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title: 'Lecture 30: Types of feedback'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Feedback in amplifiers
---

# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 30: Types of feedback
### Feedback in amplifiers

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Welcome to Lecture 30, the culminating lecture of Unit 4 on Feedback in Amplifiers. Today we provide deep, rigorous mathematical proofs of input and output impedance transformations, examine high-frequency pole trajectories under feedback, analyze instability risks in multi-pole systems, and master frequency compensation techniques like Miller pole splitting.
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# Lecture Agenda & Outline

- 1. Mathematical Proofs: Derivation of Input Impedance for Series (KVL) and Shunt (KCL) Mixing
- 2. Mathematical Proofs: Derivation of Output Impedance for Voltage and Current Sampling using Test Sources
- 3. High-Frequency Pole Movement: Root-Locus and Q-factor Dynamics in Multi-Pole Feedback Amplifiers
- 4. Stability Metrics: Gain Margin (GM) and Phase Margin (PM) Criteria on Nyquist and Bode Plots
- 5. Frequency Compensation: Dominant Pole Placement, Lag Compensation, and Miller Pole-Splitting Mechanics
- 6. Engineering Design Examples: Multi-Stage Feedback Calculations and Miller Compensation Design

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Here is our roadmap for today. We begin with mathematical proofs of impedance transformations, progress to frequency domain pole dynamics, establish stability margins (GM and PM), explore Miller compensation, and solve comprehensive design problems.
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# Mathematical Proof of Input Impedance Transformation

- Proof 1: Series Input Mixing (Voltage Comparison):
- Input Circuit Equations: Apply Kirchhoff's Voltage Law at input loop: V_s = V_i + V_f. Substitute V_f = beta * V_o = beta * (A * V_i).
- Substituting V_f: V_s = V_i + A * beta * V_i = V_i * (1 + A * beta).
- Input Current Equation: I_i = V_i / R_in (where R_in is open-loop input resistance).
- Closed-Loop Resistance Derivation: R_inf = V_s / I_i = [V_i * (1 + A * beta)] / (V_i / R_in) = R_in * (1 + A * beta). PROVED!
- Proof 2: Shunt Input Mixing (Current Comparison):
- Apply Kirchhoff's Current Law at input node: I_s = I_i + I_f. Substitute I_f = beta * I_o = beta * (A * I_i).
- Substituting I_f: I_s = I_i + A * beta * I_i = I_i * (1 + A * beta).
- Input Voltage Equation: V_i = I_i * R_in.
- Closed-Loop Resistance Derivation: R_inf = V_i / I_s = (I_i * R_in) / [I_i * (1 + A * beta)] = R_in / (1 + A * beta). PROVED!

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These two proofs are fundamental exam derivations. For Series mixing, KVL gives $V_s = V_i(1 + A\beta)$, so $R_{inf} = R_{in}(1 + A\beta)$. For Shunt mixing, KCL gives $I_s = I_i(1 + A\beta)$, so $R_{inf} = R_{in}/(1 + A\beta)$.
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# Mathematical Proof of Output Impedance Transformation

- Proof Method: Deactivate input signal source (set V_s = 0 or I_s = 0), apply an external test voltage V_x at output terminals, measure resulting current I_x. R_of = V_x / I_x.
- Proof 1: Voltage Output Sampling (Shunt Output):
- For set V_s = 0 and applied V_x: Feedback voltage V_f = beta * V_x. Series input gives error V_i = 0 - V_f = -beta * V_x.
- Internal Amplifier Current: Dependent source generates current g_m * V_i = -A * V_x / R_o.
- KCL at Output Node: I_x = (V_x / R_o) - (internal source current) = (V_x / R_o) - (-A * beta * V_x / R_o) = (V_x / R_o) * (1 + A * beta).
- Closed-Loop Output Resistance: R_of = V_x / I_x = R_o / (1 + A * beta). PROVED!
- Proof 2: Current Output Sampling (Series Output):
- Applying test current I_x through output loop with V_s = 0 gives V_f = beta * I_x => V_i = -beta * I_x.
- KVL around Output Loop: V_x = I_x * R_o - A * V_i = I_x * R_o - A * (-beta * I_x) = I_x * R_o * (1 + A * beta).
- Closed-Loop Output Resistance: R_of = V_x / I_x = R_o * (1 + A * beta). PROVED!

