Week 4 — Transient Response: First- and Second-Order Circuits on the Scope

Overview

When a circuit containing energy-storage elements is switched, it does not jump instantly to its new steady state — it transitions, and the shape of that transition is governed by a differential equation. This week is where the oscilloscope becomes the central instrument. A first-order RC or RL circuit relaxes exponentially with a single time constant \(\tau\); a second-order RLC circuit can ring, overshoot, or crawl depending on its damping. You will derive these responses from the \(i\)–\(v\) relations of Week 3, then capture them live on the Siglent and measure \(\tau\), overshoot, and ringing frequency directly off the screen. The thrill — and the discipline — is watching an exponential or a damped sinusoid appear exactly where your hand-derived equation said it would.

The conceptual payoff is large. The exponential \(e^{-t/\tau}\) and the damped sinusoid \(e^{-\alpha t}\cos(\omega_d t)\) are the time-domain fingerprints of the poles you will later find in the frequency domain (Weeks 6, 8, 10). Damping ratio \(\zeta\) and natural frequency \(\omega_0\) are the same parameters that define a filter’s resonance. So this week is simultaneously hands-on scope practice and the bridge from time-domain to frequency-domain reasoning.

It builds directly on Week 3’s differential element relations and Week 2’s network laws, and it sets up Week 5 (the steady-state sinusoidal response is what’s left after the transient dies) and Week 6 (poles, \(\omega_0\), \(Q\)).

Readings

CAD Ch. 5 (transients): First-order RC and RL circuits — the time constant \(\tau=RC\) or \(\tau=L/R\), natural response, step response, and the general first-order solution. Extract: the universal first-order form and how to read \(\tau\) off it.
CAD Ch. 6: Second-order RLC circuits — the characteristic equation, natural frequency \(\omega_0\), damping coefficient \(\alpha\) and damping ratio \(\zeta\), and the three regimes (overdamped, critically damped, underdamped). Extract: how the discriminant determines the response shape, and the formulas for damped frequency and overshoot.
PEI (time-constant / RC timing sections): Practical RC timing, the 5τ “settled” rule of thumb, and the kinds of circuits (debouncers, timers) that exploit transients. Extract: engineering rules of thumb you can sanity-check against the math.

Key Concepts

First-order response: the universal form

A single-capacitor (or single-inductor) circuit, after Thévenin-reducing everything else seen by the storage element, obeys

\[ \tau\frac{dx}{dt}+x = x(\infty), \qquad x(t)=x(\infty)+\big[x(0^+)-x(\infty)\big]e^{-t/\tau}. \]

For RC, \(\tau=R_\text{th}C\); for RL, \(\tau=L/R_\text{th}\), where \(R_\text{th}\) is the Thévenin resistance seen by the element (Week 2 paying off again). The initial value \(x(0^+)\) comes from continuity (Week 3: \(v_C\) and \(i_L\) can’t jump), and \(x(\infty)\) is the DC steady state (capacitor → open, inductor → short). After \(\tau\) the response has covered \(1-e^{-1}\approx 63\%\) of its swing; after \(5\tau\) it is within \(<1\%\) — the practical “settled” point.

Reading τ off the scope

On a captured exponential, \(\tau\) is the time to reach 63.2% of the final change, or equivalently the time constant of the exponential fit. A robust alternative: the initial slope extrapolated to the final value also intersects at \(t=\tau\). You will measure \(\tau\) both ways and back out the effective \(RC\), then compare to nominal \(R\) and the LCR-measured \(C\) from Week 3 — discrepancies are usually explained by capacitor tolerance and the source/probe resistance.

Second-order response: the characteristic equation

A series RLC circuit gives

\[ \frac{d^2x}{dt^2}+2\alpha\frac{dx}{dt}+\omega_0^2 x = \omega_0^2 x(\infty), \qquad \alpha=\frac{R}{2L},\quad \omega_0=\frac{1}{\sqrt{LC}},\quad \zeta=\frac{\alpha}{\omega_0}. \]

The roots of \(s^2+2\alpha s+\omega_0^2=0\) are \(s=-\alpha\pm\sqrt{\alpha^2-\omega_0^2}\). Three regimes by the discriminant:

Overdamped (\(\zeta>1\), \(\alpha>\omega_0\)): two real roots, sum of two decaying exponentials, no oscillation, slow approach.
Critically damped (\(\zeta=1\)): repeated real root, fastest approach without overshoot.
Underdamped (\(\zeta<1\), \(\alpha<\omega_0\)): complex roots, a damped sinusoid \(e^{-\alpha t}\cos(\omega_d t+\phi)\) with damped frequency \(\omega_d=\omega_0\sqrt{1-\zeta^2}\). This is the ringing you will see on the scope.

