# Electric Current. Note: Current has polarity. EECS 42, Spring 2005 Week 2a 1

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1 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. EECS 42, Spring 2005 Week 2a 1

2 Electric Potential (Voltage) Definition: energy per unit charge 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 b Subscript convention: v ab means the potential at a minus the potential at b. v ab v a -v b EECS 42, Spring 2005 Week 2a 2

3 Electric Power Definition: transfer of energy 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 v, it loses energy energy change = qv rate is proportional to # charges/sec EECS 42, Spring 2005 Week 2a 3

4 The Ideal Basic Circuit Element i v _ Polarity reference for voltage can be indicated by plus and minus signs Reference direction for the current is indicated by an arrow Attributes: Two terminals (points of connection) Mathematically described in terms of current and/or voltage Cannot be subdivided into other elements EECS 42, Spring 2005 Week 2a 4

5 A Note about Reference Directions A problem like Find the current or Find the voltage is always accompanied by a definition of the direction: i - v In this case, if the current turns out to be 1 ma flowing to the left, we would say i = -1 ma. 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. EECS 42, Spring 2005 Week 2a 5

6 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. a b DVM With this circuit, you are measuring v ab. The DVM indicates 1.401, so v a is lower than v b by 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. EECS 42, Spring 2005 Week 2a 6

7 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 box. If p < 0, power is being extracted from the box. EECS 42, Spring 2005 Week 2a 7

8 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 unrealistically large currents ma is more typical than A vi (W) p (W) EECS 42, Spring 2005 Week 2a 8

9 Circuit Elements 5 ideal basic circuit elements: voltage source current source resistor inductor capacitor active elements, capable of generating electric energy passive elements, incapable of generating electric energy 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 EECS 42, Spring 2005 Week 2a 9

10 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 EECS 42, Spring 2005 Week 2a 10

11 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 or dependent on a voltage or current elsewhere in the circuit, and can be constant or time-varying. Circuit symbols: v s _ v s =µ v x _ v s =ρ i x _ independent voltage-controlled current-controlled EECS 42, Spring 2005 Week 2a 11

12 EECS 42, Spring 2005 Week 2a 12

13 Other Independent Voltage Source Symbols Sinusoidal AC source v(t) = V peak sin(ωt) V effective = V peak \/2 (In US, 120 V, so V peak = 170 V) Battery (realistic source) V S EECS 42, Spring 2005 Week 2a 13

14 Realistic Voltage Source A real-life voltage source, like a battery or the function generator in lab, cannot sustain a very high current. Either a fuse blows to shut off the device, or something melts Additionally, the voltage output of a realistic source is not constant. The voltage decreases slightly as the current increases. We usually model realistic sources considering the second of these two phenomena. A realistic source is modeled by an ideal voltage source in series with an internal resistance, R S. Spring 2005 EE 42 Lecture 3 14 EECS 42, Spring 2005 Week 2a 14 V s R S

15 I-V Plot for a Real Battery EECS 42, Spring 2005 Week 2a 15

16 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: i s i s =α v x i s =β i x independent voltage-controlled current-controlled EECS 42, Spring 2005 Week 2a 16

17 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: R Units: Volts per Ampere ohms (Ω) The current flowing in the resistor is proportional to the voltage across the resistor: v = i R (Ohm s Law) where v = voltage (V), i = current (A), and R = resistance (Ω) EECS 42, Spring 2005 Week 2a 17

18 Resistance of an actual resistor Material resistivity = ρ (Ω-cm) T L W Resistance = resistivity x length/(cross-sectional area) R = ρ (L/WT) EECS 42, Spring 2005 Week 2a 18

19 Electrical Conductance Conductance is the reciprocal of resistance. Symbol: G Units: siemens (S) or mhos ( ) Ω Example: Consider an 8 Ω resistor. What is its conductance? EECS 42, Spring 2005 Week 2a 19

20 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 EECS 42, Spring 2005 Week 2a 20

21 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: EECS 42, Spring 2005 Week 2a 21

22 Kirchhoff s Laws Kirchhoff s Current Law (KCL): The algebraic sum of all the currents entering any node in a circuit equals zero. Kirchhoff s Voltage Law (KVL): The algebraic sum of all the voltages around any loop in a circuit equals zero. EECS 42, Spring 2005 Week 2a 22

23 Example: Power Absorbed by a Resistor p = vi = ( ir )i = i 2 R p = vi = v ( v/r ) = v 2 /R Note that p > 0 always, for a resistor. Example: a) Calculate the voltage v g and current i a. b) Determine the power dissipated in the 80Ω resistor. EECS 42, Spring 2005 Week 2a 23

24 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 λ = c/f >> physical dimensions of system Distance travelled by a particle travelling at the speed of light in one period Example: f = 60 Hz λ = 3 x 10 8 m/s / 60 = 5 x 10 6 m EECS 42, Spring 2005 Week 2a 24

