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title: 'Lecture 31: Voltage gain of feedback amplifier'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Feedback in amplifiers
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# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 31: Voltage gain of feedback amplifier
### Feedback in amplifiers

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Welcome to today's lecture on the voltage gain of feedback amplifiers. We will establish the exact quantitative mathematical framework governing closed-loop gain reduction, loop gain dynamics, and gain desensitivity.
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# Lecture Agenda & Systematic Outline

- System Architecture & Feedback Block Diagram Representation
- Derivation of Closed-Loop Voltage Gain Equation ($A_f = \frac{A}{1 + A\beta}$)
- Asymptotic Limits & Feedback Ratio ($\beta$) Behavior at High Loop Gain
- Loop Transmission, Return Ratio, and Amount of Feedback in Decibels
- Gain Desensitivity & Fractional Variation Analysis ($\frac{dA_f}{A_f} = \frac{1}{1 + A\beta} \frac{dA}{A}$)
- Voltage-Series vs. Voltage-Shunt Topologies Impact on Voltage Gain
- Two-Port $h$-Parameter Modeling of Loaded Feedback Networks
- Non-Inverting Op-Amp Voltage Gain Derivation ($A_{Vf} = 1 + \frac{R_2}{R_1}$)
- Inverting Op-Amp Transresistance to Voltage Gain Conversion ($A_{Vf} = -\frac{R_f}{R_1}$)
- High-Frequency Response & Gain-Bandwidth Constancy ($A_{0f} \cdot \omega_{Hf} = A_0 \cdot \omega_H$)
- Numerical Problem: Gain Variation & Desensitivity Calculation
- Circuit Analysis Example: Discrete BJT Voltage-Series Feedback Amplifier
- Summary & Engineering Synthesis

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Here is the structured agenda for our 50-minute lecture session. We will cover block diagrams, mathematical derivations, two-port parameter modeling, op-amp configurations, frequency response trade-offs, and solved numerical problems.
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# System Architecture of Feedback Amplifiers

- Basic Block Structure: Consists of an open-loop basic amplifier with gain $A = V_o / V_i$, a passive feedback network with transfer ratio $\beta = V_f / V_o$, and an input summing junction where $V_i = V_s - V_f$.
- Signal Definitions: Source signal $V_s$, net internal error signal $V_i = V_s - \beta V_o$, output voltage $V_o = A \cdot V_i$, and feedback signal $V_f = \beta V_o$.
- Negative Feedback Phase Condition: Requires feedback signal $V_f$ to subtract from input $V_s$, establishing a $180^\circ$ phase shift relative to open-loop output to stabilize total gain.
- Ideal Feedback Assumptions: Basic amplifier is unilateral ($A$ transmits forward only), feedback network is unilateral ($\beta$ transmits backward only), and $\beta$ network loading is fully accounted for.

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The basic block diagram represents any feedback system. Pay close attention to the input summer where the feedback signal V_f opposes the source signal V_s. This phase subtraction is the defining hallmark of negative feedback.
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# Mathematical Derivation of Closed-Loop Gain ($A_f$)

- Output Equation Setup: Output voltage expressed in terms of open-loop gain $A$ and net error input $V_i$: $V_o = A V_i = A (V_s - V_f)$.
- Feedback Substitution: Substituting feedback relationship $V_f = \beta V_o$ yields $V_o = A (V_s - \beta V_o) = A V_s - A \beta V_o$.
- Algebraic Rearrangement: Collecting output terms on the left side: $V_o + A \beta V_o = A V_s \implies V_o (1 + A\beta) = A V_s$.
- Closed-Loop Gain Formula: Dividing by $V_s$ gives closed-loop voltage gain $A_f = \frac{V_o}{V_s} = \frac{A}{1 + A\beta}$, where term $(1 + A\beta)$ is the desensitivity factor or amount of feedback.

