Week 6 — Frequency Response and Analog Filters: Transfer Functions, Bode, Resonance

Overview

A circuit treated as a system has a frequency response: a complex function \(H(j\omega)\) that says how much each input frequency is scaled and phase-shifted on its way to the output. This week we stop analyzing single frequencies and start thinking about the whole spectrum at once. The transfer function \(H(j\omega)\) is just the impedance divider of Week 5 written as a function of \(\omega\), and the Bode plot is the engineer’s standard way to visualize it. Filters — low-pass, high-pass, band-pass — are the canonical application, and they are everywhere downstream: anti-alias filters before an ADC (Week 9), the analog twin of every digital filter (Week 10), and the conditioning front end of any sensor (Courses 1, 4).

The bench work is the most satisfying measurement of the course so far: you will measure a Bode plot. By sweeping a sine source across decades of frequency and recording the output/input amplitude ratio and phase, you reconstruct \(|H(j\omega)|\) and \(\angle H(j\omega)\) point by point and overlay them on the hand-derived asymptotes. For the RLC band-pass you will measure resonance, bandwidth, and \(Q\) — the very same \(\omega_0\) and \(Q\) that governed the ringing in Week 4, now seen as a peak in the frequency domain. Time-domain ringing and frequency-domain resonance finally click into one picture.

This builds on Week 5’s impedance and Week 4’s poles, and it directly enables Week 8 (the transfer function is the Fourier-domain description of an LTI system) and Week 9 (anti-alias filter design).

Readings

CAD Ch. 9: Transfer functions, Bode magnitude/phase plots, first-order RC low-pass and high-pass, the RLC band-pass and band-reject, resonance, quality factor and bandwidth, and an introduction to active (op-amp) filters. Extract: the asymptotic Bode construction rules and the resonance/\(Q\)/bandwidth relations.
O-S&S Ch. 6: Magnitude–phase representation of the frequency response of LTI systems, Bode plots, and the time/frequency tradeoff (rise time vs bandwidth). Extract: the systems-level meaning of \(|H|\) and \(\angle H\), and why sharp cutoff costs phase distortion.
PEI (filters chapter): Practical passive and active filter circuits, real component effects, and filter selection guidance. Extract: what changes between the textbook filter and the breadboarded one.

Key Concepts

Transfer function and frequency response

For an LTI circuit, define \(H(j\omega)=\mathbf{V}_\text{out}/\mathbf{V}_\text{in}\) as a function of frequency. It is computed exactly like a Week-5 divider but kept symbolic in \(\omega\). For a series RC with output across C (low-pass):

\[ H(j\omega)=\frac{1/j\omega C}{R+1/j\omega C}=\frac{1}{1+j\omega RC}=\frac{1}{1+j\omega/\omega_c},\quad \omega_c=\frac{1}{RC}. \]

The magnitude \(|H|\) and phase \(\angle H\) as functions of \(\omega\) are the frequency response. The cutoff (corner) frequency \(\omega_c=1/RC=1/\tau\) ties straight back to the Week-4 time constant.

Decibels and the −3 dB point

Magnitude is plotted in decibels: \(|H|_\text{dB}=20\log_{10}|H|\). The cutoff is where \(|H|=1/\sqrt2\), i.e. \(-3\) dB, equivalently where output power is half the input — and for the first-order RC this is exactly \(\omega=\omega_c\), where the phase is \(-45°\). A first-order filter rolls off at \(\pm20\) dB/decade (\(\pm6\) dB/octave) beyond the corner.

Bode plots by asymptote

A Bode plot approximates \(|H|_\text{dB}\) and \(\angle H\) with straight-line asymptotes joined at corner frequencies. Rules: each pole contributes \(-20\) dB/decade above its corner and \(-45°\) at the corner (rolling from \(0°\) to \(-90°\) over a decade either side); each zero does the reverse (\(+20\) dB/decade, \(+45°\)). Constructing the asymptotes by hand and then sketching the smooth curve is the skill — it lets you read a circuit’s behavior across all frequencies at a glance, and it’s how you’ll sanity-check the measured sweep.

Second-order filters, resonance, and Q

A series RLC band-pass (output across R) has

\[ H(j\omega)=\frac{j\omega R C}{1-\omega^2 LC + j\omega RC},\qquad \omega_0=\frac{1}{\sqrt{LC}}. \]

It peaks at the resonant frequency \(\omega_0\), where the inductive and capacitive reactances cancel and the circuit looks purely resistive. The quality factor and bandwidth:

\[ Q=\frac{\omega_0}{\Delta\omega}=\frac{1}{R}\sqrt{\frac{L}{C}},\qquad \Delta\omega=\omega_0/Q\ \text{(−3 dB bandwidth)}. \]

High \(Q\) = narrow, sharp peak (lightly damped); low \(Q\) = broad peak. This \(Q\) and \(\omega_0\) are identical to Week 4’s: \(Q=\frac1{2\zeta}\), and a high-\(Q\) frequency peak is the same physics as long time-domain ringing. The poles you found as \(-\alpha\pm j\omega_d\) in the time domain are the peaks you measure in the frequency domain.

