electronics fundamentals circuits, devices, and applications THOMAS L. FLOYD DAVID M. BUCHLA Chapter 8 – AC Circuits

Alternating Voltage is a voltage that: Continuously varies in magnitude Periodically reverses in polarity

Symbol for a sinusoidal voltage source.

A wave is a disturbance. Unlike water waves, electrical waves cannot be seen directly but they have similar characteristics. All periodic waves can be constructed from sine waves, which is why sine waves are fundamental to alternating current.

Sine waves The sinusoidal waveform (sine wave) is the fundamental alternating current (ac) and alternating voltage waveform. Electrical sine waves are named from the mathematical function with the same shape.

Generation of a sine wave Sinusoidal voltage sources Sinusoidal voltages are produced by ac generators and electronic oscillators. When a conductor rotates in a constant magnetic field, a sinusoidal wave is generated. D B C A When the loop is moving perpendicular to the lines of flux, the maximum voltage is induced. When the conductor is moving parallel with the lines of flux, no voltage is induced.

Sine waves Sine waves are characterized by the amplitude and period. The amplitude is the maximum value of a voltage or current The period is the time interval for one complete cycle. A The amplitude (A) of this sine wave is 20 V T The period is 50.0 ms

Sine waves The period (T) of a sine wave can be measured between any two corresponding points on the waveform. T T T T A T T A By contrast, the amplitude of a sine wave is only measured from the center to the maximum point.

Frequency Frequency ( f ) is the number of cycles that a sine wave completes in one second. Frequency is measured in hertz (Hz). If 3 cycles of a wave occur in one second, the frequency is 3.0 Hz 1.0 s

Period and frequency The period and frequency are reciprocals of each other. and If the period is 50 ms, the frequency is 0.02 MHz = 20 kHz. (The 1/x key on your calculator is handy for converting between f and T.)

Sine wave voltage and current values Instantaneous value (v): Voltage or current at any point on the curve. Peak value (VP for voltage): The amplitude of a sine wave. VP v The peak voltage of this waveform is 20 V. v

Sine wave voltage and current values The average value (actually the half-wave average) is used when comparing power supplies. The average value for the sinusoidal voltage is Vavg 12.7 V.

Sine wave voltage and current values Peak to peak value: Value from positive peak to negative peak. Equation = RMS (root mean squared) value: Is the sinusoidal wave with the same heat value as a DC voltage source (known as the effective value)

Sine wave voltage and current values VP = 20 volts The peak-to-peak voltage is Vrms 40 V. VPP The rms voltage is 14.1 V. This is magnitude Vpp

Angular Measurement Degree: Angular measurement equal to 1/360 of the circumference of a circle. Radian (rad): Angle formed when the distance along the circumference of a circle equal the radius of the circle. PI ( ): The ratio of the circumference of a circle to its diameter. Has a constant value ≈ 3.1416

Angular measurement Angular measurements can be made in degrees (o) or radians. There are 360o or 2p radians in one complete revolution.

Angular measurements

Angular measurement There are 2p radians in one complete revolution and 360o in one revolution. To find the number of radians, given the number of degrees: This can be simplified to: To find the number of degrees, given the number of radians:

How many radians are in 45o? Angular measurement How many radians are in 45o? How many degrees are in 1.2 radians?

Sine wave angles

Phase of a Sine Wave Phase: Angular measurement that specifies the position on the sine wave relative to a reference point.

Phase shift Phase Shift of a sine wave is an angular measurement that specifies the position of a sine wave relative to a reference wave. Sine wave shifted right (lags) – Sine wave shifted left (leads) + where f = Phase shift

Phase shifts Occurs when a sine wave is shifted right or left in relation to the base/reference sine wave.

Phase shift – Lead/Lag Occurs when a sine wave is shifted right or left in relation to the base/reference sine wave. A - Leads B – Lags by 450 A - Lags B – Leads by 300

Phase shift Example of a wave that lags the reference (not on guided notes) …and the equation has a negative phase shift v = 30 V sin (q - 45o) Notice that a lagging sine wave is below the axis at 0o

Phase shift Example of a wave that leads the reference (not on guided notes) Notice that a leading sine wave is above the axis at 0o v = 30 V sin (q + 45o) …and the equation has a positive phase shift

PolyPhase power An important application of phase-shifted sine waves is in electrical power systems. Electrical utilities generate ac with three phases that are separated by 120o. 3-phase power is delivered to the user with three hot lines plus neutral. The voltage of each phase, with respect to neutral is 120 V. 120o 120o 120o 0o

