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Chapter 7 — Second-Order RLC Circuits

Chapter Overview (click to expand) When both a capacitor and an inductor are present in the same circuit, energy can oscillate back and forth between them, producing behaviors far richer than the simple exponential decay of first-order circuits. This chapter analyzes series and parallel RLC circuits through their characteristic equation, classifying responses as overdamped, critically damped, or underdamped based on the damping ratio. **Key Takeaways** 1. The characteristic equation of a second-order circuit has two roots whose nature — real and distinct, real and equal, or complex conjugate — determines whether the circuit is overdamped, critically damped, or underdamped. 2. An underdamped circuit oscillates at the damped natural frequency before settling, while an overdamped circuit decays to steady state without oscillation. 3. The natural frequency ω₀ and damping ratio ζ are the two key parameters that fully characterize the transient behavior of any second-order circuit.

7.1 Introduction: When Circuits Get Dramatic

If first-order RC and RL circuits are like a polite conversation — one thing leads smoothly to another — then second-order RLC circuits are like a heated debate. Things can swing back and forth, overshoot their targets, or even oscillate indefinitely.

In Chapter 6, you learned how capacitors and inductors store and release energy exponentially. But what happens when you put both energy storage elements in the same circuit? The energy sloshes back and forth between the electric field of the capacitor and the magnetic field of the inductor, like two friends tossing a ball between them. Add some resistance, and the ball gradually loses energy with each toss until everyone gets tired.

This energy exchange creates behaviors you won't see in simpler circuits:

  • Oscillations that ring like a bell
  • Overshoot that rockets past the target before settling back
  • Resonance that amplifies signals at specific frequencies

Understanding these behaviors unlocks your ability to design everything from radio tuners to shock absorbers to audio equalizers.

7.2 Second-Order Circuits: The Mathematical Upgrade

A second-order circuit is any circuit whose behavior is described by a second-order differential equation. This happens whenever a circuit contains two independent energy storage elements — typically an inductor and a capacitor.

The general form of a second-order differential equation is:

\[\frac{d^2x}{dt^2} + 2\alpha\frac{dx}{dt} + \omega_0^2 x = f(t)\]

where \(x\) is the response (voltage or current), \(\alpha\) is the damping coefficient, \(\omega_0\) is the undamped natural frequency, and \(f(t)\) is the source.

Order Energy Storage Elements Equation Type Example Response
First 1 (C or L) First-order ODE Exponential decay/rise
Second 2 (C and L) Second-order ODE Oscillatory, damped
Higher 3+ Higher-order ODE Complex multi-frequency

7.3 Series and Parallel RLC Circuits

Series RLC — all components share the same current. Applying KVL and differentiating:

\[\frac{d^2i}{dt^2} + \frac{R}{L}\frac{di}{dt} + \frac{1}{LC}i = \frac{1}{L}\frac{dV_S}{dt}\]

Series RLC: \(\quad \alpha = \dfrac{R}{2L}, \qquad \omega_0 = \dfrac{1}{\sqrt{LC}}\)

Parallel RLC — all components share the same voltage. Applying KCL:

Parallel RLC: \(\quad \alpha = \dfrac{1}{2RC}, \qquad \omega_0 = \dfrac{1}{\sqrt{LC}}\)

Notice that \(\omega_0\) is the same for both configurations — it depends only on L and C.

Diagram: RLC Circuit Interactive Explorer

7.4 The Characteristic Equation

To solve the homogeneous equation, assume \(x = Ae^{st}\). Substituting gives the characteristic equation:

\[s^2 + 2\alpha s + \omega_0^2 = 0\]
\[s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}\]

The nature of the roots determines everything about the circuit response:

Condition Root Type Response Type
\(\alpha > \omega_0\) Two distinct real roots Overdamped
\(\alpha = \omega_0\) Repeated real root Critically damped
\(\alpha < \omega_0\) Complex conjugate roots Underdamped

7.5 Natural Frequency

The natural frequency \(\omega_0\) is the frequency at which an undamped circuit would oscillate forever — the circuit's preferred rhythm:

\[\omega_0 = \frac{1}{\sqrt{LC}} \text{ rad/s} \qquad f_0 = \frac{1}{2\pi\sqrt{LC}} \text{ Hz}\]

The natural frequency depends only on L and C, not on R. Resistance controls how quickly oscillations die out, but not their frequency.

