Unit 1 - Notes

ECE221

Unit 1: Feedback Amplifiers and Frequency Response of Amplifier

1. Classification of Amplifiers

Amplifiers are classified into four broad categories based on their input and output terminal characteristics and the physical quantity (voltage or current) they amplify.

A. Voltage Amplifier

Input Signal: Voltage ( $V_s$ )
Output Signal: Voltage ( $V_o$ )
Ideal Characteristics: Infinite input impedance ( $R_i = \infty$ ), Zero output impedance ( $R_o = 0$ ).
Gain Parameter: Voltage Gain $A_v = V_o / V_i$ .
Thevenin Equivalent: Modeled as a voltage controlled voltage source.

B. Current Amplifier

Input Signal: Current ( $I_s$ )
Output Signal: Current ( $I_o$ )
Ideal Characteristics: Zero input impedance ( $R_i = 0$ ), Infinite output impedance ( $R_o = \infty$ ).
Gain Parameter: Current Gain $A_i = I_o / I_i$ .
Norton Equivalent: Modeled as a current controlled current source.

C. Transconductance Amplifier

Input Signal: Voltage ( $V_s$ )
Output Signal: Current ( $I_o$ )
Ideal Characteristics: Infinite input impedance ( $R_i = \infty$ ), Infinite output impedance ( $R_o = \infty$ ).
Gain Parameter: Transconductance $G_m = I_o / V_i$ .
Model: Voltage controlled current source.

D. Transresistance Amplifier

Input Signal: Current ( $I_s$ )
Output Signal: Voltage ( $V_o$ )
Ideal Characteristics: Zero input impedance ( $R_i = 0$ ), Zero output impedance ( $R_o = 0$ ).
Gain Parameter: Transresistance $R_m = V_o / I_i$ .
Model: Current controlled voltage source.

2. The Feedback Concept

Feedback creates a closed-loop system by returning a fraction of the output signal to the input.

Block Diagram Components

Source: Signal source ( $V_s$ or $I_s$ ).
Mixer (Comparator): Combines source signal and feedback signal.
Basic Amplifier (A): The open-loop amplifier with gain $A$ .
Sampling Network: Samples the output (voltage or current).
Feedback Network ( $\beta$ ): Attenuates the output by a factor $\beta$ to generate the feedback signal ( $V_f$ or $I_f$ ).

Types of Feedback

Positive (Regenerative) Feedback: The feedback signal is in phase with the input signal. It increases gain but reduces stability. Used in oscillators.
Negative (Degenerative) Feedback: The feedback signal is 180° out of phase with the input signal. It reduces gain but improves stability, bandwidth, and linearity.

Fundamental Equation of Negative Feedback

If $A$ is the open-loop gain and $\beta$ is the feedback factor:

A_f = \frac{A}{1 + A\beta}

$A_f$ : Closed-loop gain.
$A\beta$ : Loop gain.
$(1 + A\beta)$ : Amount of feedback (or Desensitivity Factor, $D$ ).

3. General Characteristics of Negative Feedback Amplifiers

Applying negative feedback significantly alters the performance of an amplifier.

A. Desensitivity of Gain (Stability)

The closed-loop gain is less sensitive to variations in the internal amplifier parameters (due to temperature or aging).

\frac{dA_f}{A_f} = \frac{1}{1 + A\beta} \cdot \frac{dA}{A}

The percentage change in closed-loop gain is reduced by the factor $D$ .

B. Extension of Bandwidth

The Gain-Bandwidth Product remains constant. Since gain decreases, bandwidth increases.

New Lower Cutoff: $f_{Lf} = \frac{f_L}{1 + A\beta}$
New Upper Cutoff: $f_{Hf} = f_H (1 + A\beta)$
New Bandwidth: $BW_f \approx BW (1 + A\beta)$

C. Reduction in Non-Linear Distortion

If the open-loop amplifier introduces harmonic distortion $D_{open}$ , the negative feedback reduces it.

D_{closed} = \frac{D_{open}}{1 + A\beta}

Note: This applies only to distortion generated within the feedback loop.

D. Reduction of Noise

Similar to distortion, noise generated within the amplifier is attenuated by the desensitivity factor.

4. Feedback Topologies

There are four basic feedback topologies based on the quantity sampled (Output) and the method of mixing (Input).

Naming Convention

First term (Voltage/Current): Refers to the Output (Sampling).
Second term (Series/Shunt): Refers to the Input (Mixing).

Topology Name	Also Known As	Output Sampling	Input Mixing	Signal Sampled	Feedback Signal
Voltage Series	Series-Shunt	Parallel (Voltage)	Series (Voltage)	Voltage ( $V_o$ )	Voltage ( $V_f$ )
Current Series	Series-Series	Series (Current)	Series (Voltage)	Current ( $I_o$ )	Voltage ( $V_f$ )
Current Shunt	Shunt-Series	Series (Current)	Parallel (Current)	Current ( $I_o$ )	Current ( $I_f$ )
Voltage Shunt	Shunt-Shunt	Parallel (Voltage)	Parallel (Current)	Voltage ( $V_o$ )	Current ( $I_f$ )

5. Effect of Negative Feedback on Input and Output Resistances

The effect depends on the topology used.

Shunt Mixing decreases Input Resistance.
Series Mixing increases Input Resistance.
Voltage Sampling (Parallel) decreases Output Resistance.
Current Sampling (Series) increases Output Resistance.

A. Voltage Series Feedback

Ideal for Voltage Amplifiers.

Input Resistance ( $R_{if}$ ): Increases.
$R_{if} = R_i (1 + A\beta)$
Output Resistance ( $R_{of}$ ): Decreases.
$R_{of} = \frac{R_o}{1 + A\beta}$

B. Current Series Feedback

Ideal for Transconductance Amplifiers.

