source, or a controlled or dependent current source while an ideal voltage source has zero output resistance, an ideal current source has infinite output resistance

Resistance and Ohm’s Law 12.5 Resistance and Ohm’s Law Ohm’s law constant of proportionality is the resistance R hence current through a resistor causes power dissipation

Resistors resistance of a given sample of material is determined by its electrical characteristics and its construction electrical characteristics described by its resistivity  or its conductivity  (where  = 1/)

Resistors in Series and Parallel 12.6 Resistors in Series and Parallel Resistors in series where R = (R1 + R2+…+RN).

Resistors in parallel where 1/R = 1/R1 + 1/R2 +…+ 1/RN

Kirchhoff’s Laws Node Loop Mesh Examples: 12.7 Kirchhoff’s Laws Node a point in a circuit where two or more circuit components are joined Loop any closed path that passes through no node more than once Mesh a loop that contains no other loop Examples: A, B, C, D, E and F are nodes the paths ABEFA, BCDEB and ABCDEFA are loops ABEFA and BCDEB are meshes

Current Law At any instant, the algebraic sum of all the currents flowing into any node in a circuit is zero if currents flowing into the node are positive, currents flowing out of the node are negative, then

Voltage Law At any instant the algebraic sum of all the voltages around any loop in a circuit is zero if clockwise voltage arrows are positive and anticlockwise arrows are negative then

Thévenin’s and Norton’s Theorems 12.8 Thévenin’s and Norton’s Theorems Thévenin’s Theorem As far as its appearance from outside is concerned, any two terminal network of resistors and energy sources can be replaced by a series combination of an ideal voltage source V and a resistor R, where V is the open-circuit voltage of the network and R is the voltage that would be measured between the output terminals if the energy sources were removed and replaced by their internal resistance.

Norton’s Theorem As far as its appearance from outside is concerned, any two terminal network of resistors and energy sources can be replaced by a parallel combination of an ideal current source I and a resistor R, where I is the short-circuit current of the network and R is the voltage that would be measured between the output terminals if the energy sources were removed and replaced by their internal resistance.

from the Thévenin equivalent circuit hence for either circuit

Example – see Example 12.3 from course text Determine Thévenin and Norton equivalent circuits of the following circuit.

Example (continued) if nothing is connected across the output no current will flow in R2 so there will be no voltage drop across it. Hence Vo is determined by the voltage source and the potential divider formed by R1 and R3. Hence if the output is shorted to ground, R2 is in parallel with R3 and the current taken from the source is 30V/15 k = 2 mA. This will divide equally between R2 and R3 so the output current, and so the resistance in the equivalent circuit is therefore given by

Example (continued) hence equivalent circuits are:

Superposition Principle of superposition 12.9 Superposition Principle of superposition In any linear network of resistors, voltage sources and current sources, each voltage and current in the circuit is equal to the algebraic sum of the voltages or currents that would be present if each source were to be considered separately. When determining the effects of a single source the remaining sources are replaced by their internal resistance.

Example – see Example 12.5 from course text Determine the output voltage V of the following circuit.

Example (continued) first consider the effect of the 15V source alone

Example (continued) next consider the effect of the 20V source alone

Example (continued) so, the output of the complete circuit is the sum of these two voltages

Nodal Analysis Six steps: Chose one node as the reference node 12.10 Nodal Analysis Six steps: Chose one node as the reference node Label remaining nodes V1, V2, etc. Label any known voltages Apply Kirchhoff’s current law to each unknown node Solve simultaneous equations to determine voltages If necessary calculate required currents

Example – see Example 12.8 from course text Determine the current I1 in the following circuit

Example (continued) first we pick a reference node and label the various node voltages, assigning values where these are known

Example (continued) next we sum the currents flowing into the nodes for which the node voltages are unknown. This gives solving these two equations gives V2 = 32.34 V V3 = 40.14 V and the required current is given by

Mesh Analysis Four steps: 12.11 Mesh Analysis Four steps: Identify the meshes and assign a clockwise-flowing current to each. Label these I1, I2, etc. Apply Kirchhoff’s voltage law to each mesh Solve the simultaneous equations to determine the currents I1, I2, etc. Use these values to obtain voltages if required

Example – see Example 12.9 from course text Determine the voltage across the 10  resistor

Example (continued) first assign loops currents and label voltages

Example (continued) next apply Kirchhoff’s law to each loop. This gives which gives the following set of simultaneous equations

Example (continued) these can be rearranged to give which can be solved to give

Example (continued) the voltage across the 10  resistor is therefore given by since the calculated voltage is positive, the polarity is as shown by the arrow with the left hand end of the resistor more positive than the right hand end

Solving Simultaneous Circuit Equations 12.12 Solving Simultaneous Circuit Equations Both nodal analysis and mesh analysis produce a series of simultaneous equations can be solved ‘by hand’ or by using matrix methods e.g. can be rearranged as as

Solving Simultaneous Circuit Equations 12.12 Solving Simultaneous Circuit Equations these equations can be expressed as which can be solved by hand (e.g. Cramer’s rule) or can use automated tools e.g. scientific calculators computer-based packages such as MATLAB or Mathcad 160 -20 -30 I1 50 20 -210 10 I2 = 30 -190 I3

Choice of Techniques How do we choose the right technique? 12.13 Choice of Techniques How do we choose the right technique? nodal and mesh analysis will work in a wide range of situations but are not necessarily the simplest methods no simple rules often involves looking at the circuit and seeing which technique seems appropriate see Section 12.3 of course text for an example

Key Points An electric current is a flow of charge A voltage source produces an e.m.f. which can cause a current to flow Current in a conductor is directly proportional to voltage At any instant the sum of the currents into a node is zero At any instant the sum of the voltages around a loop is zero Any two terminal network of resistors and energy sources can be replaced by a Thévenin or Norton equivalent circuit Nodal and mesh analysis provide systematic methods of applying Kirchhoff’s laws

Capacitance and Electric Fields Chapter 13 Capacitance and Electric Fields Introduction Capacitors and Capacitance Alternating Voltages and Currents The Effect of a Capacitor’s Dimensions Electric Fields Capacitors in Series and Parallel Voltage and Current Sinusoidal Voltages and Currents Energy Stored in a Charged Capacitor Circuit Symbols

13.1 Introduction We noted earlier that an electric current represents a flow of charge A capacitor can store electric charge and can therefore store electrical energy Capacitors are often used in association with alternating currents and voltages They are a key component in almost all electronic circuits

Capacitors and Capacitance 13.2 Capacitors and Capacitance Capacitors consist of two conducting surfaces separated by an insulating layer called a dielectric

A simple capacitor circuit when switch is closed electrons flow from top plate into battery and from battery onto bottom plate charge produces an electric field across the capacitor and a voltage across it

For a given capacitor the stored charge q is directly proportional to the voltage across it V The constant of proportionality is the capacitance C and thus If the charge is measured in coulombs and the voltage in volts, then the capacitance is in farads

Example – see Example 13.1 in course text A 10 F capacitor has 10 V across it. What quantity of charge is stored in it? From above

Alternating Voltages and Currents 13.3 Alternating Voltages and Currents A constant current cannot flow through a capacitor however, since the voltage across a capacitor is proportional to the charge on it, an alternating voltage must correspond to an alternating charge, and hence to current flowing into and out of the capacitor this can give the impression that an alternating current flows through the capacitor

A mechanical analogy may help to explain this consider a window - air cannot pass through it, but sound (which is a fluctuation in air pressure) can

The Effect of a Capacitor’s Dimensions 13.4 The Effect of a Capacitor’s Dimensions The capacitance of a capacitor is directly proportional to its area A, and inversely proportional to the distance between its plates d. Hence C  A/d the constant of proportionality is the permittivity  of the dielectric the permittivity is normally expressed as the product of the absolute permittivity 0 and the relative permittivity r of the dielectric used

13.5 Electric Fields The charge on the capacitor produces an electric field with an electric field strength E given by the units of E are volts/metre (V/m)

All insulating materials have a maximum value for the field strength that they can withstand the dielectric strength Em To produce maximum capacitance for a given size of capacitor we want d to be as small as possible however, as d is decreased the electric field E is increased if E exceed Em the dielectric will break down there is therefore a compromise between physical size and breakdown voltage

We also define the electric flux density D as the flux per unit area The force between positive and negative charges is described in terms of the electric flux linking them measured in coulombs (as for electric charge) a charge Q will produce a total flux of Q coulombs We also define the electric flux density D as the flux per unit area In a capacitor we can almost always ignore edge effects, and

Combining the earlier equations it is relatively easy to show that Thus the permittivity of the dielectric within a capacitor is equal to the ratio of the electric flux density to the electric field strength.

Capacitors in Series and Parallel 13.6 Capacitors in Series and Parallel Capacitors in parallel consider a voltage V applied across two capacitors then the charge on each is if the two capacitors are replaced with a single capacitor C which has a similar effect as the pair, then

Capacitors in series consider a voltage V applied across two capacitors in series the only charge that can be applied to the lower plate of C1 is that supplied by the upper plate of C2. Therefore the charge on each capacitor must be identical. Let this be Q, and therefore if a single capacitor C has the same effect as the pair, then

13.7 Voltage and Current The voltage across a capacitor is directly related to the charge on the capacitor Alternatively, since Q = CV we can see that and since dQ/dt is equal to current, it follows that

Consider the circuit shown here capacitor is initially discharged voltage across it will be zero switch is closed at t = 0 VC is initially zero hence VR is initially V hence I is initially V/R as the capacitor charges: VC increases VR decreases hence I decreases we have exponential behaviour

See Computer Simulation Exercises 13.1 and 13.2 in the course text Time constant charging current is determined by R and the voltage across it increasing R will increase the time taken to charge C increasing C will also increase time taken to charge C time required to charge to a particular voltage is determined by the product CR this product is the time constant  (greek tau) See Computer Simulation Exercises 13.1 and 13.2 in the course text

Sinusoidal Voltages and Currents 13.8 Sinusoidal Voltages and Currents Consider the application of a sinusoidal voltage to a capacitor from above I = C dV/dt current is directly proportional to the differential of the voltage the differential of a sine wave is a cosine wave the current is phase-shifted by 90 with respect to the voltage

We will return to look at frequency dependence in later lectures. Since I = C dV/dt the magnitude of the current is related to the rate of change of the voltage in sinusoidal voltages the rate of change is determined by the frequency hence capacitors are frequency dependent in their characteristics We will return to look at frequency dependence in later lectures.

