RLC Resonance: Series and Parallel Circuits

The Idea of Resonance

Add an inductor to a capacitor and a resistor and the circuit gains a new and powerful behavior: resonance. At one special frequency the energy sloshing between the inductor's magnetic field and the capacitor's electric field reinforces itself, producing a sharp peak or dip in the response. This is the principle behind tuning a radio to one station, building a bandpass filter that isolates a band of interest, matching impedances between stages, and sustaining the steady tone of an oscillator. Two arrangements matter — the series RLC driven by a voltage source and the parallel RLC driven by a current source — and although they share the same resonant frequency, they behave in opposite ways at that frequency.

Series RLC: Resonant Frequency

In a series RLC circuit the resistor, inductor, and capacitor sit in a single loop driven by a voltage source. The total impedance is the sum of the three element impedances:

On the imaginary axis the inductive reactance $+j\omega L$ and the capacitive reactance $-j/(\omega C)$ point in opposite directions. They cancel exactly when $\omega L = 1/(\omega C)$, which gives the resonant frequency:

At $\omega_0$ the reactances vanish and the impedance collapses to the resistor alone, $Z = R$. That is the defining feature of series resonance: minimum impedance, and therefore maximum current, at the resonant frequency.

Quality Factor and Bandwidth

The quality factor $Q$ measures how sharp the resonance is. High $Q$ means a tall, narrow peak; low $Q$ means a broad, gentle one. For the series RLC, three equivalent expressions follow from $\omega_0 = 1/\sqrt{LC}$:

The quality factor also fixes the −3 dB bandwidth of the resonance. The bandwidth is the width of the band between the two half-power frequencies that straddle $f_0$:

A high-$Q$ circuit responds only to frequencies clustered tightly around $f_0$, which is exactly what a radio front-end needs to separate one channel from its neighbors.

Worked Example: A 1 MHz Tuned Circuit

Design and analyze a series RLC tank with $L = 100\;\mu\text{H}$, $C = 253\;\text{pF}$, and $R = 10\;\Omega$. First the resonant frequency:

Next the quality factor. The inductive reactance at resonance is $\omega_0 L = 2\pi(10^{6})(100\times 10^{-6}) \approx 628\;\Omega$, so:

Finally the bandwidth:

This tank rings at 1 MHz, has a respectable $Q$ of about 63, and passes a 15.9 kHz band around the center — comfortably narrow enough to pick out an AM broadcast channel. At resonance the loop current is limited only by the 10 Ω resistor, and the voltage across the inductor or capacitor is $Q$ times the source voltage, a 63-fold voltage magnification that must be respected when rating components.

Parallel RLC and the Series–Parallel Contrast

In a parallel RLC circuit the three elements span the same pair of nodes and the circuit is driven by a current source. The resonant frequency is unchanged, $\omega_0 = 1/\sqrt{LC}$, but the behavior at resonance inverts. The inductor and capacitor currents cancel, leaving only the resistor branch, so the parallel circuit presents maximum impedance and maximum voltage at resonance. Its quality factor also inverts its dependence on resistance:

In the series circuit a larger $R$ lowers $Q$ (more loss), while in the parallel circuit a larger $R$ raises $Q$ (less current bled away). The table makes the symmetry explicit.

Damping and the Second-Order Response

Taking the output across the capacitor of a series RLC produces a second-order low-pass transfer function:

The denominator roots are $s = -\frac{\omega_0}{2Q} \pm j\omega_0\sqrt{1 - \frac{1}{4Q^2}}$. The damping ratio is $\zeta = 1/(2Q)$. When $Q > 0.5$ ($\zeta < 1$) the poles are complex and the step response is underdamped, overshooting and ringing. When $Q = 0.5$ the response is critically damped — the fastest settling without overshoot. When $Q < 0.5$ it is overdamped and sluggish with no oscillation. This connects resonance back to the single-pole intuition from the RC low-pass and RL high-pass articles.

The second-order response also reveals a useful design lever that the first-order filters lack. By choosing $R$, and therefore $Q$, independently of $\omega_0$, you can place the corner where you want it and then dial the sharpness of the transition separately. A designer who wants a maximally flat passband picks a damping that avoids any peaking, while one who wants a narrow, selective filter deliberately runs a high $Q$. The trade-off is always between selectivity and ringing: the same high $Q$ that sharpens the frequency response also makes the step response overshoot and oscillate for many cycles. This tension between frequency-domain sharpness and time-domain settling sits at the heart of nearly every filter and control-loop design decision.

Energy and the Meaning of Q

The quality factor is more than an algebraic shorthand; it has a direct physical meaning rooted in energy. By definition, $Q$ equals $2\pi$ times the ratio of the energy stored in the circuit to the energy dissipated per cycle of oscillation. A circuit with $Q = 100$ loses only about one hundredth of its stored energy each cycle, so once it is excited it rings for roughly a hundred cycles before the oscillation dies away. That is why a high-$Q$ resonator is both highly selective in frequency and slow to settle in time — the two views are simply different windows onto the same stored energy. In a series RLC the loss happens in the series resistor, so smaller resistance means higher $Q$; in a parallel RLC the loss is the current bled through the parallel resistor, so larger resistance means higher $Q$. Keeping the energy picture in mind makes the otherwise confusing reversal between the two topologies feel natural rather than arbitrary.

Common Mistakes

• Using the series $Q$ formula for a parallel circuit. They are reciprocals in their dependence on $R$; swapping them inverts every conclusion about selectivity.
• Confusing $\omega_0$ with $f_0$. The radian frequency is $1/\sqrt{LC}$; the hertz value carries the extra $2\pi$ in the denominator.
• Forgetting the voltage magnification. At series resonance the inductor and capacitor voltages reach $Q$ times the source, which can far exceed the input and damage under-rated parts.
• Assuming the resonant frequency is where the response peaks. For high $Q$ the peak of a capacitor-output response sits slightly below $\omega_0$; only at $Q$ well above a few is the difference negligible.
• Neglecting parasitic resistance. The inductor's winding resistance adds to $R$ and quietly lowers the achievable $Q$.

Analyze RLC Circuits in CircuitMath

Place $R$, $L$, and $C$ in series or parallel, add a source and a ground, and run the analysis. CircuitMath returns the second-order transfer function in LaTeX, and from the denominator coefficients you can read off $\omega_0$ and $Q$ directly. Open the editor and vary $R$ to watch the response move from underdamped to overdamped while $f_0$ stays put.

Related Tutorials

• RC Low-Pass Filter — the first-order building block the RLC circuit extends.
• RL High-Pass Filter — another first-order filter using inductors.
• Transfer Functions in Circuit Analysis — poles, zeros, and second-order responses.