Sine wave equation Instantaneous values of a wave are shown as v or i. The equation for the instantaneous voltage (v) of a sine wave is where Vp = q (theta) = Peak voltage Angle in rad or degrees If the peak voltage is 25 V, the instantaneous voltage at 50 degrees is 19.2 V

Sine wave equation A certain sine wave has a positive-going zero crossing at 0° and an peak value of 40V. Calculate its instantaneous voltage for the degrees listed below for the sine wave below. Vp = q (theta) = Peak voltage Angle in rad or degrees 45°, 125°, 180°, 220°,325°

Sine wave equation A plot of the example in the previous slide (peak at 25 V). The instantaneous voltage at 50o is 19.2 V as previously calculated. 4.34v 24.64v What is the voltage at 100 and 800?

Phasors Phasor (aka Phase Vector): Representation of a sine wave whose amplitude (A) and angular frequency (ω - omega) are a constant rate.

Power in resistive AC circuits A sinusoidal voltage produces a sinusoidal current

Kirchhoff’s voltage law applies to AC circuits just like DC circuits Power in resistive AC circuits Kirchhoff’s voltage law applies to AC circuits just like DC circuits

Power in resistive AC circuits Power in AC circuits is calculated using RMS values for voltage and current. The power formulas are: The dc and the ac sources produce the same power to the bulb: 120 Vdc 0 V 170 Vp = 120 Vrms 0 V WHY?

Power in resistive AC circuits Assume a sine wave with a peak value of 40 V is applied to a 100 W resistive load. What power is dissipated? Vrms = 0.707 x Vp = 0.707 x 40 V = 28.3 V 8 W

Superimposed dc and ac voltages DC and AC voltages can be combined in a waveform. Algebraically they produce a composite waveform of an AC voltage “riding” on a DC level. If dc voltage > ac voltage, the sine wave never goes negative.

Superimposed dc and ac voltages

Schematic for Superimposed DC and AC voltages

Superimposed dc and ac voltages

Superimposed dc and ac voltages VMAX = 22V VMIN = 2V VMAX = 16V VMIN = -4V

AC generator (alternator) Generators convert rotational energy to electrical energy. The armature has an induced voltage, which is connected through slip rings and brushes to a load. The armature loops are wound on a magnetic core (not shown for simplicity). Small alternators may use a permanent magnet Others use field coils to produce the magnetic flux.

AC generator (alternator) Increasing the number of poles increases the number of cycles per revolution. A four-pole generator will produce two complete cycles in each revolution.

Output Frequency of an AC Generator f – frequency (Hz) N – number of poles s - speed in RPM 1 4 2 3

Alternators In vehicles, alternators generate ac, which is converted to dc for operating electrical devices and charging the battery. AC is more efficient to produce and can be easily regulated, hence it is generated and converted to DC by diodes. The output is taken from the rotor through the slip rings.

AC Motors There are two major classifications of ac motors. Induction motor Synchronous motor. Both types use a rotating field in the stator windings. Induction motors work because current is induced in the rotor by the changing current in the stator. This current creates a magnetic field that reacts with the moving field of the stator, which develops a torque and causes the rotor to turn. Synchronous motors have a magnet for the rotor. In small motors, this can be a permanent magnet, which keeps up with the rotating field of the stator. Large motors use an electromagnet in the rotor, with external dc supplied to generate the magnetic field.

Rotating the stator produces a net magnetic field

Rotating the stator produces a net magnetic field

Induction vs. Stator Motors Squirrel Cage Rotor

Induction vs. Stator Motors

NONSINUSODIAL WAVE FORMS Pulse: Transition from one voltage/current level (baseline) to another level ( leading edge) and after an interval of time (t) transitioning back to the baseline (tailing edge) Going positive – Rising edge Going negative – Falling edge Rise Time (tr) – Time required to go from 10% amplitude to 90% amplitude Fall time (tf)– Time to go from 90% amplitude to 10% amplitude Pulse width (tw) - Interval of time from when tr is at 50% amplitude and tf is at 50% amplitude

Pulse definitions Ideal pulses

Pulse definitions Non-ideal pulses Notice that rise and fall times are measured between the 10% and 90% levels whereas pulse width is measured at the 50% level.