Energy Stored in a Charged Capacitor 13.9 Energy Stored in a Charged Capacitor To move a charge Q through a potential difference V requires an amount of energy QV As we charge up a capacitor we repeatedly add small amounts of charge Q by moving them through a voltage equal to the voltage on the capacitor Since Q = CV, it follows that Q = CV, so the energy needed E is given by

Alternatively, since V = Q/C Example – see Example 13.7 in the course text Calculate the energy stored in a 10 F capacitor when it is charged to 100 V. From above:

13.10 Circuit Symbols

Key Points A capacitor consists of two plates separated by a dielectric The charge stored on a capacitor is proportional to V A capacitor blocks DC but appears to pass AC The capacitance of several capacitors in parallel is equal to the sum of their individual capacitances The capacitance of several capacitors in series is equal to the reciprocal of the sum of the reciprocals of the individual capacitances In AC circuits current leads voltage by 90 in a capacitor The energy stored in a capacitor is ½CV2 or ½Q2/C

Inductance and Magnetic Fields Chapter 14 Inductance and Magnetic Fields Introduction Electromagnetism Reluctance Inductance Self-inductance Inductors Inductors in Series and Parallel Voltage and Current Sinusoidal Voltages and Currents Energy Storage in an Inductor Mutual Inductance Transformers Circuit Symbols The Use of Inductance in Sensors

14.1 Introduction Earlier we noted that capacitors store energy by producing an electric field within a piece of dielectric material Inductors also store energy, in this case it is stored within a magnetic field In order to understand inductors, and related components such as transformers, we need first to look at electromagnetism

14.2 Electromagnetism A wire carrying a current I causes a magnetomotive force (m.m.f) F this produces a magnetic field F has units of Amperes for a single wire F is equal to I

where l is the length of the magnetic circuit The magnitude of the field is defined by the magnetic field strength, H , where where l is the length of the magnetic circuit Example – see Example 14.1 from course text A straight wire carries a current of 5 A. What is the magnetic field strength H at a distance of 100mm from the wire? Magnetic circuit is circular. r = 100mm, so path = 2r = 0.628m

The magnetic field produces a magnetic flux,  flux has units of weber (Wb) Strength of the flux at a particular location is measured in term of the magnetic flux density, B flux density has units of tesla (T) (equivalent to 1 Wb/m2) Flux density at a point is determined by the field strength and the material present or where  is the permeability of the material, r is the relative permeability and 0 is the permeability of free space

Adding a ferromagnetic ring around a wire will increase the flux by several orders of magnitude since r for ferromagnetic materials is 1000 or more

When a current-carrying wire is formed into a coil the magnetic field is concentrated For a coil of N turns the m.m.f. (F) is given by and the field strength is

The magnetic flux produced is determined by the permeability of the material present a ferromagnetic material will increase the flux density

14.3 Reluctance In a resistive circuit, the resistance is a measure of how the circuit opposes the flow of electricity In a magnetic circuit, the reluctance, S is a measure of how the circuit opposes the flow of magnetic flux In a resistive circuit R = V/I In a magnetic circuit the units of reluctance are amperes per weber (A/ Wb)

14.4 Inductance A changing magnetic flux induces an e.m.f. in any conductor within it Faraday’s law: The magnitude of the e.m.f. induced in a circuit is proportional to the rate of change of magnetic flux linking the circuit Lenz’s law: The direction of the e.m.f. is such that it tends to produce a current that opposes the change of flux responsible for inducing the e.m.f.

When a circuit forms a single loop, the e. m. f When a circuit forms a single loop, the e.m.f. induced is given by the rate of change of the flux When a circuit contains many loops the resulting e.m.f. is the sum of those produced by each loop Therefore, if a coil contains N loops, the induced voltage V is given by where d/dt is the rate of change of flux in Wb/s This property, whereby an e.m.f. is induced as a result of changes in magnetic flux, is known as inductance

14.5 Self-inductance A changing current in a wire causes a changing magnetic field about it A changing magnetic field induces an e.m.f. in conductors within that field Therefore when the current in a coil changes, it induces an e.m.f. in the coil This process is known as self-inductance where L is the inductance of the coil (unit is the Henry)

14.6 Inductors The inductance of a coil depends on its dimensions and the materials around which it is formed

The inductance is greatly increased through the use of a ferromagnetic core, for example

Equivalent circuit of an inductor All real circuits also possess stray capacitance

Inductors in Series and Parallel 14.7 Inductors in Series and Parallel When several inductors are connected together their effective inductance can be calculated in the same way as for resistors – provided that they are not linked magnetically Inductors in Series

Inductors in Parallel

Voltage and Current Consider the circuit shown here 14.8 Voltage and Current Consider the circuit shown here inductor is initially un-energised current through it will be zero switch is closed at t = 0 I is initially zero hence VR is initially 0 hence VL is initially V as the inductor is energised: I increases VR increases hence VL decreases we have exponential behaviour

See Computer Simulation Exercises 14.1 and 14.2 in the course text Time constant we noted earlier that in a capacitor-resistor circuit the time required to charge to a particular voltage is determined by the time constant CR in this inductor-resistor circuit the time taken for the current to rise to a certain value is determined by L/R this value is again the time constant  (greek tau) See Computer Simulation Exercises 14.1 and 14.2 in the course text

Sinusoidal Voltages and Currents 14.9 Sinusoidal Voltages and Currents Consider the application of a sinusoidal current to an inductor from above V = L dI/dt voltage is directly proportional to the differential of the current the differential of a sine wave is a cosine wave the voltage is phase-shifted by 90 with respect to the current the phase-shift is in the opposite direction to that in a capacitor

Energy Storage in an Inductor 14.10 Energy Storage in an Inductor Can be calculated in a similar manner to the energy stored in a capacitor In a small amount of time dt the energy added to the magnetic field is the product of the instantaneous voltage, the instantaneous current and the time Thus, when the current is increased from zero to I

14.11 Mutual Inductance When two coils are linked magnetically then a changing current in one will produce a changing magnetic field which will induce a voltage in the other – this is mutual inductance When a current I1 in one circuit, induces a voltage V2 in another circuit, then where M is the mutual inductance between the circuits. The unit of mutual inductance is the Henry (as for self-inductance)

The coupling between the coils can be increased by wrapping the two coils around a core the fraction of the magnetic field that is coupled is referred to as the coupling coefficient

Coupling is particularly important in transformers the arrangements below give a coupling coefficient that is very close to 1

Transformers Most transformers approximate to ideal components 14.12 Transformers Most transformers approximate to ideal components that is, they have a coupling coefficient  1 for such a device, when unloaded, their behaviour is determined by the turns ratio for alternating voltages

When used with a resistive load, current flows in the secondary this current itself produces a magnetic flux which opposes that produced by the primary thus, current in the secondary reduces the output voltage for an ideal transformer

14.13 Circuit Symbols

The Use of Inductance in Sensors 14.14 The Use of Inductance in Sensors Numerous examples: Inductive proximity sensors basically a coil wrapped around a ferromagnetic rod a ferromagnetic plate coming close to the coil changes its inductance allowing it to be sensed can be used as a linear sensor or as a binary switch

Linear variable differential transformers (LVDTs) see course text for details of operation of this device

Key Points Inductors store energy within a magnetic field A wire carrying a current creates a magnetic field A changing magnetic field induces an electrical voltage in any conductor within the field The induced voltage is proportional to the rate of change of the current Inductors can be made by coiling wire in air, but greater inductance is produced if ferromagnetic materials are used The energy stored in an inductor is equal to ½LI2 When a transformer is used with alternating signals, the voltage gain is equal to the turns ratio

Alternating Voltages and Currents Chapter 15 Alternating Voltages and Currents Introduction Voltage and Current Reactance of Inductors and Capacitors Phasor Diagrams Impedance Complex Notation

Introduction From our earlier discussions we know that 15.1 Introduction From our earlier discussions we know that where Vp is the peak voltage  is the angular frequency  is the phase angle Since  = 2f it follows that the period T is given by

If  is in radians, then a time delay t is given by  / as shown below

15.2 Voltage and Current Consider the voltages across a resistor, an inductor and a capacitor, with a current of Resistors from Ohm’s law we know therefore if i = Ipsin(t)

Voltage and Current Inductors - in an inductor 15.2 Voltage and Current Inductors - in an inductor therefore if i = Ipsin(t) Capacitors - in a capacitor

Reactance of Inductors and Capacitors 15.3 Reactance of Inductors and Capacitors Let us ignore, for the moment the phase angle and consider the magnitudes of the voltages and currents Let us compare the peak voltage and peak current Resistance

Inductance Capacitance

In a resistor this is termed its resistance The ratio of voltage to current is a measure of how the component opposes the flow of electricity In a resistor this is termed its resistance In inductors and capacitors it is termed its reactance Reactance is given the symbol X Therefore Reactance Reactance

Since reactance represents the ratio of voltage to current it has units of ohms The reactance of a component can be used in much the same way as resistance: for an inductor for a capacitor

Example – see Example 15.3 from course text A sinusoidal voltage of 5 V peak and 100 Hz is applied across an inductor of 25 mH. What will be the peak current? At this frequency, the reactance of the inductor is given by Therefore

15.4 Phasor Diagrams Sinusoidal signals are characterised by their magnitude, their frequency and their phase In many circuits the frequency is fixed (perhaps at the frequency of the AC supply) and we are interested in only magnitude and phase In such cases we often use phasor diagrams which represent magnitude and phase within a single diagram

Examples of phasor diagrams (a) here L represents the magnitude and  the phase of a sinusoidal signal (b) shows the voltages across a resistor, an inductor and a capacitor for the same sinusoidal current

Phasor diagrams can be used to represent the addition of signals Phasor diagrams can be used to represent the addition of signals. This gives both the magnitude and phase of the resultant signal

Phasor diagrams can also be used to show the subtraction of signals

Phasor analysis of an RL circuit See Example 15.5 in the text for a numerical example

Phasor analysis of an RC circuit See Example 15.6 in the text for a numerical example

Phasor analysis of an RLC circuit

Phasor analysis of parallel circuits in such circuits the voltage across each of the components is the same and it is the currents that are of interest

