---
theme: default
title: 'Lecture 38: Overall gain of positive feedback amplifier'
info: |
  ## DI02011011: Electronics Circuit and Application (ECA)
  Oscillators
---

# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 38: Overall gain of positive feedback amplifier
### Oscillators

<!--
Welcome to Lecture 38. Today we focus on the overall gain of positive feedback amplifiers. We will derive the closed-loop gain equation $A_f = \frac{A}{1 - A\beta}$, examine the three operating regimes of loop gain $A\beta$, analyze gain sensitivity and bandwidth reduction, and study practical circuit implementations.
-->

---

# Agenda: Overall Gain of Positive Feedback Amplifier

- Derivation of Closed-Loop Gain $A_f = \frac{A}{1 - A\beta}$
- Operating Regimes based on Loop Gain Magnitude ($A\beta < 1$, $A\beta = 1$, $A\beta > 1$)
- Sensitivity Analysis of Closed-Loop Gain ($S_A^{A_f} = \frac{1}{1 - A\beta}$)
- Bandwidth Narrowing & Gain-Bandwidth Product Conservation
- Non-linear Distortion Multiplier & SNR Characteristics
- Quantitative Numerical Example: Gain Spikes & Sensitivity
- Circuit Case Study: Op-Amp Non-Inverting Positive Feedback Stage
- Historical Application: Armstrong Regenerative Receiver
- Comparative Matrix: Open-Loop vs Positive vs Negative Feedback
- Engineering Guidelines to Prevent Unintended Positive Feedback Oscillations
- Summary & Key Engineering Takeaways

<!--
Here is our agenda. We start with mathematical derivations of closed-loop gain, move through operating regimes, sensitivity analysis, bandwidth trade-offs, numerical examples, and practical design rules.
-->

---

# Derivation of Closed-Loop Gain Expression

- Summing Node Equation: Effective input voltage $V_i = V_s + V_f = V_s + \beta V_o$, where $V_s$ is source voltage, $V_f$ is feedback voltage, and $\beta$ is feedback factor.
- Amplifier Output Equation: Output voltage $V_o = A \cdot V_i = A (V_s + \beta V_o)$, where $A$ is open-loop voltage gain.
- Algebraic Expansion: $V_o - A\beta V_o = A V_s \implies V_o (1 - A\beta) = A V_s$.
- Closed-Loop Voltage Gain: $A_f = \frac{V_o}{V_s} = \frac{A}{1 - A\beta}$, where loop gain is defined as $T = A\beta$.

<!--
Let's derive the overall gain equation for a positive feedback amplifier. At the summing node, the feedback signal adds to the source signal. Substituting V_o = A*V_i yields closed-loop gain A_f = A / (1 - A*beta).
-->

---

# Operating Regimes based on Loop Gain Magnitude

- Regime 1 ($0 < A\beta < 1$): Stable Positive Feedback Amplification. Denominator $(1 - A\beta) < 1$, yielding closed-loop gain higher than open-loop gain ($A_f > A$).
- Regime 2 ($A\beta = 1$): Oscillation Threshold. Denominator $(1 - A\beta) = 0 \implies A_f \to \infty$. Circuit produces continuous output $V_o$ with zero input $V_s = 0$.
- Regime 3 ($A\beta > 1$): Super-Critical Instability. Output grows exponentially until active devices saturate or hit power supply rails.
- Regenerative Threshold: Keeping $A\beta$ close to 1 (e.g. $0.99$) produces enormous gain ($A_f = 100 A$) in a single amplifier stage.

<!--
The behavior of a positive feedback amplifier depends entirely on the loop gain A*beta. If A*beta is between 0 and 1, we get stable high-gain amplification. When A*beta equals 1, the circuit becomes an oscillator.
-->

---

# Sensitivity Analysis of Closed-Loop Gain

- Definition of Gain Sensitivity: Sensitivity of closed-loop gain $A_f$ with respect to open-loop gain $A$: $S_A^{A_f} = \frac{d A_f / A_f}{d A / A} = \frac{d A_f}{d A} \frac{A}{A_f}$.
- Mathematical Derivation: Differentiation yields $\frac{d A_f}{d A} = \frac{(1 - A\beta) - A(-\beta)}{(1 - A\beta)^2} = \frac{1}{(1 - A\beta)^2}$.
- Sensitivity Factor Formula: $S_A^{A_f} = \left[ \frac{1}{(1 - A\beta)^2} \right] \left[ \frac{A}{\frac{A}{1 - A\beta}} \right] = \frac{1}{1 - A\beta}$.
- Comparison with Negative Feedback: Negative feedback reduces gain sensitivity ($S = \frac{1}{1 + A\beta} < 1$), whereas positive feedback amplifies sensitivity ($S = \frac{1}{1 - A\beta} > 1$), making gain vulnerable to thermal and parameter variations.