Repetitive pulse waveforms Periodic waveforms repeat at fixed intervals. Pulse repetition frequency: Rate at which the pulses repeat. Duty Cycle – Ratio of pulse width (tw) to the period (T) Percent Duty Cycle = Vavg = baseline = (duty cycle)( amplitude)

Nonsinusoidal Wave Forms Define terms on page 359 Rise time: Fall time: Pulse width:

Voltage ramps Ramp – Linear increase or decrease in voltage or current. Slope =

Triangular and Sawtooth waves Triangular and sawtooth waveforms are formed by voltage or current ramps (linear increase/decrease) Triangular waveforms have positive-going and negative-going ramps of equal duration (same slope either increasing or decreasing). T T The sawtooth waveform consists of two ramps, one of much longer duration than the other. (unequal slopes in either direction).

Harmonics Repetitive non-sinusoidal waveforms are composed of: fundamental frequency (repetition rate of the waveform) & harmonic frequencies. Odd harmonics are frequencies that are odd multiples of the fundamental frequency. Even harmonics are frequencies that are even multiples of the fundamental frequency. Composite Waveform – Any deviation from a “normal” sine wave

Harmonics A square wave is composed only of the fundamental frequency and odd harmonics (of the proper amplitude).

Oscilloscope A device that traces the graph of a measured electrical signal on its screen.

Oscilloscopes The oscilloscope is divided into four main sections.

Video on Osciloscope http://www.cleanvideosearch.com/media/action/yt/watch?v=8VEg6L2QG5o The name of video is: AC vs Dc explain how to use an oscilloscope

Reading an Oscilloscope

Calculate for each wave: Period, Peak, Peak to Peak, RMS T= 3.0 ms Vp = 1250 mv Vpp = 2500 mv RMS = 833.8 mV T= 20 ms Vp = 1.5 v Vpp = 3.0 v RMS = 1.06 V 500 mv 0.5 ms T= 6000 µs; 6 ms Vp = 20.4 v Vpp = 40.8 v RMS = 14.4 V T= 30 µs Vp = 24 v Vpp = 48 v RMS = 16.97 V 6 v 300 µs 12 v 15 µs

Selected Key Terms Sine wave Alternating current Period (T) Frequency (f) Hertz A type of waveform that follows a cyclic sinusoidal pattern defined by the formula y = A sin q. Current that reverses direction in response to a change in source voltage polarity. The time interval for one complete cycle of a periodic waveform. A measure of the rate of change of a periodic function; the number of cycles completed in 1 s. The unit of frequency. One hertz equals one cycle per second.

Selected Key Terms Instantaneous value Peak value Peak-to-peak value rms value The voltage or current value of a waveform at a given instant in time. The voltage or current value of a waveform at its maximum positive or negative points. The voltage or current value of a waveform measured from its minimum to its maximum points. The value of a sinusoidal voltage that indicates its heating effect, also known as effective value. It is equal to 0.707 times the peak value. rms stands for root mean square.

Quiz 1. In North America, the frequency of ac utility voltage is 60 Hz. The period is a. 8.3 ms b. 16.7 ms c. 60 ms d. 60 s

Quiz 2. The amplitude of a sine wave is measured a. at the maximum point b. between the minimum and maximum points c. at the midpoint d. anywhere on the wave

Quiz 3. An example of an equation for a waveform that lags the reference is a. v = -40 V sin (q) b. v = 100 V sin (q + 35o) c. v = 5.0 V sin (q - 27o) d. v = 27 V

Quiz 4. In the equation v = Vp sin q , the letter v stands for the a. peak value b. average value c. rms value d. instantaneous value

Quiz 5. The time base of an oscilloscope is determined by the setting of the a. vertical controls b. horizontal controls c. trigger controls d. none of the above

Quiz 6. A sawtooth waveform has a. equal positive and negative going ramps b. two ramps - one much longer than the other c. two equal pulses d. two unequal pulses

Quiz 7. The number of radians in 90o are a. p/2 b. p c. 2p/3 d. 2p

Quiz 8. For the waveform shown, the same power would be delivered to a load with a dc voltage of a. 21.2 V b. 37.8 V c. 42.4 V d. 60.0 V

Quiz 9. A square wave consists of a. the fundamental and odd harmonics b. the fundamental and even harmonics c. the fundamental and all harmonics d. only the fundamental

Quiz 10. A control on the oscilloscope that is used to set the desired number of cycles of a wave on the display is a. volts per division control b. time per division control c. trigger level control d. horizontal position control

Quiz Answers: 1. b 2. a 3. c 4. d 5. b 6. b 7. a 8. c 9. a 10. b