lecture_EE1C2_lec2_frequency-response-and-transfer-function_2025-11-17

📌 Overview

This lecture recaps phasors and impedances, then introduces frequency response as how circuits behave vs. frequency (key for filters). Covers transfer functions H(ω) = Y(ω)/X(ω), poles/zeros, decibels (dB) for gain, and Bode plots for visualizing magnitude/phase on log scales. Examples include computing TF for RC circuits and sketching Bodes. Goal: Analyze steady-state sinusoidal responses and approximate TFs graphically.

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🎯Learning Objectives

• Explain frequency response and its role in filters/components.
• Define transfer functions (voltage/current gain, impedance/admittance) and compute for simple circuits.
• Identify poles (denom roots, ∞ response) and zeros (num roots, 0 response) in H(s), with s = jω for sinusoids.
• Convert gains to dB: 20 log₁₀|H| for V/I ratios.
• Construct/interpret Bode plots: mag (dB) and phase (°) vs. log ω, using standard factors (K, poles/zeros).
• Solve practice problems: TF from circuit, poles/zeros, Bode sketching.

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💡Key Concepts & Definitions

Frequency Response

Variation in circuit output (amplitude/phase) vs. input frequency ω (rad/s, ω = 2πf). For sinusoids, use phasor domain. Important for:

• Filters: Pass/block frequencies (e.g., radio channels).
• Components: e.g., amp gain, antenna impedance vs. f.

Impedances (Recap):

• $Z_R=R$ (real, no phase shift).
• $Z_L=j\omega L$ (inductive, +90° phase).
• $Z_C=\frac{1}{j\omega C}$ (capacitive, -90° phase).

Admittance $Y=1/Z$.

Transfer Function

$H(\omega )=\frac{Y(\omega )}{X(\omega )}$: Complex ratio of output Y to input X (phasors).

• Voltage gain: $H_V(\omega )=\frac{V_o(\omega )}{V_i(\omega )}$.
• Current gain: $H_I(\omega )=\frac{I_o(\omega )}{I_i(\omega )}$.
• Transfer impedance: $H_Z(\omega )=\frac{V_o(\omega )}{I_i(\omega )}$.
• Transfer admittance: $H_Y(\omega )=\frac{I_o(\omega )}{V_i(\omega )}$.

In s-domain: $H(s)=\frac{N(s)}{D(s)}$ (polynomials). For sinusoids, $s=j\omega$ (σ=0, pure tone; general s=σ+jω for exponentials $e^{st}$).

Poles & Zeros:

• Zeros: Roots of N(s)=0 → H=0 (output nulls).
• Poles: Roots of D(s)=0 → H=∞ (resonances/instabilities).
• Allowed: Stable if poles Re(s)<0. Rare: Right-half poles → unstable oscillations.

Decibels (dB)

Log scale for gain: Compresses wide ranges.

• Power: $G_{dB}=10{\log }_{10}\frac{P_2}{P_1}$.
• Voltage/Current (same R): $G_{dB}=20{\log }_{10}\frac{V_2}{V_1}$ (or I).
• For TF: $|H{|}_{dB}=20{\log }_{10}|H(\omega )|$.
• Examples: 10x gain = +20 dB; 0.1x = -20 dB. 0 dB = unity gain.

Phase: $\angle H(\omega )$ in degrees.

Bode Plots

Semilog graph: x = log₁₀ ω (decades), y = |H|_{dB} or ∠H (°).

• Approximate straight lines (asymptotic) for hand-sketching.
• Standard factors in $H(j\omega )=K\prod$ (poles/zeros):
  1. Constant K: Flat mag 20 log K dB, phase 0°.
  2. Pole at origin $1/(j\omega )$: -20 dB/dec, -90°.
  3. Zero at origin $(j\omega )$: +20 dB/dec, +90°.
  4. Simple pole $1/(1+j\omega /p)$: Flat → -20 dB/dec at ω=p (corner); phase 0° → -90° (linear -45°/dec near p).
  5. Simple zero $(1+j\omega /z)$: Flat → +20 dB/dec at ω=z; phase 0° → +90°.
  6. Quad pole $1/(1+2\zeta j\omega /{\omega }_k+(j\omega /{\omega }_k{)}^2)$: Flat → -40 dB/dec at ω=ω_k; phase 0° → -180° (steep if low ζ).
  7. Quad zero: Opposite (+40 dB/dec, +180°).
• Multiple: Multiply slopes/phases. Low ζ: Peaks in mag, sharp phase.