25 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. EECS 42, Spring 2005 Week 2a 25

26 Terminology: Nodes and Branches Node: A point where two or more circuit elements are connected Branch: A path that connects two nodes EECS 42, Spring 2005 Week 2a 26

27 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 v x. The voltage across a circuit element is defined as the difference between the node voltages at its terminals Example: a v 1 R 1 b v a c v s EECS 42, Spring 2005 Week 2a 27 R 2 v b _ REFERENCE NODE

28 Using Kirchhoff s Current Law (KCL) Consider a node connecting several branches: i 2 i 3 i 1 i 4 Use reference directions to determine whether currents are entering or leaving the node with no concern about actual current directions EECS 42, Spring 2005 Week 2a 28

29 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. EECS 42, Spring 2005 Week 2a 29

30 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 i 1 = i 2 EECS 42, Spring 2005 Week 2a 30

31 KCL Example 5 ma -10 ma i Currents entering the node: Currents leaving the node: 15 ma 3 formulations of KCL: EECS 42, Spring 2005 Week 2a 31

32 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! i 2 i 3 This could be a big chunk of a circuit, e.g., a black box i 1 i 4 EECS 42, Spring 2005 Week 2a 32

33 Generalized KCL Examples 50 ma 5µA i 2µA i EECS 42, Spring 2005 Week 2a 33

34 Using Kirchhoff s Voltage Law (KVL) Consider a branch which forms part of a loop: loop v 1 _ voltage drop loop v 2 voltage rise (negative drop) Use reference polarities to determine whether a voltage is dropped with no concern about actual voltage polarities EECS 42, Spring 2005 Week 2a 34

35 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. (Handy trick: Look at the first sign you encounter on each element when tracing the loop.) Formulation 3: Algebraic sum of voltage rises around loop = 0 Voltage drops are included with a minus sign. EECS 42, Spring 2005 Week 2a 35

36 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. v a_ v b _ Applying KVL in the clockwise direction, starting at the top: v b v a = 0 v b = v a EECS 42, Spring 2005 Week 2a 36

37 Three closed paths: KVL Example v a v 2 1 v b v 3 a b c - 2 v c 3 Path 1: Path 2: Path 3: EECS 42, Spring 2005 Week 2a 37

38 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! B v (t) v(t) 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.) EECS 42, Spring 2005 Week 2a 38

39 Resistors in Series Consider a circuit with multiple resistors connected in series. Find their equivalent resistance. I R 1 KCL tells us that the same current (I) flows through every resistor V SS R 2 R 3 KVL tells us R 4 Equivalent resistance of resistors in series is the sum EECS 42, Spring 2005 Week 2a 39

40 Voltage Divider I R 1 V 1 I = V SS / (R 1 R 2 R 3 R 4 ) V SS R 2 R 3 V 3 R 4 EECS 42, Spring 2005 Week 2a 40

41 When can the Voltage Divider Formula I be Used? I V SS R 1 R 2 R 3 V 2 V SS R 1 R 2 R 3 V 2 R 4 R 4 R 5 R R V 2 2 = V V 2 SS 2 V SS R R R R R R R R Correct, if nothing else is connected to nodes because R 5 removes condition of resistors in series EECS 42, Spring 2005 Week 2a 41

42 Resistors in Parallel I SS Consider a circuit with two resistors connected in parallel. Find their equivalent resistance. I 1 R 1 x I 2 R 2 KVL tells us that the same voltage is dropped across each resistor V x = I 1 R 1 = I 2 R 2 KCL tells us EECS 42, Spring 2005 Week 2a 42

43 General Formula for Parallel Resistors What single resistance R eq is equivalent to three resistors in parallel? I I V R 1 R 2 R 3 eq V R eq Equivalent conductance of resistors in parallel is the sum EECS 42, Spring 2005 Week 2a 43

44 Current Divider x I 1 I 2 I SS R 1 R 2 V x = I 1 R 1 = I SS R eq EECS 42, Spring 2005 Week 2a 44

45 Generalized Current Divider Formula Consider a current divider circuit with >2 resistors in parallel: I I 1 I 2 I 3 R 1 R 2 R 3 V V = 1 R I 1 R R3 I 3 = V R 3 = I 1/R 1 1/R 1/R 3 2 1/R 3 EECS 42, Spring 2005 Week 2a 45

46 EE 42: Checklist of Terms 25 Jan Charge, current, voltage, resistance, conductance, energy, power Coulomb, ampere, volt, ohm, siemen (mho), joule, watt Kirchhoff s Current Law (KCL), Kirchhoff s Voltage Law (KVL), Ohm s Law Series connection, parallel connection DC (steady), AC (time-varying) Independent and dependent ideal voltage and current source Voltage divider, current divider Analog (A/D), Digital (D/A) EECS 42, Spring 2005 Week 2a 46

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