15.5 Impedance In circuits containing only resistive elements the current is related to the applied voltage by the resistance of the arrangement In circuits containing reactive, as well as resistive elements, the current is related to the applied voltage by the impedance, Z of the arrangement this reflects not only the magnitude of the current but also its phase impedance can be used in reactive circuits in a similar manner to the way resistance is used in resistive circuits

Consider the following circuit and its phasor diagram

From the phasor diagram it is clear that that the magnitude of the voltage across the arrangement V is where Z is the magnitude of the impedance, so Z =|Z|

From the phasor diagram the phase angle of the impedance is given by This circuit contains an inductor but a similar analysis can be done for circuits containing capacitors In general and

A graphical representation of impedance

15.6 Complex Notation Phasor diagrams are similar to Argand Diagrams used in complex mathematics We can also represent impedance using complex notation where Resistors: ZR = R Inductors: ZL = jXL = jL Capacitors: ZC = -jXC =

Graphical representation of complex impedance

Series and parallel combinations of impedances impedances combine in the same way as resistors

Manipulating complex impedances complex impedances can be added, subtracted, multiplied and divided in the same way as other complex quantities they can also be expressed in a range of forms such as the rectangular, polar and exponential forms if you are unfamiliar with the manipulation of complex quantities (or would like a little revision on this topic) see Appendix D of the course text which gives a tutorial on this subject

At 50Hz, the angular frequency  = 2f = 2 50 = 314 rad/s Example – see Example 15.7 in the course text Determine the complex impedance of this circuit at a frequency of 50 Hz. At 50Hz, the angular frequency  = 2f = 2 50 = 314 rad/s Therefore

Since v = 100 sin 250t , then  = 250 Using complex impedance Example – see Section 15.6.4 in course text Determine the current in this circuit. Since v = 100 sin 250t , then  = 250 Therefore

Example (continued) The current is given by v/Z and this is easier to compute in polar form Therefore

A further example A more complex task is to find the output voltage of this circuit. The analysis of this circuit, and a numerical example based on it, are given in Section 15.6.4 and Example 15.8 of the course text

Key Points A sinusoidal voltage waveform can be described by the equation The voltage across a resistor is in phase with the current, the voltage across an inductor leads the current by 90, and the voltage across a capacitor lags the current by 90 The reactance of an inductor XL = L The reactance of a capacitor XC = 1/C The relationship between current and voltage in circuits containing reactance can be described by its impedance The use of impedance is simplified by the use of complex notation

Power in AC Circuits Introduction Power in Resistive Components Chapter 16 Power in AC Circuits Introduction Power in Resistive Components Power in Capacitors Power in Inductors Circuits with Resistance and Reactance Active and Reactive Power Power Factor Correction Power Transfer Three-Phase Systems Power Measurement

16.1 Introduction The instantaneous power dissipated in a component is a product of the instantaneous voltage and the instantaneous current p = vi In a resistive circuit the voltage and current are in phase – calculation of p is straightforward In reactive circuits, there will normally be some phase shift between v and i, and calculating the power becomes more complicated

Power in Resistive Components 16.2 Power in Resistive Components Suppose a voltage v = Vp sin t is applied across a resistance R. The resultant current i will be The result power p will be The average value of (1 - cos 2t) is 1, so where V and I are the r.m.s. voltage and current

Relationship between v, i and p in a resistor

16.3 Power in Capacitors From our discussion of capacitors we know that the current leads the voltage by 90. Therefore, if a voltage v = Vp sin t is applied across a capacitance C, the current will be given by i = Ip cos t Then The average power is zero

Relationship between v, i and p in a capacitor

16.4 Power in Inductors From our discussion of inductors we know that the current lags the voltage by 90. Therefore, if a voltage v = Vp sin t is applied across an inductance L, the current will be given by i = -Ip cos t Therefore Again the average power is zero

Relationship between v, i and p in an inductor

Circuit with Resistance and Reactance 16.5 Circuit with Resistance and Reactance When a sinusoidal voltage v = Vp sin t is applied across a circuit with resistance and reactance, the current will be of the general form i = Ip sin (t - ) Therefore, the instantaneous power, p is given by

The expression for p has two components The second part oscillates at 2 and has an average value of zero over a complete cycle this is the power that is stored in the reactive elements and then returned to the circuit within each cycle The first part represents the power dissipated in resistive components. Average power dissipation is

The average power dissipation given by is termed the active power in the circuit and is measured in watts (W) The product of the r.m.s. voltage and current VI is termed the apparent power, S. To avoid confusion this is given the units of volt amperes (VA)

From the above discussion it is clear that In other words, the active power is the apparent power times the cosine of the phase angle. This cosine is referred to as the power factor

Active and Reactive Power 16.6 Active and Reactive Power When a circuit has resistive and reactive parts, the resultant power has 2 parts: The first is dissipated in the resistive element. This is the active power, P The second is stored and returned by the reactive element. This is the reactive power, Q , which has units of volt amperes reactive or var While reactive power is not dissipated it does have an effect on the system for example, it increases the current that must be supplied and increases losses with cables

Consider an RL circuit the relationship between the various forms of power can be illustrated using a power triangle

Therefore Active Power P = VI cos  watts Reactive Power Q = VI sin  var Apparent Power S = VI VA S2 = P2 + Q2

Power Factor Correction 16.7 Power Factor Correction Power factor is particularly important in high-power applications Inductive loads have a lagging power factor Capacitive loads have a leading power factor Many high-power devices are inductive a typical AC motor has a power factor of 0.9 lagging the total load on the national grid is 0.8-0.9 lagging this leads to major efficiencies power companies therefore penalise industrial users who introduce a poor power factor

The problem of poor power factor is tackled by adding additional components to bring the power factor back closer to unity a capacitor of an appropriate size in parallel with a lagging load can ‘cancel out’ the inductive element this is power factor correction a capacitor can also be used in series but this is less common (since this alters the load voltage) for examples of power factor correction see Examples 16.2 and 16.3 in the course text

16.8 Power Transfer When looking at amplifiers, we noted that maximum power transfer occurs in resistive systems when the load resistance is equal to the output resistance this is an example of matching When the output of a circuit has a reactive element maximum power transfer is achieved when the load impedance is equal to the complex conjugate of the output impedance this is the maximum power transfer theorem

Thus if the output impedance Zo = R + jX, maximum power transfer will occur with a load ZL = R - jX

16.9 Three-Phase Systems So far, our discussion of AC systems has been restricted to single-phase arrangement as in conventional domestic supplies In high-power industrial applications we often use three-phase arrangements these have three supplies, differing in phase by 120  phases are labeled red, yellow and blue (R, Y & B)

Relationship between the phases in a three-phase arrangement

Three-phase arrangements may use either 3 or 4 conductors

16.10 Power Measurement When using AC, power is determined not only by the r.m.s. values of the voltage and current, but also by the phase angle (which determines the power factor) consequently, you cannot determine the power from independent measurements of current and voltage In single-phase systems power is normally measured using an electrodynamic wattmeter measures power directly using a single meter which effectively multiplies instantaneous current and voltage

In three-phase systems we need to sum the power taken from the various phases in three-wire arrangements we can deduce the total power from measurements using 2 wattmeter in a four-wire system it may be necessary to use 3 wattmeter in balanced systems (systems that take equal power from each phase) a single wattmeter can be used, its reading being multiplied by 3 to get the total power

Key Points In resistive circuits the average power is equal to VI, where V and I are r.m.s. values In a capacitor the current leads the voltage by 90 and the average power is zero In an inductor the current lags the voltage by 90 and the average power is zero In circuits with both resistive and reactive elements, the average power is VI cos  The term cos  is called the power factor Power factor correction is important in high-power systems High-power systems often use three-phase arrangements

Frequency Characteristics of AC Circuits Chapter 17 Frequency Characteristics of AC Circuits Introduction A High-Pass RC Network A Low-Pass RC Network A Low-Pass RL Network A High-Pass RL Network A Comparison of RC and RL Networks Bode Diagrams Combining the Effects of Several Stages RLC Circuits and Resonance Filters Stray Capacitance and Inductance

17.1 Introduction Earlier we looked at the bandwidth and frequency response of amplifiers Having now looked at the AC behaviour of components we can consider these in more detail The reactance of both inductors and capacitance is frequency dependent and we know that

We will start by considering very simple circuits Consider the potential divider shown here from our earlier consideration of the circuit rearranging, the gain of the circuit is this is also called the transfer function of the circuit

A High-Pass RC Network Consider the following circuit 17.2 A High-Pass RC Network Consider the following circuit which is shown re-drawn in a more usual form

Clearly the transfer function is At high frequencies  is large, voltage gain  1 At low frequencies  is small, voltage gain  0

Since the denominator has real and imaginary parts, the magnitude of the voltage gain is When 1/CR = 1 This is a halving of power, or a fall in gain of 3 dB

The half power point is the cut-off frequency of the circuit the angular frequency C at which this occurs is given by where  is the time constant of the CR network. Also

Substituting  =2f and CR = 1/ 2fC in the earlier equation gives This is the general form of the gain of the circuit It is clear that both the magnitude of the gain and the phase angle vary with frequency

Consider the behaviour of the circuit at different frequencies: When f >> fc fc/f << 1, the voltage gain  1 When f = fc When f << fc

The behaviour in these three regions can be illustrated using phasor diagrams At low frequencies the gain is linearly related to frequency. It falls at -6dB/octave (-20dB/decade)

Frequency response of the high-pass network the gain response has two asymptotes that meet at the cut-off frequency figures of this form are called Bode diagrams

A Low-Pass RC Network Transposing the C and R gives 17.3 A Low-Pass RC Network Transposing the C and R gives At high frequencies  is large, voltage gain  0 At low frequencies  is small, voltage gain  1

A Low-Pass RC Network A similar analysis to before gives 17.3 A Low-Pass RC Network A similar analysis to before gives Therefore when, when CR = 1 Which is the cut-off frequency

Therefore the angular frequency C at which this occurs is given by where  is the time constant of the CR network, and as before

Substituting  =2f and CR = 1/ 2fC in the earlier equation gives This is similar, but not the same, as the transfer function for the high-pass network

Consider the behaviour of this circuit at different frequencies: When f << fc f/fc << 1, the voltage gain  1 When f = fc When f >> fc

