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Lesson 97 — RLC Circuits

Differential equation of the series RLC circuit — electrical analog of the mass-spring system. Free response, forced response, and resonance.

Used in: Spécialité Maths Terminale (França) · Leistungskurs Physik Klasse 12 (Alemanha) · H3 Mathematics (Singapura)

L\frac{d^2Q}{dt^2} + R\frac{dQ}{dt} + \frac{Q}{C} = V(t)

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Rigorous notation, full derivation, hypotheses

Rigorous derivation and classification

Kirchhoff's Voltage Law

In a series $RLC$ circuit with source $V(t)$ , the sum of voltage drops equals the source:

$V_L + V_R + V_C = V(t)$

Using $V_L = L\,\dot I$ , $V_R = RI$ , $V_C = Q/C$ and $I = \dot Q$ :

L\,\ddot{Q} + R\,\dot{Q} + \frac{Q}{C} = V(t)

what this means · Differential equation of the series RLC circuit — 2nd-order linear ODE with constant coefficients.

"The equation $LQ'' + RQ' + Q/C = V(t)$ is the standard form of the RLC circuit equation and has exactly the same mathematical form as the damped mass-spring system $mx'' + cx' + kx = F(t)$ , with $L$ playing the role of mass, $R$ the damping constant, and $1/C$ the spring constant." — Lebl, Notes on Diffy Qs §2.6

Electromechanical analogy table

Complete electromechanical analogy. Every mass-spring resolution technique transfers directly to the RLC.

Classification by characteristic equation

Homogeneous equation ( $V = 0$ ): $L\lambda^2 + R\lambda + 1/C = 0$ .

$\Delta = R^2 - 4L/C$

Definition· Three free response regimes

Overdamped ( $\zeta > 1$ , $\Delta > 0$ ): two negative real $\lambda$ . Slow exponential return to equilibrium. $Q_h(t) = C_1 e^{\lambda_1 t} + C_2 e^{\lambda_2 t}$
Critically damped ( $\zeta = 1$ , $\Delta = 0$ ): double $\lambda = -R/(2L)$ . Fastest return without oscillation. $Q_h(t) = (C_1 + C_2 t)\,e^{\lambda t}$
Underdamped ( $\zeta < 1$ , $\Delta < 0$ ): $\lambda = -\zeta\omega_0 \pm i\omega_d$ . Damped oscillation. $Q_h(t) = e^{-\zeta\omega_0 t}\bigl(C_1\cos\omega_d t + C_2\sin\omega_d t\bigr)$

Steady-state response (forced)

For $V(t) = V_0\cos(\omega t)$ , particular solution:

$Q_p(t) = \frac{V_0/L}{\sqrt{(\omega_0^2-\omega^2)^2 + (2\zeta\omega_0\omega)^2}}\cos(\omega t - \phi)$

with $\tan\phi = \dfrac{2\zeta\omega_0\omega}{\omega_0^2-\omega^2}$ .

Solved examples

Example— 97.1· Regime classification — component data

Problem. Series RLC circuit: $L = 1$ H, $R = 6$ $\Omega$ , $C = 1/8$ F. Classify the regime and write the general homogeneous solution.

Strategy. Calculate $\Delta = R^2 - 4L/C$ ; identify the regime; write $Q_h$ .

Resolution.

$\Delta = 36 - 4(1)(8) = 36 - 32 = 4 > 0$ — overdamped.

Characteristic equation: $\lambda^2 + 6\lambda + 8 = 0 \Rightarrow \lambda = (-6 \pm 2)/2$ , so $\lambda_1 = -2$ , $\lambda_2 = -4$ .

$Q_h(t) = C_1 e^{-2t} + C_2 e^{-4t}$

Verification. $Q_h' = -2C_1 e^{-2t} - 4C_2 e^{-4t}$ , $Q_h'' = 4C_1 e^{-2t} + 16C_2 e^{-4t}$ . Substituting into $Q'' + 6Q' + 8Q$ : collecting $e^{-2t}$ : $4C_1 - 12C_1 + 8C_1 = 0$ ✓. Collecting $e^{-4t}$ : $16C_2 - 24C_2 + 8C_2 = 0$ ✓.

