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Capacitance

 

When a voltage is applied to a circuit containing capacitance, current flows to accumulate charge in the capacitance:
Q =
òidt = CV

Alternatively, by differentiation with respect to time:
dq/dt = i = C dv/dt
Note that the rate of change of voltage has a polarity which opposes the flow of current.

The capacitance C of a circuit is equal to the charge divided by the voltage:
C = Q / V =
òidt / V

Alternatively, the capacitance C of a circuit is equal to the charging current divided by the rate of change of voltage:
C = i / dv/dt = dq/dt / dv/dt = dq/dv

 

Capacitances in Series

When capacitances C1, C2, C3, ... are connected in series, the total capacitance CS is:
1 / CS = 1 / C1 + 1 / C2 + 1 / C3 + ...

For two capacitances C1 and C2 connected in series, the total capacitance CS is:
CS = C1C2 / (C1 + C2)
CS = product / sum

 

Voltage Division by Series Capacitances

When a total voltage ES is applied to series connected capacitances C1 and C2, the charge QS which accumulates in the series circuit is:
QS =
òiSdt = ESCS = ESC1C2 / (C1 + C2)

The voltages V1 and V2 which appear across the respective capacitances C1 and C2 are:
V1 =
òiSdt / C1 = ESCS / C1 = ESC2 / (C1 + C2)
V2 =
òiSdt / C2 = ESCS / C2 = ESC1 / (C1 + C2)

In general terms, for capacitances C1, C2, C3, ... connected in series:
QS =
òiSdt = ESCS = ES / (1 / CS) = ES / (1 / C1 + 1 / C2 + 1 / C3 + ...)
Vn =
òiSdt / Cn = ESCS / Cn = ES / Cn(1 / CS) = ES / Cn(1 / C1 + 1 / C2 + 1 / C3 + ...)
Note that the highest voltage appears across the lowest capacitance.

 

Capacitances in Parallel

When capacitances C1, C2, C3, ... are connected in parallel, the total capacitance CP is:
CP = C1 + C2 + C3 + ...

 

Charge Division by Parallel Capacitances

When a voltage EP is applied to parallel connected capacitances C1 and C2, the charge QP which accumulates in the parallel circuit is:
QP =
òiPdt = EPCP = EP(C1 + C2)

The charges Q1 and Q2 which accumulate in the respective capacitances C1 and C2 are:
Q1 =
òi1dt = EPC1 = QPC1 / CP = QPC1 / (C1 + C2)
Q2 =
òi2dt = EPC2 = QPC2 / CP = QPC2 / (C1 + C2)

In general terms, for capacitances C1, C2, C3, ... connected in parallel:
QP =
òiPdt = EPCP = EP(C1 + C2 + C3 + ...)
Qn =
òindt = EPCn = QPCn / CP = QPCn / (C1 + C2 + C3 + ...)
Note that the highest charge accumulates in the highest capacitance.