ECE2262 Electric Circuits. Chapter 1: Basic Concepts. Overview of the material discussed in ENG 1450

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1 ECE2262 Electric Circuits Chapter 1: Basic Concepts Overview of the material discussed in ENG

2 Circuit Analysis 2

3 Lab -ECE LN - ECE 2262

4 Basic Quantities: Current, Voltage, Energy, Power The study of circuit analysis involves understanding the behavior of each ideal circuit element in terms of its voltage and current Also understanding the constraints imposed on the voltage and current as a result of interconnecting the ideal elements. 4

5 Current the charge is bipolar, positive (+) charges, negative (-) charges the electric charge exists in discrete quantities, which are integral multiples of the electron charge: ! 10!19 Coulombs the electrical effects caused by charges in motion depend on the rate of charge flow. The rate of of change flow defines the electric current ( ) = dq( t) i t dt Ampere = Coulomb Second t ( ) = i! q t $ "# 5 ( )d!

6 the bipolar nature of electric charge requires that we assign polarity to the current variable. I 1 = 2A c i r c u i t I 2 =!3A c i r c u i t 2 C/sec charge moves 3 C/sec charge moves from B! T from T! B by convention the direction of current flow is in the direction of positive charge movement and opposite the direction of negative charge movement 6

7 i - positive! positive charge flowing from terminal 1 to terminal 2 i - negative! positive charge flowing from terminal 2 to terminal 1 7

8 Voltage - Potential Difference Always relative - measured between two points 8

9 ( ) = dw ( t ) v t dq Volt = Joule Coulomb dw - the potential energy loss of dq amount of charge transported from + to! terminal. A steady current through a wire requires constant application of electric pressure called the voltage The difference in voltage moves the electrons along the wire 9

10 v - positive! voltage drop from terminal 1 to terminal 2! voltage rise from terminal 2 to terminal 1 v - negative! voltage rise from terminal 1 to terminal 2! voltage drop from terminal 2 to terminal 1 10

11 Example a v b = 4V +! b c d + v d =!1V! a v! b =!4V + b c d! v d = 1V + 11

12 Example (Reference Node) v 1 = 3V +! b v =!1V 2 a c d +! Voltage drop v b = v 1! v 2 = 3! (!1) = 4V v d = v 2! v 0 =!1! 0 =!1V 12

13 v 1 = 3V! b v =!1V 2 +! a c + d! + Voltage drop v b = v 2! v 1 =!1! 3 =!4V v c = v d = 0! (!1) = 1V 13

14 Whenever the reference direction for the current in an element is in the direction of the reference voltage drop across the element (as in the figure above), use a positive sign in any expression that relates the voltage to the current. Otherwise, use a negative sign. Passive Sign Convention 14

15 The assignments of the reference polarity for voltage and the reference direction for the current are entirely arbitrary. However, once you have assigned the references, you must write all subsequent equations to agree with the chosen references! 15

16 Power The rate at which a circuit element either absorbs or supplies energy is the power absorbed or supplied ( ) = dw( t) p t dt Watt = Joule Sec ( ) = dw( t) p t dt = dw( t) dq dq dt! p( t) = v( t)i( t) Watt = Volt! Ampere 16

17 Change in the energy from time t 1 to time t 2 t 2 t 2!w = p( t)dt = v( t)i( t)dt " t 1 " t 1 17

18 Example 18

19 ( ) = dq( t) i t dt! 19

20 p( t) = i( t)! v( t) = i( t)!12! p( t) > 0 for 1! t! 2 msec and 6! t! 9 msec! the BOX is absorbing p t 20 power ( ) < 0 for 3! t! 5 msec! the BOX is supplying power to the 12-V source

21 Example p( t) = 2.5e!4t W, v( t) = 50e!t V! i( t) = p( t) v( t) = A 0.05e"3t!w = p( t)dt 0 250"10 #3 $ = mj Q = 0 250!10 "3 i( t)dt # = 8.8 mc 21