When approximate? Valid for |ω - corner| > decade (error <3 dB). Exact: Use calc/software. Rare: High-order → ripples.

➗ Formulas

Impedances:

$$\begin{array}{rl}{Z}_R& =R\\ Z_L& =j\omega L\\ Z_C& =\frac{1}{j\omega C}\end{array}$$

TF Example (Series RLC): $Z_{tot}=R+j\omega L+1/(j\omega C)$.

dB:

$$|H{|}_{dB}=20{\log }_{10}|H(j\omega )|$$

Standard Pole: $H(j\omega )=\frac{1}{1+j\omega /p}$, corner ω=p.

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✍️ Notes

Recap: Phasors & Impedances

Sinusoids $v(t)=V_m\cos (\omega t+\theta )$ → phasor $\mathbf{V}=V_m\angle \theta$.

• Circuit: Replace with impedances, solve complex eqs.
• Back to time: $|H|V_m\cos (\omega t+\theta +\angle H)$.
• Admittance $Y=1/Z$, parallel circuits.

Example: Series RC: $Z=R+1/(j\omega C)$. $H_V=Z_C/Z=\frac{1/(j\omega C)}{R+1/(j\omega C)}$.

Frequency Response

For fixed ω: Compute |H|, ∠H. Full: Plot vs. all ω → reveals passbands, cutoffs. Why? Filters block noise; e.g., low-pass RC: High f attenuated.

Visual: Bode mag for low-pass: Flat low ω, -20 dB/dec high ω.

graph TD
    A[Low ω: Pass] --> B[High ω: Attenuate]
    B --> C[Filter Application: Audio, RF]

Transfer Functions

Solve circuit in phasor domain: Nodal/mesh with Z(jω). Express $H(s)=\frac{a_m{s}^m+\cdots +a_0}{b_n{s}^n+\cdots +b_0}$.

For sinusoids: Eval at s=jω. Allowed: Linear/time-invariant circuits. Not: Nonlinear (diodes) or time-varying.

Practice Problem 1: Impedance TF (from Lecture) Circuit: Series (inferred: R=10Ω? But Z_C=20/s, so C=1/(20 s)? Wait, Z_C=1/(s C)=20/s → C=1/20 F? Unusual, but proceed. Assume total Z_tot = something + Z_C.

Stepwise:

1. Identify: Input I_i, output V_o across total Z. $H_Z(s)=V_o(s)/I_i(s)=Z_{tot}(s)$.
2. Components: Assume series R, L? But given Z_C = 20/s (C=0.05 F).
3. Z_tot = R + jωL + 20/(jω)? But slide: V_o = I_i * Z_tot, Z_C=20/s.
4. Full: From calc, Z_tot(s) = ? Slide implies Z_tot(s) = 20/s + other? Wait, practice: H(s) = Z_tot = ? Poles/zeros next slide.
5. Compute: Z_C = 1/(s C) = 20/s → C=1/20 F.
6. Assume simple: If only C, H=20/s. But circuit has more.
7. Zeros: Num roots=0 (no s in num). Poles: Denom s=0 (at origin, integrator).

Solution: $H(s)=Z_{tot}(s)=\frac{20}{s}$ (if parallel? Wait, series current I_i through all).

• Zeros: None.
• Poles: s=0 (DC block? No, integrator). Rare: Pole at 0 → infinite DC gain, but for AC ok.

Example Question 1: Compute TF for Voltage Divider Circuit: Series R1=1kΩ, C=1μF to ground. V_i across R1+C, V_o across C.

