## 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 C_{1}, C_{2}, C_{3}, ... are connected
in series, the total capacitance C_{S} is:

1 / C_{S} = 1 / C_{1} + 1 / C_{2} + 1 / C_{3}
+ ...

For
two capacitances C_{1} and C_{2} connected in series, the
total capacitance C_{S} is:

C_{S} = C_{1}C_{2} / (C_{1} + C_{2})

C_{S} = product / sum

**
**

####
Voltage
Division by Series Capacitances

When
a total voltage E_{S} is applied to series connected capacitances C_{1}
and C_{2}, the charge Q_{S} which accumulates in the series
circuit is:

Q_{S} = òi_{S}dt
= E_{S}C_{S} = E_{S}C_{1}C_{2} / (C_{1}
+ C_{2})

The
voltages V_{1} and V_{2} which appear across the respective
capacitances C_{1} and C_{2} are:

V_{1} =
òi_{S}dt
/ C_{1} = E_{S}C_{S} / C_{1} = E_{S}C_{2}
/ (C_{1} + C_{2})

V_{2} =
òi_{S}dt
/ C_{2} = E_{S}C_{S} / C_{2} = E_{S}C_{1}
/ (C_{1} + C_{2})

In
general terms, for capacitances C_{1}, C_{2}, C_{3},
... connected in series:

Q_{S} =
òi_{S}dt
= E_{S}C_{S} = E_{S} / (1 / C_{S}) = E_{S}
/ (1 / C_{1} + 1 / C_{2} + 1 / C_{3} + ...)

V_{n} =
òi_{S}dt
/
C_{n} = E_{S}C_{S} / C_{n} = E_{S}
/ C_{n}(1 / C_{S}) = E_{S} / C_{n}(1 / C_{1}
+ 1 / C_{2} + 1 / C_{3} + ...)

Note that the highest voltage appears across the lowest capacitance.

**
**

####
Capacitances
in Parallel

When
capacitances C_{1}, C_{2}, C_{3}, ... are connected
in parallel, the total capacitance C_{P} is:

C_{P} = C_{1} + C_{2} + C_{3} + ...

**
**

####
Charge
Division by Parallel Capacitances

When
a voltage E_{P} is applied to parallel connected capacitances C_{1}
and C_{2}, the charge Q_{P} which accumulates in the
parallel circuit is:

Q_{P} = òi_{P}dt
= E_{P}C_{P} = E_{P}(C_{1} + C_{2})

The
charges Q_{1} and Q_{2} which accumulate in the respective
capacitances C_{1} and C_{2} are:

Q_{1} =
òi_{1}dt
= E_{P}C_{1} = Q_{P}C_{1} / C_{P} =
Q_{P}C_{1} / (C_{1} + C_{2})

Q_{2} =
òi_{2}dt
= E_{P}C_{2} = Q_{P}C_{2} / C_{P} =
Q_{P}C_{2} / (C_{1} + C_{2})

In
general terms, for capacitances C_{1}, C_{2}, C_{3},
... connected in parallel:

Q_{P} =
òi_{P}dt
= E_{P}C_{P} = E_{P}(C_{1} + C_{2} +
C_{3} + ...)

Q_{n} =
òi_{n}dt
=
E_{P}C_{n} = Q_{P}C_{n} / C_{P} =
Q_{P}C_{n} / (C_{1} + C_{2} + C_{3}
+ ...)

Note that the highest charge accumulates in the highest capacitance.