Training Multiphysics Differential Equations RLC Circuits & Electromechanical Coupling
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RLC Circuits & Electromechanical Coupling

30 min Multiphysics Differential Equations

RLC Circuits & Electromechanical Coupling

Electrical circuits containing resistors, inductors, and capacitors obey the same second-order differential equation as the mass–spring–damper system. This mathematical analogy — current ↔ velocity, charge ↔ displacement, inductance ↔ mass, capacitance ↔ compliance, resistance ↔ damping — is the gateway to multiphysics: whenever an electrical variable drives a mechanical one (or vice versa), we get a coupled ODE system.

Examples include a loudspeaker (voltage → coil current → force on voice coil → cone motion), a DC motor (voltage → current → torque → shaft speed), and a piezoelectric transducer (voltage ↔ strain). In each case the two domains exchange energy through a coupling coefficient.

We first solve the pure series RLC equation $L\ddot q + R\dot q + q/C = v(t)$, identify under-/critical-/over-damped regimes, then couple it to a mechanical load for a DC-motor model.

Series RLC

$$L\,\ddot{q} + R\,\dot{q} + \frac{q}{C} = v(t), \qquad \omega_0 = \frac{1}{\sqrt{LC}}, \quad \zeta = \frac{R}{2}\sqrt{\frac{C}{L}}$$

DC-Motor Coupled Model

Electrical: $L\,\dot{i} + R\,i + K_e\,\omega = v(t)$   (back-EMF term $K_e\omega$).

Mechanical: $J\,\dot{\omega} + b\,\omega = K_t\,i - \tau_L$   (motor torque $K_t i$).

In SI units with consistent bookkeeping, $K_e = K_t$.

Example 1 — RLC natural frequency

$L = 10\,\text{mH}$, $C = 1\,\mu\text{F}$. Find $\omega_0$ and $f_0$.

$\omega_0 = 1/\sqrt{10^{-2}\cdot 10^{-6}} = 1/\sqrt{10^{-8}} = 10^{4}\,\text{rad/s}$. So $f_0 = 10^4/(2\pi) \approx 1.59\,\text{kHz}$.

Example 2 — Damping classification

Same $L$, $C$ as above, and $R = 100\,\Omega$. Find $\zeta$ and classify.

$\zeta = (100/2)\sqrt{10^{-6}/10^{-2}} = 50 \cdot 10^{-2} = 0.5$. Since $0 < \zeta < 1$, the circuit is under-damped (rings before settling).

Example 3 — DC motor steady-state speed

For $v = 24\,\text{V}$, $R = 2\,\Omega$, $K_t = K_e = 0.05\,\text{V·s/rad}$, $\tau_L = 0.1\,\text{N·m}$, find the steady-state angular speed (neglect $b$).

Steady state: $\dot i = \dot\omega = 0$. From the electrical equation, $i = (v - K_e\omega)/R$. From mechanics, $K_t i = \tau_L$.

Substitute: $K_t(v - K_e\omega)/R = \tau_L \Rightarrow \omega = (v - \tau_L R/K_t)/K_e = (24 - 0.1\cdot 2/0.05)/0.05 = (24 - 4)/0.05 = 400\,\text{rad/s}$ (≈ 3820 rpm).

Interactive Demo: RLC Circuit Response
ω₀ =10000rad/s
f₀ =1592Hz
ζ =0.50
Regime:Under-damped

Practice Problems

1. Give the electrical ↔ mechanical analogy for $L$, $R$, $C$.
2. Find $\omega_0$ for $L=1$ mH, $C=100$ nF.
3. If $R$ is chosen so $\zeta=1$, what is the transient response called?
4. Why is $K_e = K_t$ in SI units for an ideal DC motor?
5. What role does back-EMF play in a motor's speed regulation?
6. Give one example of a device whose model naturally couples electrical and mechanical ODEs.
Show Answer Key

1. $L \leftrightarrow m$, $R \leftrightarrow c$, $1/C \leftrightarrow k$.

2. $\omega_0 = 1/\sqrt{10^{-3}\cdot 10^{-7}} = 10^5$ rad/s.

3. Critically damped — fastest non-oscillatory return.

4. Energy conservation: mechanical power $K_t i\,\omega$ equals electrical power $K_e\omega\,i$.

5. It opposes rising speed, reducing current and limiting the terminal angular velocity.

6. Loudspeaker, microphone, DC motor, piezo actuator, galvanometer, MEMS accelerometer.

Coupled Mass–Spring–Damper Systems Coupled Heat & Mass Transfer