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To prove output impedance transformations, we set input sources to zero and attach an external test voltage $V_x$. Voltage sampling yields $R_{of} = R_o / (1 + A\beta)$, while current sampling yields $R_{of} = R_o(1 + A\beta)$.
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# Summary & Matrix of Impedance Transformation Rules

- Master Impedance Matrix:
- Topology 1 (Voltage-Series): Series Input -> R_inf = R_in * (1 + A_v * beta_v); Voltage Output -> R_of = R_o / (1 + A_v * beta_v).
- Topology 2 (Current-Series): Series Input -> R_inf = R_in * (1 + A_m * beta_z); Current Output -> R_of = R_o * (1 + A_m * beta_z).
- Topology 3 (Voltage-Shunt): Shunt Input -> R_inf = R_in / (1 + A_r * beta_g); Voltage Output -> R_of = R_o / (1 + A_r * beta_g).
- Topology 4 (Current-Shunt): Shunt Input -> R_inf = R_in / (1 + A_i * beta_i); Current Output -> R_of = R_o * (1 + A_i * beta_i).
- Signal Conditioning Matching:
- Voltage Source -> Requires Voltage-Series or Current-Series (High R_in).
- Current Source -> Requires Voltage-Shunt or Current-Shunt (Low R_in).
- Voltage Load -> Requires Voltage-Series or Voltage-Shunt (Low R_out).
- Current Load -> Requires Current-Series or Current-Shunt (High R_out).

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This slide synthesizes all 4 impedance transformations into a single engineering reference matrix. Use this to select the exact feedback topology needed for any sensor or driver interface.
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# Frequency Response Dynamics & Pole Movement under Feedback

- Single-Pole System Root Locus: Open-loop transfer function A(s) = A_0 / (1 + s / omega_p).
- Closed-Loop Transfer Function: A_f(s) = [A_0 / (1 + s / omega_p)] / [1 + beta * A_0 / (1 + s / omega_p)] = [A_0 / (1 + A_0 * beta)] / [1 + s / (omega_p * (1 + A_0 * beta))].
- Pole Trajectory: The real pole moves along the negative real axis from s = -omega_p to s = -omega_pf = -omega_p * (1 + A_0 * beta). System remains UNCONDITIONALLY STABLE for any amount of feedback!
- Two-Pole System Dynamics: Open-loop transfer function A(s) = A_0 / [(1 + s / omega_p1) * (1 + s / omega_p2)].
- Complex Conjugate Pole Pair: As feedback factor beta increases, the two real poles move toward each other, collide on the real axis, and break out into complex conjugate poles: s_1,2 = -sigma +/- j * omega_d.
- Peaking & Over-Shoot: Complex poles induce high-frequency gain peaking in frequency response and transient ringing in step response if quality factor Q = sqrt(1 + A_0 * beta) / (1 + omega_p1 / omega_p2) > 0.707.

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Single-pole feedback systems are always stable. However, two-pole systems develop complex conjugate poles under heavy feedback, causing peaking in frequency response and ringing in step response if Q > 0.707.
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# Multi-Pole Instability & High-Frequency Phase Shift Mechanics

- Three-Pole System Instability: Open-loop transfer function A(s) = A_0 / [(1 + s / omega_p1) * (1 + s / omega_p2) * (1 + s / omega_p3)].
- Phase Accumulation: Each pole contributes up to -90 degrees of phase shift at high frequencies. A 3-pole system accumulates up to -270 degrees of phase lag.
- Phase Crossover Frequency (omega_180): Frequency where total phase shift reaches -180 degrees: angle(A(j * omega_180) * beta) = -180 deg.
- Positive Feedback Transformation: At omega_180, e^(-j * 180 deg) = -1. The denominator (1 + A * beta) becomes (1 - |A * beta|). Negative feedback transforms into POSITIVE FEEDBACK!
- Barkhausen Runaway Condition: If loop gain magnitude |A(j * omega_180) * beta| >= 1 (0 dB), the system undergoes self-sustained spontaneous oscillation.

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Three-pole systems are inherently vulnerable to instability. If loop phase reaches -180 degrees while loop gain is still >= 1, the negative feedback sign flips to positive, resulting in spontaneous oscillation.
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# Gain Margin (GM) and Phase Margin (PM) Stability Criteria

- Bode Plot Stability Evaluation: Plot 20 * log10(|A * beta|) and phase angle(A * beta) versus logarithmic frequency.
- Gain Crossover Frequency (f_c): Defined where |A(j * 2 * pi * f_c) * beta| = 1 (0 dB).
- Phase Crossover Frequency (f_180): Defined where angle(A(j * 2 * pi * f_180) * beta) = -180 deg.
- Phase Margin Formula: PM = 180 deg + angle(A(j * 2 * pi * f_c) * beta).
- PM Target Values: PM > 0 deg for stability; PM >= 45 deg for acceptable transient response; PM = 60 deg for maximally flat Butterworth response (zero peaking).
- Gain Margin Formula: GM = -20 * log10(|A(j * 2 * pi * f_180) * beta|). Target: GM >= 6 dB - 10 dB.

