# Is it right to apply superposition theorem to the RC circuit with non-zero initial capacitor voltage?

I'm having confusion with applying superposition to simple RC circuit with non-zero initial capacitor voltage.

simulate this circuit – Schematic created using CircuitLab

Consider a simple RC circuit series connected with voltage source of step voltage $V_s=8u(t)$, and let's say initial voltage of the capacitor is $V_c = 2V$.

It's obvious that the value of $V_c$ over time is

$V_c = 2+(8-2)(1-e^{t/RC})$.

But if I split the voltage source $V_s=8u(t)$ with two voltage sources $V_{s1}=4u(t), V_{s2}=4u(t)$, then the corresponding $V_{c1}$ and $V_{c2}$ will be

$V_{c1} = 2+(4-2)(1-e^{t/RC})$.

$V_{c2} = 2+(4-2)(1-e^{t/RC})$, respectively.

Then, applying superposition yields $V_{c1}+V_{c2}=4+4(1-e^{t/RC})$, which is different from

$V_c = 2+(8-2)(1-e^{t/RC})$.

Where am I doing wrong?

Is it wrong to apply superposition theorem to non-zero state of capacitor?

• Draw the circuit? – MathieuL Aug 12 '15 at 15:10
• @MathieuL Edited the post – asqdf Aug 12 '15 at 15:14
• The problem here that you are using a solution to the differential equation describing the circuit, which is including the boundary conditions and is not exhibiting the linear properties by itself. To benefit the linear nature of the circuit and use the superposition rule, the differential equations should be written using it, and then solved. – Eugene Sh. Aug 12 '15 at 15:20
• A somewhat related concept: you actually can use superposition with capacitors if you are working with signals that are all at a single frequency. To read more, look up "phasor notation" and "impedance", en.wikipedia.org/wiki/Electrical_impedance. It does not apply to the unit step function though, as that is not a single frequency. – Justin Aug 12 '15 at 18:08
• @Justin You can use the superposition theorem with capacitors only when you are dealing with a sinusoidal "steady state" response of the circuit. When you are dealing with the problem of transient response of RLC circuit, you cannot apply the impedance approach and cannot apply the superposition theorem naively because the RLC circuit's differential equation can be nonlinear when capacitors or inductors in the circuit have non-zero state. – asqdf Aug 13 '15 at 7:43