# Introduction to AC Circuits (Capacitors and Inductors)

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1 Introduction to AC Circuits (Capacitors and Inductors) Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University

2 OUTLINE Previously on ELCN102 AC Circuits Capacitors Inductors Transient Analysis RC Circuits RL Circuits 2

3 Definition Previously on ELCN102 Electric circuit theorems are always beneficial to help find voltage and currents in multi loop circuits. The network theorems include: Superposition Theorem Thevenin s Theorem Norton s Theorem Maximum Power Transfer Theorem 3

4 Previously on ELCN102 Norton s Theorem A linear two-terminal circuit can be replaced by equivalent circuit consisting of a current source I N in parallel with a resistor R N 4

5 Previously on ELCN102 Steps of Norton s Theorem 1) Identify the load resistance and introduce two nodes a and b 2) Remove the load resistance between node a and b and set all the independent sources to zero (voltage sources are SC and current sources are OC) and calculate the resistance seen between nodes a and b. This resistance is R N of the Norton equivalent circuit. 3) Replace the load resistance with a short circuit and calculate the short circuit current between nodes a and b. This current is I N of the Norton equivalent circuit. 5

6 Previously on ELCN102 Thevenin and Norton equivalent circuits Thevenin equivalent circuit must be equivalent to Norton equivalent circuit R N = R th, V th = I N R N, I N = V th R R th = V th th I N 6

7 Previously on ELCN102 Maximum Power Transfer Theorem The maximum amount of power will be dissipated by a load resistance ( R L ) when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. For maximum power P RL R L = R th = R N 7

8 Definition AC Circuits An AC circuit is a combination of active elements (Voltage and current sources) and passive elements (resistors, capacitors and coils). Unlike resistance, capacitors and coils can store energy and do not dissipate it. Thus, capacitors and coils are called storage elements. 8

9 Definition AC Circuits An AC circuit is a combination of active elements (Voltage and current sources) and passive elements (resistors, capacitors and coils). The sources are usually sinusoidal voltage or current sources 9

10 Capacitors Definition and Structure A capacitor is a passive element designed to store energy in its electric field. A capacitor is an electrical device constructed of two parallel plates separated by an insulating material called the dielectric. Capacitors are the most common component beside resistors. It used in electronics, communication, and computer systems. 10

11 Capacitors Definition and Structure When a voltage source is connected to a capacitor, an electric field is generated in the dielectric and charges are accumulated on the plates. Volt Q = C V Coulomb Farad C = Q V The amount of charge (Q) that a capacitor can store per volt across the plates, is its capacitance (C). 11

12 Capacitors Definition and Structure Most capacitors in electronics have capacitance values of micro- Farad (μf = 10 6 F) to pico-farad (pf = F). 12

13 Capacitors Instantaneous Current i c t = dq t dt Q = C V For constant capacitance (c t = C) = d c t v c t dt q t = C v c t i c (t) = C dv c(t) dt 13

14 Capacitors Instantaneous Power and Energy stored Instantaneous power is given by q(t) = v t i t = v t C dv(t) dt Energy stored in the capacitor is given by w = න q t dt = න v t C dv(t) dt dt = C න v t dv(t) = 1 2 Cv2 t 14

15 Capacitors Series and Parallel Combinations Series Capacitors Q = C V Q Q Q Q Q Q Q Q v equ = v 1 + v v N Q C eq = Q C 1 + Q C Q C N 15

16 Capacitors Series and Parallel Combinations Series Capacitors Q = C V 1 C eq = 1 C C C N 16

17 Capacitors Series and Parallel Combinations Parallel Capacitors Q = C V Q eq = Q 1 + Q Q N C eq v = C 1 v + C 2 v + + C N v 17

18 Capacitors Series and Parallel Combinations Parallel Capacitors Q = C V C eq = C 1 + C C N 18

19 Example (1) Capacitors Find the equivalent capacitance seen between terminals a and b of the circuit shown. 19

20 Inductors Definition and Structure An inductor is a passive element designed to store energy in its magnetic field. An inductor is constructed of a wire wounded into a coil. Inductors are used in power supplies, transformers, radios, TVs, radars, and electric motors. 20

