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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
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4 Capacitance The voltage source transfers negative charges to one plate, and positive charges to the other plate. VSVS E
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5 Capacitance The charges establish an electric field between the plates. Energy is stored in the electric field. VSVS E
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6 Capacitance The effect of the electric field is to maintain the voltage between the two plates. VSVS E
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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.
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8 Capacitance Recall the definition of current
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9 Capacitance A capacitance in a circuit is represented by the symbol shown below. Capacitance is measured in Farads (F).
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10 Capacitance The current-voltage relationship for a capacitance is shown in the equation below.
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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.
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12 The voltage source Places a time- varying voltage across the capacitor
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13 The current through the capacitor Is proportional to the time rate-of-change of the voltage
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14 The current through the capacitor Is proportional to the slope of this waveform
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15 Slope = 0 Current = 0
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16 Slope = 4 Current = 4C
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17 Slope = 0 Current = 0
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18 Slope = 2 Current = 2C
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19 Slope = 0 Current = 0
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20 The voltage source Places a time-varying, sinusoidal voltage across the capacitor
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21 Amplitude Period = T 0
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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.
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26 The voltage source Drives a time- varying voltage across the capacitor
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27 The current through the capacitor Is proportional to the slope of this waveform
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28 The instantaneous change in voltage Causes an infinite current pulse through the capacitor
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29 An infinite pulse cannot be achieved !
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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.
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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.
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32 Capacitance Consider a current source driving a capacitance. An initial voltage v 0 is also placed across the capacitor.
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33 Capacitance The voltage-current relationship is given by:
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34 Capacitance Performing a bit of calculus:
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35 Capacitance Integrating both sides:
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36 Capacitance Completing the integration:
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37 Capacitance Solving for v(t):
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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.
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39 The current source Drives a time- varying current through the capacitance
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40 The voltage across the capacitor Is proportional to the area under the current waveform
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41 Initial voltage Accumulated area Final voltage
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42 The current source Drives a sinusoidal current through the capacitance
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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.
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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:
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48 Power and Energy in a Capacitance Leads to the instantaneous power in an capacitance: By definition, power is given by:
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49 Power and Energy in a Capacitance Consequently : and :
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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.
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51 Power and Energy in a Capacitance The energy stored in a capacitance is:
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55 Capacitors in Series
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56 Capacitors in Series
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57 Capacitors in Parallel
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58 Capacitors in Parallel
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59 Capacitors in Parallel
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