RC Circuits

An RC circuit is one that contains only resistors and capacitors. The simplest possible RC circuit is the series circuit shown below:

Since the resistor and capacitor are in series, we can immediately write the impedance of the circuit:

Z = R +X_C

We know that X_C depends on the frequency of the AC current flowing through the capacitor:

X_C =  -j/(2pf C)

We can combine these two equations to get an expression for Z in terms of f, C and R:

Z = R -j/(2pf C)

Let us consider a specific series RC circuit, shown in the figure below.

We see that the applied AC voltage has an amplitude of 12 and a frequency of 60 Hz. We use this information to compute the reactance of the capacitor:

Now we can determine the current, since we know that Z = R -j/(2pf C):

This is a series circuit so the same current flows through both components. We may use the current and Ohm's Law to determine the voltage drop across R and C:

We will now convert the current and voltage drops to polar form in order to plot them on a phasor diagram:

Thus:

I = 3Ð60

V_R = 6Ð60

V_C = 10.4Ð-30

Notice that the voltage across the resistor and the current are in phase. This is always the case, because the impedance of a resistor is purely real. The current across the capacitor leads the voltage by 90 degrees. This is always true as well and it happens because the impedance of an capacitor is purely imaginary and negative. 

From the phasor diagram, one can see that  V_C + V_R = V₀ , in accordance with KVL (Kirchoff's Voltage Law). The voltages can be added vectorially (phasors are a special kind of vector) to get V₀ using the parallelogram rule (as indicated by the dotted lines) or they can be added algebraically using the techniques described in the complex arithmetic module. 

It is possible to compute the total power delivered by the AC source and the power consumed by each component:

All the power is consumed by the resistor. No power is consumed by the capacitor because the current through it and the voltage across it are 90 degrees out of phase. 

The reactance of the capacitor is a function of frequency which makes the voltage across the capacitor a function of frequency. The graph below shows how the voltage drops across the resistor and the capacitor vary as the frequency of the applied AC varies.

As the frequency of the applied AC increases, the magnitude of the voltage across the resistor decreases. This happens because the reactance of the capacitor increases with frequency, while the resistance of the resistor stays constant. At high frequencies, most of the voltage is dropped across the capacitor.

As the frequency increases, the phase of the voltage across the resistor decreases from 90 to 0 degrees. This is also caused by the decreasing reactance of the capacitor. As the capacitive reactance decreases, the current through the RC circuit lags the applied voltage by a smaller and smaller angle. Since the voltage across R is in phase with the current, the phase of V_R steadily decreases towards 0 degrees.

The frequency-dependent behavior of the series RC circuit can be very useful. Consider what would happen if a complex audio signal, such as music or speech were applied to a series RC circuit. The signal appearing across R would contain primarily the higher frequency components. If we took our output across R, the RC circuit would behave like a high-pass filter.