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title: 'Lecture 33: Describe the effect of negative feedback on amplifier parame...'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Feedback in amplifiers
---

# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 33: Describe the effect of negative feedback on amplifier parame...
### Feedback in amplifiers

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Welcome to Lecture 33. Today we present a systematic parameter-by-parameter analysis of how negative feedback modifies the eight essential operational characteristics of analog amplifiers.
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# Lecture Agenda: Parameter-by-Parameter Breakdown

- Parameter 1: Closed-Loop Gain Reduction & Desensitivity Factor
- Parameter 2: Input Impedance Transformations ($R_{in,f}$ for Series vs. Shunt Mixing)
- Parameter 3: Output Impedance Transformations ($R_{out,f}$ for Voltage vs. Current Sampling)
- Parameter 4: System Stability Enhancement & Phase Margin
- Parameter 5: Bandwidth Extension ($f_{H,f}$ and $f_{L,f}$ Derivations)
- Parameter 6: Frequency Response Flattening & Phase Linearization
- Parameter 7: Reduction of Non-Linear Harmonic Distortion ($D_f = \frac{D}{1 + A\beta}$)
- Parameter 8: Internal Noise & Power Supply Hum Suppression
- Solved Problem: Full Parameter Evaluation Under Feedback
- Solved Problem: Impedance Transformations in Current-Shunt Feedback
- Comparative Impact Matrix Across 4 Feedback Topologies
- Summary & Key Engineering Takeaways

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Our agenda covers each of the eight parameters in detail with rigorous mathematical derivations, followed by comprehensive numerical examples and a comparative summary table.
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# Parameter 1: Gain Reduction & Desensitivity Factor

- Closed-Loop Gain Formula: $A_f = \frac{A}{1 + A\beta}$, where $A$ is open-loop gain and $\beta$ is feedback ratio.
- Gain Reduction Quantifier: Gain is reduced by the desensitivity factor $(1 + A\beta)$, equivalent to a decibel reduction $N_{dB} = 20 \log_{10}(1 + A\beta)$.
- Gain Desensitivity Function: Sensitivity $S_A^{A_f} = \frac{\partial A_f / A_f}{\partial A / A} = \frac{1}{1 + A\beta}$. Open-loop gain variations caused by temperature, aging, or supply fluctuations are suppressed by $(1 + A\beta)$.
- Asymptotic Stabilization: For $A\beta \gg 1$, $A_f \approx \frac{1}{\beta}$, establishing total gain independence from active transistor parameters.

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Parameter 1 is gain reduction. While closed-loop gain drops by (1 + A*beta), we gain immunity against temperature and transistor parameter variations.
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# Parameter 2: Input Impedance Transformations ($R_{in,f}$)

- Series Input Connection (Voltage-Series / Current-Series): Feedback voltage $V_f$ opposes input source $V_s$. Net error voltage $V_i = V_s - V_f = V_s - \beta A V_i \implies V_i(1 + A\beta) = V_s$.
- Input Current Reduction: $I_i = \frac{V_i}{R_{in}} = \frac{V_s}{R_{in}(1 + A\beta)}$.
- Series Input Impedance Formula: $R_{in,f} = \frac{V_s}{I_i} = R_{in}(1 + A\beta)$. Input impedance is BOOSTED by factor $(1 + A\beta)$.
- Shunt Input Connection (Voltage-Shunt / Current-Shunt): Feedback current $I_f$ drains source current. Total input current $I_s = I_i + I_f = I_i(1 + A\beta)$.
- Shunt Input Impedance Formula: $R_{in,f} = \frac{V_i}{I_s} = \frac{R_{in}}{1 + A\beta}$. Input impedance is REDUCED by factor $(1 + A\beta)$.