L (mH) C (μF) f₀ (Hz) Audio Equivalent
100 100 50.3 Low bass hum
10 10 503 Mid-range tone
1 1 5,033 High-pitched whistle
0.1 0.1 50,330 Ultrasonic

Diagram: Natural Frequency Calculator

7.6 Damping Ratio

The damping ratio \(\zeta\) is the single dimensionless number that classifies circuit response:

\[\zeta = \frac{\alpha}{\omega_0}\]

Series RLC: \(\zeta = \dfrac{R}{2}\sqrt{\dfrac{C}{L}}\)    Parallel RLC: \(\zeta = \dfrac{1}{2R}\sqrt{\dfrac{L}{C}}\)

Damping Ratio Condition Response Type Behavior
\(\zeta > 1\) \(\alpha > \omega_0\) Overdamped Slow, no oscillation
\(\zeta = 1\) \(\alpha = \omega_0\) Critically damped Fastest without overshoot
\(0 < \zeta < 1\) \(\alpha < \omega_0\) Underdamped Oscillation with decay
\(\zeta = 0\) \(\alpha = 0\) Undamped Endless oscillation

7.7 Overdamped Response (\(\zeta > 1\))

When \(\zeta > 1\), the characteristic equation has two distinct negative real roots. The general solution is:

\[x(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}\]

where \(s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}\) (both negative real)

The response creeps slowly toward its final value with no oscillation. Think of a door closer adjusted too tight — it closes without bouncing, but takes forever. Overdamping is preferred in precision instruments and safety systems where overshoot is unacceptable.

7.8 Underdamped Response (\(0 < \zeta < 1\))

When \(\zeta < 1\), the characteristic equation has complex conjugate roots, giving oscillatory behavior:

\[s_{1,2} = -\alpha \pm j\omega_d \qquad \omega_d = \omega_0\sqrt{1 - \zeta^2}\]
\[x(t) = Ce^{-\alpha t}\cos(\omega_d t + \phi)\]

The response oscillates at the damped natural frequency \(\omega_d\) while decaying with time constant \(1/\alpha\). Two key performance metrics:

Percent Overshoot:

\[PO = e^{-\pi\zeta/\sqrt{1-\zeta^2}} \times 100\%\]

Settling time (to within 2%): \(\displaystyle t_s \approx \frac{4}{\alpha} = \frac{4}{\zeta\omega_0}\)

7.9 Critically Damped Response (\(\zeta = 1\))

When \(\zeta = 1\), there is a repeated root \(s = -\alpha = -\omega_0\). The general solution is:

\[x(t) = (A + Bt)e^{-\omega_0 t}\]

Critical damping gives the fastest possible return to equilibrium without overshoot. It's the design target for galvanometers, analog meters, automotive suspension, and camera stabilization — any application where overshoot is unacceptable but speed matters.

7.10 Resonant Frequency

Resonance occurs when inductive reactance equals capacitive reactance:

\[X_L = X_C \implies \omega_r L = \frac{1}{\omega_r C} \implies \omega_r = \frac{1}{\sqrt{LC}} = \omega_0\]

At resonance in a series RLC circuit: impedance is minimum (= R), current is maximum, and \(V_L = V_C\) (they cancel). Voltage across L or C individually can exceed the source voltage!

At resonance in a parallel RLC circuit: impedance is maximum, current from source is minimum, and circulating current between L and C can exceed source current.

Diagram: Series vs Parallel Resonance Comparison

7.11 Quality Factor

The quality factor Q characterizes the sharpness of resonance and efficiency of energy storage:

Series RLC: \(\displaystyle Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}\)

Parallel RLC: \(\displaystyle Q = \omega_0 CR = R\sqrt{\frac{C}{L}}\)

Relationship to damping: \(\displaystyle Q = \frac{1}{2\zeta}\)

Q also defines bandwidth:

\[BW = \frac{f_0}{Q}\]
Q Bandwidth (at f₀ = 1 MHz) Application
10 100 kHz Audio filter
100 10 kHz IF stage
1000 1 kHz Crystal oscillator

Practical Q limits

Real inductors have winding resistance that caps practical Q at 100–200 for discrete LC circuits. Higher Q requires quartz crystals or mechanical resonators.

7.12 Key Formulas Summary

Parameter Series RLC Parallel RLC
Damping coefficient α \(R/2L\) \(1/2RC\)
Natural frequency ω₀ \(1/\sqrt{LC}\) \(1/\sqrt{LC}\)
Damping ratio ζ \((R/2)\sqrt{C/L}\) \((1/2R)\sqrt{L/C}\)
Quality factor Q \((1/R)\sqrt{L/C}\) \(R\sqrt{C/L}\)
Bandwidth BW \(R/L\) \(1/RC\)

Response classification:

  • Overdamped (\(\zeta > 1\)): \(x(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}\), two distinct real roots
  • Critically damped (\(\zeta = 1\)): \(x(t) = (A+Bt)e^{-\omega_0 t}\), repeated root
  • Underdamped (\(\zeta < 1\)): \(x(t) = Ce^{-\alpha t}\cos(\omega_d t + \phi)\), \(\omega_d = \omega_0\sqrt{1-\zeta^2}\)

Key relationships: \(Q = 1/(2\zeta)\), \(\quad BW = f_0/Q\), \(\quad PO = e^{-\pi\zeta/\sqrt{1-\zeta^2}} \times 100\%\)