1 ECE 3301 General Electrical Engineering Section 22 Capacitance

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3 Consider two parallel plates, separated by a non-conducting (dielectric) material as shown below. VSVS

4 Capacitance The voltage source transfers negative charges to one plate, and positive charges to the other plate. VSVS E

5 Capacitance The charges establish an electric field between the plates. Energy is stored in the electric field. VSVS E

6 Capacitance The effect of the electric field is to maintain the voltage between the two plates. VSVS E

7 Capacitance The total charge on the capacitor is related to the voltage across the capacitor by a constant of proportionality known as capacitance. q is the charge in Coulombs, C the capacitance in Farads and v the voltage in Volts.

8 Capacitance Recall the definition of current

9 Capacitance A capacitance in a circuit is represented by the symbol shown below. Capacitance is measured in Farads (F).

10 Capacitance The current-voltage relationship for a capacitance is shown in the equation below.

11 Capacitance The current through a capacitance is proportional to the time-rate-of-change of the voltage across the capacitance. The constant of proportionality is called the capacitance.

12 The voltage source Places a time- varying voltage across the capacitor

13 The current through the capacitor Is proportional to the time rate-of-change of the voltage

14 The current through the capacitor Is proportional to the slope of this waveform

15 Slope = 0 Current = 0

16 Slope = 4 Current = 4C

17 Slope = 0 Current = 0

18 Slope =  2 Current =  2C

19 Slope = 0 Current = 0

20 The voltage source Places a time-varying, sinusoidal voltage across the capacitor

21 Amplitude Period = T 0

22 The current through the capacitor Is proportional to the time rate-of-change of the voltage

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25 The current and voltage are 90 degrees “out of phase” with each other. The current leads the voltage by 90 degrees.

26 The voltage source Drives a time- varying voltage across the capacitor

27 The current through the capacitor Is proportional to the slope of this waveform

28 The instantaneous change in voltage Causes an infinite current pulse through the capacitor

29 An infinite pulse cannot be achieved !

30 Rule 1 of Capacitance Since this infinite current impulse cannot be physically realized, we conclude the first rule-of-thumb about inductance. One cannot instantaneously change the voltage across a capacitance.

31 Rule 2 of Capacitance Since the current through a capacitance is proportional to the time rate-of-change of the voltage across the capacitance, when the voltage is constant (DC), the current is zero. A current of zero through a circuit element is the definition of an open circuit. This leads to the second rule-of-thumb about capacitances. A capacitance is an open circuit to Steady State DC.

32 Capacitance Consider a current source driving a capacitance. An initial voltage v 0 is also placed across the capacitor.

33 Capacitance The voltage-current relationship is given by:

34 Capacitance Performing a bit of calculus:

35 Capacitance Integrating both sides:

36 Capacitance Completing the integration:

37 Capacitance Solving for v(t):

38 Capacitance The voltage across a capacitance is proportional to the integral of the current through the capacitance plus the initial voltage across the capacitance.

39 The current source Drives a time- varying current through the capacitance

40 The voltage across the capacitor Is proportional to the area under the current waveform

41 Initial voltage Accumulated area Final voltage

42 The current source Drives a sinusoidal current through the capacitance

43 The voltage Is proportional to the integral of the current

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46 The current and voltage are 90 degrees “out of phase” with each other. The current leads the voltage by 90 degrees.

47 Power and Energy in a Capacitance The instantaneous power absorbed by any circuit element is given by: Using the voltage-current relationship for a capacitance:

48 Power and Energy in a Capacitance Leads to the instantaneous power in an capacitance: By definition, power is given by:

49 Power and Energy in a Capacitance Consequently : and :

50 Power and Energy in a Capacitance The energy stored in a capacitance may be found by integration. Assume an initial energy of zero and an initial voltage of zero.

51 Power and Energy in a Capacitance The energy stored in a capacitance is:

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55 Capacitors in Series

56 Capacitors in Series

57 Capacitors in Parallel

58 Capacitors in Parallel

59 Capacitors in Parallel