<!--
A critical drawback of positive feedback is gain instability. Calculating the sensitivity factor S = 1 / (1 - A*beta) reveals that as loop gain approaches 1, even tiny changes in open-loop gain A result in massive percentage changes in overall closed-loop gain A_f.
-->

---

# Bandwidth Narrowing & Gain-Bandwidth Product Conservation

- Open-Loop Frequency Response: Single-pole amplifier response $A(s) = \frac{A_0}{1 + s/\omega_H}$, where $A_0$ is DC gain and $\omega_H$ is 3dB bandwidth.
- Closed-Loop Transfer Function Derivation: $A_f(s) = \frac{\frac{A_0}{1 + s/\omega_H}}{1 - \beta \frac{A_0}{1 + s/\omega_H}} = \frac{A_0}{1 - A_0\beta + s/\omega_H} = \frac{\frac{A_0}{1 - A_0\beta}}{1 + s / [\omega_H (1 - A_0\beta)]}$.
- Closed-Loop Bandwidth Expression: Closed-loop 3dB bandwidth is $\omega_{Hf} = \omega_H (1 - A_0\beta)$.
- Gain-Bandwidth Conservation: $A_{f0} \cdot \omega_{Hf} = \left( \frac{A_0}{1 - A_0\beta} \right) \left( \omega_H (1 - A_0\beta) \right) = A_0 \omega_H = GBW$ (Constant).

<!--
When we apply positive feedback, the closed-loop DC gain increases by 1/(1 - A0*beta), but the 3dB bandwidth narrows by the exact same factor (1 - A0*beta). The overall Gain-Bandwidth Product (GBW) remains constant.
-->

---

# Non-linear Distortion Multiplier & SNR Characteristics

- Harmonic Distortion Multiplier: Non-linear distortion generated in amplifier stage is multiplied by feedback factor: $D_f = \frac{D}{1 - A\beta}$.
- Reduced Dynamic Range: Near $A\beta \approx 1$, small non-linearities in $g_m$ create severe waveform clipping and intermodulation distortion.
- Noise Amplification: Internal amplifier thermal noise $v_n$ is amplified by closed-loop gain $A_f$, elevating total output noise power spectral density.
- Selectivity Benefit: Despite distortion trade-offs, positive feedback provides narrow bandpass filtering around resonant frequency $f_0$, improving frequency selectivity.

<!--
Positive feedback degrades linearity. Harmonic distortion and internal amplifier noise are amplified by the factor 1 / (1 - A*beta). Therefore, positive feedback amplifiers must be operated at very low signal levels to avoid non-linear clipping.
-->

---

# Quantitative Numerical Example: Gain Spikes & Sensitivity

- Problem Statement: Consider an amplifier with open-loop gain $A = 100$. Calculate overall closed-loop gain $A_f$ and gain sensitivity $S$ for feedback factors $\beta = 0.005, 0.008, 0.009, 0.0099$.
- Case 1 ($\beta = 0.005$): $A\beta = 0.5 \implies A_f = \frac{100}{1 - 0.5} = 200$, Sensitivity $S = \frac{1}{0.5} = 2$.
- Case 2 ($\beta = 0.008$): $A\beta = 0.8 \implies A_f = \frac{100}{1 - 0.8} = 500$, Sensitivity $S = \frac{1}{0.2} = 5$.
- Case 3 ($\beta = 0.0099$): $A\beta = 0.99 \implies A_f = \frac{100}{1 - 0.99} = 10,000$, Sensitivity $S = \frac{1}{0.01} = 100$. (A 1% change in $A$ causes a 100% change in $A_f$!).

<!--
Look at these numerical results. When beta increases from 0.005 to 0.0099, closed-loop gain jumps from 200 to 10,000. However, sensitivity jumps from 2 to 100, meaning a tiny 1% drift in transistor open-loop gain will cause a massive 100% change in closed-loop gain.
-->

---

# Circuit Case Study: Op-Amp Positive Feedback Stage

- Circuit Setup: Op-amp connected with feedback resistor $R_2$ between output and non-inverting terminal (+), and resistor $R_1$ from (+) terminal to ground.
- Feedback Factor Derivation: Voltage division at (+) terminal gives $\beta = \frac{R_1}{R_1 + R_2}$.
- Closed-Loop Gain Formulation: $A_f = \frac{A_v}{1 - A_v \frac{R_1}{R_1 + R_2}}$, where $A_v$ is op-amp open-loop gain.
- Hysteresis Thresholds in Schmitt Trigger: When $A_v \to \infty$, loop gain $A_v \beta \gg 1$ creates bi-stable switching with upper threshold $V_{UT} = +V_{sat} \frac{R_1}{R_1 + R_2}$ and lower threshold $V_{LT} = -V_{sat} \frac{R_1}{R_1 + R_2}$.

