MEMS 0031 Electric Circuits. Chapter 7-1: Energy Storage Elements- Capacitors

Transcription

1 MEMS 0031 Electric Circuits Chapter 7-1: Energy Storage Elements- Capacitors

2 Lesson Objectives Energy Storage : Capacitors Transients-RC Circuits Energy Storage: Inductors-RCL circuits

3 Capacitor A C = ε d C = Capacitance ε = dielectric permitivity (F/m) d= thickness (m) of dielectric or gap of the electrodes A= surface area (m ) 2 Capacitance is a measure of the ability of a device to store energy in the form of separate charge or electric field

4 Consider a charge +q on one place of the capacitors and -q on the other dues to the voltage v. the amount of charge stored in the plate with time is proportional to the voltage across it such that: When the capacitor is charged, current flows due to the change in charge flow per unit time such that:

5 If v is a constant after the initial charging process done, the current For v = Kt: For v = 10cos ωt: For v = Ae -t/τ i dv = C = 0 dt dv = = i C CK dt dv i = C = 10Cωcosωt dt d 1 i = C [ Ae ] = CA( ) e dt τ = CA t / τ e τ t/ τ t/ τ

6 Find voltage in term of current for a capacitor i = dv C dt 1 v= i( τ) d C v= i( ) d + i( ) d C C t 1 = i( τ ) dt + v( t0) C t 0 t t τ τ τ τ τ t t 0 The voltage across a capacitor cannot change instantaneously

7 Circuit symbols of a capacitor adhering ton PSC

8 Example EX Determine the current i(t) for t > 0 for the circuit (b) when v s (t) is given by (a) Solution: 2 2< t < 4 d ic( t) = 1 vs( t) = 1 4< t < 8 dt 0 otherwise and 2t 4 2< t < 4 ir( t) = 1vs( t) = 8 t 4< t < 8 0 otherwise so 2t 2 2< t < 4 it () = ic ( t) + ir( t) = 7 t 4< t< 8 0 otherwise

9 Capacitors is charged to vc =10 V Switch is opened at t=0

10 Set v(- ) =0

11

12 Example Sketch the capacitor current and stored energy as a function of time

13

14

15 Problem 7-3-6

16

17

18

19 Capacitors in Parallel

20

21

22

23

24 Ex 7-4-1

25

26

27

28

29 Type of capacitors Electrolytic Capacitor: They are polarized. Similar to metal film, but the electrodes are made of aluminum etched to acquire much higher surfaces, and the dielectric is soaked with liquid electrolyte Tantalum: Like electrolytic, it is polarized. It has better performance with higher frequencies Super Capacitors: Made from carbon aerogel, carbon nanotubes or highly porous electrode materials. Extremely high capacity

30

31 CLASSIFIED BASED ON THE DIELECTRIC MATERIAL Metal Film Capacitor: Made from high quality polymer film and metal foil, with a layer of metal deposited on surface. Mica Capacitor: These are very! similar to metal film. Often high voltage and suitable for high frequencies. They are Expensive. Paper Capacitor: They are usually used for high voltages. Glass Capacitor: They are used for high voltages. Stable temperature coefficient in a wide range of temperatures. They are also expensive. Ceramic Capacitor: Chips of altering layers of metal and ceramic. They often have high dissipation factor, high frequency coefficient of dissipation, their capacity depends on applied

32 Lesson Objectives Energy Storage : Capacitors Transients-RC Circuits Energy Storage: Inductors-RCL circuits

33 Inductor= An circuit element that stores energy in its magnetic field Inductors

34

35

36

37

38 d vt () = L it () dt 1 i = v( τ) dτ L 0 t i = v( τ) dτ + v( τ) dτ L L t 1 = v( τ ) dt + i( t0) L t t t t 0

39

40

41

42

43 Example 2: A current source is applied to a 5-H inductor. Sketch the voltage and energy stored in the inductors

44

45

46

47

48 Series and Parallel Inductors Series i = i = i = i 1 2 vt ( ) = v + v v ; n= di di di = L1 + L LN dt dt dt di = ( L1+ L LN ) dt = L s n= N ( L ) = n n= N ( L ) n= 1 N di dt n N

49 Parallel

50 Example

51 Example

52 P The inductor current in the circuit below is given as i(t) = 6 + 4e -8t (A) for t 0. Find v(t) for t > 0.

53 P The inductor current in the circuit below is given as i(t) = 5-3e -4t (A) for t 0. Find v(t) for t > 0.

54 Chapter 7: Lecture 7-3 Energy Storage Elements- Switches and Transients-RC, RCL Circuits

55 Lesson Objectives Energy Storage : Capacitors Transients-RC, RCL Circuits Energy Storage: Inductors-RCL circuits

56 Transient circuits have the following characteristics: Independent voltage source voltages and independent current source currents are constant function of time The circuit includes one or more switches that open or close at time t 0 Time immediately before the switch opens or closes = t _ 0 Time immediately after the switch opens or closes= t + 0 The circuit includes at least one capacitor or inductor.

57 Assume that the switches in a circuit have been in position for a long time at t = t 0 called the switching time: For such long-time, circuit is at steady state immediately before the time of switching. A circuit that contains only constant sources and is at steady state is called a dc circuit. The inductor current in the dc circuit, will be a constant function of time. The inductor voltage is v L = L(di/dt)=0. Hence, an inductor in a dc circuit behaves as a short circuit. Capacitor voltage in DC circuit will be a constant function of time. The capacitor current is proportional to the derivative of the capacitor voltage, i C = C (dv / dt)=0 Hence, the capacitor in a dc circuit acts like a open circuit.

58

59 = τ τ + c C 0 v (0 ) i( ) d v (0 ) c = 0 + v (0 ) c v (0 ) = v (0 ) c + c

60

61

62

63

64

65 Plan to analyze switched circuits Analyze the dc circuit that exists before time t 0 to determine the capacitor voltages and inductor currents. Note: Capacitors behave as open circuits and inductors behave as short circuits when they are in dc circuits. Recognize that capacitor voltages and inductor currents cannot change instantaneously, Capacitor voltages and inductor currents at time t 0 + have the same values that they had at time t 0 _

66

67

68 0

69

70

71

72

73

74 P7.8-2 Calculation Procedure: RCL Circuit

75

76 RCL Circuit P 7-8.3

77 Find the history at t=0 - Find the history at t=0 +

78 Find the history at t=

79 Example to a

80