Frequency Response of the RC Circuit

Introduction

The frequency response of an RC circuit describes how the circuit’s impedance and phase change as the frequency of the applied alternating current (AC) signal varies. RC circuits, composed of a resistor (R) and a capacitor (C) in series, are foundational in analog electronics and signal processing. Their behavior changes significantly with frequency, making them essential in filters, signal conditioning, and frequency-sensitive applications.

What Is Frequency Response?

Frequency response refers to a circuit’s behavior when an AC signal of varying frequency is applied. It shows how the magnitude and phase of the output or impedance change with frequency. In the case of a series RC circuit, the focus is often on how the total impedance and phase angle vary from low to high frequencies.

How an RC Circuit Responds to Frequency

In a series RC circuit: $$ X_C = \frac{1}{2 \pi f C} $$ where:

• (X_C) is capacitive reactance (Ω),
• (f) is the frequency (Hz),
• (C) is the capacitance (F).

As frequency increases:

Total Impedance of the RC Circuit

The total impedance ($Z_T$) of the series RC circuit combines the resistor and capacitor: $$ Z_T = R - jX_C $$ In polar form: $$ |Z_T| = \sqrt{R^2 + X_C^2} $$ $$ \theta = -\tan^{-1}\left(\frac{X_C}{R}\right) $$

• At low frequencies, ($X_C$) is large, and the circuit behaves primarily like a capacitor — the impedance magnitude is high and the phase angle approaches ($-90^\circ$).
• At high frequencies, ($X_C$) becomes small and the resistor dominates — the impedance magnitude approaches (R) and the phase angle moves closer to ($0^\circ$).

Break (Corner) Frequency

A key frequency in the RC frequency response is the break frequency ($f_1$), where: $$ X_C = R $$ Solving: $$ f_1 = \frac{1}{2 \pi R C} $$ At this point:

• The capacitive reactance equals the resistance.
• The impedance magnitude is equal parts resistive and capacitive, and the phase angle is about ($-45^\circ$).

Let's Do Some Maths

Thus far, the analysis of series circuits has been limited to a particular frequency. We will now examine the effect of frequency on the response of an RC series configuration such as that in [Fig. 1]. The magnitude of the source is fixed at 10 V, but the frequency range of analysis will extend from zero to 20 kHz. Let us first determine how the impedance of the circuit $Z_T$ will vary with frequency for the specified frequency range of interest. Before getting into specifics, however, let us first develop a sense for what we should expect by noting the impedance-versus-frequency curve of each element, as drawn in [Fig. 2]. At low frequencies the reactance of the capacitor will be quite high and considerably more than the level of the resistance R, suggesting that the total impedance will be primarily capacitive in nature. At high frequencies the reactance XC will drop below the $R = 5kΩ$ level, and the network will start to shift toward one of a purely resistive nature (at 5 kΩ). The frequency at which $X_C = R$ can be determined in the following manner: and which for the network of interest is For frequencies less than $f_1$, $X_C > R$, and for frequencies greater than $f_1$, $R > X_C$, as shown in Fig. 2.
Now for the details. The total impedance is determined by the following equation: The magnitude and angle of the total impedance can now be found at any frequency of interest by simply substituting into Eq. (1). The presence of the capacitor suggests that we start from a low frequency (100 Hz) and then open the spacing until we reach the upper limit of interest (20 kHz).

f = 100 Hz

and with and which compares very closely with $Z_C = 159.16 kΩ \angle -90^\circ$ if the circuit were purely capacitive (R = 0 Ω). Our assumption that the circuit is primarily capacitive at low frequencies is therefore confirmed.

f = 1 kHz

and with and A noticeable drop in the magnitude has occurred, and the impedance angle has dropped almost $17^\circ$ from the purely capacitive level. Continuing: Note how close the magnitude of $Z_T$ at $f = 20 kHz$ is to the resistance level of $5 kΩ$. In addition, note how the phase angle is approaching that associated with a pure resistive network ($0^\circ$).
A plot of $Z_T$ versus frequency in [Fig. 3] completely supports our assumption based on the curves of [Fig. 2].