The behaviour in these three regions can again be illustrated using phasor diagrams At high frequencies the gain is linearly related to frequency. It falls at 6dB/octave (20dB/decade)

Frequency response of the low-pass network the gain response has two asymptotes that meet at the cut-off frequency you might like to compare this with the Bode Diagram for a high-pass network

17.4 A Low-Pass RL Network Low-pass networks can also be produced using RL circuits these behave similarly to the corresponding CR circuit the voltage gain is the cut-off frequency is

17.5 A High-Pass RL Network High-pass networks can also be produced using RL circuits these behave similarly to the corresponding CR circuit the voltage gain is the cut-off frequency is

A Comparison of RC and RL Networks 17.6 A Comparison of RC and RL Networks Circuits using RC and RL techniques have similar characteristics for a more detailed comparison, see Figure 17.10 in the course text

17.7 Bode Diagrams Straight-line approximations

Creating more detailed Bode diagrams

Combining the Effects of Several Stages 17.8 Combining the Effects of Several Stages The effects of several stages ‘add’ in bode diagrams

Multiple high- and low-pass elements may also be combined this is illustrated in Figure 17.14 in the course text

RLC Circuits and Resonance 17.9 RLC Circuits and Resonance Series RLC circuits the impedance is given by if the magnitude of the reactance of the inductor and capacitor are equal, the imaginary part is zero, and the impedance is simply R this occurs when

This situation is referred to as resonance the frequency at which is occurs is the resonant frequency in the series resonant circuit, the impedance is at a minimum at resonance the current is at a maximum at resonance

The resonant effect can be quantified by the quality factor, Q this is the ratio of the energy dissipated to the energy stored in each cycle it can be shown that and

The series RLC circuit is an acceptor circuit the narrowness of bandwidth is determined by the Q combining this equation with the earlier one gives

Parallel RLC circuits as before

The parallel arrangement is a rejector circuit in the parallel resonant circuit, the impedance is at a maximum at resonance the current is at a minimum at resonance in this circuit

17.10 Filters RC Filters The RC networks considered earlier are first-order or single-pole filters these have a maximum roll-off of 6 dB/octave they also produce a maximum of 90 phase shift Combining multiple stages can produce filters with a greater ultimate roll-off rates (12 dB, 18 dB, etc.) but such filters have a very soft ‘knee’

An ideal filter would have constant gain and zero phase shift for frequencies within its pass band, and zero gain for frequencies outside this range (its stop band) Real filters do not have these idealised characteristics

LC Filters Simple LC filters can be produced using series or parallel tuned circuits these produce narrow-band filters with a centre frequency fo

Active filters combining an op-amp with suitable resistors and capacitors can produce a range of filter characteristics these are termed active filters

see Section 17.10.3 of the course text for more information on these Common forms include: Butterworth optimised for a flat response Chebyshev optimised for a sharp ‘knee’ Bessel optimised for its phase response see Section 17.10.3 of the course text for more information on these

Stray Capacitance and Inductance 17.11 Stray Capacitance and Inductance All circuits have stray capacitance and stray inductance these unintended elements can dramatically affect circuit operation for example: (a) Cs adds an unintended low-pass filter (b) Ls adds an unintended low-pass filter (c) Cs produces an unintended resonant circuit and can produce instability

Key Points The reactance of capacitors and inductors is dependent on frequency Single RC or RL networks can produce an arrangement with a single upper or lower cut-off frequency. In each case the angular cut-off frequency o is given by the reciprocal of the time constant  For an RC circuit  = CR, for an RL circuit  = L/R Resonance occurs when the reactance of the capacitive element cancels that of the inductive element Simple RC or RL networks represent single-pole filters Active filters produce high performance without inductors Stray capacitance and inductance are found in all circuits

Transient Behaviour Introduction Chapter 18 Transient Behaviour Introduction Charging Capacitors and Energising Inductors Discharging Capacitors and De-energising Inductors Response of First-Order Systems Second-Order Systems Higher-Order Systems

18.1 Introduction So far we have looked at the behaviour of systems in response to: fixed DC signals constant AC signals We now turn our attention to the operation of circuits before they reach steady-state conditions this is referred to as the transient response We will begin by looking at simple RC and RL circuits

Charging Capacitors and Energising Inductors 18.2 Charging Capacitors and Energising Inductors Capacitor Charging Consider the circuit shown here Applying Kirchhoff’s voltage law Now, in a capacitor which substituting gives

Assuming VC = 0 at t = 0, this can be solved to give The above is a first-order differential equation with constant coefficients Assuming VC = 0 at t = 0, this can be solved to give see Section 18.2.1 of the course text for this analysis Since i = Cdv/dt this gives (assuming VC = 0 at t = 0) where I = V/R

Thus both the voltage and current have an exponential form

A similar analysis of this circuit gives Inductor energising A similar analysis of this circuit gives where I = V/R – see Section 18.2.2 for this analysis

Thus, again, both the voltage and current have an exponential form

Discharging Capacitors and De-energising Inductors 18.3 Discharging Capacitors and De-energising Inductors Capacitor discharging Consider this circuit for discharging a capacitor At t = 0, VC = V From Kirchhoff’s voltage law giving

Solving this as before gives where I = V/R – see Section 18.3.1 for this analysis

In this case, both the voltage and the current take the form of decaying exponentials

Inductor de-energising A similar analysis of this circuit gives where I = V/R – see Section 18.3.1 for this analysis

And once again, both the voltage and the current take the form of decaying exponentials

A comparison of the four circuits

Response of First-Order Systems 18.4 Response of First-Order Systems Initial and final value formulae increasing or decreasing exponential waveforms (for either voltage or current) are given by: where Vi and Ii are the initial values of the voltage and current where Vf and If are the final values of the voltage and current the first term in each case is the steady-state response the second term represents the transient response the combination gives the total response of the arrangement

Example – see Example 18.3 from course text The input voltage to the following CR network undergoes a step change from 5 V to 10 V at time t = 0. Derive an expression for the resulting output voltage.

Here the initial value is 5 V and the final value is 10 V Here the initial value is 5 V and the final value is 10 V. The time constant of the circuit equals CR = 10  103 20  10-6 = 0.2s. Therefore, from above, for t  0

The nature of exponential curves

Response of first-order systems to a square waveform see Section 18.4.3

Response of first-order systems to a square waveform of different frequencies see Section 18.4.3

18.5 Second-Order Systems Circuits containing both capacitance and inductance are normally described by second-order differential equations. These are termed second-order systems for example, this circuit is described by the equation

When a step input is applied to a second-order system, the form of the resultant transient depends on the relative magnitudes of the coefficients of its differential equation. The general form of the response is where n is the undamped natural frequency in rad/s and  (Greek Zeta) is the damping factor

Response of second-order systems  =0 undamped  <1 under damped  =1 critically damped  >1 over damped

18.6 Higher-Order Systems Higher-order systems are those that are described by third-order or higher-order equations These often have a transient response similar to that of the second-order systems described earlier Because of the complexity of the mathematics involved, they will not be discussed further here

Key Points The charging or discharging of a capacitor, and the energising and de-energising of an inductor, are each associated with exponential voltage and current waveforms Circuits that contain resistance, and either capacitance or inductance, are termed first-order systems The increasing or decreasing exponential waveforms of first-order systems can be described by the initial and final value formulae Circuits that contain both capacitance and inductance are usually second-order systems. These are characterised by their undamped natural frequency and their damping factor

Semiconductor Diodes Introduction Diodes Chapter 19 Semiconductor Diodes Introduction Diodes Electrical Properties of Solids Semiconductors pn Junctions Semiconductor Diodes Special-Purpose Diodes Diode Circuits

19.1 Introduction This course adopts a top-down approach to the subject and so far we have taken a ‘black-box’ view of active components (such as op-amps) It is now time to look ‘inside the box’ we will start by looking at diodes and semiconductors then progress to transistors later we will look at more detailed aspects of circuit design

19.2 Diodes An ideal diode passing electricity in one direction but not the other

One application of diodes is in rectification the example below shows a half-wave rectifier In practice, no real diode has ideal characteristics but semiconductor pn junctions make good diodes To understand such devices we need to look at some properties of materials

Electrical Properties of Solids 19.3 Electrical Properties of Solids Conductors e.g. copper or aluminium have a cloud of free electrons (at all temperatures above absolute zero). If an electric field is applied electrons will flow causing an electric current Insulators e.g. polythene electrons are tightly bound to atoms so few can break free to conduct electricity

Semiconductors e.g. silicon or germanium at very low temperatures these have the properties of insulators as the material warms up some electrons break free and can move about, and it takes on the properties of a conductor - albeit a poor one however, semiconductors have several properties that make them distinct from conductors and insulators

Semiconductors Pure semiconductors 19.4 Semiconductors Pure semiconductors thermal vibration results in some bonds being broken generating free electrons which move about these leave behind holes which accept electrons from adjacent atoms and therefore also move about electrons are negative charge carriers holes are positive charge carriers At room temperatures there are few charge carriers pure semiconductors are poor conductors this is intrinsic conduction

Doping the addition of small amounts of impurities drastically affects its properties some materials form an excess of electrons and produce an n-type semiconductor some materials form an excess of holes and produce a p-type semiconductor both n-type and p-type materials have much greater conductivity than pure semiconductors this is extrinsic conduction

The dominant charge carriers in a doped semiconductor (e. g The dominant charge carriers in a doped semiconductor (e.g. electrons in n-type material) are called majority charge carriers. Other type are minority charge carriers The overall doped material is electrically neutral

19.5 pn Junctions When p-type and n-type materials are joined this forms a pn junction majority charge carriers on each side diffuse across the junction where they combine with (and remove) charge carriers of the opposite polarity hence around the junction there are few free charge carriers and we have a depletion layer (also called a space-charge layer)

The diffusion of positive charge in one direction and negative charge in the other produces a charge imbalance this results in a potential barrier across the junction

Potential barrier the barrier opposes the flow of majority charge carriers and only a small number have enough energy to surmount it this generates a small diffusion current the barrier encourages the flow of minority carriers and any that come close to it will be swept over this generates a small drift current for an isolated junction these two currents must balance each other and the net current is zero

Forward bias if the p-type side is made positive with respect to the n-type side the height of the barrier is reduced more majority charge carriers have sufficient energy to surmount it the diffusion current therefore increases while the drift current remains the same there is thus a net current flow across the junction which increases with the applied voltage

