Introduction to Electronics - Filters: Voltage division in AC using reactance

Voltage division in AC circuits is an extension of the classic resistive voltage divider concept, incorporating the frequency-dependent behaviour of reactive components. In AC, capacitors and inductors offer reactance—denoted by $X_C$ for capacitors and $X_L$ for inductors—that affects how the input voltage is distributed across circuit elements. Unlike purely resistive circuits, the voltage division in AC depends on the magnitude and phase of each component’s impedance, making it essential to consider both the amplitude and phase angle when analysing signal behaviour.

Reactance and Impedance

Reactance is the frequency-dependent opposition to current flow presented by capacitors and inductors. It is a component of impedance, which is the total opposition that a circuit element offers to alternating current (AC).

Impedance, denoted by $Z$, is a complex quantity that combines resistance ($R$) and reactance ($X$) in the following way:

$Z = R + jX$

Here, $j$ is the imaginary unit ($\sqrt{-1}$). The reactance $X$ can be positive (for inductive reactance, $X_L$) or negative (for capacitive reactance, $X_C$).

In a purely resistive circuit, $X = 0$, so the impedance is real. In a circuit with reactive components, the impedance is complex, and both the magnitude and phase of the voltage and current are affected. The magnitude of the impedance is given by:

$|Z| = \sqrt{R^2 + X^2}$

This relationship is important in voltage division for AC circuits, where voltage is divided not just by resistances, but by the magnitudes of impedances. For example, in an RC voltage divider, the output voltage depends on the relative magnitudes of $R$ and $X_C$, both of which determine how the input voltage is split.

Understanding the role of reactance as a component of impedance helps explain how capacitors and inductors influence circuit behaviour at different frequencies, and why voltage division formulas in AC analysis use $Z$ instead of just $R$.

Frequency dependence of reactance

The reactance of capacitors and inductors changes significantly depending on the frequency of the input signal.

Capacitive Reactance ($X_C$)

The reactance of a capacitor is given by:

$X_C = \frac{1}{2\pi f C}$

From this formula, notice the following:

• At very low frequencies ($f \rightarrow 0$), $X_C$ becomes very large. This means a capacitor behaves like an open circuit to DC and low-frequency signals.
• At very high frequencies ($f \rightarrow \infty$), $X_C$ approaches zero. The capacitor behaves like a short circuit, allowing high-frequency signals to pass through easily.

Inductive Reactance ($X_L$)

The reactance of an inductor is given by:

$X_L = 2\pi f L$

As with the capacitive reactance formula, notice the following points for the reactance of an inductor:

• At very low frequencies ($f \rightarrow 0$), $X_L$ becomes very small. An inductor behaves like a short circuit to DC and low-frequency signals.
• At very high frequencies ($f \rightarrow \infty$), $X_L$ becomes very large. The inductor acts like an open circuit, blocking high-frequency signals.

These frequency-dependent behaviours are fundamental in the design of filters, where components are chosen to either pass or block specific ranges of frequencies. Understanding how $X_C$ and $X_L$ change with frequency is essential to predicting and controlling circuit behaviour in the frequency domain.

RC circuit reactance

Consider a simple AC voltage divider composed of a resistor and a capacitor in series. The output voltage is taken across the capacitor. The schematic is as follows:

In this circuit, the resistor (R) has a constant impedance, while the capacitor (C) has an impedance given by $Z_C=\frac{1}{j\omega C}$, where $\Omega$ is the angular frequency. The voltage division across the capacitor changes with frequency due to its reactive nature. As you can probably imagine, when the angular frequency becomes very large, the impedance of the capacitor becomes very small, until eventually, for practical purposes, it becomes very close to zero.

The voltage divider rule in AC circuits is similar to the resistive case, but it uses complex impedances. For the above circuit, the output voltage across the capacitor is given by:

$V_{out}=V_{in}\times\frac{Z_C}{R+Z_C}$

Substituting the capacitor's impedance, we have:

$V_{out}=V_{in}\times\frac{\frac{1}{j\omega C}}{R+\frac{1}{j\omega C}}$

This expression can be simplified to analyse both the magnitude and phase of the output voltage. The magnitude of the voltage division ratio is:

$\left|\frac{V_{out}}{V_{in}}\right|=\frac{1}{\sqrt{1+( \omega RC)^2}}$

This formula shows that at low frequencies (small $\Omega$), the capacitor’s impedance is high, and most of the input voltage appears across it. As the frequency increases, the capacitor’s impedance decreases, reducing the output voltage magnitude.

A similar approach applies to RL circuits, where the inductor’s impedance is $Z_L=j\omega L$, leading to a voltage division ratio when the output is taken across the resistor or the inductor.