Input Resistance ( $R_{if}$ ): Increases.
$R_{if} = R_i (1 + A\beta)$
Output Resistance ( $R_{of}$ ): Increases.
$R_{of} = R_o (1 + A\beta)$

C. Current Shunt Feedback

Ideal for Current Amplifiers.

Input Resistance ( $R_{if}$ ): Decreases.
$R_{if} = \frac{R_i}{1 + A\beta}$
Output Resistance ( $R_{of}$ ): Increases.
$R_{of} = R_o (1 + A\beta)$

D. Voltage Shunt Feedback

Ideal for Transresistance Amplifiers.

Input Resistance ( $R_{if}$ ): Decreases.
$R_{if} = \frac{R_i}{1 + A\beta}$
Output Resistance ( $R_{of}$ ): Decreases.
$R_{of} = \frac{R_o}{1 + A\beta}$

6. Distortion in Amplifier

Ideally, the output waveform should be an exact replica of the input waveform (amplified). Any deviation is distortion.

Types of Distortion

Amplitude (Non-linear) Distortion: Occurs when the amplifier operates in the non-linear region of the transistor characteristics. This generates new frequency components (harmonics) not present in the input.
- Harmonic Distortion: Expressed as % harmonic distortion ( $D_2, D_3...$ ). Total Harmonic Distortion (THD) is:
  $D = \sqrt{D_2^2 + D_3^2 + ...}$
Frequency Distortion: Occurs when different frequency components of a signal are amplified by different amounts (non-flat frequency response).
Phase Distortion: Occurs when phase shift is not proportional to frequency (signals take different times to pass through).

Effect of Feedback on Distortion

Negative feedback reduces non-linear distortion significantly. By feeding back the distorted output, the pre-distortion signal subtracts from the input, effectively "linearizing" the transfer curve.

7. Frequency Response of an Amplifier

The frequency response is the plot of Gain ( $A$ ) versus Frequency ( $f$ ). It is usually plotted on a semi-log scale (Gain in dB vs Log Frequency).

Regions of Response

Low Frequency Region: Gain drops due to coupling capacitors ( $C_c$ ) and bypass capacitors ( $C_E$ or $C_S$ ). These act as high-pass filters.
Mid Frequency Region: Gain is constant. Capacitors are effectively short circuits (coupling) or open circuits (stray).
High Frequency Region: Gain drops due to internal transistor capacitances ( $C_{gs}, C_{gd}$ ) and stray wiring capacitance. These act as low-pass filters.

Key Metrics

Cut-off Frequencies ( $f_L, f_H$ ): Frequencies where gain drops to $70.7\%$ ( $-3\text{dB}$ ) of maximum gain.
Bandwidth ( $BW$ ): $f_H - f_L$ .

8. Low Frequency Response of an RC Coupled Stage

In an RC coupled amplifier (e.g., Common Emitter or Common Source), the low-frequency response is determined by external capacitors.

The Effect of Capacitors

At low frequencies, the reactance $X_C = \frac{1}{2\pi f C}$ becomes large.

Input/Output Coupling Capacitors: Form a voltage divider with the input/load resistance, attenuating the signal as frequency drops.
Bypass Capacitor: Its reactance increases, failing to bypass the emitter/source resistor effectively. This introduces negative feedback at the emitter/source, reducing gain.

Analysis

The voltage gain at low frequencies ( $A_{low}$ ) relative to mid-band gain ( $A_{mid}$ ) is given by:

\frac{A_{low}}{A_{mid}} = \frac{1}{1 - j(f_L / f)}

Magnitude: $|A_{low}| = \frac{A_{mid}}{\sqrt{1 + (f_L/f)^2}}$
Phase: $\theta = \tan^{-1}(f_L/f)$

Tilt (Sag)

When a square wave is applied to an RC coupled amplifier at low frequencies, the top of the waveform tilts downward.

Percentage Tilt ( $P$ ):
$P = \frac{V_{drop}}{V_{amp}} \times 100 \approx \frac{\pi f_L}{f} \times 100$
Where $f$ is the square wave frequency.

9. High Frequency Response of an FET Stage

At high frequencies, the external capacitors act as short circuits. The response is limited by internal parasitic capacitances of the FET.

FET Internal Capacitances

$C_{gs}$ : Gate-to-Source capacitance (Input).
$C_{gd}$ : Gate-to-Drain capacitance (Feedback path).
$C_{ds}$ : Drain-to-Source capacitance (Output).

The Miller Effect

The feedback capacitance $C_{gd}$ connects the input and output. In a Common Source amplifier (inverting, gain $-A$ ), this capacitance is "magnified" when viewed from the input.

Miller Input Capacitance ( $C_{Mi}$ ):

C_{Mi} = C_{gd} (1 + |A_v|)

This makes the total input capacitance very large:

C_{in} = C_{gs} + C_{Mi}

Miller Output Capacitance ( $C_{Mo}$ ):

C_{Mo} = C_{gd} (1 + \frac{1}{|A_v|}) \approx C_{gd}

High Frequency Analysis

The circuit acts as a Low-Pass RC filter.

Input Circuit: Determined by $R_{sig}$ (signal resistance) and $C_{in}$ .
Output Circuit: Determined by $R_{L}'$ (effective load) and $C_{out}$ .

The high frequency gain ( $A_{high}$ ) relative to mid-band gain ( $A_{mid}$ ):

\frac{A_{high}}{A_{mid}} = \frac{1}{1 + j(f / f_H)}

The upper cut-off frequency $f_H$ is dominated by the input pole (due to the Miller Effect):

f_H = \frac{1}{2\pi R_{sig} C_{in(total)}}

Where $C_{in(total)} = C_{gs} + C_{gd}(1 + g_m R_L')$ .