Unit 4 - Notes

ECE182 9 min read

Unit 4: Feedback amplifiers and power amplifier

1. Introduction to Feedback

In electronics, feedback is the process whereby a portion of the output signal (voltage or current) of an amplifier is returned (fed back) to the input. This feedback signal is then combined with the external input signal to produce the actual signal applied to the amplifier. Feedback plays a critical role in stabilizing amplifier characteristics, adjusting impedances, and reducing distortion, which is essential in robotic sensor processing and actuator control.

Basic Block Diagram

A standard feedback amplifier consists of:

Basic Amplifier: Provides the open-loop gain ( $A$ ).
Feedback Network: A passive network (usually resistors) that determines the feedback factor ( $\beta$ ), the fraction of the output fed back to the input.
Sampling Network: Connects to the output to sample either voltage (parallel connection) or current (series connection).
Mixer Network: Connects to the input to mix the feedback signal with the source signal either in series (voltage) or in shunt (current).

2. Positive and Negative Feedback

Positive (Regenerative) Feedback

Definition: The feedback signal is in phase with the input signal. The feedback signal adds to the input, increasing the effective input signal.
Equation: $A_f = \frac{A}{1 - A\beta}$ (where $A_f$ is the closed-loop gain).
Characteristics: Increases gain but significantly reduces stability. Increases distortion and noise.
Applications: Primarily used in oscillators (e.g., clock generators for robotic microcontrollers). If $A\beta = 1$ , the gain becomes infinite, and the circuit generates its own signal without external input (Barkhausen criterion).

Negative (Degenerative) Feedback

Definition: The feedback signal is out of phase (usually 180° shifted) with the input signal. It subtracts from the input, decreasing the effective input signal.
Equation: $A_f = \frac{A}{1 + A\beta}$
Characteristics: Decreases overall gain, but significantly improves amplifier stability, frequency response, and linearity.
Applications: Widely used in almost all practical amplifier circuits, operational amplifiers, and robotic control systems to ensure predictable performance.

3. Effect of Negative Feedback on Amplifier Characteristics

Negative feedback fundamentally alters the performance of an amplifier. The term $(1 + A\beta)$ is known as the desensitivity factor.

Effect on Gain

Negative feedback reduces the overall voltage gain.

Formula: $A_f = \frac{A}{1 + A\beta}$
Gain Stability: The primary advantage is gain desensitivity. If the open-loop gain $A$ is very large ( $A\beta \gg 1$ ), then $A_f \approx \frac{1}{\beta}$ . The closed-loop gain becomes solely dependent on the passive feedback network ( $\beta$ ), which is highly stable against temperature and aging, unlike active BJT parameters.

Effect on Bandwidth

Negative feedback extends the bandwidth of the amplifier.

The product of gain and bandwidth (Gain-Bandwidth Product, GBP) remains constant for a given amplifier.
Since gain is reduced by the factor $(1 + A\beta)$ , both the lower cut-off frequency ( $f_L$ ) and upper cut-off frequency ( $f_H$ ) are improved.
$f_{Hf} = f_H \times (1 + A\beta)$ (Upper cut-off increases)
$f_{Lf} = \frac{f_L}{1 + A\beta}$ (Lower cut-off decreases)
New Bandwidth: $BW_f = f_{Hf} - f_{Lf} \approx BW \times (1 + A\beta)$

Effect on Noise and Nonlinear Distortion

Noise: If noise is introduced inside the amplifier stages (not at the input), negative feedback reduces the noise at the output by the factor $(1 + A\beta)$ .
Distortion: Nonlinear distortion caused by the non-linear transfer characteristics of BJTs (e.g., clipping, harmonic distortion) is similarly reduced: $D_f = \frac{D}{1 + A\beta}$ .

Effect on Input and Output Impedances

The effect on impedance depends entirely on the topology of the feedback network (how the signal is sampled and mixed).