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This standard derivation is fundamental to all analog engineering. Notice how open-loop gain A is divided by (1 + A*beta). The term (1 + A*beta) is critical as it dictates all performance trade-offs.
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# Asymptotic Limits & Feedback Ratio ($\beta$) Behavior

- High Loop Gain Limit ($A\beta \gg 1$): When open-loop gain product $A\beta$ is extremely large ($A\beta \rightarrow \infty$), $1 + A\beta \approx A\beta$.
- Asymptotic Closed-Loop Gain: $A_f = \frac{A}{1 + A\beta} \approx \frac{A}{A\beta} = \frac{1}{\beta}$. Closed-loop gain depends exclusively on feedback factor $\beta$.
- Passive Component Precision: Since $\beta$ is constructed using stable passive resistors ($R_1, R_2$), closed-loop gain becomes virtually immune to transistor non-linearities, temperature drift, and manufacturing tolerances.
- Feedback Ratio Bounds: Passive feedback networks constrain $0 < \beta \le 1$; for a pure unity voltage follower, $\beta = 1$, giving $A_f \approx 1$ V/V.

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When the loop gain A*beta is much greater than 1, closed-loop gain simplifies to 1/beta. This is why op-amps with massive open-loop gains achieve precise closed-loop gains set purely by external resistors.
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# Loop Gain ($A\beta$), Return Ratio, and Feedback Measurement

- Loop Gain Definition: $T = A\beta$ represents the total voltage transmission around the feedback loop when broken at a specified node.
- Return Ratio Determination: Disconnect feedback path, inject test voltage $V_t$ at break point with source ground $V_s=0$; returned voltage $V_r = -A\beta V_t$. Return ratio $RR = -V_r / V_t = A\beta$.
- Amount of Feedback in Decibels ($N_{dB}$): Defined quantitatively as $N_{dB} = 20 \log_{10} |1 + A\beta|$.
- Decibel Example: A desensitivity factor $(1 + A\beta) = 100$ corresponds to $20 \log_{10}(100) = 40\text{ dB}$ of feedback, reducing voltage gain by $40\text{ dB}$ while enhancing stability equally.

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Loop gain T = A*beta determines stability and performance. We measure it by opening the loop and injecting a test signal. The amount of feedback in dB quantifies gain reduction.
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# Gain Desensitivity & Fractional Variation Analysis

- Sensitivity Derivative: Differentiating closed-loop gain $A_f = \frac{A}{1 + A\beta}$ with respect to open-loop gain $A$ yields $\frac{dA_f}{dA} = \frac{(1 + A\beta) - A\beta}{(1 + A\beta)^2} = \frac{1}{(1 + A\beta)^2}$.
- Fractional Gain Sensitivity Equation: Expressing relative change: $\frac{dA_f}{A_f} = \frac{dA}{A} \cdot \frac{A}{A_f} \cdot \frac{dA_f}{dA} = \frac{dA}{A} \cdot (1 + A\beta) \cdot \frac{1}{(1 + A\beta)^2} = \frac{1}{1 + A\beta} \frac{dA}{A}$.
- Desensitivity Factor Signification: Closed-loop gain fractional variation $\frac{dA_f}{A_f}$ is reduced by factor $(1 + A\beta)$ compared to open-loop variation $\frac{dA}{A}$.
- Sensitivity Function $S_A^{A_f}$: Defined as $S_A^{A_f} = \frac{\% \Delta A_f}{\% \Delta A} = \frac{1}{1 + A\beta}$, proving that high loop gain renders the system desensitized to internal variations.

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This differential derivation shows why feedback is so powerful. If open-loop gain A shifts by 20%, but your desensitivity factor is 100, the closed-loop gain shifts by only 0.2%!
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# Voltage-Series vs. Voltage-Shunt Topology Impact on Gain

- Voltage-Series (Series-Shunt) Topology: Samples output voltage $V_o$ and mixes feedback voltage $V_f$ in series with source $V_s$. Ideal pure voltage amplifier with closed-loop voltage gain $A_{Vf} = \frac{A_V}{1 + \beta_v A_V}$.
- Voltage-Shunt (Shunt-Shunt) Topology: Samples output voltage $V_o$ and mixes feedback current $I_f$ in shunt with source current $I_s$. Basic amplifier has transresistance gain $R_m = V_o / I_i$.
- Closed-Loop Transresistance ($R_{mf}$): $R_{mf} = \frac{R_m}{1 + \beta_g R_m}$, where $\beta_g = I_f / V_o$ is feedback transconductance in Siemens.
- Terminal Voltage Gain Conversion: For voltage-shunt feedback with source resistance $R_s$, effective voltage gain $A_{Vf} = \frac{V_o}{V_s} = \frac{V_o}{I_s R_s} = \frac{R_{mf}}{R_s} = \frac{R_m / (1 + \beta_g R_m)}{R_s}$.

