Lecture 17b: The RLC Circuit and Damped Oscillation

What this lecture covers

This lecture answers a question the previous lecture left open: what happens when you add a resistor to an LC circuit that was oscillating forever? It builds on the demo of two inverters where adding a small resistance produced strange ringing on the output, and on the prior lecture's analysis of the pure LC circuit. Here the instructor adds the resistor back in, derives the governing second-order differential equation for the series RLC circuit, and solves it for a step input with zero initial conditions.

After watching, you can write and solve the characteristic equation for a series RLC circuit, classify its response as overdamped, underdamped, or critically damped, and sketch the underdamped step response from just a few circuit-derived parameters without solving a differential equation from scratch each time.

Key ideas

• Series RLC step response: adding a resistor to an LC circuit causes its sinusoidal response to decay over time instead of oscillating forever, settling at the input voltage.
• Characteristic equation: written in canonical form s^2 + 2*alpha*s + omega_0^2 = 0, where omega_0 = 1/sqrt(LC) is the natural oscillation frequency and alpha = R/2L for the series RLC circuit.
• Three damping cases: overdamped (alpha > omega_0, sum of two decaying exponentials), critically damped (alpha = omega_0), and underdamped (alpha < omega_0, a decaying sinusoid).
• Underdamped solution: the capacitor voltage settles to v(t) = VI - VI*e^(-alpha*t)*cos(omega_d*t + phase), where omega_d = sqrt(omega_0^2 - alpha^2) is the damped ringing frequency.
• Quality factor (Q): defined as omega_0 / (2*alpha), it approximates how many cycles the circuit rings before settling; high Q means long ringing, low Q means fast settling.
• Sketching by inspection: once you know the initial value, final value, omega_d, and Q, you can sketch the underdamped response without solving the full differential equation.
• The Feynman trick: check a derived equation by setting a parameter (here, R) to zero and confirming it collapses back to the simpler case already solved (the LC circuit).
• Capacitor voltage can exceed the input: in an LC or lightly damped RLC circuit, the capacitor voltage can briefly rise above the source voltage, a property behind DC-to-DC voltage converters.

Walkthrough

Websim demo and course context (0:00)

The lecture opens with an optional demo of a web-based circuit simulator built by the instructor, where students can set MOSFET or RC/RLC parameters, apply inputs including music, and view or listen to the output. This section is explicitly not required course material.

Review of LC circuit results (7:10)

The instructor reviews the two-inverter demo that motivated this unit and the prior lecture's LC circuit analysis, including its zero-state step response, which oscillates indefinitely as a sinusoid around the final value VI. He notes that the capacitor voltage overshooting the input is a real, useful property, referenced later in DC-to-DC converter design, and previews that adding a resistor will damp this oscillation, matching the earlier demo's odd behavior.

Setting up and solving the RLC node equations (15:15)

Using the element rules for the inductor, capacitor, resistor, and voltage source, the instructor writes two node equations for the series RLC circuit and combines them into a single second-order differential equation for the capacitor voltage, differing from the LC case only by the added resistor term. He applies the "Feynman trick" of setting R to zero to confirm the new equation reduces correctly to the earlier LC result.

The four-step method and characteristic equation (21:27)

Following the same three-step (particular, homogeneous, total solution) and four-step homogeneous-solution pattern used for RC and LC circuits earlier in the course, the instructor finds the particular solution (a constant, VI), then assumes a homogeneous solution of the form A*e^(s*t) to derive the characteristic equation, rewritten in canonical form using alpha = R/2L and omega_0^2 = 1/LC.

Overdamped, underdamped, and critically damped cases (29:42)

Depending on whether alpha is greater than, equal to, or less than omega_0, the roots of the characteristic equation are real and distinct, repeated, or complex, giving the overdamped, critically damped, and underdamped cases respectively. The instructor works through the underdamped case in detail, introducing omega_d = sqrt(omega_0^2 - alpha^2) and using Euler's relationship to reduce the solution to sines and cosines.

Applying initial conditions and simplifying to a single cosine (34:50)

With zero initial capacitor voltage and current, the instructor solves for the unknown constants and, using the trigonometric identity that a scaled sum of sines and cosines of the same frequency is itself a single sinusoid, reduces the underdamped solution to the compact form v(t) = VI - VI*e^(-alpha*t)*cos(omega_d*t + phase). He again applies the Feynman check, confirming that setting alpha to zero recovers the undamped LC sinusoid.

The quality factor and physical meaning of ringing (43:06)

The instructor names the oscillation "ringing" and explains that its frequency is omega_d, its decay rate is governed by alpha, and the quality factor Q = omega_0 / (2*alpha) estimates how many cycles it rings before settling. He emphasizes that all of these quantities come directly from the characteristic equation, so once you have that equation you do not need to re-derive the full time-domain solution.

Sketching the underdamped response by inspection (46:12)

Using a worked example with nonzero initial voltage and current, the instructor shows how to sketch the underdamped response directly from the initial value, final value, omega_d, and Q, without solving the differential equation each time. He leaves the direction of the first swing (up or down) as an open question for students to reason through before the next lecture.

Before you watch

• Review the appendix on complex algebra and the trigonometric identity for combining sines and cosines of the same frequency, both referenced directly in this lecture.
• Be familiar with the node method and element rules for inductors, capacitors, and resistors from earlier lectures.
• Watch the prior lecture on the LC circuit's step response, since this lecture builds directly on its results and repeatedly compares back to them.

Check your understanding

1. Why does setting the resistance R to zero in the RLC characteristic equation recover the LC circuit's result, and how does this check help verify a derivation?
2. What determines whether a series RLC circuit is overdamped, underdamped, or critically damped, and how does each case look when plotted?
3. How are omega_d, alpha, and Q defined in terms of R, L, and C, and what does each one tell you about the circuit's step response?
4. Given the initial and final values of the capacitor voltage along with omega_d and Q, what information would you still need to fully sketch the underdamped response?