Voltage-Series Feedback: (Samples voltage, mixes in series)
- Input Impedance: Increases ( $Z_{inf} = Z_{in}(1 + A\beta)$ )
- Output Impedance: Decreases ( $Z_{outf} = \frac{Z_{out}}{1 + A\beta}$ )
- Ideal for Voltage Amplifiers.
Voltage-Shunt Feedback: (Samples voltage, mixes in parallel)
- Input Impedance: Decreases
- Output Impedance: Decreases
- Ideal for Transresistance Amplifiers.
Current-Series Feedback: (Samples current, mixes in series)
- Input Impedance: Increases
- Output Impedance: Increases
- Ideal for Transconductance Amplifiers.
Current-Shunt Feedback: (Samples current, mixes in parallel)
- Input Impedance: Decreases
- Output Impedance: Increases
- Ideal for Current Amplifiers.

4. RC-Coupled BJT Amplifiers

RC (Resistor-Capacitor) coupling is the most common method for connecting multi-stage BJT amplifiers. It is widely used for audio and small-signal amplification in robotic sensors.

Circuit Components

Coupling Capacitors ( $C_{in}$ , $C_{out}$ , $C_c$ ): Block DC signals to prevent the biasing of one stage from affecting the next, while allowing AC signals to pass.
Bypass Capacitor ( $C_E$ ): Connected across the emitter resistor ( $R_E$ ). It provides a low-reactance path for AC signals, preventing AC negative feedback and preserving high voltage gain.
Biasing Resistors ( $R_1, R_2, R_C, R_E$ ): Typically configured as a voltage divider to set a stable DC operating point (Q-point) in the active region.

Frequency Response Curve

The gain of an RC-coupled amplifier varies with frequency, typically plotted on a semi-log logarithmic graph (Bode plot).

Low-Frequency Range: Gain decreases because coupling and bypass capacitors exhibit high reactance at low frequencies, causing voltage drops.
Mid-Frequency Range: Gain remains constant. Capacitors act as short circuits, and internal transistor capacitances are negligible.
High-Frequency Range: Gain decreases due to the internal parasitic capacitances of the BJT (junction capacitances $C_{be}$ and $C_{bc}$ ) and stray wiring capacitance, which act as shunts to ground.

SPICE

* Basic SPICE netlist structure for an RC-Coupled BJT stage
Vcc 1 0 12V
Vin 2 0 AC 10mV
Cin 2 3 10uF
R1 1 3 47k
R2 3 0 10k
Rc 1 4 4.7k
Re 5 0 1k
Ce 5 0 47uF
Q1 4 3 5 2N2222  * Collector Base Emitter BJT
Cout 4 6 10uF
Rload 6 0 10k
.AC DEC 10 10Hz 1MHz

5. Cascaded Systems

In robotics, a single amplifier stage rarely provides sufficient gain (e.g., amplifying microvolts from a thermocouple or strain gauge to standard logic levels). Stages are connected in series (cascaded).

Characteristics of Cascaded Amplifiers

Overall Gain: The total voltage gain is the product of the individual stage gains.
- $A_{total} = A_1 \times A_2 \times A_3 \times ... \times A_n$
- In decibels (dB): $A_{total}(dB) = A_1(dB) + A_2(dB) + ... + A_n(dB)$
Loading Effect: When stage 2 is connected to stage 1, the input impedance of stage 2 acts as the load for stage 1. This reduces the effective gain of stage 1 compared to its open-circuit gain. Calculations must account for this loaded gain.
Overall Bandwidth: Cascading amplifiers reduces the overall bandwidth. If identical stages are cascaded, the overall lower and upper cutoff frequencies become:
- $f_{H(cascaded)} = f_H \times \sqrt{2^{1/n} - 1}$
- $f_{L(cascaded)} = \frac{f_L}{\sqrt{2^{1/n} - 1}}$

6. Power Amplifiers

While voltage amplifiers are designed to increase the amplitude of a small-signal without drawing much current, Power Amplifiers (Large-Signal Amplifiers) are designed to deliver significant current and power to a low-impedance load. In robotics, power amplifiers are essential for driving actuators, DC motors, servo motors, and speakers.

Key Differences from Voltage Amplifiers

Load Impedance: Operates with low load impedances (e.g., $4\Omega$ to $8\Omega$ speakers, fractional ohm motors).
Heat Dissipation: Handles large currents; requires heat sinks and power transistors (e.g., Darlington pairs, MOSFETs).
Efficiency: A primary metric. Power efficiency ( $\eta$ ) is the ratio of AC output power delivered to the load to the DC power drawn from the supply: $\eta = \frac{P_{AC(out)}}{P_{DC(in)}} \times 100\%$

7. Classes of Power Amplifiers

Power amplifiers are classified based on the portion of the input cycle during which the transistor conducts (conduction angle).

