Announcements Information you’d like to get from this course. Think of one or more things you’re curious about and would like to learn about in this course. Put them on a blank card I’ll supply. (For example, how does radio work? What is a carbon nanotube and how might one use it?) We’ll try to give you answers. Since people in EE42 are supposed to take P/NP 1-unit EE43 we will not include a lab grade in the EE42 scoring. The new weights for grading in EE42 will be: MT1 22%; MT2 22%; HW 12%; Final 44%. Homeworks – including the first one – will be due at 12:00 noon on Thursdays (not Wednesdays). We’ll add some new lab sections; check the web site. EE 42 and 100, Fall 2005 Week 1

Electric Charge Definition: The quantity of electricity that flows in a given time or is held in a component such as a capacitor Symbol: q Unit: Coulomb (C) Notes: Charges may be positive or negative Like charges (e.g., + and +) repel each other, unlike charges (+ and -) attract each other EE 42 and 100, Fall 2005 Week 1

Note: Current has polarity (flow direction). Electric Current Definition: rate of positive charge flow Symbol: i Units: Coulombs per second ≡ Amperes (A) i = dq/dt where q = charge (in Coulombs), t = time (in seconds) Note: Current has polarity (flow direction). EE 42 and 100, Fall 2005 Week 1

Note: Potential is always referenced to some point. Electric Potential (Voltage) Definition: energy per unit charge expended in moving a charge from one place to another Symbol: v Units: Joules/Coulomb ≡ Volts (V) v = dw/dq where w = energy (in Joules), q = charge (in Coulombs) Note: Potential is always referenced to some point. a Subscript convention: vab means the potential at a minus the potential at b. vab ≡ va - vb b EE 42 and 100, Fall 2005 Week 1

p = dw/dt = (dw/dq)(dq/dt) = vi Electric Power Definition: amount of energy transferred per unit time Symbol: p Units: Joules per second ≡ Watts (W) p = dw/dt = (dw/dq)(dq/dt) = vi Concept: As a positive charge q moves through a drop in voltage of amount v, it loses energy energy change = qv rate is proportional to # charges/sec EE 42 and 100, Fall 2005 Week 1

The Ideal Basic Circuit Element Polarity reference for voltage can be indicated by plus and minus signs Reference direction for the current is indicated by an arrow Examples: resistor, capacitor, inductor, diode, … + v _ Attributes: Two terminals (points of connection) Cannot be subdivided into other elements Mathematically described by its current-voltage relationship EE 42 and 100, Fall 2005 Week 1

A Note about Reference Directions A problem like “Find the current” or “Find the voltage” is always accompanied by a definition of the direction: - v + i In this case, if the current turns out to be 1 mA flowing to the left, we would say i = -1 mA. (See prefixes in Hambley, p. 18.) In order to perform circuit analysis to determine the voltages and currents in an electric circuit, you need to specify reference directions. There is no need to guess the reference direction so that the answers come out positive, however. EE 42 and 100, Fall 2005 Week 1

Sign Convention Example Suppose you have an unlabelled battery and you measure its voltage with a digital voltmeter (DVM). It will tell you the magnitude and sign of the voltage. With this circuit, you are measuring vab. The DVM indicates 1.401, so va is lower than vb by 1.401 V. Which is the positive battery terminal? Note that we have used the “ground” symbol ( ) for the reference node on the DVM. Often it is labeled “C” for “common.” EE 42 and 100, Fall 2005 Week 1

Sign Convention for Power Passive sign convention p = vi p = -vi i i i i _ v + _ v + + v _ + v _ If p > 0, power is being delivered to the gray box. If p < 0, power is being extracted from the box. EE 42 and 100, Fall 2005 Week 1

Power Calculation Example Find the power absorbed by each element: Conservation of energy  total power delivered equals total power absorbed Aside: For electronics these are un- realistically large currents – mA is more typical than A (1 mA = 0.001A) vi (W) 918 - 810 - 12 - 400 - 224 1116 p (W) EE 42 and 100, Fall 2005 Week 1

