Impedance of a Series RLC Circuit

Finds the impedance of an AC circuit as Z = √(R² + (X_L − X_C)²), together with the inductive and capacitive reactances and the phase difference between voltage and current.

Explanation

In an alternating-current circuit, coils and capacitors oppose the current alongside the resistor. The combined opposition is the impedance, written $Z$ and measured in ohms.

Z = \sqrt{R^2 + (X_L - X_C)^2}

X_L = 2\pi f L \qquad X_C = \dfrac{1}{2\pi f C}

Here $X_L$ is the inductive reactance and $X_C$ the capacitive one. The phase difference is

\varphi = \arctan \dfrac{X_L - X_C}{R}

Example

Take $R = 100$ Ω, $L = 10$ mH, $C = 1$ μF at 1000 Hz.

X_L = 2\pi \times 1000 \times 0.01 = 62.8\ \Omega \qquad X_C = \dfrac{1}{2\pi \times 1000 \times 10^{-6}} = 159.2\ \Omega

Z = \sqrt{100^2 + (62.8 - 159.2)^2} = \sqrt{10000 + 9278} = 138.8\ \Omega

The phase is $-43.9$ degrees. Being negative, the current leads the voltage: the circuit is capacitive.

Why they do not simply add

Because the three components put voltage and current out of step with each other by different amounts.

Resistor — voltage and current stay in step
Coil — the current lags the voltage by 90 degrees
Capacitor — the current leads the voltage by 90 degrees

Coil and capacitor are exact opposites, so $X_L$ and $X_C$ cancel. What remains, $(X_L - X_C)$ , sits 90 degrees away from $R$ , so the two combine as the legs of a right triangle. That is where Pythagoras enters.

The role of frequency

$X_L = 2\pi f L$ — grows with frequency, so a coil blocks high frequencies
$X_C = \dfrac{1}{2\pi f C}$ — falls with frequency, so a capacitor passes them

This single contrast is the foundation of every filter circuit ever built.