Series RLC Ringing That Does Not Match the Component Values
A final-year lab group characterises a series RLC low-pass network on a breadboard: L = 10.0 mH (toroidal choke, DC resistance measured on a bench DMM as 12 Ω), C = 1.5 µF metallised polypropylene, and an external 47 Ω carbon-film resistor. The network is driven from a unity-gain power op-amp buffer whose output resistance is below 0.5 Ω, so the source is effectively ideal. The input is a 1 V peak square wave at 200 Hz and the capacitor voltage is captured with a 10× probe on a 100 MHz digital oscilloscope. The first overshoot measures 18% of the final value and the ringing frequency, taken from successive zero crossings, is 1.15 kHz. A simulation using exactly the nominal component values predicts about 30% overshoot at essentially the same ringing frequency. The group's draft report blames the oscilloscope probe for 'adding damping'. The demonstrator replies that the frequency data can be trusted but the damping estimate cannot, and asks for a quantitative reconciliation before the report is marked.
1. From the measured 18% overshoot and 1.15 kHz ringing, extract the damping ratio and the undamped natural frequency, and test them against L = 10 mH and C = 1.5 µF.
Read the worked answer
For a second-order underdamped step response, percentage overshoot is Mp = exp(−πζ/√(1−ζ²)). Inverting with Mp = 0.18:
ζ = ln(1/0.18) / √(π² + ln²(1/0.18)) = 1.715 / √(9.870 + 2.941) = 1.715 / 3.579 = 0.479
The ringing frequency is the damped frequency: ωd = 2π(1150) = 7226 rad/s, so
ωn = ωd / √(1−ζ²) = 7226 / √(1 − 0.229) = 7226 / 0.878 = 8230 rad/s
Now the independent check from the components: ωn = 1/√(LC) = 1/√(10×10⁻³ × 1.5×10⁻⁶) = 1/√(1.5×10⁻⁸) = 8165 rad/s, i.e. 0.8% below the measured value, and the corresponding damped frequency is 8165 × 0.878 / 2π = 1141 Hz against the measured 1150 Hz (0.8% high).
That agreement is far inside the tolerances of a toroidal choke (±10%) and a film capacitor (±5%), so the LC product is sound and the frequency data is trustworthy. The damping, however, is a separate parameter and cannot be inferred from the frequency alone - which is exactly why the demonstrator accepts one number and rejects the other.
2. Reconcile the damping with the circuit. Compute the total series resistance implied by the measurement, test the probe hypothesis and the 'function generator source impedance' hypothesis, and say where the missing resistance most likely sits.
Read the worked answer
For a series RLC, ζ = (R/2)√(C/L), so
R = 2ζ√(L/C) = 2(0.479)√(10×10⁻³ / 1.5×10⁻⁶) = 0.958 × 81.65 = 78.2 Ω
The accounts show only 47 Ω + 12 Ω = 59 Ω, so about 19 Ω is unaccounted for.
Test the probe hypothesis. A 10× probe adds roughly 10–15 pF across C. That is 15 pF against 1.5 µF, a 1% increase in C. Since ζ ∝ √C the damping ratio would move from 0.479 to 0.481 and the ringing frequency would fall by 0.5%. Neither is measurable; the probe cannot explain a 12-point overshoot discrepancy. Rejected.
Test the source-impedance hypothesis. If a 50 Ω generator output were in series, total R = 109 Ω, giving ζ = (109/2)√(1.5×10⁻⁶/10×10⁻³) = 54.5/81.65 = 0.667, and Mp = exp(−π(0.667)/√(1−0.445)) = 6%. The measurement is 18%, so a 50 Ω source is also excluded - and in any case the buffer is specified below 0.5 Ω.
What remains is loss inside the network. Breadboard contact and lead resistance is a few tenths of an ohm at most; polypropylene ESR is milliohms. The credible suspect is the choke's effective series resistance at 1.15 kHz, which for a high-permeability core includes core loss and can far exceed its DC value. Confirm by measuring the current with a 1 Ω shunt and comparing the envelope decay, or by a swept impedance measurement of the choke alone.
3. The specification requires overshoot no greater than 10%. Using the measured 78 Ω total, determine what change to the external resistor achieves it, and state the side effect on the transient.
Read the worked answer
For Mp ≤ 10%: ζ = ln(10)/√(π² + ln²(10)) = 2.303/√(9.870 + 5.302) = 2.303/3.896 = 0.591 minimum.
With L and C fixed, ζ scales linearly with R, so required total resistance is
R = 2(0.591)√(L/C) = 1.182 × 81.65 = 96.5 Ω
Against the 78.2 Ω now present, that is an extra ≈18 Ω in the external resistor, i.e. 47 Ω → 65 Ω (a 68 Ω 1% part with the measured 19 Ω loss would land near 99 Ω and give roughly 8% overshoot, still inside spec).
The side effects are worth stating, because students often assume damping is free. The damped frequency falls only slightly - ωd goes from 0.878 × 8165 = 7170 rad/s to 0.807 × 8165 = 6590 rad/s, i.e. 1141 Hz to 1049 Hz - but the rise time and settling time both grow, and the peak capacitor voltage for a given input is reduced. The LC product, and therefore the ringing frequency the group is measuring, is unchanged, which is the useful sanity check: adding series R must not move the frequency measurably, only the envelope. If it does, the added component is not purely resistive.