lecture_EE1C2_lec3_resonant-circuits-and-filters_2025-11-17

📌 Overview

This lecture covers filter networks (low-pass, high-pass, band-pass, band-stop) and resonant RLC circuits (series/parallel). Key parameters: resonant frequency ${\omega }_0$, quality factor $Q$, bandwidth $B$. Discusses passive vs. active filters, with Bode plots, transfer functions, and examples. Builds on frequency response from prior lectures.

Filters shape signals by passing/rejecting frequencies; resonant circuits create peaks/dips via LC interaction.

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🎯Learning Objectives

• Understand filter types and ideal responses (pass/stop bands).
• Analyze RC low/high-pass filters: transfer functions, cutoff ${\omega }_0=1/(RC)$.
• Derive resonant frequency ${\omega }_0=1/\sqrt{LC}$, $Q$, $B$ for series/parallel RLC.
• Differentiate passive (R,L,C) vs. active (op-amps) filters: pros/cons.
• Sketch Bode plots for filters; identify high/low/band-pass/stop from $H(j\omega )$.
• Solve example problems: compute ${\omega }_0$, $Q$, $B$; verify filter type via limits.

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💡Key Concepts & Definitions

Filter Networks

Circuits that pass (passband) or reject (stopband) frequency ranges. Ideal: sharp transitions; real: gradual (-3 dB cutoff).

• Low-pass: Passes $\omega <{\omega }_0$, attenuates high $\omega$.
• High-pass: Passes $\omega >{\omega }_0$, attenuates low $\omega$.
• Band-pass: Passes narrow band around ${\omega }_0$.
• Band-stop (notch): Rejects narrow band around ${\omega }_0$.

Cutoff frequency ${\omega }_0$: Where $|H(j\omega )|=1/\sqrt{2}\approx 0.707$ (linear) or -3 dB.
Why -3 dB? Power halves ($10{\log }_{10}(1/2)=-3$ dB); voltage $\sqrt{1/2}$.

Bandwidth $B$: ${\omega }_2-{\omega }_1$ (half-power points, -3 dB). For high $Q\ge 10$, symmetric: ${\omega }_{1,2}={\omega }_0\pm B/2$.

Quality factor $Q$: Sharpness of resonance; $Q={\omega }_0/B$. High $Q$: narrow peak (low damping).

Resonant Circuits

LC combo creates resonance at ${\omega }_0=1/\sqrt{LC}$ (imag. part of Z/Y = 0).

• Series RLC: Min. impedance at ${\omega }_0$; band-pass if $V_R/V_s$.
• Parallel RLC: Max. impedance at ${\omega }_0$; band-stop if $V_{LC}/V_s$.

Passive Filters: R,L,C only. Simple, reliable, but bulky inductors, gain $\le 1$, poor low-freq.

Active Filters: Add op-amps for gain $>1$, no L, tunable. But stability issues, power needed.

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➗ Formulas

RC Filters (1st Order)

Low-pass (RC series, $V_o$ across C):

$$H(j\omega )=\frac{V_o}{V_i}=\frac{1}{1+j\omega RC},{\omega }_0=\frac{1}{RC}$$

Bode: Flat to ${\omega }_0$ (-20 dB/decade roll-off).

High-pass (CR series, $V_o$ across R):

$$H(j\omega )=\frac{V_o}{V_i}=\frac{j\omega RC}{1+j\omega RC},{\omega }_0=\frac{1}{RC}$$

Bode: 0 at DC, rises to flat (+20 dB/decade).

Series RLC (Band-pass: $V_R/V_s$)

Impedance: $Z(j\omega )=R+j(\omega L-1/(\omega C))$
Resonance: ${\omega }_0=1/\sqrt{LC}$ (Im(Z)=0, min |Z|=R).

$$H(j\omega )=\frac{V_R}{V_s}=\frac{j\omega {\tau }_R}{(j\omega {\tau }_R{)}^2+j\omega {\tau }_R+1/{\tau }_{LC}},{\tau }_R=L/R,{\tau }_{LC}=LC$$

$Q={\omega }_{0}L/R=1/({\omega }_{0}RC)$, $B=R/L={\omega }_0/Q$.
(Allowed for underdamped; if $Q<1$, no sharp peak—overdamped.)

Parallel RLC (Band-stop: $V_{LC}/V_s$)

Admittance: $Y(j\omega )=1/R+j(\omega C-1/(\omega L))$
${\omega }_0=1/\sqrt{LC}$ (Im(Y)=0, max |Z|=R).
$Q={\omega }_{0}RC=R/({\omega }_{0}L)$, $B=1/(RC)={\omega }_0/Q$.
(Rare: If R very low, approximates short; high R, open circuit.)

Active Filters (Op-amp, ideal: infinite gain, zero input current)

General: $H(j\omega )=-Z_f/Z_i$.
1st-order Low-pass: $Z_i=R_i$, $Z_f=R_f||(1/(j\omega C_f))$ → $H=-(R_f/R_i)/(1+j\omega C_f{R}_f)$, ${\omega }_c=1/(C_f{R}_f)$.
High-pass: Swap, ${\omega }_c=1/(C_i{R}_i)$.
Band-pass: Combine RC paths (no L needed).
Band-stop: Similar, but parallel paths.
(Limits: $\omega \to 0$ or $\infty$ to verify type; assumes ideal op-amp—real: finite gain, offsets.)

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✍️ Notes

Recap: Frequency Response

From Lec 2: $H(j\omega )=Y(j\omega )/X(j\omega )$; poles/zeros shape Bode. dB: $20{\log }_{10}|H|$ (voltage). Synthesis: Sketch Bode → infer $H$ → components.