Overshoot, ringing, and Q

For the underdamped step response, the fractional overshoot depends only on \(\zeta\):

\[ M_p=\exp\!\left(\frac{-\pi\zeta}{\sqrt{1-\zeta^2}}\right). \]

The quality factor \(Q=\omega_0/(2\alpha)=\frac1{2\zeta}=\frac1R\sqrt{L/C}\) (series RLC) measures how many cycles the ringing persists — high \(Q\) means light damping and long ringing. These exact same \(\omega_0\), \(\zeta\), and \(Q\) reappear in Week 6 as the resonant frequency, bandwidth, and peak sharpness of a filter. Time-domain ringing and frequency-domain resonance are two views of the same poles.

Poles: the unifying idea

The roots \(s=-\alpha\pm j\omega_d\) are the system’s poles. Real part \(\to\) decay rate; imaginary part \(\to\) oscillation frequency. The entire course’s frequency-domain machinery (Laplace, transfer functions, the s-plane, later the z-plane) is bookkeeping for where these poles sit. Seeing them as the exponents of your measured waveform now makes the abstraction concrete later.

Theory Exercises

Derive the first-order universal solution \(x(t)=x(\infty)+[x(0^+)-x(\infty)]e^{-t/\tau}\) from \(\tau\,dx/dt + x = x(\infty)\), and show \(\tau=R_\text{th}C\) for RC and \(L/R_\text{th}\) for RL.
For an RC circuit with \(R=10\,\text{k}\Omega\), \(C=0.1\,\mu\text{F}\): compute \(\tau\), the 10–90% rise time, and the time to settle within 1%.
Derive the series-RLC characteristic equation and express \(\alpha\), \(\omega_0\), \(\zeta\), \(Q\) in terms of \(R\), \(L\), \(C\).
Solve the underdamped step response fully, deriving \(\omega_d=\omega_0\sqrt{1-\zeta^2}\) and the overshoot formula \(M_p=e^{-\pi\zeta/\sqrt{1-\zeta^2}}\).
Given target \(\omega_0\) and \(\zeta\), choose \(R\), \(L\), \(C\) from kit values to realize an underdamped response with ~20% overshoot. Identify the poles in the s-plane.
Show that as \(R\to0\) the series RLC becomes a lossless oscillator at \(\omega_0\), and relate this to energy sloshing between \(\tfrac12CV^2\) and \(\tfrac12LI^2\).

Lab / Bench Work

RC step response: Drive an RC circuit with a square wave (from the function source — see syllabus signal-source note) whose period is \(\gg 5\tau\) so each edge is a clean step. Capture \(v_C(t)\) on the Siglent. Measure \(\tau\) via the 63% point and via the scope’s automatic rise-time / cursor tools. Back out \(RC\) and compare to nominal × LCR-measured values.

RL step response (optional/dual): Repeat with an RL circuit, measuring the current waveform (via voltage across a small sense resistor) to see \(i_L\) rise with \(\tau=L/R\).

RLC three regimes: Build a series RLC and, by changing \(R\) (use the potentiometer), drive it through overdamped → critically damped → underdamped. Capture all three step responses. For the underdamped case, measure \(\omega_d\) (ringing period) and overshoot \(M_p\) off the scope, then solve for \(\zeta\) and compare to \(\zeta=R/(2)\sqrt{C/L}\).

Qucs-S (ngspice): Build each circuit and run a .tran (transient) analysis, overlay simulated and captured waveforms, and confirm \(\tau\), \(\omega_d\), and \(M_p\) agree.

Measurement Methodology

Probe loading: a 10× scope probe presents ~10 MΩ / ~10–15 pF. The capacitance can shift fast RC time constants; account for it when \(C\) is small. Compensate the probe (square-wave cal) before precision timing measurements.
Square-wave period: must be long enough (\(T\gg10\tau\)) that the circuit fully settles each half-cycle, or you measure a truncated exponential.
Triggering: trigger on the driving edge; use single-shot capture for clean one-time transients.
Source impedance: the function source’s output resistance adds to \(R\) in \(\tau\); measure or account for it (often \(50\,\Omega\), negligible against k\(\Omega\) but not against small \(R\)).
Reconcile measured \(\tau\)/\(\omega_d\)/\(M_p\) against hand calculation (using LCR-measured L, C) and Qucs-S. Attribute residuals to tolerance, probe capacitance, or ESR/winding resistance.

Expected baselines: Measured \(\tau\) within ~5–10% of \(R\times C\) using nominal values, tightening when you use LCR-measured C. Underdamped ringing frequency within a few percent of \(\omega_d/2\pi\); overshoot matching \(M_p\) once real damping (ESR, winding resistance) is included — ideal-component predictions slightly under-damp compared to reality.

Connections

The exponential and damped-sinusoid responses captured here are the time-domain signature of poles, which Week 6 locates in the frequency domain as filter resonance, and Week 8 formalizes via the transfer function and impulse response. \(\omega_0\), \(\zeta\), and \(Q\) defined here are identical to the resonance parameters of next-phase filters. The continuity conditions and Thévenin reduction used to set up each transient tie directly back to Weeks 2–3. The scope-measurement discipline (probe compensation, triggering, settling time) is reused for every captured waveform from here to the capstone.