21 Inductors Definition and Structure When the current flowing through an inductor changes, the magnetic field induces a voltage in the conductor, according to Faraday s law of electromagnetic induction, to resist this change in the current. v L t = L di L t dt L is the inductance in Henri (H) 21

22 Inductors Definition and Structure Most inductors in electronics have inductance values of mille Henri (mh = 10 3 H) to micro- Henri (μh = 10 6 H). 22

23 Inductors Series and Parallel Combinations Series Inductors v equ = v 1 + v v N di L eq dt = L di 1 dt + L di 2 dt + + L N di dt 23

24 Inductors Series and Parallel Combinations Series Inductors L eq = L 1 + L L N 24

25 Inductors Series and Parallel Combinations Parallel Inductors v = L di dt i = i 1 + i i N 1 න v dt = 1 න v dt + 1 න v dt න v dt L eq L 1 L 2 L N 25

26 Inductors Series and Parallel Combinations Parallel Inductors 1 = L eq L 1 L 2 L N 26

27 Example (2) Inductors Find the equivalent inductance seen between terminals a and b of the circuit shown. 27

28 Definition Transient Analysis The transient response of the circuit is the response when the input is changed suddenly or a switches status is changed. v t the same v t = v 1 t, t 0 < t < t 1 v 2 t, t 1 < t < t 2 v n t, t n 1 < t < t n 28

29 Definition Transient Analysis The transient response of the circuit is the response when the input is changed suddenly or a switches status is changed. v t the same 29

30 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits At t = 0, capacitor is initially charged to V 0. v c 0 = V 0 At t = 0, the switch is closed. From the resistor From the capacitor i t = v c t R i t = C dv c t dt 30

31 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits C dv c t dt dv c t dt = v c t R = v c t CR This is a first order differential equation with an initial condition v c 0 = V 0 Hint: d dt ae bt = ab e bt 31

32 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits dv c t dt = v c t CR v c t = αe t RC Using the initial condition v c 0 = αe 0 RC = V 0 α = V 0 32

33 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits v c t = V 0 e t RC RC is called the time constant of the circuit (τ) The time constant (τ) of a circuit is the time required for the response to decay to 1/e or 36.8% of its initial value (a change of 63%). 33

34 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits v c t = V 0 e t RC v c t Change τ V 0 63% 2τ V 0 86% 3τ V 0 95% 4τ V 0 98% 5τ V 0 99% It takes about 5τ to change the voltage by 99%. This period of 5τ is called transient time. 34

35 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits v R t = v c t = V 0 e t RC i t = V 0 R e t RC Power dissipated in the resistor p t = v R t i t = V 0 2 e 2t RC R 35

36 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits p t = v R t i t = V 0 2 e 2t RC R Energy dissipated by the resistor t w t = න p t dt 0 = V 0 2 t R න e 2t RC dt = V R RC e 2t RC 2 t 0 = V 0 2 C 2 1 e 2t RC 36

37 Example (3) Transient Analysis For the circuit shown, v C (0) = 15 V. R 1 = 5kΩ, R 2 = 8kΩ, R 3 = 12kΩ, C = 100μF. Find v C t, v x t, and i x t for t > 0. 37

38 Transient Analysis Time Domain Analysis 1 st Order Systems Step response of an RC circuits The step response is the response of the circuit due to a sudden change of voltage or current source. i R t = E v c t R i C t = C dv c t dt 38

39 Transient Analysis Time Domain Analysis 1 st Order Systems Step response of an RC circuits C dv c t dt i C t = i R t C d E v c t dt d E v c t dt = E v c t R = E v c t R = E v c t RC 39

40 Transient Analysis Time Domain Analysis 1 st Order Systems Step response of an RC circuits d E v c t dt = E v c t RC v c 0 = V 0 E v c t = αe t RC E V 0 = αe 0 RC α = E V 0 40

41 Transient Analysis Time Domain Analysis 1 st Order Systems Step response of an RC circuits E v c t = E V 0 e t RC v c t = E E V 0 e t RC If V 0 = 0 v c t = E Ee t RC = E 1 E e t RC 41

42 Transient Analysis Time Domain Analysis 1 st Order Systems Step response of an RC circuits E v c t = E V 0 e t RC v c t = E E V 0 e t RC If V 0 = 0 v c t = E Ee t RC = E 1 E e t RC 42