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We measure stability margins at two critical crossover points: Phase Margin at the gain crossover frequency (where gain = 0 dB), and Gain Margin at the phase crossover frequency (where phase = -180 deg).
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# Frequency Compensation: Dominant Pole & Miller Pole Splitting

- Need for Frequency Compensation: High-gain multistage op-amps have multiple poles, leading to PM < 0 (unstable). Frequency compensation alters loop response to force |A * beta| < 1 before phase reaches -180 degrees.
- Dominant Pole Compensation: Adding a large capacitor C_D across an internal high-resistance node pulls the first pole down to a very low frequency f_p1' << f_p1.
- Drawback of Simple Dominant Pole: Drastically reduces open-loop bandwidth and limits amplifier Slew Rate (SR = dV/dt).
- Miller Compensation (Pole Splitting): Connecting a small compensation capacitor C_C across the high-gain inverting stage (Stage 2).
- Miller Pole-Splitting Action:
- 1. Dominant pole f_p1 is pulled LOW to f_p1' approx 1 / [2 * pi * (g_m2 * R_1 * R_2 * C_C)] via Miller input capacitance multiplication.
- 2. Non-dominant pole f_p2 is pushed HIGH to f_p2' approx g_m2 / [2 * pi * (C_1 + C_2)] via negative feedback around Stage 2!
- Result: Widely separates poles, ensuring 20 dB/decade roll-off through gain crossover frequency and achieving PM approx 60 - 90 degrees!

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Miller pole splitting is an elegant compensation technique used in virtually all op-amps (such as the 741). It uses a small capacitor $C_C$ to simultaneously push the primary pole down and the secondary pole up, ensuring stability.
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# Effect of Feedback on Noise & Internal Interference

- Noise Source Classification: Input-referred noise V_ni (from sensors/first stage) vs Internal stage noise V_n2 (from power supply ripple, second stage drivers).
- Signal-to-Noise Ratio (SNR) Equation: Output signal V_o = A_1 * A_2 * V_s; Output noise from stage 2 V_on = A_2 * V_n2.
- Closed-Loop SNR Transformation: With feedback factor beta applied around overall system:
- V_of = (A_1 * A_2 * V_s) / (1 + A_1 * A_2 * beta), and V_onf = (A_2 * V_n2) / (1 + A_1 * A_2 * beta).
- System SNR with Feedback: SNR_f = V_of / V_onf = (A_1 * A_2 * V_s) / (A_2 * V_n2) = A_1 * (V_s / V_n2).
- Noise Suppression Conclusion: Negative feedback dramatically suppresses internal noise V_n2 injected AFTER high-gain stage A_1. However, noise V_ni present at the primary input is amplified along with the signal!

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Negative feedback suppresses internal power supply hum and driver noise generated in later stages by (1 + A_1*A_2*beta). However, it cannot improve the SNR of noise corrupting the primary input signal.
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# Design Example 1: Multi-Stage Feedback Impedance Analysis

- Problem Statement: A two-stage BJT Series-Shunt (Voltage-Series) feedback amplifier has open-loop voltage gain A_v = 2,000 (66 dB), open-loop input resistance R_in = 4 k-Ohm, open-loop output resistance R_o = 8 k-Ohm. The feedback factor is beta_v = 0.025.
- Step 1: Calculate Loop Gain T: T = A_v * beta_v = 2,000 * 0.025 = 50 (33.98 dB).
- Step 2: Calculate Amount of Feedback F: F = 1 + T = 1 + 50 = 51 (34.15 dB).
- Step 3: Calculate Closed-Loop Gain A_vf: A_vf = A_v / (1 + T) = 2,000 / 51 = 39.215 (31.87 dB). (Note: Asymptotic 1 / beta_v = 1 / 0.025 = 40!).
- Step 4: Calculate Closed-Loop Input Resistance R_inf: Series input mixing boosts input resistance: R_inf = R_in * (1 + T) = 4 k-Ohm * 51 = 204 k-Ohm!
- Step 5: Calculate Closed-Loop Output Resistance R_of: Voltage output sampling lowers output resistance: R_of = R_o / (1 + T) = 8,000 Ohm / 51 = 156.86 Ohms!
- Conclusion: Feedback boosted input resistance from 4 k-Ohm to 204 k-Ohm, and dropped output resistance from 8 k-Ohm to 156.86 Ohms.