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Remember the golden rule for input impedance: Series input connection INCREASES input impedance by (1 + A*beta). Shunt input connection DECREASES input impedance by (1 + A*beta).
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# Parameter 3: Output Impedance Transformations ($R_{out,f}$)

- Voltage (Shunt) Output Sampling (Voltage-Series / Voltage-Shunt): Output voltage $V_o$ sampled. Deactivate source ($V_s = 0$) and apply test voltage $V_x$ at output port.
- Error Voltage Induced: $V_i = -V_f = -\beta V_x$. Test current $I_x = \frac{V_x - A V_i}{R_{out}} = \frac{V_x - A(-\beta V_x)}{R_{out}} = \frac{V_x(1 + A\beta)}{R_{out}}$.
- Voltage Sampling Output Impedance: $R_{out,f} = \frac{V_x}{I_x} = \frac{R_{out}}{1 + A\beta}$. Output impedance is REDUCED by factor $(1 + A\beta)$ (behaves as ideal voltage source).
- Current (Series) Output Sampling (Current-Series / Current-Shunt): Output current $I_o$ sampled.
- Current Sampling Output Impedance: $R_{out,f} = R_{out}(1 + A\beta)$. Output impedance is BOOSTED by factor $(1 + A\beta)$ (behaves as ideal current source).

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Remember the golden rule for output impedance: Voltage (shunt) sampling REDUCES output impedance by (1 + A*beta). Current (series) sampling INCREASES output impedance by (1 + A*beta).
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# Parameter 4: System Stability Enhancement

- Operating Point Drift Stabilization: Temperature changes cause DC bias point shifts $\Delta I_C$ and gain drift $\Delta A$. Closed-loop gain drift is restricted to $\Delta A_f = \frac{\Delta A}{(1 + A\beta)^2}$.
- Nyquist Criterion Criterion: Negative feedback suppresses open-loop gain magnitude at critical high frequencies where phase shift approaches $-180^\circ$.
- Gain Margin ($GM$) Expansion: $GM = -20\log_{10}|A(j\omega_{180})\beta|$. Increasing feedback desensitivity expands gain margin, preventing runaway oscillations.
- Transient Overshoot Control: Enhances phase margin $\phi_m$, reducing ringing and settling time in closed-loop step response.

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Parameter 4 is stability. Negative feedback stabilizes both DC operating points and AC transient responses, expanding gain margin and preventing self-oscillation.
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# Parameter 5: Bandwidth Extension Derivations

- High-Frequency Response ($f_{H,f}$): Open-loop gain $A(f) = \frac{A_0}{1 + j(f / f_H)}$. Closed-loop response $A_f(f) = \frac{A(f)}{1 + \beta A(f)} = \frac{A_0 / (1 + A_0\beta)}{1 + j \frac{f}{f_H (1 + A_0\beta)}}$. Upper cutoff frequency becomes $f_{H,f} = f_H (1 + A_0\beta)$.
- Low-Frequency Response ($f_{L,f}$): Open-loop gain $A(f) = \frac{A_0}{1 - j(f_L / f)}$. Closed-loop lower cutoff frequency becomes $f_{L,f} = \frac{f_L}{1 + A_0\beta}$.
- Total Bandwidth ($BW_f$): $BW_f = f_{H,f} - f_{L,f} \approx f_H(1 + A_0\beta)$. Bandwidth expands by exact desensitivity factor $(1 + A_0\beta)$!
- Gain-Bandwidth Product Invariance: $A_f \times BW_f = \left( \frac{A_0}{1 + A_0\beta} \right) \times \left[ f_H(1 + A_0\beta) \right] = A_0 \times f_H = f_T$.

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Parameter 5 is bandwidth extension. Upper 3-dB frequency f_H increases by (1 + A0*beta) while lower cutoff f_L decreases by (1 + A0*beta), expanding overall bandwidth dramatically.
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# Parameter 6: Frequency Response Flattening & Phase Linearization

- Midband Gain Flatness: Within operating passband $f_{L,f} < f < f_{H,f}$, gain fluctuations caused by reactive coupling and stray capacitances are suppressed by factor $(1 + A\beta)$.
- Phase Shift Linearization: Open-loop phase lag $\theta(f) = -\arctan(f / f_H)$ causes non-linear phase delay. Feedback reduces net phase distortion across signal frequencies.
- Ripple Suppression: Smooths out midband ripples and peakings caused by multi-stage reactive networks.
- High-Fidelity Audio Impact: Essential for audio amplifiers to ensure uniform amplification across the full human audible spectrum ($20\text{ Hz} - 20\text{ kHz}$).

