Unit 1 - Notes

ECE221 8 min read

Unit 1: Feedback Amplifiers and Frequency Response of Amplifier

1. Classification of Amplifiers

Before introducing feedback, it is essential to classify amplifiers based on their input and output signal quantities (voltage or current). Amplifiers are broadly classified into four types:

Voltage Amplifier: Takes a voltage input and produces a voltage output. Ideally, it has an infinite input resistance ( $R_i = \infty$ ) and zero output resistance ( $R_o = 0$ ). Represents a dependent voltage source controlled by an input voltage ( $A_v = V_o / V_i$ ).
Current Amplifier: Takes a current input and produces a current output. Ideally, it has zero input resistance ( $R_i = 0$ ) and infinite output resistance ( $R_o = \infty$ ). Represents a dependent current source controlled by an input current ( $A_i = I_o / I_i$ ).
Transconductance Amplifier: Takes a voltage input and produces a current output. Ideally, it has infinite input resistance ( $R_i = \infty$ ) and infinite output resistance ( $R_o = \infty$ ). The gain is measured in Siemens or mhos ( $G_m = I_o / V_i$ ).
Transresistance Amplifier: Takes a current input and produces a voltage output. Ideally, it has zero input resistance ( $R_i = 0$ ) and zero output resistance ( $R_o = 0$ ). The gain is measured in Ohms ( $R_m = V_o / I_i$ ).

2. The Feedback Concept

Feedback is the process of taking a portion of the output signal (voltage or current) and returning it to the input to modify the amplifier's behavior.

A basic feedback amplifier consists of three main blocks:

Basic Amplifier (Gain $A$ ): Amplifies the input signal.
Feedback Network (Feedback factor $\beta$ ): Samples the output and produces a feedback signal ( $X_f = \beta X_o$ ).
Mixer / Comparator: Combines the external input signal ( $X_s$ ) with the feedback signal ( $X_f$ ) to produce the actual input to the basic amplifier ( $X_i$ ).

Types of Feedback:

Positive (Regenerative) Feedback: The feedback signal is in phase with the input signal ( $X_i = X_s + X_f$ ). It increases overall gain but leads to instability. Primarily used in oscillators.
Negative (Degenerative) Feedback: The feedback signal is out of phase (usually by 180°) with the input signal ( $X_i = X_s - X_f$ ). It decreases the overall gain but significantly improves the amplifier's performance characteristics.

The overall closed-loop gain ( $A_f$ ) of a negative feedback amplifier is given by:
$A_f = \frac{A}{1 + A\beta}$
Where $A$ is the open-loop gain, and $1 + A\beta$ is the amount of feedback (desensitivity factor).

3. General Characteristics of Negative Feedback Amplifiers

Applying negative feedback trades raw gain for improvements in almost all other amplifier parameters:

Gain Desensitivity: The fractional change in closed-loop gain is drastically reduced compared to the open-loop gain. $\frac{dA_f}{A_f} = \frac{1}{1+A\beta} \frac{dA}{A}$ . The gain becomes highly stable against variations in temperature, aging, and device parameters.
Bandwidth Extension: Negative feedback increases the bandwidth of the amplifier. The product of gain and bandwidth remains constant (Gain-Bandwidth Product). $BW_f = BW \times (1 + A\beta)$ .
Reduction of Non-linear Distortion: Large-signal non-linearities generated within the basic amplifier are reduced by the factor $(1 + A\beta)$ .
Noise Reduction: Internal noise generated by the active devices in the amplifier is attenuated, provided the noise is introduced at an intermediate stage and not exactly at the input.
Modification of Input/Output Impedance: Depending on the feedback topology, input and output resistances can be increased or decreased to approach ideal amplifier characteristics.

4. Effect of Negative Feedback upon Output and Input Resistances

The changes in input resistance ( $R_{if}$ ) and output resistance ( $R_{of}$ ) depend strictly on how the signal is sampled at the output and how it is mixed at the input.

Mixing at the Input:
- Series Mixing: Increases the input resistance ( $R_{if} = R_i (1 + A\beta)$ ). Ideal for voltage inputs.
- Shunt Mixing: Decreases the input resistance ( $R_{if} = R_i / (1 + A\beta)$ ). Ideal for current inputs.
Sampling at the Output:
- Voltage Sampling (Shunt connection at output): Decreases the output resistance ( $R_{of} = R_o / (1 + A\beta)$ ). Ideal for creating a stiff voltage source.
- Current Sampling (Series connection at output): Increases the output resistance ( $R_{of} = R_o (1 + A\beta)$ ). Ideal for creating a stiff current source.

5. Topologies of Negative Feedback

Voltage Series Feedback

Also known as: Series-Shunt feedback.
Basic Amplifier Type: Voltage Amplifier.
Mechanism: Samples output voltage (shunt at output) and mixes in series with the input voltage.
Effects:
- $R_{in}$ increases: $R_{if} = R_i (1 + A_v\beta)$
- $R_{out}$ decreases: $R_{of} = \frac{R_o}{1 + A_v\beta}$
Application: Ideal voltage amplifier (e.g., Non-inverting Op-Amp configuration).

