After finding the zero-input response and the zero-state response of an RC series circuit, you can easily find the total response of the circuit. Remember that a first-order RC series circuit has one resistor (or network of resistors) and one capacitor connected in series.

Here is a sample RC circuit shown with zero-input response and zero-state response. The top-right diagram shows the zero-input response, which you get by setting the input to 0. The bottom-right diagram shows the zero-state response, which you get by setting the initial conditions to 0.

The first-order differential equation you need to find the zero-input response *v*_{ZI}*(t)** *for this circuit.

After applying your math skills, you find the zero-input response of the circuit:

Now to find the zero-state response, you need to study the circuit under zero initial conditions (when there’s no voltage across the capacitor at time *t *= 0). The sample circuit has *zero initial conditions* and an input voltage of V_{T}*(t)* = *u(t)*, where *u(t)* is a unit step input.

After applying your math skills again, you find the zero-state response of the circuit:

Suppose that *RC** *= 1 second and initial voltage *V*_{0} = 5 volts. Then this chart plots the decaying exponential, showing that it takes about 5 time constants, or 5 seconds, for the capacitor voltage to decay to 0.

Now you are ready to finally add up the zero-input response *v*_{ZI}*(t)* and the zero-state response *v*_{Z}_{S}*(t)* to get the total response *v(t)*:

Time to verify whether the solution is reasonable. When *t* = 0, the initial voltage across the capacitor is

You bet this is a true statement! But you can check out when the initial conditions die out after a long period of time if you feel unsure about your solution. The output should just be related to the input voltage or step voltage.

After a long period of time (or after 5 time constants), you get the following:

Another true statement. The output voltage follows the step input with strength *V** _{A}* after a long time. In other words, the capacitor voltage charges to a value equal to the strength

*V*

*of the step input after the initial condition dies out in about 5 time constants.*

_{A}Try this example with these values: *V*_{0} = 5 volts, *V** _{A}* = 10 volts, and RC = 1 second. You should get the capacitor voltage charging from an initial voltage of 5 volts and a final voltage of 10 volts after 5 seconds (5 time constants). Using the given values, you get the plot shown here.

The plot starts at 5 volts, and you end up at 10 volts after 5 time constants (5 seconds = 5*RC*). So this example shows how changing voltage states takes time. Circuits with capacitors don’t change voltages instantaneously. A large resistor also slows things down. That’s why the time constant *RC* takes into account how the capacitor voltage will change from one voltage state to another.

The total capacitor voltage consists of the zero-input response and a zero-state response:

This equation shows that the total response is a combination of two outputs added together: one output due only to the initial voltage *V*_{0} = 5 volts (at time *t* = 0) and the other due only to the step input with strength *V** _{A}* = 10 volts (after time

*t*= 0).