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title: 'Lecture 28: Concept of feedback in amplifiers: Negative and Positive'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Feedback in amplifiers
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# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 28: Concept of feedback in amplifiers: Negative and Positive
### Feedback in amplifiers

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Welcome to Lecture 28. Building upon our foundational feedback equations, today we delve into the rigorous circuit implementation of feedback amplifiers. We will examine how signals are sampled at the output, mixed at the input, represented via two-port parameter matrices, and how loading effects are accounted for in real transistor circuits.
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# Lecture Agenda & Outline

- 1. Output Signal Sampling: Voltage (Node/Shunt) vs. Current (Loop/Series) Sampling Mechanics
- 2. Input Signal Summing: Series (Voltage Comparison) vs. Shunt (Current Comparison) Mixing Mechanics
- 3. Two-Port Parameter Matrices: Selection of h, g, y, and z Parameters for Feedback Networks
- 4. Feedback Loading Analysis: Derivation of Input/Output Loading Resistances (r_11 and r_22)
- 5. The Systematic 5-Step Feedback Analysis Algorithm for Discrete Transistor Circuits
- 6. Step-by-Step Circuit Calculations: Voltage-Series and Current-Series Transistor Amplifier Case Studies

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Today's agenda provides a step-by-step engineering methodology. We begin by categorizing sampling and mixing mechanisms, progress to two-port network loading derivations, present our 5-step analytical algorithm, and apply it to discrete transistor feedback circuits.
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# Output Signal Sampling: Voltage (Shunt) vs. Current (Series)

- Voltage (Shunt) Sampling Mechanism: The feedback network is connected in PARALLEL across output load terminals to sense output voltage V_o.
- Ideal Voltage Sampler Properties: To avoid loading the output, an ideal voltage sampler presents infinite input impedance looking into the feedback network (R_in,beta -> infinity).
- Effect on Output Impedance: Voltage sampling stabilizes output voltage V_o, resulting in a DRAMATIC REDUCTION of closed-loop output resistance: R_of = R_o / (1 + A * beta).
- Current (Series) Sampling Mechanism: The feedback network is connected in SERIES with output load loop to sense output current I_o.
- Ideal Current Sampler Properties: To avoid loading the output loop, an ideal current sampler presents zero insertion impedance looking into the feedback network (R_in,beta -> 0).
- Effect on Output Impedance: Current sampling stabilizes output current I_o, resulting in a DRAMATIC INCREASE of closed-loop output resistance: R_of = R_o * (1 + A * beta).

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Remember this rule of thumb for output sampling: Voltage sampling is connected in parallel (shunt) to measure voltage across nodes, reducing output resistance. Current sampling is connected in series to measure current through a loop, increasing output resistance.
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# Input Signal Summing: Series (Voltage) vs. Shunt (Current)

- Series (Voltage Comparison) Summing: The feedback signal V_f is connected in SERIES with signal source V_s at input: V_i = V_s - V_f.
- Ideal Series Summer Properties: Series mixing requires high input impedance for voltage sensing. Closed-loop input resistance INCREASES: R_inf = R_in * (1 + A * beta).
- Voltage Source Matching: Ideal for signal sources with low source resistance R_s (Thevenin equivalent voltage sources).
- Shunt (Current Comparison) Summing: The feedback signal I_f is connected in PARALLEL (shunt) with signal source I_s at input node: I_i = I_s - I_f.
- Ideal Shunt Summer Properties: Shunt mixing draws current from input node. Closed-loop input resistance DECREASES: R_inf = R_in / (1 + A * beta).
- Current Source Matching: Ideal for signal sources with high source resistance R_s (Norton equivalent current sources).

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For input mixing: Series mixing subtracts feedback voltage from source voltage, boosting input resistance by (1 + A*beta). Shunt mixing subtracts feedback current from source current, reducing input resistance by (1 + A*beta).
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# The Four Basic Feedback Topologies & Ideal Amplifier Models

- 1. Voltage-Series (Series-Shunt) Topology: Samples V_o, sums V_f in series. Ideal Voltage Amplifier (V_o / V_s). High R_inf, Low R_of.
- 2. Current-Series (Series-Series) Topology: Samples I_o, sums V_f in series. Ideal Transconductance Amplifier (I_o / V_s). High R_inf, High R_of.
- 3. Voltage-Shunt (Shunt-Shunt) Topology: Samples V_o, sums I_f in shunt. Ideal Transresistance Amplifier (V_o / I_s). Low R_inf, Low R_of.
- 4. Current-Shunt (Shunt-Series) Topology: Samples I_o, sums I_f in shunt. Ideal Current Amplifier (I_o / I_s). Low R_inf, High R_of.
- System Design Selection: Choice of topology depends entirely on source impedance R_s and load impedance R_L requirements.

