--- theme: default title: 'Lecture 27: Concept of feedback in amplifiers: Negative and Positive' info: | ## DI02011011: Electronics Circuit and Application (ECA) Feedback in amplifiers --- # DI02011011: Electronics Circuit and Application (ECA) ## Lecture 27: Concept of feedback in amplifiers: Negative and Positive ### Feedback in amplifiers --- # Lecture Agenda & Outline - 1. Universal Feedback Architecture: Open-Loop Amplifier, Feedback Network, and Summing Junctions - 2. Closed-Loop Gain Derivation: Mathematical Formulation of A_f = A / (1 + A * beta) and Loop Gain T - 3. Negative Feedback Desensitization: Immunity to Parameter Variations, Temperature Drift, and Aging - 4. Performance Modifications: Bandwidth Extension, Gain-Bandwidth Invariance, and Harmonic Distortion Reduction - 5. Positive Feedback Mechanics: Regenerative Action, Latching, and Barkhausen Stability Criterion (|A * beta| = 1, phase = 0 deg) - 6. Quantitative Engineering Design Examples and Comparative Performance Analysis Matrix --- # General Feedback System Model & Block Diagram Topology - System Components: A feedback amplifier consists of a basic open-loop forward amplifier with transfer function A(s), a feedback network with transfer function beta(s), a sampling network, and a mixing/summing junction. - Signal Variables: Input signal x_s (voltage or current), feedback signal x_f = beta * x_o, net input signal x_i = x_s - x_f (for negative feedback) or x_i = x_s + x_f (for positive feedback). - Forward Path Amplifier (A): Idealized open-loop gain A = x_o / x_i. Internal parameters are sensitive to temperature, DC bias shifts, and manufacturing tolerances. - Feedback Network (beta): Passive resistive/reactive attenuation network beta = x_f / x_o (typically beta <= 1). Designed with precise, temperature-stable passive components (resistors/capacitors). - Mixer/Summing Node: Combines source signal x_s and feedback signal x_f to generate error/effective drive signal x_i. --- # Derivation of Closed-Loop Gain A_f & Loop Gain Dynamics - Algebraic Derivation for Negative Feedback: Output equation x_o = A * x_i. Substituting net input x_i = x_s - x_f gives x_o = A * (x_s - beta * x_o). - Rearranging Terms: x_o + A * beta * x_o = A * x_s => x_o * (1 + A * beta) = A * x_s. - Closed-Loop Gain Formula: A_f = x_o / x_s = A / (1 + A * beta). - Loop Gain (T): The product T = A * beta represents the total gain around the feedback loop (from mixer through forward path and feedback network back to mixer). - Amount of Feedback (F): Defined as F = 1 + A * beta = 1 + T. Expressed in decibels as F_dB = 20 * log10(|1 + A * beta|). - Asymptotic Gain Limit: Under heavy negative feedback (where A * beta >> 1): A_f approx A / (A * beta) = 1 / beta! Closed-loop gain becomes virtually independent of open-loop gain A. --- # Mechanisms of Negative Feedback: Desensitization of Gain - Sensitivity Definition: Fractional change in closed-loop gain A_f relative to fractional change in open-loop gain A: S_A^{A_f} = (dA_f / A_f) / (dA / A). - Derivation of Sensitivity Reduction: Differentiating A_f with respect to A gives dA_f / dA = d/dA [A / (1 + A * beta)] = 1 / (1 + A * beta)^2. - Sensitivity Equation: dA_f / A_f = (1 / (1 + A * beta)) * (dA / A). Fractional gain variation is reduced by the factor (1 + A * beta). - Thermal & Aging Immunity: Temperature-induced variations in BJT beta_0 or MOSFET g_m (which cause 50% - 100% drift in open-loop gain A) cause negligible drift (< 1%) in closed-loop gain A_f. - Nonlinear Stabilization: Linearizes overall amplifier transfer characteristics, even when transistor open-loop gain A fluctuates across large signal swings. --- # Bandwidth Extension via Negative Feedback - High-Frequency Response with Feedback: Single-pole open-loop amplifier A(s) = A_0 / (1 + s / omega_H), where A_0 is open-loop midband gain and omega_H = 2 * pi * f_H is upper cutoff frequency. - Closed-Loop Frequency Transfer Function: A_f(s) = [A_0 / (1 + s / omega_H)] / [1 + beta * A_0 / (1 + s / omega_H)] = A_0 / (1 + A_0 * beta + s / omega_H). - Dividing by (1 + A_0 * beta): A_f(s) = [A_0 / (1 + A_0 * beta)] / [1 + s / (omega_H * (1 + A_0 * beta))]. - Closed-Loop Cutoff Frequency: New upper 3-dB frequency omega_Hf = omega_H * (1 + A_0 * beta) => f_Hf = f_H * (1 + A_0 * beta). Upper cutoff frequency is extended by (1 + A_0 * beta)! - Closed-Loop Lower Cutoff Frequency: Lower 3-dB frequency is lowered: f_Lf = f_L / (1 + A_0 * beta). - Gain-Bandwidth Product Invariance: A_f0 * f_Hf = [A_0 / (1 + A_0 * beta)] * [f_H * (1 + A_0 * beta)] = A_0 * f_H = Constant (GBP). --- # Nonlinear Distortion & Noise Reduction Dynamics - Harmonic Distortion Reduction: Transistor non-linear transfer functions generate output harmonic distortion D. Applying negative feedback reduces output harmonic distortion to D_f = D / (1 + A * beta). - Linearity Enhancement: By suppressing second and third harmonic amplitudes, negative feedback transforms non-linear amplifiers into high-fidelity linear stages. - Signal-to-Noise Ratio (SNR) Analysis: Consider noise voltage V_n injected inside an amplifier stage: V_o = A_1 * A_2 * V_s + A_2 * V_n. - Closed-Loop Output with Noise: V_of = (A_1 * A_2 * V_s) / (1 + A_1 * A_2 * beta) + (A_2 * V_n) / (1 + A_1 * A_2 * beta). - Noise Limitation: Negative feedback reduces output noise generated WITHIN the feedback loop, but CANNOT improve SNR for noise present at the primary input source (V_ni)! --- # Mechanisms of Positive Feedback & Regenerative Action - Positive Feedback Transfer Function: Feedback signal x_f is added in-phase at the mixer: x_i = x_s + x_f = x_s + beta * x_o. - Closed-Loop Gain Formulation: A_f = A / (1 - A * beta). - Regenerative Gain Enhancement: For 0 < A * beta < 1, the denominator (1 - A * beta) < 1, causing closed-loop gain A_f to EXCEED open-loop gain A (gain enhancement). - Instability & Saturation Region: As loop gain A * beta approaches +1, closed-loop gain A_f -> infinity. The system becomes unstable; any microscopic noise pulse triggers exponential growth until circuit supply rails latch. - Applications of Positive Feedback: Controlled positive feedback is utilized in oscillators (sinusoidal/waveform generation), Schmitt triggers (bistable multivibrators with hysteresis), and regenerative receivers. --- # Barkhausen Criterion for Sustained Oscillations - Oscillator Definition: An electronic circuit that converts DC power into a periodic AC output signal without requiring any external AC input (x_s = 0). - Barkhausen Condition 1 (Loop Gain Magnitude): The magnitude of the loop transmission around the feedback loop must equal unity at the desired frequency of oscillation f_0: |T(j * omega_0)| = |A(j * omega_0) * beta(j * omega_0)| = 1. - Barkhausen Condition 2 (Loop Phase Shift): The total phase shift around the feedback loop must equal zero degrees or an integer multiple of 360 degrees: angle(A * beta) = 2 * n * pi (where n = 0, 1, 2...). - Practical Oscillation Build-Up: To guarantee initial startup from thermal noise, practical designs set initial loop gain magnitude slightly higher: |A * beta| approx 1.05 - 1.1. - Amplitude Stabilization: Non-linear amplitude limiting (such as FET AGC circuits, thermistors, or diode clamps) reduces effective gain to |A * beta| = 1 precisely at steady state to prevent signal clipping. --- # Frequency Response & Stability Margins (Gain & Phase Margins) - Potential Instability in Negative Feedback Amplifiers: At high frequencies, internal amplifier poles introduce phase lag. If phase lag reaches 180 degrees while |A * beta| >= 1, negative feedback turns into positive feedback! - Gain Crossover Frequency (f_c): The frequency at which loop gain magnitude equals unity (0 dB): |A(j * omega_c) * beta(j * omega_c)| = 1 (0 dB). - Phase Crossover Frequency (f_180): The frequency at which loop phase shift equals -180 degrees: angle(A(j * omega_180) * beta(j * omega_180)) = -180 deg. - Phase Margin (PM): PM = 180 deg + angle(A(j * omega_c) * beta(j * omega_c)). For stable operation without ringing, PM >= 45 deg (60 deg optimal). - Gain Margin (GM): GM = -20 * log10(|A(j * omega_180) * beta(j * omega_180)|). Indicates how much loop gain can increase before instability occurs (typically GM >= 6 dB - 10 dB). --- # Quantitative Analysis of Negative Feedback Amplifier - Problem Statement: An open-loop amplifier has midband voltage gain A_0 = 10,000 (80 dB), upper 3-dB frequency f_H = 1 kHz, and second-harmonic distortion D = 10%. Calculate negative feedback properties with beta = 0.009. - Step 1: Calculate Loop Gain & Feedback Factor: Loop gain T = A_0 * beta = 10,000 * 0.009 = 90. Amount of feedback F = 1 + A_0 * beta = 1 + 90 = 91 (39.18 dB). - Step 2: Calculate Closed-Loop Gain (A_f): A_f = A_0 / (1 + A_0 * beta) = 10,000 / 91 = 109.89 (40.82 dB). Note: 1 / beta = 1 / 0.009 = 111.11. - Step 3: Calculate Gain Sensitivity Reduction: If open-loop gain A drops by 20% (dA/A = -0.20), closed-loop gain variation is dA_f / A_f = (-0.20) / 91 = -0.0022 = -0.22%! - Step 4: Calculate Closed-Loop Bandwidth: f_Hf = f_H * (1 + A_0 * beta) = 1 kHz * 91 = 91 kHz. - Step 5: Calculate Reduced Distortion: D_f = D / (1 + A_0 * beta) = 10% / 91 = 0.11%. --- # Quantitative Oscillator Design using Barkhausen Criterion - Problem Statement: Design an RC Phase-Shift Oscillator core using a 3-stage RC ladder network in the feedback path. Determine the required open-loop amplifier gain A and frequency of oscillation f_0. - RC Ladder Feedback Transfer Function: beta(s) = (s * R * C)^3 / [(s * R * C)^3 + 6 * (s * R * C)^2 + 5 * (s * R * C) + 1]. - Step 1: Setting Imaginary Part to Zero for Phase Shift: Setting s = j * omega and equating imaginary terms to zero gives: -omega^3 * R^3 * C^3 + 5 * omega * R * C = 0. - Oscillation Frequency Derivation: omega_0^2 * R^2 * C^2 = 6 => omega_0 = 1 / (sqrt(6) * R * C) => f_0 = 1 / (2 * pi * sqrt(6) * R * C). - Step 2: Evaluating Feedback Factor at f_0: Substituting omega_0 into beta(j * omega_0) yields beta(j * omega_0) = -1 / 29. - Step 3: Applying Barkhausen Gain Condition: For sustained oscillations, |A * beta| = 1 => A * (-1 / 29) = -1 => A = +29. - Conclusion: The forward amplifier must provide an inverting gain of at least A = -29 (or non-inverting gain +29 with net 180 deg ladder phase shift) to sustain oscillations. --- # Comparative Performance Matrix: Negative vs. Positive Feedback - Voltage Gain: Negative feedback reduces gain [A_f = A / (1 + A * beta)]; Positive feedback increases gain [A_f = A / (1 - A * beta)]. - Gain Stability: Negative feedback increases stability [dA_f/A_f = (dA/A)/(1 + A * beta)]; Positive feedback decreases stability (leads to saturation/latching). - System Bandwidth: Negative feedback expands bandwidth [f_Hf = f_H * (1 + A * beta)]; Positive feedback narrows bandwidth. - Nonlinear Distortion: Negative feedback suppresses distortion [D_f = D / (1 + A * beta)]; Positive feedback increases distortion. - Terminal Noise (Internal): Negative feedback reduces internal noise; Positive feedback amplifies noise. - Primary Applications: Negative feedback is used in audio/RF linear amplifiers and operational amplifiers; Positive feedback is used in sinusoidal oscillators, relaxation oscillators, and Schmitt triggers. --- # Summary & Core Takeaways of Feedback Concepts - Closed-Loop Transfer Function: A_f = A / (1 +/- A * beta). Negative feedback uses '+', Positive feedback uses '-'. - Desensitization & Independence: Under heavy negative feedback (A * beta >> 1), closed-loop gain depends solely on passive components: A_f approx 1 / beta. - Performance Enhancements: Negative feedback increases upper cutoff frequency (f_Hf = f_H * (1 + A * beta)), maintains constant Gain-Bandwidth Product, and lowers distortion (D_f = D / (1 + A * beta)). - Barkhausen Criteria: Sustained oscillation requires |A * beta| = 1 and loop phase shift angle(A * beta) = 0 deg or 360 deg. - Stability Margin Requirements: Amplifier stability requires positive Phase Margin (PM >= 45 deg) and Gain Margin (GM >= 6 dB) to prevent unwanted oscillations.