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Different feedback topologies modify different primary gain metrics. Voltage-series feedback regulates voltage gain directly, whereas voltage-shunt feedback regulates transresistance Rm, which converts to voltage gain through source resistance R_s.
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# Two-Port Parameter Modeling of Loaded Feedback Networks

- Loading Effect Realization: Practical feedback networks load both input and output ports of the basic amplifier, altering open-loop gain from $A$ to modified open-loop gain $A'$.
- Two-Port $h$-Parameter Extraction: For voltage-series feedback, $\beta$-network is modeled as an $h$-parameter two-port block: $h_{11} = Z_{in,\beta}|_{V_o=0}$, $h_{22}^{-1} = Z_{out,\beta}|_{I_i=0}$, $\beta = h_{12} = \frac{V_f}{V_o}\Big|_{I_i=0}$.
- Modified Basic Amplifier Construction: Include $h_{11}$ in series with input impedance $R_{in}$ and $h_{22}^{-1}$ in parallel with output load $R_L$ to form modified amplifier $A'$.
- Loaded Voltage Gain Equation: Compute $A' = \frac{V_o'}{V_i'}\Big|_{loaded}$; exact closed-loop gain is then $A_{Vf} = \frac{A'}{1 + h_{12} A'}$.

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To analyze real discrete feedback circuits accurately, we cannot ignore feedback network loading. Using two-port h-parameters, we attach h11 to the input and h22 to the output before calculating open-loop gain A'.
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# Voltage Gain Analysis of Non-Inverting Op-Amp Amplifier

- Circuit Topology: Input voltage $V_s$ applied to non-inverting terminal $(V_+ = V_s)$; feedback voltage divider $R_1, R_2$ connected between output $V_o$ and inverting terminal $(V_- = V_f)$.
- Feedback Factor Calculation: $\beta = \frac{V_f}{V_o} = \frac{R_1}{R_1 + R_2}$. Op-amp open-loop voltage gain is $A_{OL}$.
- Exact Voltage Gain Equation: $A_{Vf} = \frac{A_{OL}}{1 + A_{OL} \beta} = \frac{A_{OL}}{1 + A_{OL} \frac{R_1}{R_1 + R_2}}$.
- Ideal Op-Amp Simplification ($A_{OL} \rightarrow \infty$): $A_{Vf} \approx \frac{1}{\beta} = \frac{R_1 + R_2}{R_1} = 1 + \frac{R_2}{R_1}$. Voltage gain is completely determined by passive resistor ratios.

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The non-inverting op-amp is the quintessential voltage-series feedback amplifier. Since beta = R1/(R1+R2), when open-loop gain A_OL is very large, gain simplifies to 1 + R2/R1.
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# Voltage Gain Analysis of Inverting Feedback Amplifier

- Circuit Topology: Non-inverting terminal grounded ($V_+ = 0\text{ V}$); input $V_s$ connected via $R_1$ to virtual ground node $V_-$; feedback resistor $R_f$ connected from output $V_o$ to $V_-$.
- Voltage-Shunt Analysis: Operates as voltage-shunt feedback. Transresistance gain $R_{mf} = \frac{V_o}{I_s} = -R_f$. Source current $I_s = \frac{V_s}{R_1}$.
- Closed-Loop Voltage Gain Derivation: $A_{Vf} = \frac{V_o}{V_s} = \frac{V_o}{I_s R_1} = \frac{-R_f}{R_1}$.
- Finite Op-Amp Gain Correction: Including finite $A_{OL}$, exact voltage gain is $A_{Vf} = \frac{-R_f / R_1}{1 + \frac{1 + R_f / R_1}{A_{OL}}}$, revealing gain error when $A_{OL}$ is small.

