13 · LC Oscillations, Damped RLC Circuits, and AC Response

1. Ideal LC oscillations

Choose signed capacitor charge \(q\) and current \(i=dq/dt\). The ideal circuit's energy is

\[ U=\frac{q^2}{2C}+\frac12Li^2. \]

Energy conservation, or the loop equation directly, gives

\[ \boxed{L\ddot q+\frac qC=0,\qquad \omega_0=\frac1{\sqrt{LC}},\qquad T_0=2\pi\sqrt{LC}.} \]

The loop equation remains valid at instants when \(i=0\); an energy derivation should not divide by zero there. The general solution is

\[ q=q_m\cos(\omega_0t+\alpha),\qquad i=-\omega_0q_m\sin(\omega_0t+\alpha). \]

Initial charge and initial current determine both constants:

\[ q(0)=q_m\cos\alpha,\qquad i(0)=-\omega_0q_m\sin\alpha. \]

Current and charge are a quarter cycle apart. When charge is maximal, current is zero; when charge is zero, current magnitude is maximal. The energy exchange is

\[ U_E=U_0\cos^2(\omega_0t+\alpha),\qquad U_B=U_0\sin^2(\omega_0t+\alpha),\qquad U_0=\frac{q_m^2}{2C}. \]

Mechanical oscillator	LC circuit
Displacement \(x\)	Charge \(q\)
Velocity \(\dot x\)	Current \(i\)
Mass \(m\)	Inductance \(L\)
Spring constant \(k\)	Inverse capacitance \(1/C\)
\(m\ddot x+kx=0\)	\(L\ddot q+q/C=0\)

The scale \(\sqrt{LC}\) is \(1/\omega_0\), not the full oscillation period.

2. Free damped RLC response（阻尼振荡）

A series resistor removes energy at rate \(i^2R\):

\[ \frac{dU}{dt}=-Ri^2,\qquad \boxed{L\ddot q+R\dot q+\frac qC=0.} \]

Define \(\beta=R/(2L)\) and try \(q\propto e^{st}\). The characteristic roots are

\[ s_\pm=-\beta\pm\sqrt{\beta^2-\omega_0^2}. \]

Regime	Condition	Charge response
Underdamped（欠阻尼）	\(\beta<\omega_0\)	\(q=Ae^{-\beta t}\cos(\omega_dt+\alpha)\), \(\omega_d=\sqrt{\omega_0^2-\beta^2}\)
Critical（临界阻尼）	\(\beta=\omega_0\)	\(q=(A+Bt)e^{-\beta t}\)
Overdamped（过阻尼）	\(\beta>\omega_0\)	\(q=Ae^{s_+t}+Be^{s_-t}\); both roots real and negative

The critical resistance is \(R_c=2\sqrt{L/C}\). Critical damping gives the boundary between oscillatory and nonoscillatory relaxation and the fastest nonoscillatory settling in the standard comparison with the same initial displacement and zero velocity.

Underdamped, critical, and overdamped RLC responses with the same initial charge and zero initial current

The underdamped amplitude decays on time \(1/\beta=2L/R\), different from the RL time constant \(L/R\). In weak damping, the cycle-averaged energy envelope falls approximately as \(e^{-2\beta t}\); the exact instantaneous statement is \(\dot U=-Ri^2\).

The slides also use complex frequency: with \(q=\operatorname{Re}(\widetilde q e^{j\widetilde\omega t})\) and \(j^2=-1\), decay requires \(\widetilde\omega=\omega_d+j\beta\). This sign depends on the chosen exponential convention.

3. Driven oscillations and phasors（相量）

A rotating \(N\)-turn coil in a uniform field has \(\Lambda=NBA\cos\Omega t\), hence generator emf \(\mathcal E=NBA\Omega\sin\Omega t\). Choosing a shifted time origin gives the cosine drive below; its frequency is imposed by the generator.

For a source \(\mathcal E(t)=\mathcal E_m\cos\Omega t\),

\[ \boxed{L\ddot q+R\dot q+\frac qC=\mathcal E_m\cos\Omega t.} \]

The full solution is a decaying free transient plus a forced solution. For \(R>0\), the long-time response oscillates at the drive frequency \(\Omega\), not at the free damped frequency \(\omega_d\).