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Each topology matches a specific ideal amplifier model: Voltage-Series creates an ideal voltage amplifier (high R_in, low R_out), Current-Series creates a transconductance amplifier (high R_in, high R_out), Voltage-Shunt creates transresistance, and Current-Shunt creates current amplification.
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# Two-Port Network Representations of Feedback Networks

- Purpose of Two-Port Formulation: Practical feedback networks are 2-port resistive networks. Two-port analysis decouples forward transfer from feedback loading effects.
- Voltage-Series (h-parameters): Hybrid matrix where V_1 = h_11 * I_1 + h_12 * V_2 and I_2 = h_21 * I_1 + h_22 * V_2. Feedback factor beta = h_12 = (V_1 / V_2) |_{I_1=0}.
- Current-Series (z-parameters): Impedance matrix V_1 = z_11 * I_1 + z_12 * I_2 and V_2 = z_21 * I_1 + z_22 * I_2. Feedback factor beta = z_12 = (V_1 / I_2) |_{I_1=0}.
- Voltage-Shunt (y-parameters): Admittance matrix I_1 = y_11 * V_1 + y_12 * V_2 and I_2 = y_21 * V_1 + y_22 * V_2. Feedback factor beta = y_12 = (I_1 / V_2) |_{V_1=0}.
- Current-Shunt (g-parameters): Inverse hybrid matrix I_1 = g_11 * V_1 + g_12 * I_2 and V_2 = g_21 * V_1 + g_22 * I_2. Feedback factor beta = g_12 = (I_1 / I_2) |_{V_1=0}.

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To analyze real feedback circuits, we represent the passive feedback network using 2-port parameters. Each topology uses a specific parameter matrix: h for Voltage-Series, z for Current-Series, y for Voltage-Shunt, and g for Current-Shunt.
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# Feedback Network Loading Analysis: r_11 and r_22 Resistances

- Concept of Loaded Open-Loop Circuit: Real feedback networks load both the amplifier input and output nodes. We construct a basic amplifier WITHOUT feedback (A') by incorporating loading resistances r_11 and r_22.
- Input Loading Resistance (r_11): Resistance looking into port 1 of feedback network with port 2 properly terminated:
- For Voltage Output (V_o sampled): Port 2 is SHORT-CIRCUITED (V_2 = 0). r_11 = Input resistance looking into feedback network with output shorted.
- For Current Output (I_o sampled): Port 2 is OPEN-CIRCUITED (I_2 = 0). r_11 = Input resistance looking into feedback network with output open.
- Output Loading Resistance (r_22): Resistance looking into port 2 of feedback network with port 1 properly terminated:
- For Series Input (V_f mixed): Port 1 is OPEN-CIRCUITED (I_1 = 0). r_22 = Output resistance looking into feedback network with input open.
- For Shunt Input (I_f mixed): Port 1 is SHORT-CIRCUITED (V_1 = 0). r_22 = Output resistance looking into feedback network with input shorted.

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Pay close attention to these termination rules for r_11 and r_22! To find input loading r_11: short the output if voltage sampled; open it if current sampled. To find output loading r_22: open the input if series mixed; short it if shunt mixed.
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# Systematic 5-Step Algorithm for Analyzing Feedback Amplifiers

- Step 1 (Identify Topology): Determine if output sampling is Voltage or Current, and if input mixing is Series or Shunt. Identify the 4 basic topologies.
- Step 2 (Determine Feedback Factor beta): Find beta = Signal_fed_back / Output_signal (e.g., V_f / V_o, V_f / I_o, I_f / V_o, or I_f / I_o) with input signal set to zero.
- Step 3 (Construct Loaded Open-Loop Circuit A'): Draw basic amplifier without feedback. Include source resistance R_s, load R_L, input loading r_11, and output loading r_22.
- Step 4 (Calculate Open-Loop Gain A and Loop Gain T): Compute open-loop gain A = Output / Input of the loaded circuit. Compute loop gain T = A * beta.
- Step 5 (Compute Closed-Loop Parameters): Calculate closed-loop gain A_f = A / (1 + A * beta), closed-loop input resistance R_inf, and closed-loop output resistance R_of.