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Parameter 6 is frequency response flattening. Negative feedback creates a flat magnitude response and linear phase profile across the operational bandwidth.
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# Parameter 7: Non-Linear Harmonic Distortion Reduction ($D_f$)

- Distortion Mechanism: Large signal swings push transistors into non-linear transfer regions, generating harmonic distortion voltage $D$ at the output.
- Feedback Output Equation: Output voltage with distortion: $V_o = A V_i + D$. Substituting $V_i = V_s - \beta V_o$ gives $V_o = A(V_s - \beta V_o) + D$.
- Algebraic Derivation: $V_o(1 + A\beta) = A V_s + D \implies V_o = \frac{A V_s}{1 + A\beta} + \frac{D}{1 + A\beta}$.
- Closed-Loop Distortion Formula: $V_o = A_f V_s + D_f$, where closed-loop distortion $D_f = \frac{D}{1 + A\beta}$. Non-linear harmonic distortion is reduced by exact factor $(1 + A\beta)$!

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Parameter 7 is harmonic distortion reduction. Notice the algebraic step: output distortion D is divided by (1 + A*beta). A feedback factor of 100 drops THD from 5% down to 0.05%!
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# Parameter 8: Noise Reduction & Signal-to-Noise Ratio (SNR)

- Internal Stage Noise Suppression: Power supply hum ($50/100\text{ Hz}$ ripple) and thermal noise $N$ generated within intermediate driver or output stages are reduced: $N_f = \frac{N}{1 + A\beta}$.
- Input Stage Noise Constraint: Noise voltage $N_i$ generated at the primary input stage (first transistor) is amplified along with signal $V_s$.
- Output Signal & Noise Scaling: Output signal $V_{os} = A_f V_s$; output input-noise $V_{on} = A_f N_i$.
- SNR Invariance Rule: Signal-to-Noise Ratio at output $SNR_{out} = \frac{V_{os}}{V_{on}} = \frac{A_f V_s}{A_f N_i} = \frac{V_s}{N_i} = SNR_{in}$. Negative feedback CANNOT improve input stage SNR.

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Parameter 8 is noise. Feedback attenuates hum and noise generated in internal stages, but cannot improve SNR for noise originating at the input stage because both signal and input noise are scaled equally by A_f.
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# Numerical Problem: Full Parameter Evaluation Under Feedback

- Given Open-Loop Parameters: $A_0 = 1000$, $R_{in} = 2\text{ k}\Omega$, $R_{out} = 40\text{ k}\Omega$, $f_L = 50\text{ Hz}$, $f_H = 50\text{ kHz}$, harmonic distortion $D = 10\%$, internal noise $N = 20\text{ mV}$. Feedback ratio $\beta = 0.039$ (Voltage-Series topology).
- Desensitivity Factor Calculation: $(1 + A_0\beta) = 1 + (1000 \times 0.039) = 1 + 39 = 40$ ($32\text{ dB}$ feedback).
- Calculated Closed-Loop Parameters:
- 1. Gain: $A_f = \frac{1000}{40} = 25$ V/V.
- 2. Input Impedance: $R_{in,f} = R_{in}(1 + A_0\beta) = 2k \times 40 = 80\text{ k}\Omega$.
- 3. Output Impedance: $R_{out,f} = \frac{R_{out}}{1 + A_0\beta} = \frac{40k}{40} = 1\text{ k}\Omega$.
- 4. Upper Cutoff Frequency: $f_{H,f} = 50\text{ kHz} \times 40 = 2.0\text{ MHz}$.
- 5. Lower Cutoff Frequency: $f_{L,f} = \frac{50\text{ Hz}}{40} = 1.25\text{ Hz}$.
- 6. Distortion: $D_f = \frac{10\%}{40} = 0.25\%$. 7. Internal Noise: $N_f = \frac{20\text{ mV}}{40} = 0.5\text{ mV}$.