The time–frequency tradeoff

A filter cannot have both an arbitrarily sharp frequency cutoff and a clean transient response — sharper magnitude rolloff buys more phase distortion and worse ringing/overshoot in the time domain (O-S&S Ch. 6). Rise time and bandwidth are inversely related (\(t_r \approx 0.35/f_{-3\text{dB}}\) for a first-order system). This tension governs every real filter-design choice and is why filter type (Butterworth flat magnitude, Bessel flat delay, Chebyshev sharp cutoff) is a deliberate decision.

Active filters (op-amp preview)

Passive RC/RLC filters load down and can’t provide gain. Op-amp active filters (Sallen–Key, multiple-feedback) buffer the output, set gain, and realize sharp higher-order responses without inductors — built next week once the op-amp is in hand. For now, note that the divider intuition still applies; the op-amp just removes loading and adds gain.

Theory Exercises

Derive \(H(j\omega)=1/(1+j\omega/\omega_c)\) for the RC low-pass; find \(|H|\), \(\angle H\), and confirm \(-3\) dB and \(-45°\) at \(\omega=\omega_c\).
Derive the RC high-pass transfer function and sketch its Bode asymptotes.
State the Bode asymptote rules for a single pole and a single zero (slope and phase), and construct the Bode plot for \(H(s)=\frac{\omega_c}{s+\omega_c}\) and for a two-pole low-pass.
Derive the series-RLC band-pass \(H(j\omega)\), find \(\omega_0\), \(Q=\frac1R\sqrt{L/C}\), and the −3 dB bandwidth \(\Delta\omega=\omega_0/Q\).
Show the relation \(Q=\frac1{2\zeta}\) between the frequency-domain quality factor and the Week-4 damping ratio.
Using the time–frequency tradeoff, estimate the rise time of a first-order low-pass with \(f_{-3\text{dB}}=10\) kHz, and explain why a sharper filter rings more.

Lab / Bench Work

Measure an RC Bode plot: Sweep a sine source over ~3 decades (e.g. 100 Hz–100 kHz) around \(f_c=1/(2\pi RC)\). At each frequency record input and output amplitude (ratio → dB) and the phase shift. Plot measured \(|H|_\text{dB}\) and \(\angle H\) vs log frequency and overlay the hand-derived asymptotes and the exact curve. Confirm \(f_c\), the −3 dB point, the −45° phase, and the −20 dB/decade slope.

RLC band-pass resonance: Build a series RLC band-pass. Sweep frequency to find the peak (\(\omega_0\)), measure the −3 dB bandwidth, and compute measured \(Q=\omega_0/\Delta\omega\). Compare to \(\frac1R\sqrt{L/C}\) (using LCR-measured L, C) and to Week 4’s ringing-derived \(Q\) for the same components — they should match.

Automate (optional): Use the scope’s PyVISA interface from the laptop to step the source frequency and log amplitude/phase automatically into a CSV, then plot in Python. This is the seed of the Week 9–10 acquisition tooling.

Qucs-S (ngspice): Build the filter schematic and run an .ac dec analysis over the same range, overlaying simulated and measured Bode curves.

Measurement Methodology

Settling per point: at each frequency, let the steady state establish (several periods) before reading amplitude/phase.
Amplitude reading: use peak-to-peak from the scope’s automatic measurement; keep the input amplitude constant across the sweep (verify, since some sources sag at high frequency).
Phase wrap: track phase continuously through the sweep; a first-order low-pass goes \(0°\to-90°\), don’t let the cursor method alias it.
Probe & source loading: probe capacitance adds a stray pole at high frequency — note where it would appear (\(f\sim1/2\pi R_\text{src}C_\text{probe}\)) so you don’t mistake it for the filter’s own behavior.
Reconcile measured Bode, hand asymptotes, and Qucs-S .ac. Resonant \(Q\) measured three ways (frequency bandwidth, time-domain ringing, \(\frac1R\sqrt{L/C}\)) should agree within tolerance + ESR.

Expected baselines: RC corner within ~5–10% of \(1/2\pi RC\); slope −20 dB/decade; phase −45° at corner. Measured RLC \(Q\) slightly below the ideal \(\frac1R\sqrt{L/C}\) because real ESR and winding resistance add loss — quantify the gap.

Connections

The transfer function \(H(j\omega)\) measured here is exactly the frequency response of an LTI system formalized in Week 8 — there, it is the Fourier transform of the impulse response. Resonance, \(Q\), and bandwidth unify with Week 4’s poles and ringing. The −3 dB anti-alias requirement for Week 9’s sampling front end is a direct application: you’ll design a low-pass to attenuate above Nyquist. Active filters built next week extend this to sharp, buffered, gain-bearing responses. The PyVISA sweep tooling foreshadows the automated acquisition used in the Weeks 9–10 DSP work, and the time–frequency tradeoff reappears as the window/resolution tradeoff in the DFT (Week 10).