Circuit Elements 5 ideal basic circuit elements: voltage source current source resistor inductor capacitor Many practical systems can be modeled with just sources and resistors The basic analytical techniques for solving circuits with inductors and capacitors are the same as those for resistive circuits active elements, capable of generating electric energy passive elements, incapable of generating electric energy EE 42 and 100, Fall 2005 Week 1

Electrical Sources An electrical source is a device that is capable of converting non-electric energy to electric energy and vice versa. Examples: battery: chemical electric dynamo (generator/motor): mechanical electric Electrical sources can either deliver or absorb power EE 42 and 100, Fall 2005 Week 1

Ideal Independent and Dependent Voltage Sources Circuit element that maintains a prescribed voltage across its terminals, regardless of the current flowing in those terminals. Voltage is known, but current is determined by the circuit to which the source is connected. The voltage can be either independent of dependent on a voltage or current elsewhere in the circuit, and can be constant or time-varying. Circuit symbols: _ vs vs=m vx vs=r ix + + + independent voltage-controlled current-controlled EE 42 and 100, Fall 2005 Week 1

Other Independent Voltage Source Symbols Sinusoidal AC source v(t) = Vpeaksin(wt) (In US, veff = 120 V, so Vpeak = 170 V) Battery (realistic source) + VS EE 42 and 100, Fall 2005 Week 1

EE 42 and 100, Fall 2005 Week 1

I-V Plot for a Real Battery EE 42 and 100, Fall 2005 Week 1

Ideal Independent and Dependent Current Sources Circuit element that maintains a prescribed current through its terminals, regardless of the voltage across those terminals. Current is known, but voltage is determined by the circuit to which the source is connected. The current can be either independent or dependent on a voltage or current elsewhere in the circuit, and can be constant or time-varying. Circuit symbols: is is=a vx is=b ix independent voltage-controlled current-controlled EE 42 and 100, Fall 2005 Week 1

Electrical Resistance Resistance: Electric field is proportional to current density, within a resistive material. Thus, voltage is proportional to current. The circuit element used to model this behavior is the resistor. Circuit symbol: Unit for R: Volts per Ampere ≡ ohms (W) The current flowing in the resistor is proportional to the voltage across the resistor: v = i R where v = voltage (V), i = current (A), and R = resistance (W) R (Ohm’s Law) EE 42 and 100, Fall 2005 Week 1

Water analogies for electrical elements Since you can’t see electricity, it is useful to think of the flow of water as being like that of electric charge. We’ll see water analogies for a number of electrical devices – capacitors, diodes, etc. Water analogy for charge: a quantity of water (say, 1 gallon is like 1 coulomb of charge) Water analogy for current flow: flow rate of water (say, 1 gallon per second is like 1 coulomb per second = 1 ampere) Electric potential difference (voltage) is like the difference in pressure in a hydraulic system The model for a resistor (an increased resistance reduces the current that flows for a given voltage) is like the effect of putting a sponge in a water pipe or crimping it EE 42 and 100, Fall 2005 Week 1

Water Models for Currents and Resistors EE 42 and 100, Fall 2005 Week 1

Resistance of an actual resistor Material resistivity = r (W-cm) T L W Resistance = resistivity x length/(cross-sectional area) R = r (L/WT) EE 42 and 100, Fall 2005 Week 1

Electrical Conductance Conductance is the reciprocal of resistance. Symbol: G Units: siemens (S) or mhos ( ) Example: Consider an 8 W resistor. What is its conductance? W EE 42 and 100, Fall 2005 Week 1

Short Circuit and Open Circuit Wire (“short circuit”): R = 0  no voltage difference exists (all points on the wire are at the same potential) Current can flow, as determined by the circuit Air (“open circuit”): R =   no current flows Voltage difference can exist, as determined by the circuit EE 42 and 100, Fall 2005 Week 1

Circuit Nodes and Loops A node is a point where two or more circuit elements are connected. A loop is formed by tracing a closed path in a circuit through selected basic circuit elements without passing through any intermediate node more than once Example: EE 42 and 100, Fall 2005 Week 1