Passive Filters: RC Basics

Simple 1st-order. Visualize:

Low-pass: $V_o$ across C (blocks high $\omega$).

Example: RC low-pass, R=1kΩ, C=1μF. ${\omega }_0=1/(10^3\cdot 10^{-6})=10^3$ rad/s ($f_0\approx 159$ Hz).
At $\omega =10^3$, |H|=0.707 (-3 dB).

Resonant Circuits

LC resonance: Energy oscillates between L (magnetic) and C (electric). Damping via R.

Series RLC

Min Z at ${\omega }_0$; current max. Band-pass for $V_R$.
Bode: Peak at ${\omega }_0$, width $B$ (narrow for high Q).
Rare: If R=0 (superconducting), infinite Q—sustained oscillation (not practical, unstable).

Parallel RLC

Max Z at ${\omega }_0$; voltage max across. Band-stop for $V_{LC}$.
Q inversely to series (R in parallel damps differently).

High Q ($\ge 10$): Symmetric half-power; low Q: Arithmetic mean ${\omega }_0=\sqrt{{\omega }_1{\omega }_2}$.

Active Filters

Op-amp: $V_{+}=V_{-}$, no input current. Gain from feedback.
Pros: Amplification, integrable (no L). Cons: Bandwidth limited by op-amp GBW; noise.

Example Bode for active low-pass: DC gain $-R_f/R_i$, rolls off at ${\omega }_c$.

Example Questions & Solutions

Exam(ple) 1: Series RLC Band-pass

Circuit: Series R=100Ω, L=1mH, two C=0.5μF parallel (total C=1μF). $V_o=V_R$.
Stepwise Plan:

1. Identify: Series RLC, band-pass ($V_R/V_s$). Total C=1/(1/0.5 + 1/0.5)=1μF? Wait, parallel C: C_eq=0.5+0.5=1μF.
2. ${\omega }_0=1/\sqrt{LC}=1/\sqrt{10^{-3}\cdot 10^{-6}}=1/\sqrt{10^{-9}}=10^{4.5}\approx 31623$ rad/s ($f_0\approx 5$ kHz).
3. $Q={\omega }_{0}L/R=31623\cdot 10^{-3}/100=0.316$ (low Q, broad band).
4. $B={\omega }_0/Q\approx 31623/0.316\approx 10^5$ rad/s.
   (Allowed: Standard formulas; low Q means no sharp peak—use exact ${\omega }_{1,2}=\sqrt{({\omega }_0^2+B^2/4)\pm B/2}$ for precision, but approx OK for Q>>1.)

Exam(ple) 2: Active High-pass Filter

Circuit: Op-amp, $V_i$ to $C_i=1\mu F$ series R_i=1kΩ to $V_{-}$, feedback R_f=2.5kΩ from $V_o$ to $V_{-}$, $V_{+}$ grounded. Show high-pass, find ${\omega }_c$, gain.

Stepwise Plan:

1. KCL at $V_{-}$: $(V_i-V_{-})/Z_i+(V_o-V_{-})/R_f=0$, but ideal: $V_{-}=0$ (virtual ground).
2. Current: $I_i=V_i/(R_i+1/(j\omega C_i))$, but wait—standard inverting: $Z_i=1/(j\omega C_i)+R_i$? From PDF: Likely $V_i$ — C — node — R_i to ground, but PDF messy. Assume standard: $Z_i=R_i||(1/(j\omega C_i))$? No, for high-pass: Capacitor in series to $V_{-}$, R_i feedback? PDF: High-pass is swap of low-pass.
3. From low-pass swap: $H(j\omega )=-(R_f/R_i)\cdot (j\omega C{R}_i)/(1+j\omega C{R}_i)$, wait—standard high-pass: $Z_i=1/(j\omega C_i)$, $Z_f=R_f$. $H=-R_f/(1/(j\omega C_i))=-j\omega C_i{R}_f$, but with parallel R_i on input for 1st order. Actual: Input: $V_i$ — C_i — $V_{-}$ , feedback R_f; but to ground R_i from $V_{-}$? PDF example has specifics.
4. General: $H=-Z_f/Z_i$. For high-pass: $Z_i=R_i/(1+j\omega R_i{C}_i)$? Derive limits.
5. Low $\omega \to 0$: Capacitor open, H→0 (high-pass). High $\omega \to \infty$: Capacitor short, H→ -R_f/R_i (flat gain).
6. ${\omega }_c=1/(R_i{C}_i)$. Given R=500Ω (likely R_i), ${\omega }_c=1/(500\cdot C)$ but C not given—assume from context.
7. Gain = -R_f / R_i = -2.5k / 500 = -5.
   (Rare: If op-amp saturation, distorts; assume $\omega <<$ op-amp BW.)

Exam(ple) 3: Verify & Compute

From PDF: Given H, show high-pass: Lim $\omega =0$: H=0; $\omega =\infty$: H= constant. ${\omega }_c=1/(C{R}_f)$, R_f=2.5kΩ, etc. (As above.)

For band-pass/stop: Check peak/dip at ${\omega }_0$, zeros/poles.

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🔗 Resources

• Presentation:
• Book: Derivations for parallel Q/B.
• Online: Mini-Circuits filters (real apps).

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❓ Post lecture

• Why Q>10 for approx symmetry? (Arithmetic vs geometric mean.)
• Active vs passive: When use active? (Low freq, integration.)
• Simulate: Use CircuitJS for RLC Bode.

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📖 Homework

• SGH: Compute ${\omega }_0$, Q for given circuits.
• Seminar: Analyze filter responses.
• Prep: Power concepts next (RMS, avg power).