Source. Lebl, Notes on Diffy Qs §2.6, Example 2.6.1 — CC-BY-SA

Example— 97.2· Underdamped regime with initial conditions

Problem. $L = 1$ H, $R = 2$ $\Omega$ , $C = 1/2$ F, $V = 0$ . Conditions: $Q(0) = 1$ C, $I(0) = 0$ A. Find $Q(t)$ .

Strategy. Calculate $\Delta$ , identify underdamped regime, apply IC to determine $C_1, C_2$ .

Resolution.

$\Delta = 4 - 4(1)(2) = 4 - 8 = -4 < 0$ — underdamped.

$\lambda = -1 \pm i$ , so $\alpha = 1$ , $\omega_d = 1$ .

$Q_h(t) = e^{-t}(C_1 \cos t + C_2 \sin t)$

IC: $Q(0) = C_1 = 1$ .

$\dot Q = e^{-t}[(-C_1 + C_2\omega_d)\cos t + (-C_2 - C_1\omega_d)\sin t]$ ... with $\omega_d = 1$ :

$\dot Q(0) = -C_1 + C_2 = 0 \Rightarrow C_2 = 1$ .

$Q(t) = e^{-t}(\cos t + \sin t) = \sqrt{2}\,e^{-t}\cos\!\left(t - \frac{\pi}{4}\right)$

Verification. $Q(0) = 1$ ✓; $\dot Q(0) = -1 + 1 = 0$ ✓. Amplitude decays as $\sqrt{2}e^{-t}$ ✓.

Source. Lebl, Notes on Diffy Qs §2.6, Exercise 2.6.2 — CC-BY-SA

Example— 97.3· Forced response — steady state

Problem. $L = 0.1$ H, $R = 10$ $\Omega$ , $C = 10^{-3}$ F, $V(t) = 50\cos(100t)$ V. Find the steady-state current $I_p(t) = \dot Q_p$ .

Strategy. Calculate $Z(j\omega)$ and use $\hat I = \hat V/Z$ .

Resolution.

$\omega = 100$ rad/s. $\omega_0 = 1/\sqrt{0.1 \times 10^{-3}} = 100$ rad/s — resonance!

$Z(j100) = 10 + j(100 \times 0.1 - 1/(100 \times 10^{-3})) = 10 + j(10 - 10) = 10$ $\Omega$ .

$I_p(t) = \frac{50}{10}\cos(100t) = 5\cos(100t)$ A.

At resonance, the impedance is purely resistive and the current reaches its maximum value.

Verification. $|Z|_{\omega=100} = 10$ $\Omega$ ✓; $I_{\max} = V_0/R = 50/10 = 5$ A ✓.

Source. OpenStax University Physics Vol. 2, §14.5 Example 14.5 — CC-BY

Example— 97.4· Quality factor and bandwidth

Problem. AM radio receiver: $L = 0.5$ mH, $C = 25$ pF, $R = 5$ $\Omega$ . Calculate $f_0$ , $Q_f$ and $\Delta f$ (bandwidth).

Strategy. Apply the formulas for $f_0$ , $Q_f$ and $BW$ directly.

Resolution.

$\omega_0 = 1/\sqrt{0.5 \times 10^{-3} \times 25 \times 10^{-12}} = 1/\sqrt{1.25 \times 10^{-14}} \approx 8.94 \times 10^6$ rad/s.

$f_0 = \omega_0/(2\pi) \approx 1.42$ MHz (AM band).

$Q_f = \omega_0 L/R = (8.94 \times 10^6 \times 0.5 \times 10^{-3})/5 = 4470/5 = 894$ .

$\Delta f = f_0/Q_f = 1.42 \times 10^6/894 \approx 1.59$ kHz.

AM channel has 10 kHz bandwidth — $Q_f = 894$ is too selective for AM, but suitable for stereo FM.

Verification. $\zeta = 1/(2Q_f) = 1/1788 \approx 5.6 \times 10^{-4} \ll 1$ — strongly underdamped ✓.

