Capacitors 3: In Series and Parallel

by gaussmarkov

The Capacitor “Law”

As described previously, capacitors are fundamentally different from resistors. They do not conduct DC current, but capacitors do transmit AC current because they can repeatedly accumulate and release a voltage potential like a rechargeable battery.

Charging and discharging are responses to changes in the voltage across the leads of a capacitor. There is a simple equation that describes accurately the relationship between the current and the change in voltage for a capacitor with a capacitance of C farads:

I(t) = C ⋅
dV(t)
dt

where I(t) is the current at time t, V(t) is the voltage at time t, and dV(t)/dt is the rate of change of voltage at time t. I am using the calculus notation for a derivative to denote the rate of change of voltage.

This relationship plays the same role for analyzing capacitors that Ohm’s law plays for resistors. That’s why I called this section The Capacitor “Law.” Indeed, if we rearrange terms, this equation is similar to Ohm’s law:

dV(t)
dt
= I(t)⋅
1
C
versus V(t) = I(t) ⋅ R

The differences between Ohm’s law and the capacitor law are that voltage V(t) has been replaced with the rate of change in voltage dV(t)/dt and the constant R has been replaced by the constant 1/C. In what follows, voltage and rate of change in voltage can be treated analogously. The reciprocal notation for the constants of proportionality explains the differences between the formulas for combining capacitors and the formulas for combining resistors.

Capacitors in Parallel

The same argument applies to capacitors in parallel as resistors, so what is written here will be a briefer version of what I have written for resistors in parallel.

The voltage across two parallel capacitors is the same for both capacitors. This applies to the change in voltage as well. So we can also write the capacitor law that applies to each capacitor as a function of a single voltage rate of change:

I1 = C1
dVP(t)
dt
and I2 = C2
dVP(t)
dt

where C1 is the capacitance of capacitor C1, C2 is the capacitance of capacitor C2, and VP(t) denotes the common voltage for the pair of capacitors.

Kirchoff’s Current Law says that the currents flowing through each of the capacitors sum to the current flowing through them as a parallel pair:

IP = I1 + I2

where IP denotes the combined current, I1 is the current through C1, and I2 is the current through C2. If we combine these relationships then we can figure out the effective capacitance of the capacitors. Substituting the capacitor law expressions into the current equation above gives

IP = C1
dVP(t)
dt
+ C2
dVP(t)
dt
= (C1 + C2) ⋅
dVP(t)
dt

This last equation tells us the combined capacitance of the parallel capacitors. It is the capacitor law for a capacitor with capacitance equal to the sum of the individual capacitances, C1 + C2.

Capacitors in Series

Again, following the argument for resistors, Let VS be the total voltage difference across two capacitors in series. Kirchoff’s Voltage Law says that the voltage across the individual resistors, V1 and V2, sums to VS. This also applies to rates of change:

dVS(t)
dt
=
dV1(t)
dt
+
dV2(t)
dt

At the same time, the currents through the capacitors are equal:

IS(t) = I1(t) = I2(t).

Combining these equations with the capacitors law for each capacitor,

dVS(t)
dt
=
IS(t)
C1
+
IS(t)
C2
= IS(t) ⋅ (1/C1 + 1/C2)

or

IS(t) =
C1 C2
C1 + C2
dVS(t)
dt

we obtain the capacitor law for a capacitance of C1⋅C2/(C1 + C2), which we can interpret as the capacitance of two capacitors in series.

Capacitors versus Resistors

It appears as though the formulas for combining resistors are the reverse of the formulas for combining capacitors. Resistors in series have a resistance
equal to their sum whereas capacitors in series have a capacitance equal to their product divided by their sum

R1 + R2 versus
C1 C2
C1 + C2

while resistors in parallel have a combined resistance equal to their product divided by their sum whereas capacitors in parallel have a combined capacitance equal to their sum

R1 R2
R1 + R2
versus C1 + C2

However, if we were to measure the value of a capacitor as the reciprocal of capacitance these calculations would be identical. This is what is suggested by the comparison of Ohm’s law and the capacitor law above. In words, the reciprocal capacitance of two capacitors in series is the sum of the individual reciprocal capacitances:

1/C1 + 1/C2 =
C1 + C2
C1 C2

Similarly, the reciprocal capacitance of two capacitors in parallel is the product of the individual reciprocal capacitances divided by the sum of the individual reciprocal capacitances:

1/C1 1/C2
1/C1 + 1/C2
=
1
C1 + C2

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One Response to “Capacitors 3: In Series and Parallel”

  1. Mads said:

    Thanks for this simplified theory lesson on dV/dt on parallel and series capacitors, just what I needed at this moment to twist my head around my thoughts for a project 🙂

    Posted 15.11.2011 at 3:08 pm