22 Example 22

23 ( ) = p( t) v( t) = 1 v( t) i t dw( t) dt = 1 10 dw( t) dt?? 23

24 Q = 12 # i( t)dt = 2! ! ("0.25) +1! " ( ) + 2! = 0! Try problems: 1.9, 1.13, 1.14, 1.15,

25 Sign Convention for Power: absorbing and delivering power circuit elements Energy Supplier Energy Absorber 25

26 26

27 (a) Energy absorbing by a circuit element P = 2A! 3V = 6W The element is absorbing 6 J of energy / sec. 27

28 (b) Energy supplied by a circuit element P = (!2A) " 3V =!6W The element is supplying energy to terminals A-B! This is consistent with the Passive Sign Convention! 28

29 Sign Convention for Power The current must enter the component at the positive terminal of the voltage If the power is positive ( p( t) > 0) then power is being delivered to the circuit element inside the box. The element absorbs power. If the power is negative ( p( t) < 0) then power is being extracted from the circuit element inside the box. The element delivers power 29

30 Polarity references and the expression for power 30

31 Example (a) (b) 31

32 (a) P = (!4A) "12V =!48W - the element is supplying power (b) P = 2A! 4V = 8W - the element is absorbing power 32

33 Example (a) the element is absorbing power: (!2A) " V 1 = 40W! V 1 =!20V (b) the element is supplying power: I!10 = "50! I =!5 33

34 Example 34

35 P 1 = 16V!1A = 16W, P 2 = 4V!1A = 4W, P 3 = 12V!1A = 12W, P 4 = 8V! 2A = 16W, P 12V = 2A!12V = 24W - all these elements are absorbing power P 24V = (!3A) " 24V =!72W - the element is supplying power P 1 + P 2 + P 3 + P 4 + P 12V + P 24V = 0! Tellegen s Theorem 35

36 Tellegen s Theorem If we follow the sign convention, the algebraic sum of the powers of all circuit components is equal to zero. This theorem has been introduced in the year of 1952 by Dutch Electrical Engineer Bernard D.H. Tellegen. 36

37 Example 37

38 ( ) " 25V =!125W, P Ix = (!I x A) "10V =!10I x W ( ) "15V =!15I x W, P 3 = 2A!15V = 30W, P 10V = 2A!10V = 20W P 1 = 1A! 25V = 25W, P 5 A =!5A P 2 =!I x A P 1 + P 5 A + P Ix + P 2 + P 3 + P 10V = 0!25I x! 50 = 0! I x =!2A, P Ix = 20W 38

39 Circuit Elements voltage sources current sources resistors inductors capacitors active elements passive elements 39

40 Chapter II Chapter VI voltage sources current sources resistors inductors capacitors active elements passive elements 40

41 Voltage and Current Sources Independent Sources Independent Voltage Source 41

42 Independent Current Source the v-i characteristic of a resistor? 42

43 Dependent Sources - Controlled Sources Dependent Voltage Sources units for µ, r? 43

44 Dependent Current Sources units for g,!? 44

45 Example 45

46 P 24V = (!6A) " 24V =!144W ; P 4 Ix = (!3A) " 4I x V = (!3A) " 8V =!24W P 1 = 6A! 8V = 48W ; P 2 = 4A!10V = 40W ; P 3 = I 0 A! 6V = 6I 0 W P 4 = 2A!16V = 32 W ; P 5 = 1A! 6V = 6 W ; P 6 = 3A! 8V = 24 W! P i = 0! 6I = 168! 6I 0 = 18! I 0 = 3A 46

47 Example: Testing Interconnections of Ideal Sources Valid connection; each source supplies voltage across the same pair of terminals and they supply the same voltage with the same polarity. Valid connection; each source supplies current through the same pair of terminals and they supply the same current in the same direction. 47

48 Invalid connection; each source supplies supplies voltage across the same pair of terminals but they supply the voltage of different values. Invalid connection; each source supplies supplies current through the same pair of terminals but they supply the current of different values. 48

49 Valid connection; the voltage source supplies voltage across the pair of terminals. The current source supplies current through the same pair of terminals. Because an ideal voltage source supplies the same voltage regardless of the current, and an ideal current source supplies the same current regardless of the voltage, this is a permissible connection. 49

50 50

51 Invalid Valid Valid Invalid 51

52 DC and AC Electricity 52

53 RMS Values: 230 V, 110 V RMS = Peak Value 2 53

54 54

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