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This 5-step algorithm solves ANY feedback amplifier problem deterministically! Always follow these steps in order: topology -> beta -> loaded A' circuit -> open-loop gain A -> closed-loop metrics.
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# Voltage-Series (Series-Shunt) Topology Formulation

- Signal Variables: Source voltage V_s, feedback voltage V_f, error voltage V_i = V_s - V_f, output voltage V_o.
- Feedback Factor: beta_v = (V_f / V_o) |_{I_1=0} (dimensionless ratio).
- Loading Resistances: r_11 = h_11 = (V_1 / I_1) |_{V_2=0} (Port 2 shorted); r_22 = 1 / h_22 = (V_2 / I_2) |_{I_1=0} (Port 1 open).
- Loaded Open-Loop Circuit (A_v): Basic amplifier voltage gain including loading: A_v = (V_o / V_i)|_{loaded}. Input resistance R_in = R_s + r_11 + r_pi; Output resistance R_o = R_C || r_22 || R_L.
- Closed-Loop Gain: A_vf = V_o / V_s = A_v / (1 + A_v * beta_v).
- Closed-Loop Resistances: Input resistance R_inf = R_in * (1 + A_v * beta_v); Output resistance R_of = R_o / (1 + A_v * beta_v).

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For Voltage-Series feedback, both beta_v and A_v are dimensionless voltage ratios. The closed-loop input resistance increases by (1 + A_v*beta_v), while output resistance decreases by the exact same factor.
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# Current-Series (Series-Series) Topology Formulation

- Signal Variables: Source voltage V_s, feedback voltage V_f, error voltage V_i = V_s - V_f, output current I_o.
- Feedback Factor: beta_z = (V_f / I_o) |_{I_1=0} (has units of Ohms, Ohms = V / A).
- Loading Resistances: r_11 = z_11 = (V_1 / I_1) |_{I_2=0} (Port 2 open); r_22 = z_22 = (V_2 / I_2) |_{I_1=0} (Port 1 open).
- Loaded Open-Loop Transconductance Gain (A_m): A_m = (I_o / V_i)|_{loaded} (has units of Siemens, A / V).
- Loop Gain Product: T = A_m * beta_z (dimensionless, Siemens x Ohms = 1).
- Closed-Loop Parameters: Transconductance A_mf = A_m / (1 + A_m * beta_z). Voltage gain A_vf = V_o / V_s = -A_mf * R_L.
- Closed-Loop Resistances: Input resistance R_inf = R_in * (1 + A_m * beta_z); Output resistance R_of = R_o * (1 + A_m * beta_z).

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In Current-Series feedback, open-loop gain A_m is a transconductance (A/V), and feedback factor beta_z is a feedback impedance (V/A). Their product A_m*beta_z is dimensionless loop gain T. Both input and output resistances INCREASE!
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# Voltage-Shunt & Current-Shunt Formulations

- Voltage-Shunt Topology (Shunt-Shunt):
- Feedback factor beta_g = I_f / V_o (Siemens, A/V). Open-loop transresistance A_r = V_o / I_i (Ohms, V/A). Loop gain T = A_r * beta_g.
- Closed-Loop Gain: A_rf = A_r / (1 + A_r * beta_g). Voltage gain A_vf = V_o / V_s = -A_rf / R_s.
- Resistances: BOTH input and output resistances DECREASE: R_inf = R_in / (1 + A_r * beta_g), R_of = R_o / (1 + A_r * beta_g).
- Current-Shunt Topology (Shunt-Series):
- Feedback factor beta_i = I_f / I_o (dimensionless current ratio). Open-loop current gain A_i = I_o / I_i (dimensionless).
- Closed-Loop Current Gain: A_if = A_i / (1 + A_i * beta_i).
- Resistances: Input resistance DECREASES, Output resistance INCREASES: R_inf = R_in / (1 + A_i * beta_i), R_of = R_o * (1 + A_i * beta_i).

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Voltage-Shunt decreases both input and output resistances, making it ideal for transresistance buffers. Current-Shunt decreases input resistance and increases output resistance, creating an ideal current amplifier.
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# Step-by-Step Analysis: Voltage-Series BJT Feedback Stage