<!--
When we apply positive feedback around an op-amp using resistors R1 and R2 at the non-inverting terminal, open-loop gain A_v is extremely high ($10^5$), driving $A_v \beta \gg 1$. This transforms the amplifier into a Schmitt trigger with sharp hysteretic switching thresholds.
-->

---

# Historical Application: Armstrong Regenerative Receiver

- Historical Significance: Invented by Edwin Howard Armstrong in 1912; revolutionized radio communications.
- Operating Principle: Controlled positive feedback (regeneration) applied via a tickler coil back to grid circuit, maintaining loop gain $A\beta \approx 0.99$.
- Performance Breakthrough: Single vacuum tube achieved $1000\times$ voltage amplification and razor-sharp selectivity, receiving distant RF signals.
- Oscillation Threshold Risk: If operator adjusted tickler coil past $A\beta = 1$, receiver burst into self-oscillation, radiating interference to nearby radios.

<!--
The most famous historical application of positive feedback gain is Armstrong's regenerative receiver. By tuning positive feedback via a tickler coil just below oscillation ($A\beta = 0.99$), early radio operators achieved incredible sensitivity and selectivity from a single vacuum tube.
-->

---

# Comparative Matrix: Open-Loop vs Positive vs Negative Feedback

- Gain Parameter: Open-Loop = $A$; Negative Feedback = $A_f = \frac{A}{1 + A\beta} < A$; Positive Feedback = $A_f = \frac{A}{1 - A\beta} > A$.
- Bandwidth Parameter: Open-Loop = $BW$; Negative Feedback = $BW(1 + A\beta)$ (Wider); Positive Feedback = $BW(1 - A\beta)$ (Narrower).
- Gain Stability / Sensitivity: Open-Loop = High; Negative Feedback = Extremely Low ($S = \frac{1}{1+A\beta}$); Positive Feedback = Extremely High ($S = \frac{1}{1-A\beta}$).
- Primary Applications: Open-Loop = Comparators; Negative Feedback = Linear Amplifiers; Positive Feedback = Oscillators, Schmitt Triggers, Regenerative Receivers.

<!--
This comparative table summarizes the trade-offs. Negative feedback sacrifices gain to achieve high bandwidth and gain stability. Positive feedback boosts gain and narrows bandwidth, but increases sensitivity and risks instability.
-->

---

# Engineering Guidelines to Prevent Unintended Oscillations

- Parasitic Feedback Isolation: Minimize parasitic capacitance $C_{gd}$ or $C_{\mu}$ between amplifier output and input traces on PCB.
- Power Supply Decoupling: Install ceramic ($0.1\ \mu\text{F}$) and electrolytic ($10\ \mu\text{F}$) decoupling capacitors directly at amplifier supply pins to prevent supply line feedback.
- Ground Plane Design: Utilize continuous low-impedance ground plane to prevent common ground impedance coupling.
- Shielding & Component Placement: Separate input and output stage components physically to prevent radiative electromagnetic positive feedback.

<!--
In linear amplifier design, unintended positive feedback causes unwanted oscillations. We prevent this by proper PCB layout, short signal traces, power supply decoupling capacitors, and low-impedance ground planes.
-->

---

# Summary: Key Takeaways on Overall Gain

- Closed-Loop Gain Formulation: Positive feedback increases overall gain to $A_f = \frac{A}{1 - A\beta}$.
- Operating Regimes: $0 < A\beta < 1$ yields stable gain enhancement; $A\beta = 1$ triggers self-sustained oscillation ($A_f \to \infty$).
- Gain-Bandwidth Trade-off: Gain increases by $\frac{1}{1 - A\beta}$ while 3dB bandwidth narrows by $(1 - A\beta)$, keeping $GBW$ constant.
- Design Caution: High sensitivity $S = \frac{1}{1 - A\beta}$ and elevated distortion require strict control of feedback factor $\beta$ and parasitic coupling.

<!--
In conclusion, understanding the overall gain of positive feedback amplifiers gives us the insight needed to exploit gain enhancement in regenerative circuits while avoiding unintended parasitic oscillations.
-->