43 Transient Analysis Time Domain Analysis 1 st Order Systems Step response of an RC circuits v c t = E E V 0 e t RC i t = C dv c t dt = E V 0 R e t RC = E v c t R 43

44 Transient Analysis Time Domain Analysis 1 st Order Systems Steady State Response Steady state response of a circuit is the behavior of the circuit a long time after an external excitation is applied. Theoretically, the steady state period starts at t =. However, practically, it starts at t = 5τ. In steady state, we deal with DC analysis. In DC, the capacitor is consider open circuit (i c = 0) and the coil is consider short circuit (v L = 0). i c (t) = C dv c(t) dt v L t = L di L t dt 44

45 Transient Analysis Time Domain Analysis 1 st Order Systems Steady State Response At steady state, v c = E i = 0 v c t = E E V 0 e t RC i t = E V 0 R e t RC t= t= v c = E i = 0 45

46 Transient Analysis Time Domain Analysis 1 st Order Systems Transient response Transient response of a circuit is the circuit s temporary response that will die out with time. The transient period starts at t = 0 and ends at t = 5τ Generally, For a first order RC circuit v c t = V f V f V i e t τ V i and V f are the initial and final capacitor voltages, respectively. τ = R eq C, R eq is the resistance seen between the capacitor nodes while all sources are switched off. 46

47 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits v c t = V f V f V i e t τ V i = v c 0 = V 0 V f = v c = 0 v c t = 0 0 V 0 e t RC = V 0 e t RC τ = RC 47

48 Transient Analysis Time Domain Analysis 1 st Order Systems Steady State Response v c t = V f V f V i e t τ V i = v c 0 = V 0 V f = v c = E v c t = E E V 0 e t RC τ = RC 48

49 Example (4) Transient Analysis For the shown circuit, switch has been in position a for a long time. At t = 0, the switch moves to b. Determine v c (t) for t > 0 and calculate its value at t = 1, 4, and 20ms. 49

50 Example (5) Transient Analysis For the shown circuit, switch has been open for a long time and is closed at t = 0. Calculate v c (t) at t = 0.5 and 4ms. 50

51 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RL Circuits v L t = L di L t dt v R t = Ri L t v L t = v R t L di L t dt di L t dt = Ri L t = R L i L t 51

52 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RL Circuits di L t dt = R L i L t i L 0 = I 0 i L t = I 0 e t τ, τ = L R Generally, i L t = I f I f I i e t τ, τ = L R 52

53 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RL Circuits i L t = I f I f I i e t τ, τ = L R I i = i L 0 = I 0 Then I f = 0 τ = L R i L t = 0 0 I 0 e t τ = I 0 e t τ 53

54 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RL Circuits i L t = I 0 e t τ v R t = i L t R = I 0 Re t τ Power dissipated in the resistor p t = v R t i L t = I 2 0 Re 2t τ 54

55 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RL Circuits p t = v R t i L t = I 2 0 Re 2t τ Energy dissipated by the resistor t w t = න p t dt 0 t = I 2 0 R න e 2t τ dt = I 2 0 R L e 2t τ 0 2R t 0 = I 0 2 L 2 1 e 2t τ 55

56 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits i L t = I f I f I i e t τ I i = i L 0 = I 0 I f = E R τ = L R i L t = E R E R I 0 e t τ i L t = E R 1 e t τ 56

57 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits i L t = I f I f I i e t τ I i = i L 0 = I 0 I f = E R τ = L R i L t = E R E R I 0 e t τ i L t = E R 1 e t τ 57

58 Transient Analysis Time Domain Analysis 1 st Order Systems Source Free RC Circuits i L t = E R 1 e t τ v R t = R i L t = E 1 e t τ v L t = L di L t dt = E v R t = Ee t τ 58

59 Example (6) Transient Analysis For the shown circuit, switch has been closed for a long time and is opened at t = 0. Calculate i L (t) for t > 0. 59

60 Example (7) Transient Analysis For the shown circuit, switch has been opened for a long time and is closed at t = 0. find i R, v o, and i L for all time. 60

61 Example (8) Transient Analysis For the shown circuit, switch has been closed for a long time and is opened at t = 0. find i L (t) for t > 0. 61

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