- Circuit Parameters: Two-stage BJT amplifier with feedback resistor R_F = 10 kOhm from Stage 2 emitter to Stage 1 emitter. R_E1 = 100 Ohm, R_C1 = 4 kOhm, R_C2 = 2 kOhm, r_pi1 = r_pi2 = 1 kOhm, h_fe1 = h_fe2 = 100.
- Step 1 (Topology): Output voltage V_o is sampled at Stage 2 collector/emitter node; feedback voltage V_f across R_E1 is in series with Stage 1 base-emitter input => Voltage-Series Feedback.
- Step 2 (Feedback Factor beta_v): beta_v = V_f / V_o = R_E1 / (R_E1 + R_F) = 100 / (100 + 10,000) = 0.0099 approx 0.01.
- Step 3 (Loading Resistances): r_11 (output shorted) = R_E1 || R_F = 100 || 10,000 approx 99 Ohm.
- r_22 (input open) = R_E1 + R_F = 100 + 10,000 = 10.1 kOhm.
- Step 4 (Open-Loop Gain A_v): Loaded first stage gain A_v1 = -h_fe1 * R_C1 / (r_pi1 + (1+h_fe1)*r_11) = -100 * 4k / (1k + 101*99) = -36.3.
- Loaded second stage gain A_v2 = -h_fe2 * (R_C2 || r_22) / r_pi2 = -100 * (2k || 10.1k) / 1k = -165.3.
- Total Open-Loop Gain A_v = A_v1 * A_v2 = (-36.3) * (-165.3) = +6000 (75.56 dB).
- Step 5 (Closed-Loop Gain A_vf): A_vf = A_v / (1 + A_v * beta_v) = 6000 / (1 + 6000 * 0.01) = 6000 / 61 = 98.36 (39.85 dB).

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Walk through this 5-step numerical example carefully. Point out how $r_{11} = R_{E1} \parallel R_F$ is added to the emitter of stage 1, and $r_{22} = R_{E1} + R_F$ is added to stage 2 collector load. Open-loop gain $A_v = 6000$, resulting in $A_{vf} = 98.36$.
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# Step-by-Step Analysis: Current-Series Transconductance Stage

- Circuit Parameters: Single-stage BJT Common-Emitter amplifier with unbypassed emitter resistor R_E = 500 Ohm. Transistor parameters: g_m = 40 mS, r_pi = 2.5 kOhm, R_C = 5 kOhm, R_s = 1 kOhm.
- Step 1 (Topology): Output current I_c is sampled through emitter loop; feedback voltage V_f = I_e * R_E is in series with base input => Current-Series Feedback.
- Step 2 (Feedback Impedance beta_z): V_f = I_o * R_E => beta_z = V_f / I_o = R_E = 500 Ohm.
- Step 3 (Loading Resistances): Output open (I_o = 0) => r_11 = R_E = 500 Ohm; Input open (I_i = 0) => r_22 = R_E = 500 Ohm.
- Step 4 (Open-Loop Transconductance A_m): Loaded open-loop transconductance (without feedback loop): A_m = I_o / V_i = g_m = 40 mS.
- Loop Gain T: T = A_m * beta_z = 40 mS * 500 Ohm = 20.
- Step 5 (Closed-Loop Parameters): Closed-loop transconductance A_mf = A_m / (1 + T) = 40 mS / (1 + 20) = 1.905 mS.
- Closed-Loop Voltage Gain: A_vf = -A_mf * R_C = -1.905 mS * 5 kOhm = -9.525. (Note: -R_C / R_E = -5000 / 500 = -10!).
- Input Resistance Transformation: R_inf = (R_s + r_pi) * (1 + T) = (1k + 2.5k) * (1 + 20) = 3.5 kOhm * 21 = 73.5 kOhm!

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Notice how an unbypassed emitter resistor $R_E$ represents Current-Series feedback! Loop gain is $T = g_m R_E = 20$. The closed-loop voltage gain becomes $A_{vf} \approx -R_C / R_E = -10$, and input resistance jumps from 3.5 kOhm to 73.5 kOhm!
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# Comprehensive Summary & Two-Port Matrix Overview

- Feedback Topologies Summary:
- 1. Voltage-Series: beta_v = V_f / V_o, R_inf = R_in * (1 + A_v * beta_v), R_of = R_o / (1 + A_v * beta_v) [Voltage Amp].
- 2. Current-Series: beta_z = V_f / I_o, R_inf = R_in * (1 + A_m * beta_z), R_of = R_o * (1 + A_m * beta_z) [Transconductance Amp].
- 3. Voltage-Shunt: beta_g = I_f / V_o, R_inf = R_in / (1 + A_r * beta_g), R_of = R_o / (1 + A_r * beta_g) [Transresistance Amp].
- 4. Current-Shunt: beta_i = I_f / I_o, R_inf = R_in / (1 + A_i * beta_i), R_of = R_o * (1 + A_i * beta_i) [Current Amp].
- Two-Port Rule: Input series mixing INCREASES R_in; Input shunt mixing DECREASES R_in. Voltage output sampling DECREASES R_out; Current output sampling INCREASES R_out.
- Methodology: Always execute the systematic 5-step algorithm incorporating r_11 and r_22 loading resistances.

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To wrap up Lecture 28: memorize the 2-port parameter matrix associations, the termination rules for r_11 and r_22 loading resistances, and the systematic 5-step algorithm.
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