Current Series Feedback

Also known as: Series-Series feedback.
Basic Amplifier Type: Transconductance Amplifier.
Mechanism: Samples output current (series at output) and mixes in series with the input voltage.
Effects:
- $R_{in}$ increases: $R_{if} = R_i (1 + G_m\beta)$
- $R_{out}$ increases: $R_{of} = R_o (1 + G_m\beta)$
Application: Voltage-to-current converters, unbypassed emitter/source resistors.

Current Shunt Feedback

Also known as: Shunt-Series feedback.
Basic Amplifier Type: Current Amplifier.
Mechanism: Samples output current (series at output) and mixes in shunt (parallel) with the input current.
Effects:
- $R_{in}$ decreases: $R_{if} = \frac{R_i}{1 + A_i\beta}$
- $R_{out}$ increases: $R_{of} = R_o (1 + A_i\beta)$
Application: Ideal current amplifiers.

Voltage Shunt Feedback

Also known as: Shunt-Shunt feedback.
Basic Amplifier Type: Transresistance Amplifier.
Mechanism: Samples output voltage (shunt at output) and mixes in shunt (parallel) with the input current.
Effects:
- $R_{in}$ decreases: $R_{if} = \frac{R_i}{1 + R_m\beta}$
- $R_{out}$ decreases: $R_{of} = \frac{R_o}{1 + R_m\beta}$
Application: Current-to-voltage converters (e.g., Inverting Op-Amp configuration).

6. Distortion in Amplifier

Distortion occurs when the output signal is not an exact, scaled replica of the input signal.

Amplitude (Non-linear) Distortion: Occurs when the amplifier gain depends on the amplitude of the input signal, typically due to the non-linear transfer characteristics of BJTs or FETs. This results in the generation of harmonics (frequencies that are integer multiples of the fundamental frequency) not present in the original signal.
Frequency Distortion: Occurs when different frequency components of the input signal are amplified by different amounts. This happens because the amplifier's reactive components (capacitors, internal junction capacitances) cause the gain to roll off at low and high frequencies.
Phase (Delay) Distortion: Occurs when the phase shift introduced by the amplifier is not a linear function of frequency. Different frequency components experience different time delays, causing a complex waveform to change shape at the output.

7. Frequency Response of an Amplifier

The frequency response of an amplifier is a graph of its gain (and phase) versus frequency, typically plotted on a logarithmic scale (Bode plot).

Mid-band Region: The range of frequencies where the gain is relatively constant and maximum ( $A_{mid}$ ).
Cut-off Frequencies: The frequencies at which the gain drops by 3 dB (or to of its mid-band value, roughly 70.7%).
- $f_L$ : Lower cut-off frequency.
- $f_H$ : Upper cut-off frequency.
Bandwidth (BW): The difference between the upper and lower cut-off frequencies: $BW = f_H - f_L$ . For DC amplifiers, $f_L = 0$ , so $BW = f_H$ .

8. Low Frequency Response of an RC Coupled Stage

At low frequencies, the gain of an RC coupled amplifier drops due to the presence of external capacitors:

Coupling Capacitors ( $C_C$ ): Block DC to isolate biasing, but at low frequencies, their reactance ( $X_c = 1 / 2\pi f C$ ) becomes significant, acting as a voltage divider with the input/output resistance and dropping the signal.
Bypass Capacitors ( $C_E$ or $C_S$ ): Used to short AC signals to ground to prevent unwanted current-series negative feedback. At low frequencies, their reactance increases, reducing the bypassing effect, introducing negative feedback, and thus reducing the gain.

The lower cut-off frequency ( $f_L$ ) is determined by finding the pole frequency contributed by each capacitor. The dominant lower cut-off frequency is generally the highest of these individual pole frequencies.
$f_{L} \approx \sum \frac{1}{2\pi R_{eq} C}$

9. High Frequency Response of an FET Stage

At high frequencies, coupling and bypass capacitors act as perfect short circuits. However, the gain drops due to internal device parasitic capacitances:

FET Parasitic Capacitances: $C_{gs}$ (gate-to-source), $C_{gd}$ (gate-to-drain), and $C_{ds}$ (drain-to-source), plus any external wiring capacitance.
Miller Effect: The capacitance connecting the input and output () is multiplied by the gain of the amplifier when viewed from the input.
- $C_{in(Miller)} = C_{gd}(1 - A_v)$
- Since $A_v$ is usually a large negative number for a common-source amplifier, the input capacitance becomes very large, heavily shunting high-frequency input signals to ground.
Upper Cut-off Frequency ( $f_H$ ): Formed by the interaction of the signal source resistance and the total equivalent input capacitance (including Miller capacitance), acting as a low-pass filter.
$f_H = \frac{1}{2\pi R_{sig} (C_{gs} + C_{in(Miller)})}$

Unit 2