Represent sinusoidal signals by

\[ v(t)=\operatorname{Re}(\widetilde V e^{j\Omega t}),\qquad i(t)=\operatorname{Re}(\widetilde I e^{j\Omega t}),\qquad \widetilde Z=\frac{\widetilde V}{\widetilde I}. \]

Differentiation becomes multiplication by \(j\Omega\). The complex impedance combines magnitude and phase:

Element	Impedance	Current relative to its voltage
Resistor	\(R\)	In phase
Inductor	\(j\Omega L\)	Lags by \(\pi/2\)
Capacitor	\(1/(j\Omega C)=-j/(\Omega C)\)	Leads by \(\pi/2\)

Series impedances add; parallel admittances add:

\[ \widetilde Z_{\mathrm{series}}=\sum_j\widetilde Z_j,\qquad \frac1{\widetilde Z_{\mathrm{parallel}}}=\sum_j\frac1{\widetilde Z_j}. \]

Add the complex values, not their magnitudes. These relations assume lumped elements and negligible unmodeled mutual coupling.

4. Series RLC impedance and phase

\[ \boxed{\widetilde Z=R+j\left(\Omega L-\frac1{\Omega C}\right),\qquad |Z|=\sqrt{R^2+\left(\Omega L-\frac1{\Omega C}\right)^2}.} \]

Let \(\phi=\arg\widetilde Z\), so \(\tan\phi=(\Omega L-1/(\Omega C))/R\) for \(R>0\). With the source phase set to zero,

\[ \boxed{i(t)=I_m\cos(\Omega t-\phi),\qquad I_m=\frac{\mathcal E_m}{|Z|}.} \]

\(\Omega<\omega_0\): capacitive response; \(\phi<0\), current leads.
\(\Omega>\omega_0\): inductive response; \(\phi>0\), current lags.
\(\Omega=\omega_0\): reactive contributions cancel; voltage and current are in phase.

中文提示：本页把 \(\phi\) 定义为“电压相位减电流相位”，所以电流写成 \(\cos(\Omega t-\phi)\)。若改写为加号，那个相位变量就应当换成 \(-\phi\)，不能混用两套定义。

5. Resonance and selectivity（共振与选频）

For a fixed-amplitude voltage drive and constant \(R\),

\[ \boxed{\Omega_{\mathrm{res}}=\omega_0,\qquad I_{m,\max}=\frac{\mathcal E_m}{R}.} \]

This is current resonance in the series circuit. Other observables, such as capacitor-voltage amplitude, need not peak at exactly the same frequency when damping is appreciable. Ideal \(R=0\) at resonance has no finite steady-state current amplitude; the resonant response grows rather than being described by division by zero.

Series RLC current resonance for three resistance values, showing increased peak height and selectivity at smaller resistance

The start of class 14 adds a resonance-width interpretation. Define the quality factor and half-power width by

\[ \boxed{\mathcal Q=\frac{\omega_0L}{R} =\frac1R\sqrt{\frac LC},\qquad \Delta\Omega=\frac RL,\qquad \mathcal Q=\frac{\omega_0}{\Delta\Omega}.} \]

At the two half-power frequencies, \(I_m=I_{m,\max}/\sqrt2\), not \(I_{m,\max}/2\). Small resistance gives high \(\mathcal Q\) and a narrow peak, useful for selecting a frequency. The formula uses angular-frequency width; frequency width in hertz is \(\Delta f=\Delta\Omega/(2\pi)\).

Power accounting

For sinusoidal steady state,

\[ V_{\mathrm{rms}}=\frac{V_m}{\sqrt2},\quad I_{\mathrm{rms}}=\frac{I_m}{\sqrt2},\quad \boxed{\langle P\rangle=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi =\frac12I_m^2R.} \]

Ideal inductors and capacitors exchange energy with the circuit but absorb zero net energy over a full cycle. This energy check also explains the half-power width. Mechanical resonance, frequency selection, and resonant electromagnetic absorption are the classroom's qualitative applications; detailed MRI or quantum-transition theory is not developed here.

Sources and checks

Lecture 13 pp. 3–14: LC dynamics and analogy; pp. 15–20: complex form and damping; pp. 21–28: AC impedance and resonance. Class 14's first approximately ten minutes supplies quality factor/selectivity. Keep \(\omega_0\), \(\omega_d\), and \(\Omega\) distinct; check phase conventions and distinguish amplitude decay, energy decay, and half-power width.