Reverse bias if the p-type side is made negative with respect to the n-type side the height of the barrier is increased the number of majority charge carriers that have sufficient energy to surmount it rapidly decreases the diffusion current therefore vanishes while the drift current remains the same thus the only current is a small leakage current caused by the (approximately constant) drift current the leakage current is usually negligible (a few nA)

Currents in a pn junction

Forward and reverse currents pn junction current is given approximately by where I is the current, e is the electronic charge, V is the applied voltage, k is Boltzmann’s constant, T is the absolute temperature and  (Greek letter eta) is a constant in the range 1 to 2 determined by the junction material for most purposes we can assume  = 1

Thus If V > +0.1 V If V < -0.1 V at room temperature e/kT ~ 40 V-1 If V > +0.1 V If V < -0.1 V IS is the reverse saturation current

19.6 Semiconductor Diodes Forward and reverse currents

Silicon diodes generally have a turn-on voltage of about 0.5 V generally have a conduction voltage of about 0.7 V have a breakdown voltage that depends on their construction perhaps 75 V for a small-signal diode perhaps 400 V for a power device have a maximum current that depends on their construction perhaps 100 mA for a small-signal diode perhaps many amps for a power device

Turn-on and breakdown voltages for a silicon device

Special-Purpose Diodes 19.7 Special-Purpose Diodes Light-emitting diodes discussed earlier when we looked at light actuators

Zener diodes uses the relatively constant reverse breakdown voltage to produce a voltage reference breakdown voltage is called the Zener voltage, VZ output voltage of circuit shown is equal to VZ despite variations in input voltage V a resistor is used to limit the current in the diode

Schottky diodes formed by the junction between a layer of metal (e.g. aluminium) and a semiconductor action relies only on majority charge carriers much faster in operation than a pn junction diode has a low forward voltage drop of about 0.25 V used in the design of high-speed logic gates

Tunnel diodes high doping levels produce a very thin depletion layer which permits ‘tunnelling’ of charge carriers results in a characteristic with a negative resistance region used in high-frequency oscillators, where they can be used to ‘cancel out’ resistance in passive components

Varactor diodes a reversed-biased diode has two conducting regions separated by an insulating depletion region this structure resembles a capacitor variations in the reverse-bias voltage change the width of the depletion layer and hence the capacitance this produces a voltage-dependent capacitor these are used in applications such as automatic tuning circuits

Diode Circuits Half-wave rectifier 19.8 Diode Circuits Half-wave rectifier peak output voltage is equal to the peak input voltage minus the conduction voltage of the diode reservoir capacitor used to produce a steadier output

Full-wave rectifier use of a diode bridge reduces the time for which the capacitor has to maintain the output voltage and thus reduced the ripple voltage

Signal rectifier used to demodulate full amplitude modulated signals (full-AM) also known as an envelope detector found in a wide range of radio receivers from crystal sets to superheterodynes

Signal clamping a simple form of signal conditioning circuits limit the excursion of the voltage waveform can use a combination of signal and Zener diodes

Catch diode used when switching inductive loads the large back e.m.f. can cause problems such as arcing in switches catch diodes provide a low impedance path across the inductor to dissipate the stored energy the applied voltage reverse-biases the diode which therefore has no effect when the voltage is removed the back e.m.f. forward biases the diode which then conducts

Key Points Diodes allow current to flow in only one direction At low temperatures semiconductors act like insulators At higher temperatures they begin to conduct Doping of semiconductors leads to the production of p-type and n-type materials A junction between p-type and n-type semiconductors has the properties of a diode Silicon semiconductor diodes approximate the behaviour of ideal diodes but have a conduction voltage of about 0.7 V There are also a wide range of special purpose diodes Diodes are used in a range of applications

Field-Effect Transistors Chapter 20 Field-Effect Transistors Introduction An Overview of Field-Effect Transistors Insulated-Gate Field-Effect Transistors Junction-Gate Field-Effect Transistors FET Characteristics Summary of FET Characteristics FET Amplifiers Other FET Applications

20.1 Introduction Field-effect transistors (FETs) are probably the simplest form of transistor widely used in both analogue and digital applications they are characterised by a very high input resistance and small physical size, and they can be used to form circuits with a low power consumption they are widely used in very large-scale integration two basic forms: insulated gate FETs junction gate FETs

An Overview of Field-Effect Transistors 20.2 An Overview of Field-Effect Transistors Many forms, but basic operation is the same a voltage on a control input produces an electric field that affects the current between two other terminals when considering amplifiers we looked at a circuit using a ‘control device’ a FET is a suitable control device

Notation FETs are 3 terminal devices the gate is the control input drain (d) source (s) gate(g) the gate is the control input diagram illustrates the notation used for labelling voltages and currents

Insulated-Gate Field-Effect Transistors 20.3 Insulated-Gate Field-Effect Transistors Such devices are sometimes called IGFETs (insulated-gate field-effect transistors) or sometimes MOSFETs (metal oxide semiconductor field-effect transistors) Digital circuits constructed using these devices are usually described as using MOS technology Here we will describe them as MOSFETs

Construction two polarities: n-channel and p-channel

Operation gate volt controls the thickness of the channel consider an n-channel device making the gate more positive attracts electrons to the gate and makes the gate region thicker – reducing the resistance of the channel. The channel is said to be enhanced making the gate more negative repels electrons from the gate and makes the gate region thinner – increasing the resistance of the channel. The channel is said to be depleted

the effect of varying the gate voltage

gates as described above are termed Depletion-Enhancement MOSFETs or simply DE MOSFETs some MOSFETs are constructed so that in the absence of any gate voltage there is no channel such devices can be operated in an enhancement mode, but not in a depletion mode (since there is no channel to deplete) these are called Enhancement MOSFETs both forms of MOSFET are available as either n-channel or p-channel devices

MOSFET circuit symbols

Junction-Gate Field-Effect Transistors 20.4 Junction-Gate Field-Effect Transistors Sometimes known as a JUGFET Here we will use another common name – the JFET Here the insulated gate of a MOSFET is replaced with a reverse-biased pn junction Since the gate junction is always reverse-biased no current flows into the gate and it acts as if it were insulated

Construction two polarities: n-channel and p-channel

Operation the reverse-biased gate junction produced a depletion layer in the region of the channel the gate volt controls the thickness of the depletion layer and hence the thickness of the channel consider an n-channel device the gate will always be negative with respect to the source to keep the junction between the gate and the channel reverse-biased making the gate more negative increases the thickness of the depletion layer, reducing the width of the channel – increasing the resistance of the channel.

the effect of varying the gate voltage

JFET circuit symbols

20.5 FET Characteristics While MOSFETs and JFETs operate in different ways, their characteristics are quite similar Input characteristics in both MOSFETs and JFETs the gate is effectively insulated from the remainder of the device Output characteristics consider n-channel devices usually the drain is more positive than the source the drain voltage affects the thickness of the channel

FET output characteristics

Transfer characteristics similar shape for all forms of FET – but with a different offset not a linear response, but over a small region might be considered to approximate a linear response

Normal operating ranges for FETs

When operating about its operating point we can describe the transfer characteristic by the change in output that is caused by a certain change in the input this corresponds to the slope of the earlier curves this quantity has units of current/voltage, which is the reciprocal of resistance (this is conductance) since this quantity described the transfer characteristics it is called the transconductance, gm Note:

Small-signal equivalent circuit of a FET models the behaviour of the device for small variations of the input about the operating point

Summary of FET Characteristics 20.6 Summary of FET Characteristics FETS have three terminals: drain, source and gate The gate is the control input Two polarities of device: n-channel and p-channel Two main forms of FET: MOSFET and JFET In each case the drain current is controlled by the voltage applied to the gate with respect to the source Behaviour is characterised by the transconductance The operating point differs between devices

FET circuit symbols:

FET Amplifiers A simple DE MOSFET amplifier 20.7 FET Amplifiers A simple DE MOSFET amplifier RG is used to ‘bias’ the gate at its correct operating point (which for a DE MOSFET is 0 V) C is a coupling capacitor and is used to couple the AC signal while preventing externals circuits from affecting the bias this is an AC-coupled amplifier

AC-coupled amplifier input resistance – equal to RG output resistance – approximately equal to RD gain – approximately –gmRD (the minus sign shows that this is an inverting amplifier) C produced a low-frequency cut-off at a frequency fc given by where R is the input resistance of the amplifier (which in this case is equal to RG)

Negative feedback amplifier reduces problems of variability of active components voltage across Rs is proportional to drain current, which is directly proportional to the output voltage this voltage is subtracted from input voltage to gate hence negative feedback

Source follower similar to earlier circuit, but output is now taken from the source feedback causes the source to follow the input voltage produces a unity-gain amplifier also called a source follower

Other FET Applications 20.8 Other FET Applications A voltage controlled attenuator for small drain-to-source voltages FETs resemble voltage-controlled resistors the gate voltage VG is used to control this resistance and hence the gain of the potential divider used, for example, in automatic gain control in radio receivers

A FET as an analogue switch

A FET as a logical switch

Key Points FETs are widely used in both analogue and digital circuits They have high input resistance and small physical size There are two basic forms of FET: MOSFETs and JFETs MOSFETs may be divided into DE and Enhancement types In each case the gate voltage controls the current from the drain to the source The characteristics of the various forms of FET are similar except that they require different bias voltages The use of coupling capacitors prevents the amplification of DC and produced AC amplifiers FETs can be used to produce various forms of amplifier and a range of other circuit applications

Bipolar Transistors Introduction An Overview of Bipolar Transistors Chapter 21 Bipolar Transistors Introduction An Overview of Bipolar Transistors Bipolar Transistor Operation Bipolar Transistor Characteristics Summary of Bipolar Transistor Characteristics Bipolar Transistor Amplifiers Other Bipolar Transistor Applications

21.1 Introduction Bipolar transistors are one of the main ‘building-blocks’ in electronic systems They are used in both analogue and digital circuits They incorporate two pn junctions and are sometimes known as bipolar junction transistors or BJTs Here will refer to them simply as bipolar transistors

An Overview of Bipolar Transistors 21.2 An Overview of Bipolar Transistors While control in a FET is due to an electric field, control in a bipolar transistor is generally considered to be due to an electric current current into one terminal determines the current between two others as with a FET, a bipolar transistor can be used as a ‘control device’