Stepwise Plan:

1. Phasor: Z_R = 1000, Z_C = 1/(jω * 10^{-6}).
2. H_V(jω) = V_o / V_i = Z_C / (Z_R + Z_C) = \frac{1/(jω C)}{R + 1/(jω C)} = \frac{1}{1 + jω R C}.
3. RC = 10^{-3} s, corner ω=1/RC=10^3 rad/s.
4. Standard: Simple pole at p=10^3.
5. Allowed: Steady-state sinusoid. Not: Transients (use Laplace full s).
6. Poles: s=-1/RC (stable). Zeros: None.

Math:

$$H(j\omega )=\frac{1}{1+j\omega /(1/RC)}|H|=\frac{1}{\sqrt{1+(\omega RC{)}^2}}$$

Visual: Low-Pass Filter Bode Use pgfplots for approx.

(Flat 0 dB low ω, -20 dB/dec after ω=10^3 → logω=3.)

Phase: 0° low, -90° high, -45° at corner.

Poles & Zeros

Factor H(s): Zeros where N=0 (block freq), poles D=0 (resonate). For sinusoids: Plot |H(jω)|, peaks at imag poles.

Example Question 2: Find Poles/Zeros H(s) = \frac{s+1}{(s+2)(s^2 + 3s + 2)}.

Stepwise:

1. Num: s+1=0 → zero s=-1.
2. Denom: s+2=0 → pole s=-2; s^2+3s+2=(s+1)(s+2)=0 → poles s=-1,-2 (repeated? No).
3. Zeros: s=-1. Poles: s=-2 (simple), s=-1 (cancels zero? Net simple pole s=-2).
4. Allowed: Factor fully. Rare: Cancellation → reduce order, but check if exact.
5. Bode: Zero at 1 rad/s (+20 dB/dec), poles at 1 & 2 (-20 each, net flat high?).

Rare: Complex poles → resonance peak if low damping.

Decibels

Compress: 3 dB ≈ √2 ≈1.4x voltage (double power).

Example: |H|=10 → 20 dB. Phase=45°.

Bode Plots

Hand-sketch: Identify factors, add slopes at corners.

• Start low ω: DC gain 20 log K.
• Each simple pole/zero: ±20 dB/dec change.
• Quad: ±40 dB/dec.
• Phase: ±90° total, linear transition over decade.

Practice Problem 2: Sketch Bode for H(ω)= \frac{10 jω}{(1 + jω/2)(1 + jω/10)} Stepwise (from Lecture):

1. Standard form: Already (K=10, zero origin, poles p=2,10).
2. Low ω: Slope +20 dB/dec (zero), starts at 20 log10=20 dB? Wait, at ω=1: adjust.
3. At ω=2: Pole → slope 0 dB/dec.
4. At ω=10: Pole → -20 dB/dec.
5. Phase: +90° (zero) -45° transitions at poles.
6. Allowed: Asymptotic approx. Error max 3 dB near corners. Rare: If corners close (<decade), use exact.

Visual: Asymptotic Mag

log ω	Slope Contribution	Cumulative Slope	Mag at ω=1
<0	+20 (zero)	+20	20 dB
0-0.3	+20	+20	-
0.3-1	0 (after p=2)	0	20 dB
>1	-20 (after p=10)	-20	-

For phase: +90° low, drops to 0° mid, -90° high.

Example Question 3: From Bode to TF Bode mag: +40 dB flat to ω=10, then -20 dB/dec.

Stepwise:

1. Low ω flat → no origin pole/zero. Gain: 40 dB = 20 log K → K=100.
2. At ω=10: Slope -20 → simple pole p=10.
3. High: -20 dB/dec confirms single pole.
4. TF: H(ω) = \frac{100}{1 + jω/10}.
5. Allowed: Identify changes in slope. Rare: If peak, quad pole (low ζ).

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🔗 Resources

• Presentation:

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❓ Post lecture

• Why Bode approx? Quick sketch vs. exact calc (MATLAB).
• Difference pole vs. zero? Pole amplifies, zero attenuates at that freq.
• When use s vs. jω? Sinusoids: jω; transients: full Laplace.

Gaps: Lecture assumes phasor comfort; review Week 2.1 if needed. No circuits drawn—visualize series/parallel.

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📖 Homework

• SGH assignments (self-graded).
• Practice: Sketch Bode for RC low-pass, compute cutoff.
• Seminar: Tue/Fri—apply to filters (next week).