Kirchhoff’s Laws Kirchhoff’s Current Law (KCL): The algebraic sum of all the currents entering any node in a circuit equals zero. (An expression of the conservation of charge.) Kirchhoff’s Voltage Law (KVL): The algebraic sum of all the voltages around any loop in a circuit equals zero. (As a result of conservation of energy.) EE 42 and 100, Fall 2005 Week 1

Note that p > 0 always, for a resistor. Example: Power Absorbed by a Resistor p = vi = ( iR )i = i2R p = vi = v ( v/R ) = v2/R Note that p > 0 always, for a resistor. Example: Calculate the voltage vg and current ia. Determine the power dissipated in the 80W resistor. EE 42 and 100, Fall 2005 Week 1

“Lumped Element” Circuit Modeling (Model = representation of a real system which simplifies analysis) In circuit analysis, important characteristics are grouped together in “lumps” (separate circuit elements) connected by perfect conductors (“wires”) An electrical system can be modeled by an electric circuit (combination of paths, each containing 1 or more circuit elements) if the dimensions of the circuit are small compared with the wavelength of any electromagnetic waves in the vicinity (i.e., wavelength l = c/f >> physical dimensions of system, where c = velocity of EM waves and f = frequency of wave). For example, at 60 Hz, the powerline frequency in the U. S., l = 3 x 108 (m/s)/60 (1/s) = 5 x 106 m ~ 3000 miles! so we can do lumped element circuit modeling of ordinary circuits at 60 Hz. EE 42 and 100, Fall 2005 Week 1

Construction of a Circuit Model The electrical behavior of each physical component is of primary interest. We need to account for undesired as well as desired electrical effects. Simplifying assumptions should be made wherever reasonable. EE 42 and 100, Fall 2005 Week 1

Terminology: Nodes and Branches Node: A point where two or more circuit elements are connected Branch: A path that connects two nodes EE 42 and 100, Fall 2005 Week 1

Notation: Node and Branch Voltages Use one node as the reference (the “common” or “ground” node) – label it with a symbol The voltage drop from node x to the reference node is called the node voltage vx. The voltage across a circuit element is defined as the difference between the node voltages at its terminals Example: – v1 + a R1 b + va _ + vb _ + vs R2 _ c  REFERENCE NODE EE 42 and 100, Fall 2005 Week 1

Using Kirchhoff’s Current Law (KCL) Consider a node connecting several branches: i2 i3 i1 i4 Use reference directions to determine whether currents are “entering” or “leaving” the node – with no concern about actual current directions EE 42 and 100, Fall 2005 Week 1

Alternative Formulations of Kirchhoff’s Current Law (Charge stored in node is zero.) Formulation 1: Sum of currents entering node = sum of currents leaving node Formulation 2: Algebraic sum of currents entering node = 0 Currents leaving are included with a minus sign. Formulation 3: Algebraic sum of currents leaving node = 0 Currents entering are included with a minus sign. EE 42 and 100, Fall 2005 Week 1

EE 42 and 100, Fall 2005 Week 1

A Major Implication of KCL KCL tells us that all of the elements in a single branch carry the same current. We say these elements are connected in series. Current entering node = Current leaving node i1 = i2 EE 42 and 100, Fall 2005 Week 1

KCL Example Currents entering the node: -10 mA i Currents leaving the node: -10 mA i 5 mA 15 mA 3 formulations of KCL: 1. 2. 3. EE 42 and 100, Fall 2005 Week 1

Generalization of KCL The sum of currents entering/leaving a closed surface is zero. Circuit branches can be inside this surface, i.e. the surface can enclose more than one node! i2 i3 This could be a big chunk of a circuit, e.g., a “black box” i4 i1 EE 42 and 100, Fall 2005 Week 1