Notation Notation bipolar transistors are 3 terminal devices collector (c) base (b) emitter (e) the base is the control input diagram illustrates the notation used for labelling voltages and currents Notation

Relationship between the collector current and the base current in a bipolar transistor characteristic is approximately linear magnitude of collector current is generally many times that of the base current the device provides current gain

Construction two polarities: npn and pnp

Bipolar Transistor Operation 21.3 Bipolar Transistor Operation We will consider npn transistors pnp devices are similar but with different polarities of voltage and currents when using npn transistors collector is normally more positive than the emitter VCE might be a few volts device resembles two back-to-back diodes – but has very different characteristics with the base open-circuit negligible current flows from the collector to the emitter

Now consider what happens when a positive voltage is applied to the base (with respect to the emitter) this forward biases the base-emitter junction the base region is light doped and very thin because it is likely doped, the current produced is mainly electrons flowing from the emitter to the base because the base region is thin, most of the electrons entering the base get swept across the base-collector junction into the collector this produces a collector current that is much larger than the base current – this gives current amplification

Transistor action

Bipolar Transistor Characteristics 21.4 Bipolar Transistor Characteristics Behaviour can be described by the current gain, hfe or by the transconductance, gm of the device

Transistor configurations transistors can be used in a number of configurations most common is as shown emitter terminal is common to input and output circuits this is a common-emitter configuration we will look at the characteristics of the device in this configuration

Input characteristics the input takes the form of a forward-biased pn junction the input characteristics are therefore similar to those of a semiconductor diode

Output characteristics region near to the origin is the saturation region this is normally avoided in linear circuits slope of lines represents the output resistance

Transfer characteristics can be described by either the current gain or by the transconductance DC current gain hFE or  is given by IC / IB AC current gain hfe is given by ic / ib transconductance gm is given approximately by gm  40IC  40 IE siemens

Equivalent circuits for a bipolar transistor

Summary of Bipolar Transistor Characteristics 21.5 Summary of Bipolar Transistor Characteristics Bipolar transistors have three terminals: collector, base and emitter The base is the control input Two polarities of device: npn and pnp The collector current is controlled by the base voltage/current IC = hFEIB Behaviour is characterised by the current gain or the transconductance

Bipolar Transistor Amplifiers 21.6 Bipolar Transistor Amplifiers A simple transistor amplifier RB is used to ‘bias’ the transistor by injecting an appropriate base current C is a coupling capacitor and is used to couple the AC signal while preventing external circuits from affecting the bias this is an AC-coupled amplifier

AC-coupled amplifier VB is set by the conduction voltage of the base-emitter junction and so is about 0.7 V voltage across RB is thus VCC – 0.7 this voltage divided by RB gives the base current IB the collector current is then given by IC = hFEIB the voltage drop across RC is given by IC RC the quiescent output voltage is therefore Vo = VCC - IC RC output is determined by hFE which is very variable

Negative feedback amplifiers

Example – see Example 21.2 from course text Determine the quiescent output voltage of this circuit

Base current is small, so Emitter voltage VE = VB – VBE = 2.7 – 0.7 = 2.0 V Emitter current Since IB is small, collector current IC  IE = 2 mA Output voltage = VCC – ICRC = 10 - 2 mA 2.2 k = 5.6 V

A common-collector amplifier unity gain high input resistance low output resistance a very good buffer amplifier

Other Bipolar Transistor Applications 21.7 Other Bipolar Transistor Applications A phase splitter

A voltage regulator

A logical switch

Key Points Bipolar transistors are widely used in both analogue and digital circuits They can be considered as either voltage-controlled or current-controlled devices Their characteristics may be described by their gain or by their transconductance Feedback can be used to overcome problems of variability The majority of circuits use transistors in a common-emitter configuration where the input is applied to the base and the output is taken from the collector Common-collector circuits make good buffer amplifiers Bipolar transistors are used in a wide range of applications

Power Electronics Introduction Bipolar Transistor Power Amplifiers Chapter 22 Power Electronics Introduction Bipolar Transistor Power Amplifiers Classes of Amplifier Four-layer Devices Power Supplies and Voltage Regulators

22.1 Introduction Amplifiers that produce voltage amplification or current amplification also produce power amplification However, the term power amplifier is normally reserved for circuits whose main function is to deliver large amounts of power These can be produced using FETs or bipolar transistors, or using special purpose devices such as thyristors and triacs

Bipolar Transistor Power Amplifiers 22.2 Bipolar Transistor Power Amplifiers When designing a power amplifier we normally require a low output resistance so that the circuit can deliver a high output current we often use an emitter-follower this does not produce voltage gain but has a low output resistance in many cases the load applied to a power amplifier is not simply resistive but also has an inductive or capacitive element

Current sources and loads when driving a reactive load we need to supply current at some times (the output acts as a current source) at other times we need to absorb current (the output acts as a current sink)

the circuit above is a good current source but a poor current sink (stored charge must be removed by RE) an alternative circuit using pnp transistors (below) is a good current sink but a poor current source

Push-pull amplifiers combining these circuits can produce an arrangement that is both a good current source and a good current sink this is termed a push-pull amplifier

Driving a push-pull stage

Distortion in push-pull amplifiers

Improved push-pull output stage arrangements

Amplifier efficiency an important consideration in the design of power amplifiers is efficiency efficiency determines the power dissipated in the amplifier itself power dissipation is important because it determines the amount of waste heat produced excess heat may require heat sinks, cooling fans, etc.

Classes of Amplifier Class A 22.3 Classes of Amplifier Class A active device conducts for complete cycle of input signal example shown here poor efficiency (normally less than 25%) low distortion

Class B active devices conducts for half of the complete cycle of input signal example shown here good efficiency (up to 78%) considerable distortion

Class AB active devices conducts for more than half but less than the complete cycle of input signal example shown here (with appropriate Rbias) efficiency depends on bias distortion depends on bias

Class C active devices conducts for less than half the complete cycle of input signal example shown here high efficiency (approaching 100%) gross distortion

Class D in class D amplifiers the active devices are switches and are either ON or OFF an ideal switch would dissipate no power since either the current or the voltage is zero even real devices make good switches amplifiers of this type are called switching amplifiers or switch-mode amplifiers efficiency is very high

22.4 Four-layer Devices Although transistors make excellent switches, they have limitations when it comes to switching high currents at high voltages In such situations we often use devices that are specifically designed for such applications These are four-layer devices these are not transistors, but have a great deal in common with bipolar transistors

The thyristor a four-layer device with a pnpn structure three terminals: anode, cathode and gate gate is the control input

Thyristor operation construction resembles two interconnected bipolar transistors turning on T2 holds on T1 device then conducts until the current goes to zero

Use of a thyristor in AC power control once triggered the device conducts for the remainder of the half cycle varying firing time determines output power allows control from 0-50% of full power

Full-wave power control using thyristors full-wave control required two devices allows control from 0-100% of full power requires two gate drive circuits opto-isolation often used to insulate circuits from AC supply

The triac resembles a bidirectional thyristor allows full-wave control using a single device often used with a bidirectional trigger diode (a diac) to produce the necessary drive pulses this breaks down at a particular voltage and fires the triac

A simple lamp-dimmer using a triac

Power Supplies and Voltage Regulators 22.5 Power Supplies and Voltage Regulators Unregulated DC power supplies

Regulated DC power supplies

Voltage regulators

Switch-mode power supplies uses a switching regulator output voltage is controlled by the duty-cycle of the switch uses an averaging circuit to ‘smooth’ output

An LC averaging circuit

Using feedback in a switching regulator

Key Points Power amplifiers are designed to deliver large amounts of power to their load Bipolar circuits often use an emitter follower circuit Many power amplifiers use a push-pull arrangement The efficiency of an amplifier is greatly affected by its class While transistors make excellent switches, in high power applications we often use special-purpose devices such as thyristors or triacs A transformer, a rectifier and a capacitor can be used to form a simple unregulated supply A more constant output voltage can be produced by adding a regulator. This can use linear or switching techniques

Electric Motors and Generators Chapter 23 Electric Motors and Generators Introduction A Simple AC Generator A Simple DC Generator DC Generators or Dynamos AC Generators or Alternators DC Motors AC Motors Universal Motors Electrical Machines – A Summary

23.1 Introduction In this lecture we consider various forms of rotating electrical machines These can be divided into: generators – which convert mechanical energy into electrical energy motors – which convert electrical energy into mechanical energy Both types operate through the interaction between a magnetic field and a set of windings

23.2 A Simple AC Generator We noted earlier that Faraday’s law dictates that if a coil of N turns experiences a change in magnetic flux, then the induced voltage V is given by If a coil of area A rotates with respect to a field B, and if at a particular time it is at an angle  to the field, then the flux linking the coil is BAcos, and the rate of change of flux is given by

Thus for the arrangement shown below

Therefore this arrangement produces a sinusoidal output as shown below

Wires connected to the rotating coil would get twisted Therefore we use circular slip rings with sliding contacts called brushes

23.3 A Simple DC Generator The alternating signal from the earlier AC generator could be converted to DC using a rectifier A more efficient approach is to replace the two slip rings with a single split slip ring called a commutator this is arranged so that connections to the coil are reversed as the voltage from the coil changes polarity hence the voltage across the brushes is of a single polarity adding additional coils produces a more constant output

Use of a commutator

A simple generator with two coils

The ripple can be further reduced by the use of a cylindrical iron core and by shaping the pole pieces this produces an approximately uniform field in the narrow air gap the arrangement of coils and core is known as the armature

DC Generators or Dynamos 23.4 DC Generators or Dynamos Practical DC generators or dynamos can take a number of forms depending on how the magnetic field is produced can use a permanent magnet more often it is generated electrically using field coils current in the field coils can come from an external supply this is known as a separately excited generator but usually the field coils are driven from the generator output this is called a self-excited generator often use multiple poles held in place by a steel tube called the stator

A four-pole DC generator

Field coil excitation sometimes the field coils are connected in series with the armature, sometimes in parallel (shunt) and sometimes a combination of the two (compound) these different forms produce slightly different characteristics diagram here shows a shunt-wound generator

DC generator characteristics vary slightly between forms examples shown here are for a shunt-wound generator

AC Generators or Alternators 23.5 AC Generators or Alternators Alternators do not require commutation this allows a simpler construction the field coils are made to rotate while the armature windings are stationary Note: the armature windings are those that produce the output thus the large heavy armature windings are in the stator the lighter field coils are mounted on the rotor and direct current is fed to these by a set of slip rings

A four-pole alternator

As with DC generators multiple poles and sets of windings are used to improve efficiency sometimes three sets of armature windings are spaced 120 apart around the stator to form a three-phase generator The e.m.f. produced is in sync with rotation of the rotor so this is a synchronous generator if the generator has a single set of poles the output frequency is equal to the rotation frequency if additional pole-pairs are used the frequency is increased accordingly

Example – see Example 23.2 from course text A four-pole alternator is required to operate at 60 Hz. What is the required rotation speed? A four-pole alternator has two pole pairs. Therefore the output frequency is twice the rotation speed. Therefore to operate at 60Hz, the required speed must be 60/2 = 30Hz. This is equivalent to 30  60 = 1800 rpm.