RC circuit demo Python script

Below is a Python script that demonstrates voltage division in an RC circuit by plotting the reactance of a capacitor and a resistor over a range of frequencies. The script compares the reactance behaviour of the resistor, which depends on the frequency, against the resistor.

import numpy as np
import matplotlib.pyplot as plt

# Define component values
R = 100 # Resistance in Ohms
C = 1e-6 # Capacitance in Farads

# Frequency range: 10 Hz to 1 MHz (logarithmic spacing)
frequencies = np.logspace(1, 6, 500) # 10^1 to 10^6 Hz

# Calculate reactances
X_R = np.full_like(frequencies, R) # Constant for resistor
X_C = 1 / (2 * np.pi * frequencies * C) # Capacitive reactance

# Plotting
plt.figure(figsize=(10, 6))
plt.semilogx(frequencies,
X_R,
label='Resistor (100 $\\Omega$)',
linestyle='--')
plt.semilogx(frequencies,
X_C,
label='Capacitor (1 $\\mu$F)',
linewidth=2)

# Decorations
plt.title('Reactance vs Frequency for Resistor and Capacitor')
plt.xlabel('Frequency (Hz)')
plt.ylabel('Reactance ($\\Omega$)')
plt.grid(True, which='both', linestyle=':')
plt.legend()
plt.tight_layout()
plt.show()

Here’s the plot:

In this plot, you will see how reactance behaves differently for a resistor and a capacitor as the frequency of the signal increases. The flat dashed line shows the reactance of a resistor. It stays constant at 100 ohms no matter what the frequency is. This means that a resistor always offers the same amount of opposition to current. It doesn’t care if the signal is slow or fast—it resists the same way.

The curved line shows the reactance of a capacitor. At low frequencies, this line is high, which means the capacitor resists the current strongly. But as the frequency increases, the line drops, showing that the capacitor offers less opposition to the current. In other words, the faster the signal, the more easily it passes through the capacitor.

The key lesson here is that a resistor’s reactance doesn’t change with frequency, but a capacitor’s does. This helps explain why capacitors are used to filter out certain frequencies in electronic circuits.

RL circuit reactance

Let us now consider an AC voltage divider composed of a resistor and an inductor in series. This time, the output voltage is taken across the inductor. The schematic is as follows:

In this circuit, the resistor (R) once again provides a constant impedance. The inductor (L), however, introduces a frequency-dependent impedance given by:

$Z_L = j\omega L$

This impedance grows as the angular frequency ($\omega$) increases. In other words, inductors resist changes in current more strongly at higher frequencies. When the frequency is very low (close to zero), the inductor behaves almost like a short circuit, offering minimal opposition to current. But at very high frequencies, the inductive reactance becomes large, impeding the current flow significantly.

We use the same voltage divider rule as before, this time applying it to the inductor. The output voltage across the inductor is:

$V_{out} = V_{in} \times \frac{Z_L}{R + Z_L}$

Substituting the impedance of the inductor, we get:

$V_{out} = V_{in} \times \frac{j\omega L}{R + j\omega L}$

From this, we can determine the magnitude of the voltage division ratio:

$\left|\frac{V_{out}}{V_{in}}\right| = \frac{\omega L}{\sqrt{R^2 + (\omega L)^2}}$

This equation tells us that at low frequencies (small $\omega$), the inductor’s impedance is small, so the voltage drop across it is also small. But as the frequency increases, the inductive impedance increases, and the output voltage rises accordingly.

This frequency-dependent behaviour makes inductors useful in applications where we want to allow high-frequency signals to pass, while blocking lower frequencies—such as in certain types of filter circuits.

RL circuit demo Python script

Below is a Python script that plots the reactance of both the resistor and the inductor across a wide frequency range. This will help you visualise how the inductor's reactance changes with frequency, and how it compares to the resistor’s constant impedance.

import numpy as np
import matplotlib.pyplot as plt

# Define component values
R = 100 # Resistance in Ohms
L = 1e-3 # Inductance in Henrys (1 mH)

# Frequency range: 10 Hz to 1 MHz (logarithmic spacing)
frequencies = np.logspace(1, 6, 500) # 10^1 to 10^6 Hz

# Calculate reactances
X_R = np.full_like(frequencies, R) # Constant for resistor
X_L = 2 * np.pi * frequencies * L # Inductive reactance

# Plotting
plt.figure(figsize=(10, 6))
plt.semilogx(frequencies,
X_R,
label='Resistor (100 $\\Omega$)',
linestyle='--')
plt.semilogx(frequencies,
X_L,
label='Inductor (1 mH)',
linewidth=2)

# Decorations
plt.title('Reactance vs Frequency for Resistor and Inductor')
plt.xlabel('Frequency (Hz)')
plt.ylabel('Reactance ($\\Omega$)')
plt.grid(True, which='both', linestyle=':')
plt.legend()
plt.tight_layout()
plt.show()

Here’s the plot:

In the plot, the flat dashed line again represents the resistor’s constant impedance of 100 ohms, unaffected by frequency. The curved line represents the inductor’s reactance. At low frequencies, it is close to zero, meaning the inductor does not resist the current much. As the frequency increases, the inductor’s reactance rises sharply, showing that it opposes higher-frequency currents more strongly.

The key insight here is that while a resistor always resists current equally, an inductor resists changes in current depending on how fast those changes are. This makes inductors valuable tools for managing frequency-dependent behaviour in AC circuits.

Key takeaways

So, what should you remember?

• In AC circuits, the division of voltage across components depends on both resistance and reactance, the latter being frequency-dependent.
• Reactive components introduce a phase angle, so voltage division must consider both magnitude and phase.
• The output voltage across a reactive component in a series circuit is given by $V_{out}=V_{in}\times\frac{Z_{reactive}}{R+Z_{reactive}}$, where $Z_{reactive}$ varies with frequency.

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