Source. OpenStax University Physics Vol. 2, §14.6 — CC-BY

Example— 97.5· LC circuit without resistance — pure oscillation

Problem. $L = 4$ H, $C = 1/16$ F, $R = 0$ , $V = 0$ . $Q(0) = 0$ , $I(0) = 2$ A. Find $Q(t)$ and total energy.

Strategy. Undamped case: $\ddot Q + \omega_0^2 Q = 0$ , general sine/cosine solution.

Resolution.

$\omega_0 = 1/\sqrt{4 \times 1/16} = 1/\sqrt{1/4} = 2$ rad/s.

$Q(t) = A\cos(2t) + B\sin(2t)$ .

$Q(0) = A = 0 \Rightarrow Q(t) = B\sin(2t)$ .

$\dot Q(0) = 2B = 2 \Rightarrow B = 1$ .

$Q(t) = \sin(2t), \quad I(t) = 2\cos(2t)$

Energy: $E = \frac{1}{2}L I^2 + \frac{Q^2}{2C} = \frac{1}{2}(4)(4\cos^2 2t) + \frac{\sin^2 2t}{2(1/16)} = 8\cos^2 2t + 8\sin^2 2t = 8$ J (constant ✓).

Source. Trench, Elementary Differential Equations §6.2, Exercise 6.2.3 — open

Exercise list

34 exercises · 8 with worked solution (25%)