23.6 DC Motors When current flows in a conductor it produces a magnetic field about it - as shown in (a) below when the current-carrying conductor is within an externally generated magnetic field, the fields interact and a force is exerted on the conductor - as in (b)

Therefore if a conductor lies within a magnetic field: motion of the conductor produces an electric current an electric current in the conductor will generate motion The reciprocal nature of this relationship means that, for example, the DC generator above will function as a DC motor although machines designed as motors are more efficient in this role Thus the four-pole DC generator shown earlier could equally well be a four-pole DC motor

DC motor characteristics many forms – each with slightly different characteristics again can be permanent magnet, or series-wound, shunt-wound or compound wound figure below shows a shunt-wound DC motor

AC Motors AC motors can be divided into two main forms: 23.7 AC Motors AC motors can be divided into two main forms: synchronous motors induction motors High-power versions of either type invariably operate from a three-phase supply, but single-phase versions of each are also widely used – particularly in a domestic setting

Synchronous motors just as a DC generator can be used as a DC motor, so AC generators (or alternators) can be used as synchronous AC motors three phase motors use three sets of stator coils the rotating magnetic field drags the rotor around with it single phase motors require some starting mechanism torque is only produced when the rotor is in sync with the rotating magnetic field not self-starting – may be configured as an induction motor until its gets up to speed, then becomes a synchronous motor

Induction motors these are perhaps the most important form of AC motor rather than use slip rings to pass current to the field coils in the rotor, current is induced in the rotor by transformer action the stator is similar to that in a synchronous motor the rotor is simply a set of parallel conductors shorted together at either end by two conducting rings

A squirrel-cage induction motor

In a three-phase induction motor the three phases produce a rotating magnetic field (as in a three-phase synchronous motor) a stationary conductor will see a varying magnetic field and this will induce a current current is induced in the field coils in the same way that current is induced in the secondary of a transformer this current turns the rotor into an electromagnet which is dragged around by the rotating magnetic field the rotor always goes slightly slower than the magnetic field – this is the slip of the motor

In single-phase induction motors other techniques must be used to produce the rotating magnetic field various techniques are used leading to various forms of motor such as capacitor motors shaded-pole motors such motors are inexpensive and are widely used in domestic applications

23.8 Universal Motors While most motors operate from either AC or DC, some can operate from either These are universal motors and resemble series-wound DC motors, but are designed for both AC and DC operation typically operate at high speed (usually > 10,000 rpm) offer high power-to-weight ratio ideal for portable equipment such as hand drills and vacuum cleaners

Electrical Machines – A Summary 23.9 Electrical Machines – A Summary Power generation is dominated by AC machines range from automotive alternators to the synchronous generators used in power stations efficiency increases with size (up to 98%) Both DC and AC motors are used high-power motors are usually AC, three-phase domestic applications often use single-phase induction motors DC motors are useful in control applications

Key Points Electrical machines include both generators and motors Motors can usually function as generators, and vice versa Electrical machines can be divided into AC and DC forms The rotation of a coil in a uniform magnetic field produces a sinusoidal e.m.f. This is the basis of an AC generator A commutator can be used to produce a DC generator The magnetic field in an electrical machine is normally produced electrically using field coils DC motors are often similar in form to DC generators Some forms of AC generator can also be used as motors The most widely used form of AC motor is the induction motor

Positive Feedback, Oscillators and Stability Chapter 24 Positive Feedback, Oscillators and Stability Introduction Oscillators Stability

Introduction Earlier we looked at feedback in general terms 24.1 Introduction Earlier we looked at feedback in general terms in particular we concentrated on negative feedback In this chapter we will consider positive feedback this is used in both analogue and digital circuits it is used in the production of oscillators positive feedback can occur unintentionally within circuits when it has implications for stability

Oscillators Earlier we looked at a generalised feedback system 24.2 Oscillators Earlier we looked at a generalised feedback system We also derived the closed-loop gain G of this

Looking at the expression we note that when AB = -1, the gain is infinite this represents the condition for oscillation The requirements for oscillation are described by the Baukhausen criterion: The magnitude of the loop gain AB must be 1 The phase shift of the loop gain AB must be 180, or 180 plus an integer multiple of 360

RC or phase-shift oscillator one way of producing a phase shift of 180 is to use an RC ladder network this gives a phase shift of 180 when at this frequency the gain of the network is

Therefore the complete oscillator is

Wien-bridge oscillator uses a Wien-bridge network this produces a phase-shift of 0 at a single frequency, and is used with an inverting amplifier the selected frequency is when the gain is 1/3

A complete oscillator might look like

Amplitude stabilisation in both the oscillators above, the loop gain is set by component values in practice the gain of the active components is very variable if the gain of the circuit is too high it will saturate if the gain of the circuit is too low the oscillation will die real circuits need some means of stabilising the magnitude of the oscillation to cope with variability in the gain of the circuit see Section 24.2.3 in the course text for more discussion of this topic

Digital oscillators many examples, for example the relaxation oscillator

Crystal oscillators frequency stability is determined by the ability of the circuit to select a particular frequency in tuned circuits this is described by the quality factor, Q piezoelectric crystals act like resonant circuits with a very high Q – as high as 100,000

A typical crystal oscillator

24.3 Stability Earlier we used a general expression for the gain of a feedback network So far we have assumed that A and B are real gains the gain of a real amplifier has not only a magnitude, but also a phase angle a phase shift of 180 represents an inversion and so the gain changes polarity this can turn negative feedback into positive feedback

The gain of all real amplifiers falls at high frequencies and this also produces a phase shift All multi-stage amplifiers will produce 180 of phase shift at some frequency To ensure stability we must ensure that the Baukhausen conditions for oscillation are not met to guarantee this we must ensure that the gain falls below unity before the phase shift reaches 180

Gain and phase margins these are a measure of the stability of a circuit

Unintended feedback stability can also be affected by unintended feedback within a circuit this might be caused by stray capacitance or stray inductance if these produce positive feedback they can cause instability a severe problem in high-frequency applications must be tackled by careful design

Key Points Positive feedback is used in analogue and digital systems A primary use is in the production of oscillators The requirement for oscillation is that the loop gain AB must have a magnitude of 1, and a phase shift of 180 (or 180 plus some integer multiple of 360) This can be achieved using a circuit that produces a phase shift of 180 together with a non-inverting amplifier Alternatively, it can be achieved using a circuit that produces a phase shift of 0 with an inverting amplifier For good frequency stability we often use crystals Care must be taken to ensure the stability of all feedback systems

Digital Components Introduction Gate Characteristics Logic Families Chapter 25 Digital Components Introduction Gate Characteristics Logic Families Logic Family Characteristics A Comparison of Logic Families Complementary Metal Oxide Semiconductor Transistor-Transistor Logic

25.1 Introduction Earlier we looked at a range of digital applications based on logic gates – at that time we treated the gates as ‘black boxes’ We will now consider the construction of such gates, and their characteristics In this lecture we will concentrate on small- and medium-scale integration circuits containing just a handful of gates typical gates are shown on the next slide

Typical logic device pin-outs

Gate Characteristics The inverter or NOT gate 25.2 Gate Characteristics The inverter or NOT gate consider the characteristics of a simple inverting amplifier as shown below we normally use only the linear region

We can use an inverting amplifier as a logical inverter but using only the non-linear region

we choose input values to ensure that we are always outside of the linear region – as in (a) unlike linear amplifiers, we use circuits with a rapid transition between the non-linear regions – as in (b)

Logic levels the voltage ranges representing ‘0’ and ‘1’ represent the logic levels of the circuit often logic 0 is represented by a voltage close to 0 V but the allowable voltage range varies considerably the voltage used to represent logic 1 also varies greatly. In some circuits it might be 2-4 V, while in others it might be 12-15 V in order for one gate to work with another the logic levels must be compatible

Noise immunity noise is present in all real systems this adds random fluctuations to voltages representing logic levels to cope with noise, the voltage ranges defining the logic levels are more tightly constrained at the output of a gate than at the input thus small amounts of noise will not affect the circuit the maximum noise voltage that can be tolerated by a circuit is termed its noise immunity, VNI

Transistors as switches both FETs and bipolar transistors make good switches neither form produce ideal switches and their characteristics are slightly different both forms of device take a finite time to switch and this produces a slight delay in the operation of the gate this is termed the propagation delay of the circuit

The FET as a logical switch

Rise and fall times because the waveforms are not perfectly square we need a way of measuring switching times we measure the rise time, tr and fall time, tf as shown below

The bipolar transistor as a logical switch

when the input voltage to a bipolar transistor is high the transistor turns ON and the output voltage is driven down to its saturation voltage which is about 0.1 V however, saturation of the transistor results in the storage of excess charge in the base region this increases the time taken to turn OFF the device – an effect known as storage time this makes the device faster to turn ON than OFF some switching circuits increase speed by preventing the transistors from entering saturation

Timing considerations all gates have a certain propagation delay time, tPD this is the average of the two switching times

25.3 Logic Families We have seen that different devices use different voltages ranges for their logic levels They also differ in other characteristics In order to assure correct operation when gates are interconnected they are normally produced in families The most widely used families are: complementary metal oxide semiconductor (CMOS) transistor-transistor logic (TTL) emitter-coupled logic (ECL)

