Series RLC Circuit Simulator — AC Impedance and Resonance

A series RLC circuit places resistance $R$, inductance $L$ and capacitance $C$ on a single loop driven by an AC source $V(t) = V_0\sin(\omega t)$ at frequency $f$ (angular frequency $\omega = 2\pi f$). Each element has a frequency-dependent impedance: the resistor's $Z_R = R$ is frequency-independent, the inductor's $Z_L = j\omega L$ leads voltage by 90°, and the capacitor's $Z_C = 1/(j\omega C) = -j/(\omega C)$ lags by 90°. Because the elements are in series, the impedances add as complex numbers, giving the loop $Z = R + j(\omega L - 1/(\omega C))$.

The magnitude $|Z| = \sqrt{R^2 + (\omega L - 1/(\omega C))^2}$ and the phase angle $\varphi = \arctan((\omega L - 1/(\omega C))/R)$ are the measured quantities. At the resonance frequency $f_0 = 1/(2\pi\sqrt{LC})$ the reactive terms cancel ($\omega_0 L = 1/(\omega_0 C)$) so $|Z| = R$ at its minimum and $\varphi = 0$ — the circuit looks purely resistive. The quality factor $Q = (1/R)\sqrt{L/C} = \omega_0 L/R = 1/(\omega_0 R C)$ controls the sharpness; the half-power bandwidth is $\Delta f = f_0/Q$.

With the default values $R = 100$ Ω, $L = 10$ mH, $C = 1.0$ μF, $f = 1592$ Hz the simulator reports $|Z| = 100.0$ Ω, $f_0 = 1591.5$ Hz, $\varphi = 0.0°$ and $Q = 1.00$. Because $f = 1592$ Hz is almost exactly the resonance frequency 1591.5 Hz, both reactances are about 100 Ω, they cancel, and the magnitude reduces to $R$.

Radio tuners: An AM radio uses a variable capacitor to move f0 through the 530–1600 kHz band. When f0 matches a station's carrier the circuit current is at its maximum and the station is heard cleanly. Other stations have higher |Z| and produce very little current. The Q value of the tank determines selectivity — adjacent-channel separation depends directly on how sharp the resonance is.

Loudspeaker crossovers: Two-way and three-way speakers split the audio band among woofer, mid-range and tweeter using LC and RLC band-pass filters. High frequencies pass through a capacitor-led high-pass to the tweeter, low frequencies pass through an inductor-led low-pass to the woofer, and the mid-band is routed through an RLC band-pass. f0 and Q of each network determine the crossover frequency and the slope.

Power-line EMI filters: Switching-mode power supplies emit broadband noise between a few kHz and several MHz. π- and T-section filters made of series inductors and shunt capacitors pass the 50/60 Hz mains while attenuating the noise band. Too high a Q can produce a resonant peak that amplifies a specific noise tone, so dampening resistors are added to keep Q under control.

Power-factor correction: Industrial loads such as induction motors are inductive — the current lags the voltage and the power factor falls below 1. Adding a parallel capacitor introduces a capacitive reactance that cancels part of the inductive reactance, restoring the power factor toward 1. Over-correction can swing the system into the capacitive regime, so careful measurement of the load is essential.

The most common mistake is to assume that the current becomes infinite at resonance. Ideal L and C alone would indeed give |Z| = 0 at resonance, but every real circuit has resistance — coil winding resistance, capacitor ESR and wire losses — so the resonant impedance is the finite value R. The simulator does not allow R = 0 for that reason. The smaller R is, the higher the Q, so the resonant current can still exceed component ratings; this is why high-Q designs need explicit current limiting.

Next, confusing series and parallel resonance. A series RLC has minimum |Z| at resonance (maximum current), whereas a parallel RLC has maximum |Z| at resonance (minimum source current). The two behave in opposite ways, so the same f0 expression can lead to opposite circuit-level effects. Antenna feed matching often uses series resonance, while trap circuits that reject a single frequency use parallel resonance. This simulator covers only the series case, so remember that the V-shaped dip is the resonance.

Finally, forgetting that L and C jointly determine f0 but separately determine Q. Because f0 depends on the product LC, you can double L and halve C and keep f0 unchanged. But Q = (1/R)√(L/C) depends on the ratio L/C, so the same f0 can be reached with different selectivities. Designers therefore choose f0 and Q independently, then pick L, C and R to satisfy both. RF design treats √(L/C) as a separate characteristic impedance to manage explicitly.