Application 18Understanding 4Modeling 9Challenge 1Proof 2

Ex. 97.1ApplicationAnswer key
RLC circuit with $L = 1$ H, $R = 4$ $\Omega$ , $C = 1/4$ F, $V = 0$ . Identify the regime and write the general homogeneous solution.
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Ex. 97.2Application
$L = 1$ H, $R = 1$ $\Omega$ , $C = 1/2$ F, $V = 0$ . Classify and write $Q_h$ .
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Ex. 97.3Application
$L = 1$ H, $R = 2$ $\Omega$ , $C = 1/2$ F, $V = 0$ , $Q(0) = 1$ C, $I(0) = 2$ A. Solve the IVP.
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Ex. 97.4ApplicationAnswer key
Calculate the natural frequency $\omega_0$ and $f_0$ of an LC circuit with $L = 2$ H and $C = 0.02$ F.
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Ex. 97.5ApplicationAnswer key
What condition on $R$ , $L$ and $C$ guarantees critical damping?
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Ex. 97.6Application
$L = 0.1$ H, $R = 10$ $\Omega$ , $C = 10^{-3}$ F. Calculate $\zeta$ and classify the regime.
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Ex. 97.7ApplicationAnswer key
$V_0 = 100$ V, $R = 20$ $\Omega$ . What is the maximum current at resonance?
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Ex. 97.8Application
$L = 10$ mH, $C = 100$ $\mu$ F, $R = 5$ $\Omega$ . Calculate the quality factor $Q_f$ and the bandwidth.
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Ex. 97.9Application
At a certain instant: $L = 0.5$ H, $I = 4$ A, $C = 1$ $\mu$ F, $Q = 2$ mC. Calculate the total stored energy.
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Ex. 97.10Application
$L = 1$ H, $R = 6$ $\Omega$ , $C = 1/8$ F, $V = 0$ , $Q(0) = Q_0$ , $I(0) = 0$ . Sketch the solution $Q(t)$ and explain why it does not oscillate.
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Ex. 97.11Application
$\omega_0 = 4$ rad/s, $\zeta = 0.5$ . Calculate the damped oscillation frequency $\omega_d$ .
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Ex. 97.12Application
Underdamped circuit with $\alpha = 1$ s $^{-1}$ and $\omega_d = 3$ rad/s. What is the oscillation period and by what factor does the amplitude decay each cycle?
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Ex. 97.13ModelingAnswer key
Derive the general expression for the particular solution $Q_p(t)$ for $V(t) = V_0\cos(\omega t)$ .
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Ex. 97.14Modeling
For $V(t) = 120\cos(2\pi \times 60\,t)$ V and $|Z| = 40$ $\Omega$ with a 30-degree phase angle, calculate the average dissipated power.
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Ex. 97.15Modeling
AM radio: $L = 0.25$ mH. What capacitance tunes 1000 kHz?
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Ex. 97.16Modeling
RC filter: $R = 10$ k $\Omega$ , $C = 10$ $\mu$ F, $V_s = 5$ V. How long until $V_C = 3.16$ V?
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Ex. 97.17ModelingAnswer key
RL circuit: $L = 2$ H, $R = 4$ $\Omega$ , DC source $V_0 = 12$ V, $I(0) = 0$ . Find $I(t)$ and the steady-state value.
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Ex. 97.18ModelingAnswer key
Ideal LC circuit ( $R = 0$ ) with $L = 0.1$ H, $C = 100$ pF, $I(0) = 5$ mA, $Q(0) = 0$ . What is the maximum charge on the capacitor?
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Ex. 97.19Understanding
What happens to the amplitude of the forced response of an LC circuit (without resistance) when $\omega \to \omega_0$ ?
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Ex. 97.20Understanding
To increase the free oscillation period of an underdamped RLC circuit, what should be done?
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Ex. 97.21Understanding
What is the correct expression for the quality factor $Q_f$ and what does it physically represent?
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Ex. 97.22UnderstandingAnswer key
In the electromechanical analogy between the RLC circuit and the mass-spring system, which electrical component corresponds to mass $m$ ?
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Ex. 97.23Application
Find the poles of the RLC circuit with $R = 5$ $\Omega$ , $L = 0.5$ H, $C = 0.02$ F. Represent in the complex plane.
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Ex. 97.24Application
$R = 10$ $\Omega$ , $L = 0.1$ H, $C = 100$ $\mu$ F, $f = 50$ Hz. Calculate the impedance $|Z|$ .
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Ex. 97.25Application
Underdamped RLC circuit with $L = 1$ H, $R = 2$ $\Omega$ . How long until the oscillation amplitude drops by half?
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Ex. 97.26Application
ODE: $\ddot Q + 6\dot Q + 25Q = 0$ , $Q(0) = 0$ , $\dot Q(0) = 2$ . Solve.
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Ex. 97.27Application
$\ddot Q + 4\dot Q + 13Q = 0$ . Find the general solution and the damped oscillation frequency.
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Ex. 97.28Modeling
Show that the RLC circuit with $L, R, C > 0$ and $V = 0$ is always asymptotically stable (all transients decay to zero).
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Ex. 97.29Modeling
Why does the complete response of an RLC circuit with $R > 0$ always converge to the steady state $Q_p$ regardless of initial conditions?
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Ex. 97.30Modeling
FM receiver (88–108 MHz), $L = 0.1$ mH. What is the required capacitance range? Discuss feasibility.
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Ex. 97.31Application
$\ddot Q + 4\dot Q + 13Q = 10\cos(2t)$ . Find the particular solution by the method of undetermined coefficients.
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Ex. 97.32Challenge
Show that the voltage across the capacitor of an RLC circuit at resonance can exceed the source voltage by a factor $Q_f$ . Calculate for $Q_f = 100$ , $V_0 = 5$ V.
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Ex. 97.33Proof
Demonstrate that the average power dissipated by an RLC circuit with a sinusoidal source is maximized at resonance and equals $V_0^2/(2R)$ .
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Ex. 97.34Proof
Prove that the energy $E(t) = \frac{1}{2}LI^2 + \frac{Q^2}{2C}$ is monotonically non-increasing in the free RLC circuit ( $V = 0$ , $R > 0$ ).

Sources

Lebl, Jiří. Notes on Diffy Qs: Differential Equations for Engineers. Version 6.4. CC-BY-SA. jirka.org/diffyqs — Main reference; §2.6 covers RLC as an application of 2nd-order ODEs.
Trench, William F. Elementary Differential Equations with Boundary Value Problems. Brooks-Cole (open). digitalcommons.trinity.edu/mono/9 — Cap. 6 treats RL, RC, and RLC circuits with a classical approach.
OpenStax. University Physics Volume 2. CC-BY. openstax.org/details/books/university-physics-volume-2 — §14.5–14.6: resonance, quality factor, bandwidth, physical perspective.