Logic Family Characteristics 25.4 Logic Family Characteristics Complementary metal oxide semiconductor (CMOS) most widely used family for large-scale devices combines high speed with low power consumption usually operates from a single supply of 5 – 15 V excellent noise immunity of about 30% of supply voltage can be connected to a large number of gates (about 50) many forms – some with tPD down to 1 ns power consumption depends on speed (perhaps 1 mW)

Transistor-transistor logic (TTL) based on bipolar transistors one of the most widely used families for small- and medium-scale devices – rarely used for VLSI typically operated from 5V supply typical noise immunity about 1 – 1.6 V many forms, some optimised for speed, power, etc. high speed versions comparable to CMOS (~ 1.5 ns) low-power versions down to about 1 mW/gate

Emitter-coupled logic (ECL) based on bipolar transistors, but removes problems of storage time by preventing the transistors from saturating very fast operation - propagation delays of 1ns or less high power consumption, perhaps 60 mW/gate low noise immunity of about 0.2-0.25 V used in some high speed specialist applications, but now largely replaced by high speed CMOS

A Comparison of Logic Families 25.5 A Comparison of Logic Families Parameter CMOS TTL ECL Basic gate NAND/NOR NAND OR/NOR Fan-out >50 10 25 Power per gate (mW) 1 @ 1 MHz 1 - 22 4 - 55 Noise immunity Excellent Very good Good tPD (ns) 1 - 200 1.5 – 33 1 - 4

Complementary Metal Oxide Semiconductor 25.6 Complementary Metal Oxide Semiconductor A CMOS inverter

CMOS gates

CMOS logic levels and noise immunity

Transistor-Transistor Logic 25.7 Transistor-Transistor Logic Discrete TTL inverter and NAND gate circuits

A basic integrated circuit TTL NAND gate

A standard TTL NAND gate

A TTL NAND gate with open collector output

Key Points Physical gates are not ideal components Logic gates are manufactured in a range of logic families The ability of a gate to ignore noise is its ‘noise immunity’ Both MOSFETs and bipolar transistors are used in gates All logic gates exhibit a propagation delay when responding to changes in their inputs The most widely used logic families are CMOS and TTL CMOS is available in a range of forms offering high speed or very low power consumption TTL logic is also produced in many versions, each optimised for a particular characteristic

Data Acquisition and Conversion Chapter 26 Data Acquisition and Conversion Introduction Sampling Signal Reconstruction Data Converters Sample and Hold Gates Multiplexing

26.1 Introduction Digital techniques have several advantages over analogue methods: they are less affected by noise processing, transmission and storage is often easier However, we often produce or use analogue signals Therefore, we often have the need to translate between analogue and digital representations

26.2 Sampling In order to obtain a picture of a varying quantity we need to take regular measurements this process is called sampling but how often do we need to sample? The answer is given by the Nyquist sampling theorem which says that: the sampling rate must be greater than twice the highest frequency present in the signal being sampled this minimum sampling rate is the Nyquist rate

The effects of sampling rate are illustrated here: (a) shows the original signal (b) shows the effects of sampling at a rate above the Nyquist rate (c) shows the effects of sampling at a rate below the Nyquist rate

Note: the sampling rate is determined by the highest frequency present in the signal, not the highest frequency of interest If a signal contains unwanted high frequency components these should be removed before sampling this is done using a low-pass filter such a filter is called an anti-aliasing filter It is common to sample at about 20% above the Nyquist rate to allow for imperfect filtering

Signal Reconstruction 26.3 Signal Reconstruction In many cases it is necessary to reconstruct an analogue signal from a series of samples typically after they have been processed, transmitted or stored This requires the removal of the step transitions in the sampled waveform Reconstruction is achieved using a low-pass filter to remove these unwanted frequencies this filter is called a reconstruction filter

26.4 Data Converters Sampling involves taking a series of instantaneous measurements of a signal and converting these into a digital form Reconstruction involves taking a series of digital readings and converting these back into their analogue equivalents These two operations are performed by data converters which can be of two basic types: analogue-to-digital converters (ADCs) digital-to-analogue converters (DACs)

Resolution of data converters a range of converters is available, each providing conversion to a particular resolution this determines the number of quantisation levels used an n-bit converter uses 2n discrete steps e.g. an 8-bit converter uses 28 or 256 levels a 10-bit converter uses 210 or 1024 levels an 8-bit converter gives a resolution of about 0.25% where greater accuracy is required converters with up to 20-bit resolution or more are available

Speed of conversion conversions of either form take a finite time this is referred to as the conversion time or settling time of the converter the time taken depends on the converter DACs are usually faster than ADCs

Digital-to-analogue converters (DACs) available with a wide range of resolutions and in general conversion time increases with resolution a typical general-purpose 8-bit DAC would have a settling time of between 100 ns and 1 s a typical 16-bit converter would have a settling time of a few milliseconds for specialist applications high-speed converters have settling times of a few nanoseconds a video DAC might have a resolution of 8 bits and a maximum sampling rate of 100 MHz

Analogue-to-digital converters (ADCs) again available in a range of resolutions and speeds most general-purpose devices use a successive approximation approach a typical 8-bit converter might have a settling time of between 1 and 10 s a typical 12-bit converter might have a settling time of 10 to 100 s high speed flash converters can exceed 150 million samples per second

26.5 Sample and Hold Gates It is often useful to be able to sample a signal and then hold its value constant this is useful when performing analogue-to-digital conversion so that the signal does not change during conversion it is also useful when doing digital-to-analogue conversion to maintain the output voltage constant between conversions This task is performed by a sample and hold gate

Sample and hold gate circuits

Most sample and hold gates are constructed using integrated circuits Typical devices require a few microseconds to sample the incoming waveform, which then decays (or droops) at a rate of a few millivolts per millisecond High speed devices, such as those used for video applications, can sample an input in a few nanoseconds, but may experience a droop of a few millivolts per microsecond

26.6 Multiplexing While some systems have a single input and a single output, often there are multiple inputs and outputs Rather than have separate converters for each input and output, we often use multiplexing multiplexers make use of electrically operated switches to control the routing of signals these can be used at the input or output of a system normally separate anti-aliasing and/or reconstruction filters would be used with each input

Input multiplexing

Input multiplexing with sample and hold gates

Output multiplexing

Single-chip data acquisition systems these are a combination of an ADC and a multiplexer within a single integrated circuit usually designed for use with microprocessor-based systems, and provide all the control lines necessary for simple interfacing

Key Points Converting an analogue signal to a digital form is achieved by sampling the waveform and then performing analogue to digital conversion As long as the sampling rate is above the Nyquist rate, no information is lost as a result of sampling When sampling broad spectrum signals we make use of anti-aliasing filters to remove unwanted components When reconstructing signals, filters are used to remove the effects of the sampling A wide range of ADCs and DACs is available Sample and hold gates may be useful at the input or output Multiplexers can reduce the number of converters required

Implementing Digital Systems Chapter 27 Implementing Digital Systems Introduction Semiconductor Memory Array Logic Microprocessors Programmable Logic Controllers Selecting an Implementation Method

27.1 Introduction In this lecture we will look at the techniques used to implement complex digital systems We will begin by looking at the evolution of complex integrated circuits, and then progress to look at implementation strategies Many terms are used to describe integration level Available integration level increases exponentially with time (Moore’s Law)

Integration level Number of transistors Zero scale integration (ZSI) 1 Small scale integration (SSI) 2–30 Medium scale integration (MSI) 30 - 103 Large scale integration (LSI) 103 - 105 Very large scale integration (VLSI) 105 – 107 Ultra large scale integration (ULSI) 107 – 109 Giga-scale integration (GSI) 109 – 1011 Tera-scale integration (TSI) 1011 – 1013

Integration densities of Intel microprocessors

Semiconductor Memory Random access memory (RAM) 27.2 Semiconductor Memory Random access memory (RAM) this is read-write memory write describes the process of storing information read described the process of retrieval RAM is volatile in nature several forms: static RAM - uses circuitry similar to a bistable dynamic RAM – uses charge on capacitors, needs refreshing battery backup can be used to provide non-volatility

Read-only memory (ROM) this can be read from, but not written to is inherently non-volatile (useful for programs, etc) many forms available some are programmed by the manufacturer (such as masked programmed devices) others are user programmable (such as EPROM, and EEPROM) memory such as EEPROM can be written to (programmed) as well as read, but it is not RAM it can only be programmed relatively slowly

Memory organisation

Array Logic Array logic has two major forms: 27.3 Array Logic Array logic has two major forms: programmable logic devices (PLDs) field programmable gate arrays (FPGAs) Programmable logic devices (PLDs) these are examples of uncommitted logic forms include: PLA – programmable logic array PAL – programmable array logic GAL – generic array logic EPLD – erasable programmable logic device CPLD – complex programmable logic device

Programmable logic array (PLA) has an array of inverters, AND gates and OR gates can implement any logic function (given limits on numbers of inputs and outputs) Example: consider a system with four inputs A, B, C and D and three output X, Y and Z, where

The structure of a simple PLA

The PLA programmed to give the required output functions the device is programmed by blowing fusible links at the various interconnection points

Field programmable gate arrays a programmable device using more complex cells

Microprocessors A microcomputer system 27.4 Microprocessors A microcomputer system the CPU take the form of a microprocessor

Communication within the microcomputer

Registers fundamental building blocks within computers can be constructed using D flip-flops some are used for storage, others for input/output

Programmable Logic Controllers 27.5 Programmable Logic Controllers Programmable logic controllers (PLCs) are self-contained microcomputers that are optimised for industrial control They consist of one or more processors together with power supply and interface circuitry A range of input and output modules are available to allow the units to be used in a range of situations Facilities are also provided for programming and for system development

Selecting an Implementation Method 27.6 Selecting an Implementation Method The implementation method will depend on the complexity of the required functionality applications requiring just a handful of gates might use CMOS or TTL devices slightly more complex applications will often make use of array logic complex digital applications will probably use either complex programmable devices (such as CPLDs or FPGAs) or a microprocessor

Key Points Technologies can be categorised into a number of levels of integration from ‘zero-scale’ to ‘tera-scale’ integration The available complexity doubles every couple of years Semiconductor memory can be divided into RAM and ROM Array logic integrates large numbers of gates within a single package that is then configured for a particular application Complex digital systems can also be implemented using a microcomputer A programmable logic controller is a self-contained microcomputer that is optimised for industrial control The implementation method used will depend on the complexity of the required system