Series RLC Circuit Behavior

What “series RLC behavior” means under sine and pulse excitation — resonance, ringing, damping, and why the simulator is a safe analytic reference

Series RLC is the simplest “true resonator”: it has two energy stores (magnetic in \(L\), electric in \(C\)). That single fact explains almost everything you observe: phase flips, impedance dips at resonance, ringing under pulses, and energy oscillating between \(L\) and \(C\).

1) What changes when you add \(C\) in series

• RL (one store): energy only in \(L\) → current relaxes exponentially (one time constant \(\tau\)).
• Series RLC (two stores): energy can oscillate between \(L\) and \(C\) → you get resonance and ringing.
• Stored energies: \[ W_L=\frac{1}{2}Li^2,\qquad W_C=\frac{1}{2}Cv_C^2 \] and energy consistency uses: \[ \Delta W = \int v(t)i(t)\,dt \] (same accounting definitions used by the tool).

2) Sine excitation (phasor steady state)

With \(v(t)=V_{pk}\sin(\omega t)\), steady state is transparent in impedance form:

\[ Z=R+j\left(\omega L-\frac{1}{\omega C}\right),\quad |Z|=\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2} \]

Phase and current amplitude:

\[ \phi=\tan^{-1}\!\left(\frac{\omega L-\frac{1}{\omega C}}{R}\right),\qquad I_{pk}=\frac{V_{pk}}{|Z|} \]

Resonance and detuning

Ideal series resonance occurs when the reactive terms cancel:

\[ \omega=\omega_0=\frac{1}{\sqrt{LC}} \]

A useful normalized detuning measure:

\[ \delta=\frac{\omega-\omega_0}{\omega_0} \]

Power quantities for sine (P/Q/S/PF)

For a sinusoid, RMS + phasor relations are:

\[ P=V_{rms}I_{rms}\cos\phi,\qquad Q=V_{rms}I_{rms}\sin\phi,\qquad S=V_{rms}I_{rms},\qquad PF=\frac{P}{S} \]

In series RLC, reactive power is a net result:

\[ Q_{net}=Q_L+Q_C,\qquad Q_L>0,\quad Q_C<0 \]

3) Pulse excitation (ringing + steady-periodic pulse)

A pulse train repeatedly “kicks” the resonator. Because there are two energy stores, a fast edge can excite the natural mode — you see ringing with an envelope set by damping. The model’s natural frequency and damping ratio are:

\[ \omega_0=\frac{1}{\sqrt{LC}},\qquad \zeta=\frac{R}{2}\sqrt{\frac{C}{L}} \]

• \(\zeta < 1\): underdamped → oscillatory ringing.
• \(\zeta = 1\): critically damped → fastest non-oscillatory response.
• \(\zeta > 1\): overdamped → slow, no oscillation.

Why pulse mode can still be “steady”

In steady-periodic pulse operation, the state repeats every period:

\[ x(T^-)=x(0)\quad\text{with}\quad x=\begin{bmatrix} i \\ v_C \end{bmatrix} \]

4) Worked example (with calculation steps)

We’ll do one sine example (resonance + phase) and one pulse-edge example (ringing frequency + envelope). These are the two most common “trust-building” checks you can do against the simulator.

Example A — Sine series RLC near resonance

Given: \(L=10\,\text{mH}\), \(C=10\,\mu\text{F}\), \(R=1\,\Omega\), \(V_{pk}=10\,\text{V}\), \(f=500\,\text{Hz}\)