Generalized KCL Examples 50 mA 5mA 2mA i i EE 42 and 100, Fall 2005 Week 1

Using Kirchhoff’s Voltage Law (KVL) Consider a branch which forms part of a loop: – v2 + + v1 _ voltage “drop” voltage “rise” (negative drop) loop loop Use reference polarities to determine whether a voltage is dropped – with no concern about actual voltage polarities EE 42 and 100, Fall 2005 Week 1

Formulations of Kirchhoff’s Voltage Law (Conservation of energy) Formulation 1: Sum of voltage drops around loop = sum of voltage rises around loop Formulation 2: Algebraic sum of voltage drops around loop = 0 Voltage rises are included with a minus sign. Formulation 3: Algebraic sum of voltage rises around loop = 0 Voltage drops are included with a minus sign. (Handy trick: Look at the first sign you encounter on each element when tracing the loop.) EE 42 and 100, Fall 2005 Week 1

A Major Implication of KVL KVL tells us that any set of elements that are connected at both ends carry the same voltage. We say these elements are connected in parallel. + va _ + vb _ Applying KVL in the clockwise direction, starting at the top: vb – va = 0  vb = va EE 42 and 100, Fall 2005 Week 1

KVL Example Three closed paths: v2 v3 +  1 2 +  va vb vc - 3 Path 1: +  vb v3 v2 - b a c Path 1: Path 2: Path 3: EE 42 and 100, Fall 2005 Week 1

An Underlying Assumption of KVL No time-varying magnetic flux through the loop Otherwise, there would be an induced voltage (Faraday’s Law) Note: Antennas are designed to “pick up” electromagnetic waves; “regular circuits” often do so undesirably. +  Avoid these loops! How do we deal with antennas (EECS 117A)? Include a voltage source as the circuit representation of the induced voltage or “noise”. (Use a lumped circuit model rather than a distributed (wave) model.) EE 42 and 100, Fall 2005 Week 1

Resistors in Series Consider a circuit with multiple resistors connected in series. Find their “equivalent resistance”. KCL tells us that the same current (I) flows through every resistor KVL tells us I R1 R2 VSS  + R3 R4 Equivalent resistance of resistors in series is the sum of the individual resistances EE 42 and 100, Fall 2005 Week 1

Voltage Divider I = VSS / (R1 + R2 + R3 + R4) + – + – I R1 V1 R2 VSS  + + – R3 V3 R4 EE 42 and 100, Fall 2005 Week 1

When can the Voltage Divider Formula be Used? + – + – R2 V2 R2 V2  +  + VSS VSS R3 R3 R4 R4 R5 SS 4 3 2 1 V R × + = SS 4 3 2 1 V R × + ≠ Correct, if nothing else is connected to nodes because R5 removes condition of resistors in series EE 42 and 100, Fall 2005 Week 1

Resistors in Parallel Consider a circuit with two resistors connected in parallel. Find their “equivalent resistance”. x KVL tells us that the same voltage is dropped across each resistor Vx = I1 R1 = I2 R2 KCL tells us I1 I2 ISS R1 R2 EE 42 and 100, Fall 2005 Week 1

Equivalent conductance of resistors in parallel is the sum General Formula for Parallel Resistors What single resistance Req is equivalent to three resistors in parallel? +  V I R3 R2 R1 Req eq  Equivalent conductance of resistors in parallel is the sum EE 42 and 100, Fall 2005 Week 1

EE 42 and 100, Fall 2005 Week 1

Current Divider x Vx = I1 R1 = ISS Req I1 I2 ISS R1 R2 EE 42 and 100, Fall 2005 Week 1

Generalized Current Divider Formula Consider a current divider circuit with >2 resistors in parallel: +  V EE 42 and 100, Fall 2005 Week 1

Checklist of Terms Introduced Charge, current, voltage, resistance, conductance, energy, power Coulomb (C), ampere (A), volt (V), ohm (W), siemens (S) (mho), joule (J), watt (W) Kirchhoff’s Current Law (KCL), Kirchhoff’s Voltage Law (KVL), Ohm’s Law Series connection, parallel conncection DC (steady), AC (time-varying) Independent and dependent ideal voltage and current sources EE 42 and 100, Fall 2005 Week 1