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The inverting amplifier utilizes voltage-shunt feedback. While its transresistance gain is -R_f, converting to voltage gain gives -R_f/R_1. Notice the finite gain error term in the denominator.
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# Numerical Problem: Gain Variation & Desensitivity Calculation

- Problem Statement: An amplifier has an open-loop voltage gain $A = 10,000 \pm 20\%$. Design a voltage-series feedback network to achieve a closed-loop gain $A_{Vf} = 100\text{ V/V}$. Calculate required feedback factor $\beta$, desensitivity factor, and final percentage variation in $A_{Vf}$.
- Step 1 - Calculate Required Feedback Factor ($\beta$): From $A_{Vf} = \frac{A}{1 + A\beta} \implies 100 = \frac{10,000}{1 + 10,000\beta} \implies 1 + 10,000\beta = 100 \implies \beta = \frac{99}{10,000} = 0.0099$ ($0.99\%$).
- Step 2 - Determine Desensitivity Factor: $(1 + A\beta) = 1 + (10,000 \times 0.0099) = 100$ ($40\text{ dB}$ feedback).
- Step 3 - Calculate Closed-Loop Gain Variation: $\frac{dA_{Vf}}{A_{Vf}} = \frac{1}{1 + A\beta} \frac{dA}{A} = \frac{1}{100} \times (\pm 20\%) = \pm 0.20\%$. Closed-loop gain varies by only $\pm 0.20\%$, bounded between $99.8$ and $100.2$ V/V!

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Walk through this step-by-step calculation. Notice how a massive 20% swing in open-loop gain is suppressed down to a tiny 0.2% variation in closed-loop gain by choosing beta = 0.0099.
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# Circuit Example: Single-Stage BJT Voltage-Series Amplifier

- Circuit Parameters: Common-Emitter stage with $R_C = 4.7\text{ k}\Omega$, un-bypassed emitter resistor $R_E = 470\ \Omega$, $h_{fe} = 120$, $h_{ie} = 2.4\text{ k}\Omega$, source $R_S = 600\ \Omega$. Feedback sampled at collector and fed to emitter.
- Step 1 - Unloaded Open-Loop Voltage Gain ($A_V$): Transconductance $g_m = \frac{h_{fe}}{h_{ie}} = \frac{120}{2400\ \Omega} = 50\text{ mA/V}$. Open-loop gain $A_V = -g_m R_C = -(0.050) \times 4700 = -235\text{ V/V}$.
- Step 2 - Feedback Factor ($\beta$): Local emitter resistor provides feedback ratio $\beta = \frac{R_E}{R_C} = \frac{470}{4700} = 0.10$.
- Step 3 - Closed-Loop Gain Calculation: Desensitivity factor $(1 + |A_V| \beta) = 1 + (235 \times 0.10) = 24.5$. Closed-loop gain $A_{Vf} = \frac{-235}{24.5} = -9.59\text{ V/V}$.

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In this discrete BJT example, the un-bypassed emitter resistor RE creates local voltage-series feedback. Open-loop gain of -235 is stabilized to a closed-loop gain of -9.59 V/V.
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# Summary & Key Takeaways: Voltage Gain of Feedback Amplifiers

- Closed-Loop Voltage Gain Formula: $A_f = \frac{A}{1 + A\beta}$ quantitatively describes gain reduction by desensitivity factor $(1 + A\beta)$.
- Asymptotic Precision: For high loop gain ($A\beta \gg 1$), gain simplifies to $A_f \approx \frac{1}{\beta}$, trading raw gain magnitude for passive component precision.
- Gain Desensitivity: Relative gain variations are reduced by factor $(1 + A\beta)$ according to $\frac{dA_f}{A_f} = \frac{1}{1 + A\beta} \frac{dA}{A}$.
- Topological Modeling: Voltage-series feedback directly stabilizes voltage gain $A_{Vf}$, whereas voltage-shunt feedback stabilizes transresistance $R_{mf}$ converted via source resistance $R_s$.

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To summarize, negative feedback allows us to trade raw, unstable open-loop gain for highly stable, precise closed-loop gain governed by 1/beta. Make sure you memorize these core formulas.
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