1. Natural frequency: \[ \omega_0=\frac{1}{\sqrt{LC}} =\frac{1}{\sqrt{0.01\cdot 10^{-5}}} =3162.277\,\text{rad/s} \] \[ f_0=\frac{\omega_0}{2\pi}=\frac{3162.277}{6.283185}=503.29\,\text{Hz} \] (matches the series resonance definition.)
2. Driving angular frequency: \[ \omega=2\pi f=2\pi\cdot 500=3141.593\,\text{rad/s} \] (same convention.)
3. Reactive term: \[ X=\omega L-\frac{1}{\omega C} =3141.593\cdot 0.01-\frac{1}{3141.593\cdot 10^{-5}} \] \[ X=31.416-31.831=-0.415\,\Omega \]
4. Impedance magnitude: \[ |Z|=\sqrt{R^2+X^2}=\sqrt{1^2+(-0.415)^2}=\sqrt{1.172}=1.083\,\Omega \]
5. Phase: \[ \phi=\tan^{-1}\!\left(\frac{X}{R}\right) =\tan^{-1}(-0.415)=-22.6^\circ \] Negative phase here means the branch looks slightly capacitive at 500 Hz (because we are below \(f_0\)).
6. Peak current: \[ I_{pk}=\frac{V_{pk}}{|Z|}=\frac{10}{1.083}=9.23\,\text{A} \]
7. RMS values: \[ V_{rms}=\frac{10}{\sqrt2}=7.071\,\text{V},\qquad I_{rms}=\frac{9.23}{\sqrt2}=6.53\,\text{A} \]
8. Power: \[ P=V_{rms}I_{rms}\cos\phi =7.071\cdot 6.53\cdot \cos(-22.6^\circ) \approx 42.6\,\text{W} \] \[ Q=V_{rms}I_{rms}\sin\phi =7.071\cdot 6.53\cdot \sin(-22.6^\circ) \approx -17.7\,\text{var} \] (power relations as defined in the math reference.)

Example B — Pulse edge excites ringing (underdamped case)

Given: same \(L=10\,\text{mH}\), \(C=10\,\mu\text{F}\), \(R=1\,\Omega\), step/pulse amplitude \(V=10\,\text{V}\)

1. Compute damping ratio: \[ \zeta=\frac{R}{2}\sqrt{\frac{C}{L}} =\frac{1}{2}\sqrt{\frac{10^{-5}}{10^{-2}}} =0.5\sqrt{10^{-3}}=0.01581 \] Since \(\zeta \ll 1\), the response is strongly underdamped → ringing is expected.
2. Exponential decay rate (useful physical intuition): \[ \alpha=\frac{R}{2L}=\frac{1}{2\cdot 0.01}=50\,\text{s}^{-1} \] So the envelope decays roughly like \(e^{-50t}\) (time constant ~20 ms).
3. Damped ringing frequency: \[ \omega_d=\omega_0\sqrt{1-\zeta^2}\approx \omega_0 \approx 3162.277\,\text{rad/s} \] \[ f_d\approx \frac{\omega_d}{2\pi}\approx 503\,\text{Hz} \] So after a sharp pulse edge you should see ~503 Hz ringing in \(i(t)\), \(v_L(t)\), and \(v_C(t)\).
4. A compact underdamped step-current form (zero initial energy) is: \[ i(t)\approx \frac{V}{L\omega_d}e^{-\alpha t}\sin(\omega_d t) \] This explains two key observations:
   • Fast edge → energy injection → oscillation at ~\(f_0\).
   • Resistor \(R\) sets decay speed (bigger \(R\) → faster damping, less ringing).

5) Why this makes the simulator “safe” and trustworthy for series RLC

• Explicit state model: the true states are \(\{i, v_C\}\); derivatives are not treated as independent states.
• Pulse steady state is defined correctly: periodic closure uses the true state vector \(x(T^-)=x(0)\).
• Analytic steady-periodic pulse solution: no “run-until-stable” numerical integration drift.
• Energy accounting is explicit: \(W_L\), \(W_C\), and \(\int v i\,dt\) are consistent checks.
• Resonance is not hand-wavy: in sine mode it is the clean condition \(\operatorname{Im}(Z)=0\).

For the full derivation, state-space matrix form, periodicity criteria, and limitations, see: Mathematical Foundations.