Class A Power Amplifier

Conduction Angle: $360^{\circ}$ . The transistor is biased exactly in the center of the active region (Q-point). It conducts during the entire input cycle.
Distortion: Lowest distortion among all classes (high fidelity).
Efficiency: Very poor. The transistor continually draws DC power even when there is no input signal.
- Series-fed Class A: Maximum theoretical efficiency is 25%.
- Transformer-coupled Class A: Maximum theoretical efficiency is 50%.
Applications: Sensitive audio pre-amps, radio receivers; rarely used for robotic motor drivers due to massive heat generation.

Class B Power Amplifier

Conduction Angle: $180^{\circ}$ . The transistor is biased strictly at cutoff. It only conducts during one-half of the input AC cycle (either positive or negative).
Efficiency: Greatly improved because it draws zero quiescent current when there is no input. Maximum theoretical efficiency is 78.5% ( $\frac{\pi}{4}$ ).
Push-Pull Configuration: To amplify the full cycle, two complementary transistors (NPN and PNP) are used. One handles the positive half, the other the negative half.
Distortion: Suffers from Cross-over Distortion. Because BJTs require about 0.7V ( $V_{BE}$ ) to turn on, there is a dead zone between -0.7V and +0.7V where neither transistor conducts, flattening the output waveform at the zero-crossing.

Class AB Power Amplifier

Conduction Angle: Between $180^{\circ}$ and $360^{\circ}$ (typically just over $180^{\circ}$ ).
Biasing: Solves the cross-over distortion of Class B. The transistors are given a slight forward bias (usually via diodes in the base circuit) so they sit just on the threshold of conduction.
Efficiency: Compromise between Class A and Class B, typically around 60% to 70%.
Applications: The most common linear power amplifier topology. Used in robotic audio outputs, linear servo drivers, and operational amplifier output stages.

(Note: In modern robotics, Class D amplifiers (PWM-based switching amplifiers) are mostly used for motor driving due to >90% efficiencies, though A, B, and AB remain foundational analog topologies).

8. Voltage Regulators

A robotic system relies on various internal voltage levels (e.g., 5V for logic, 3.3V for sensors, 12V+ for motors). Battery voltage fluctuates as it discharges. A Voltage Regulator is a circuit that maintains a constant DC output voltage despite variations in the input voltage (line) or load current (load).

Key Performance Metrics

Line Regulation: The ability to maintain a constant output voltage despite changes in the input (supply) voltage.
- $\text{Line Regulation} = \frac{\Delta V_{out}}{\Delta V_{in}}$
Load Regulation: The ability to maintain a constant output voltage despite changes in the load current (e.g., when a robot motor stalls and draws heavy current).
- $\text{Load Regulation} = \frac{V_{NL} - V_{FL}}{V_{FL}} \times 100\%$ (where $V_{NL}$ is no-load voltage and $V_{FL}$ is full-load voltage).

Types of Voltage Regulators

Zener Diode Regulator:
- The simplest form of a shunt regulator.
- Uses a Zener diode reverse-biased in its breakdown region, where it maintains a stable voltage across its terminals.
- Used only for very low-power, low-current reference voltages.
Linear Integrated Circuit Regulators:
- Use an active pass transistor operated in its linear region to drop excess voltage.
- 78XX Series: Standard positive voltage regulators (e.g., 7805 gives +5V, 7812 gives +12V).
- 79XX Series: Standard negative voltage regulators.
- Characteristics: Clean, low-noise output. Low efficiency because excess power is dissipated as heat ( $P_{loss} = (V_{in} - V_{out}) \times I_{load}$ ). Requires heat sinks for high current.
Switching Regulators (DC-DC Converters):
- Transistors operate as high-speed switches (cutoff and saturation) using Pulse Width Modulation (PWM), combined with inductors and capacitors to smooth the output.
- Buck Converters: Step-down voltage.
- Boost Converters: Step-up voltage.
- Characteristics: Extremely high efficiency (often >90%), crucial for battery-powered robots. Can generate noise/ripple due to high-frequency switching.

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