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Equation 44 is the de ning circuit law for a capacitor If the differential equation that de nes the i-v relationship for a capacitor is integrated, one can obtain the following relationship for the voltage across a capacitor: vC (t) = 1 C
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Equation 45 indicates that the capacitor voltage depends on the past current through the capacitor, up until the present time, t Of course, one does not usually have precise information regarding the ow of capacitor current for all past time, and so it is useful to de ne the initial voltage (or initial condition) for the capacitor according to the following, where t0 is an arbitrary initial time: V0 = vC (t = t0 ) = 1 C
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The capacitor voltage is now given by the expression vC (t) = 1 C
t t0
1 1 1 1 + + C1 C2 C3 Capacitances in series combine like resistors in parallel CEQ =
iC dt + V0
t t0
(47)
C1 C2 C3
The signi cance of the initial voltage, V0 , is simply that at time t0 some charge is stored in the capacitor, giving rise to a voltage, vC (t0 ), according to the relationship Q = CV Knowledge of this initial condition is suf cient to account for the entire past history of the capacitor current Capacitors connected in series and parallel can be combined to yield a single equivalent capacitance The rule of thumb, which is illustrated in Figure 42, is the following:
CEQ = C1 + C2 + C3 Capacitances in parallel add
Figure 42 Combining capacitors in a circuit
Capacitors in parallel add Capacitors in series combine according to the same rules used for resistors connected in parallel
4
AC Network Analysis
EXAMPLE 41 Calculating Capacitor Current from Voltage
Problem
Calculate the current through a capacitor from knowledge of its terminal voltage
Solution
Known Quantities: Capacitor terminal voltage; capacitance value Find: Capacitor current Schematics, Diagrams, Circuits, and Given Data: v(t) = 5 e t/10
V t 0 s;
C = 01 F The terminal voltage is plotted in Figure 43
4 iC (t) v (t), V 3
v (t)
2 1
2 Time, s
Figure 43 Assumptions: The capacitor is initially discharged: v(t = 0) = 0 Analysis: Using the de ning differential relationship for the capacitor, we may obtain
the current by differentiating the voltage: dv(t) 5 6 6 = 10 7 6 e t/10 A t 0 = 05e t/10 dt 10 A plot of the capacitor current is shown in Figure 44 Note how the current jumps to 05 A instantaneously as the voltage rises exponentially: The ability of a capacitor s current to change instantaneously is an important property of capacitors iC (t) = C
Comments: As the voltage approaches the constant value 5 V, the capacitor reaches its
maximum charge-storage capability for that voltage (since Q = CV ) and no more current ows through the capacitor The total charge stored is Q = 05 10 6 C This is a fairly small amount of charge, but it can produce a substantial amount of current for a brief period of time For example, the fully charged capacitor could provide 100 mA of current
Part I
Circuits
iC (t), A
Time, s
for a period of time equal to 5 s: I= Q 05 10 6 = = 01 A t 5 10 6
There are many useful applications of this energy-storage property of capacitors in practical circuits
Focus on Computer-Aided Tools: The MatlabTM m- les used to generate the plots of
Figures 43 and 44 may be found in the CD-ROM that accompanies this book
EXAMPLE 42 Calculating Capacitor Voltage from Current and Initial Conditions
Problem
Calculate the voltage across a capacitor from knowledge of its current and initial state of charge
Solution
Known Quantities: Capacitor current; initial capacitor voltage; capacitance value Find: Capacitor voltage Schematics, Diagrams, Circuits, and Given Data:
iC (t) =
0 I = 10 mA 0
t <0s 0 t 1s t >1s
vC (t = 0) = 2 V; C = 1,000 F The capacitor current is plotted in Figure 45(a)
4
AC Network Analysis
10 9 8 7 6 5 4 3 2 1 0 02
02 04 06 08 Time (s) (a)
12 11 10 9 8 7 6 5 4 3 2 02
ic (t) mA
vc (t) V
02 04 06 08 Time (s) (b)
Assumptions: The capacitor is initially charged such that vC (t = t0 = 0) = 2 V Analysis: Using the de ning integral relationship for the capacitor, we may obtain the
voltage by integrating the current: vC (t) = 1 C
iC (t ) dt +vC (t0 )
t t0
1 I 1 I dt + V0 = t + V0 = 10t + 2 V C 0 C vC (t) = 12 V
0 t 1s t >1s
further but remains now at the maximum value it reached at t = 1 s: vC (t = 1) = 12 V The nal value of the capacitor voltage after the current source has stopped charging the capacitor depends on two factors: (1) the initial value of the capacitor voltage, and (2) the history of the capacitor current Figure 45(a) and (b) depicts the two waveforms
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