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Walk through this step-by-step design calculation. Note how loop gain $T = 50$ pulls closed-loop gain down to 39.2, while transforming input resistance to 204 k-Ohm and output resistance to 156.86 Ohms.
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# Design Example 2: Miller Compensation & Phase Margin Calculation

- Problem Statement: An uncompensated op-amp has open-loop gain A_0 = 10^5, primary pole f_p1 = 100 Hz, secondary pole f_p2 = 1 MHz, and tertiary pole f_p3 = 10 MHz. Calculate Phase Margin with beta = 0.1, and design Miller capacitor C_C for PM = 60 deg.
- Step 1: Uncompensated Phase Margin at Gain Crossover: Loop gain T(0) = 10^5 * 0.1 = 10^4 (80 dB). Uncompensated crossover frequency f_c approx 100 Hz * 10^4 = 1 MHz.
- At f_c = 1 MHz: Phase lag angle(T) = -arctan(1M / 100) - arctan(1M / 1M) - arctan(1M / 10M) = -90 deg - 45 deg - 5.7 deg = -140.7 deg.
- Uncompensated Phase Margin: PM = 180 deg - 140.7 deg = 39.3 deg (Inadequate! Target PM >= 60 deg).
- Step 2: Miller Pole-Splitting Design for PM = 60 deg: For PM = 60 deg, new gain crossover f_c' must equal non-dominant pole f_p2' = 1 MHz.
- Required New Dominant Pole f_p1': f_c' = f_p1' * T(0) => 1 MHz = f_p1' * 10^4 => f_p1' = 100 Hz / 10 = 10 Hz.
- Step 3: Calculating Miller Capacitor C_C: Given stage 1 output resistance R_1 = 1 M-Ohm and stage 2 transconductance g_m2 = 5 mS:
- f_p1' = 1 / [2 * pi * g_m2 * R_1 * R_2 * C_C] => C_C = 1 / [2 * pi * 5mS * 1M-Ohm * 10k-Ohm * 10Hz] = 3.18 pF.

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This design example illustrates Miller compensation in IC design. Adding a tiny 3.18 pF capacitor $C_C$ across Stage 2 pulls the dominant pole down to 10 Hz, establishing a stable Phase Margin of 60 degrees.
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# Multi-Variable Design Trade-offs: Gain vs Bandwidth vs Stability

- 1. Voltage Gain vs Bandwidth Trade-Off: Increasing feedback factor beta lowers closed-loop voltage gain A_f, but expands system bandwidth f_Hf = f_H * (1 + A_0 * beta) proportionally.
- 2. Gain Margin / Phase Margin vs Loop Gain Trade-Off: High loop gain T = A * beta provides superior gain desensitization and distortion reduction, but pushes multi-pole systems toward phase crossover, degrading Phase Margin.
- 3. Input/Output Impedance Matching: Series input boosts R_in (ideal for voltage sensing); Shunt input lowers R_in (ideal for current sensing). Voltage sampling lowers R_out; Current sampling boosts R_out.
- 4. Power & Area Overhead: Miller compensation requires silicon area for capacitor C_C and increases high-frequency power dissipation during large-signal slewing.
- Engineering Synthesis: Active circuit design demands balancing gain requirements, load impedance matching, noise figure, and Phase Margin >= 60 degrees.

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Engineering design is about managing trade-offs. Increasing feedback improves gain stability, lowers distortion, and expands bandwidth, but demands careful frequency compensation to maintain stability margins.
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# Unit 4 Comprehensive Summary & Master Takeaways

- Closed-Loop Fundamental Formula: A_f = A / (1 + A * beta). Under heavy feedback (A * beta >> 1), A_f approx 1 / beta.
- Universal Impedance Modification: All input and output impedances are transformed by factor (1 + Loop Gain) = (1 + A * beta).
- Series Input -> R_inf = R_in * (1 + A * beta); Shunt Input -> R_inf = R_in / (1 + A * beta).
- Voltage Output -> R_of = R_o / (1 + A * beta); Current Output -> R_of = R_o * (1 + A * beta).
- Stability Criterion: Stable amplifier operation requires Phase Margin PM >= 45 deg (60 deg optimal) and Gain Margin GM >= 6 dB.
- Miller Compensation: Uses pole splitting via capacitor C_C to pull primary pole down and push secondary pole up, guaranteeing unconditional stability.

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Congratulations on completing Unit 4 on Feedback in Amplifiers! You have mastered closed-loop transfer functions, two-port loading derivations, 4-topology impedance transformations, and Miller frequency compensation.
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