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Walk through this comprehensive numerical problem step-by-step. Notice how every parameter is scaled by the desensitivity factor (1 + A0*beta) = 40 according to our quantitative rules.
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# Numerical Problem: Impedance Transformation in Current-Shunt Feedback

- Given Circuit: Current-Shunt feedback amplifier with open-loop current gain $A_i = 400$, input impedance $R_{in} = 1.5\text{ k}\Omega$, output impedance $R_{out} = 10\text{ k}\Omega$, feedback factor $\beta_i = 0.09$.
- Desensitivity Factor: $(1 + A_i \beta_i) = 1 + (400 \times 0.09) = 1 + 36 = 37$.
- Input Impedance Analysis: Input connection is SHUNT $\implies R_{in,f} = \frac{R_{in}}{1 + A_i \beta_i} = \frac{1500\ \Omega}{37} = 40.54\ \Omega$.
- Output Impedance Analysis: Output connection is SERIES (current sampling) $\implies R_{out,f} = R_{out}(1 + A_i \beta_i) = 10\text{ k}\Omega \times 37 = 370\text{ k}\Omega$.
- Engineering Function: Transforms stage into an ideal current amplifier with low $R_{in}$ ($40.54\ \Omega$) and ultra-high $R_{out}$ ($370\text{ k}\Omega$).

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In current-shunt feedback, input is shunt (low R_in = 40.54 ohms) and output is series current sampling (high R_out = 370 k-ohms), forming an ideal current buffer.
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# Comparative Impact Matrix Across 4 Feedback Topologies

- Voltage-Series (Series-Shunt): Gain $A_{vf} = \frac{A_v}{1+A\beta}$; $R_{in,f} = R_{in}(1+A\beta)$ (High); $R_{out,f} = \frac{R_{out}}{1+A\ensuremath{\beta}}$ (Low). Function: Ideal Voltage Amplifier.
- Voltage-Shunt (Shunt-Shunt): Gain $R_{mf} = \frac{R_m}{1+A\beta}$; $R_{in,f} = \frac{R_{in}}{1+A\beta}$ (Low); $R_{out,f} = \frac{R_{out}}{1+A\beta}$ (Low). Function: Ideal Transresistance Amplifier.
- Current-Series (Series-Series): Gain $G_{mf} = \frac{G_m}{1+A\beta}$; $R_{in,f} = R_{in}(1+A\beta)$ (High); $R_{out,f} = R_{out}(1+A\beta)$ (High). Function: Ideal Transconductance Amplifier.
- Current-Shunt (Shunt-Series): Gain $A_{if} = \frac{A_i}{1+A\beta}$; $R_{in,f} = \frac{R_{in}}{1+A\beta}$ (Low); $R_{out,f} = R_{out}(1+A\beta)$ (High). Function: Ideal Current Amplifier.

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This matrix is your master reference for exams and practical circuit design. Match your desired input/output impedance needs to the corresponding feedback topology.
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# Summary & Key Takeaways: Parameter Effects of Negative Feedback

- Gain & Desensitivity: Open-loop gain is reduced to $A_f = \frac{A}{1 + A\beta}$, suppressing gain variations by factor $(1 + A\beta)$.
- Impedance Control: Series mixing boosts input impedance; shunt mixing reduces input impedance. Voltage sampling reduces output impedance; current sampling boosts output impedance.
- Bandwidth & Linearity: Bandwidth expands by $(1 + A\beta)$ ($BW_f = BW \cdot (1 + A\beta)$), while non-linear harmonic distortion drops to $D_f = \frac{D}{1 + A\beta}$.
- Noise Limits: Internal stage noise and power supply hum are reduced to $N_f = \frac{N}{1 + A\beta}$, but primary input stage SNR remains unchanged.

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To wrap up: negative feedback trades raw gain for superior stability, wider bandwidth, lower distortion, and full control over input and output impedances.
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