Circuit Characterization & Filter Design

1.0 Overview

Two linked projects run on the same three passive components: a 100 Ω resistor, a 0.1 µF capacitor, and a 1 mH inductor. The first characterized the step response of RC and RL circuits and pulled their time constants off oscilloscope captures. The second built RC low-pass, RC high-pass, and series RLC bandpass filters, measured their frequency response from 1 kHz to 500 kHz on a network analyzer, and compared the result against analytical transfer functions and LTspice.

The useful result is the one that links the two. The capacitance implied by the RC time constant, applied with no further fitting, predicts the filter stopbands to within 0.01 dB.

2.0 Transient response

RC circuit

With R = 100 Ω and C = 0.1 µF, the theoretical time constant is:

τ = RC = 100 × 0.1×10⁻⁶ = 10 µs

The turn-on transient across the resistor follows V_R(t) = V_s·e^−t/τ, so at t = τ, V_R = 0.368 V_s.

Measured: τ = 9.573 µs against 10 µs nominal, a 4.3% shortfall. That sits inside the ±10% tolerance typical of a ceramic capacitor. Solving τ = RC with R at nominal gives an effective capacitance of 95.7 nF, about 4% below its marking.

RL circuit

With R = 100 Ω and L = 1 mH:

τ = L/R = 0.001 / 100 = 10 µs

Here the transient follows V_R(t) = V_s·(1 − e^−t/τ), giving 0.632 V_s at t = τ.

Measured: τ = 8.969 µs, a 10.3% shortfall. That one has a physical cause. A real inductor carries winding resistance in series with its inductance, so the loop time constant is L/(R + R_series), not L/R:

R_total = L/τ = 1×10⁻³ / 8.969×10⁻⁶ = 111.5 Ω

That is about 11.5 Ω beyond the resistor, plausible for a small through-hole 1 mH inductor. Breadboard contact resistance and inductor tolerance push the same direction and cannot be separated from this data alone.

The RL capture recorded both channels at the second cursor, so the threshold could be checked directly: 631.9 / 995.1 = 0.635 against a target of 0.632. The cursor sat very slightly past one time constant, putting the true value nearer 8.89 µs. That correction is smaller than the gap to theory and changes no conclusion.

3.0 Filter design

RC low-pass and high-pass

f_c = 1 / (2πRC) = 15.92 kHz

Both filters share that corner, sit 3 dB down at it, and roll off at 20 dB/decade beyond it. The low-pass phase is −45° at the corner and the high-pass is +45°.

RLC bandpass

With R = 100 Ω, L = 1 mH, C = 0.1 µF, the reactances cancel at resonance:

f₀ = 1 / (2π√(LC)) = 15,915 Hz

B = R / (2πL) = 15,915 Hz  ·  Q = f₀ / B = (1/R)√(L/C) = 1.00

Because f₀ = B here, the symmetric shortcut f₀ ± B/2 would give 7.96 and 23.87 kHz. That shortcut only holds at high Q. The exact half-power frequencies are:

f_c1,2 = f₀[√(1 + 1/4Q²) ± 1/2Q] = 9.84 kHz, 25.75 kHz

Their geometric mean is f₀, as it has to be for a second-order bandpass.

4.0 Measured against theory

The network analyzer swept a 1 V sine from 1 kHz to 500 kHz in 151 logarithmic steps. All three filters followed their predicted shapes across the full range.

Measured Q of 0.98 agrees with the design value of 1.00, which makes this a broadband filter rather than a selective one. The geometric mean of the two measured half-power points, √(9.93 × 26.32) = 16.17 kHz, agrees with the measured zero crossing, which is an internal check on the digitization.

5.0 What the simulation missed

Each filter was simulated in LTspice with an AC analysis from 1 kHz to 500 kHz. The directive was .ac dec 1 1k 500k, which evaluates the circuit at only 1 kHz, 10 kHz, 100 kHz and 500 kHz and joins those four points with straight lines. The visible kinks in every simulated trace fall exactly on the decade frequencies.

That matters most for the bandpass filter: the highest simulated point is −2.85 dB at 10 kHz, so the plot never shows the 0 dB peak at 15.9 kHz. The models were correct and the simulated values match the analytical transfer functions at all four sample frequencies. The sweep resolution was the problem, not the circuit. Simulation resolution has to be chosen for the feature being studied rather than left at a setting that happens to draw a plausible curve.

6.0 Above 200 kHz

The low-pass phase does not settle at −90°. It bottoms out near −79° around 170 kHz, then climbs back to −71.4° at 490 kHz, and the magnitude there sits 0.5 dB above the effective-C prediction. Both are consistent with a small resistance in series with the capacitor from ESR, lead resistance, and breadboard contacts, which adds a zero to the response. Adding 1.0 Ω in series gives −29.13 dB and −71.6° at 490.55 kHz, within 0.2 dB and 0.2° of measurement. A parasitic that size is invisible at low frequency and dominant once the capacitor's reactance falls to a few ohms. It sets a practical floor of roughly −30 dB on what this breadboard filter can attenuate.

7.0 Limitations

• Component values were not measured. R, L and C were taken at nominal markings. The effective capacitance and implied series resistance are inferred from the data, not read off an LCR meter.
• One capture per measurement. Each time constant comes from a single cursor placement on a single capture, so repeatability was not quantified.
• Coarse simulation sweep. One point per decade cannot resolve the bandpass peak or the shape of any curve between decades.
• Digitized phase data. Phase crossings came from screenshots rather than exported data files, limiting resolution to roughly ±3%.
• Loading and parasitics. Generator output impedance, scope input capacitance, capacitor ESR and inductor self-capacitance were not modeled explicitly.