Part III: Analog Electronics | Chapter 7

Amplifier Fundamentals

Voltage gain, current gain, impedances, common-emitter and common-source topologies, small-signal analysis, frequency response, and the gain-bandwidth product

1. Amplifier Parameters

Any two-port amplifier is completely characterized by four quantities: voltage gain, current gain, input impedance, and output impedance. In decibels, voltage gain is\( A_v\,[\text{dB}] = 20\log_{10}|A_v| \), while power gain uses\( G\,[\text{dB}] = 10\log_{10}(P_{out}/P_{in}) \).

Voltage Gain

\( A_v = V_{out}/V_{in} \)

Current Gain

\( A_i = I_{out}/I_{in} \)

Input Impedance

\( Z_{in} = V_{in}/I_{in} \)

Output Impedance

\( Z_{out} = V_{oc}/I_{sc} \)

The ideal amplifier has infinite \( Z_{in} \) (draws no input current) and zero\( Z_{out} \) (drives any load without loss). Real amplifiers approximate these ideals through careful design.

2. Common-Emitter Amplifier

The common-emitter (CE) configuration is the workhorse BJT amplifier. Using the small-signal hybrid-π model, the voltage gain (with emitter bypass capacitor) is\( A_v = -g_m R_C \), where the transconductance\( g_m = I_C / V_T \) (\( V_T \approx 26\,\text{mV} \) at room temperature).

Common-emitter amplifier with biasing resistors R_B1, R_B2, emitter bypass, and coupling capacitors

The minus sign in \( A_v = -g_m R_C \) indicates a 180° phase inversion — a hallmark of the common-emitter stage. The input impedance is\( Z_{in} = r_\pi \| R_{B1} \| R_{B2} \) where\( r_\pi = \beta / g_m \).

With an unbypassed emitter resistor \( R_E \), the gain becomes\( A_v \approx -R_C / R_E \), trading gain for improved linearity and temperature stability.

3. Common-Source MOSFET Amplifier

The MOSFET common-source (CS) amplifier is the FET analogue of the CE stage. Using the small-signal model with transconductance \( g_m = 2I_D / (V_{GS} - V_{th}) \), the voltage gain is \( A_v = -g_m (R_D \| r_{ds}) \).

CS Voltage Gain

\( A_v = -g_m R_D \)

(large r_ds)

CS Input Impedance

\( Z_{in} \approx \infty \)

(gate draws no DC current)

Transconductance

\( g_m = \sqrt{2\mu_n C_{ox}(W/L)I_D} \)

4. Frequency Response & Gain-Bandwidth Product

Every amplifier has a frequency-dependent gain \( A_v(j\omega) \). For a single-pole amplifier, the transfer function is:

\[ A_v(j\omega) = \frac{A_0}{1 + j\omega/\omega_p} \]

The magnitude falls to \( A_0/\sqrt{2} \) (i.e., −3 dB) at the pole frequency\( \omega_p = 1/RC \). At higher frequencies, gain rolls off at −20 dB/decade.

A fundamental trade-off exists: the gain-bandwidth product (GBW) is approximately constant for a given amplifier technology:

\[ \text{GBW} = A_0 \cdot f_{-3\text{dB}} = \text{constant} \]

Increasing DC gain \( A_0 \) proportionally reduces bandwidth. Negative feedback exploits this: by reducing closed-loop gain by factor \( (1+A_0\beta) \), bandwidth increases by the same factor.

5. Python: Bode Plot & Gain-Bandwidth Product

Plot the frequency response of a single-pole amplifier (A₀ = 60 dB, f_p = 1 kHz), demonstrate the −3 dB bandwidth, and show how feedback trades gain for bandwidth.

Python

script.py96 lines

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec

plt.style.use('dark_background')
fig = plt.figure(figsize=(14, 10))
fig.patch.set_facecolor('#0a0a0a')
gs = gridspec.GridSpec(2, 2, figure=fig, hspace=0.45, wspace=0.35)

# --- Single-pole amplifier parameters ---
A0_dB = 60       # DC gain in dB => A0 = 1000 V/V
A0   = 10**(A0_dB / 20)
f_p  = 1e3       # pole frequency 1 kHz (open-loop -3 dB)
GBW  = A0 * f_p  # gain-bandwidth product = 1 MHz

f = np.logspace(1, 8, 2000)   # 10 Hz to 100 MHz
omega = 2 * np.pi * f

# Transfer function H(jw) = A0 / (1 + jw/wp)
H = A0 / (1 + 1j * f / f_p)
mag_dB = 20 * np.log10(np.abs(H))
phase_deg = np.degrees(np.angle(H))

# -3 dB frequency
f_3dB = f_p   # for single pole amp, -3dB at pole

# ---- Plot 1: Magnitude Bode (open-loop) ----
ax1 = fig.add_subplot(gs[0, :])
ax1.set_facecolor('#0f0f0f')
ax1.semilogx(f, mag_dB, color='#34d399', linewidth=2, label='Open-loop |A(f)|')
ax1.axhline(A0_dB - 3, color='#fbbf24', linewidth=1.2, linestyle='--', label=f'-3 dB = {A0_dB - 3} dB')
ax1.axvline(f_3dB, color='#f87171', linewidth=1.2, linestyle='--', label=f'f_p = {f_p:.0f} Hz (-3 dB BW)')
ax1.axvline(GBW, color='#a78bfa', linewidth=1.2, linestyle='--', label=f'GBW = {GBW/1e6:.1f} MHz')
# Annotate gain-bandwidth
ax1.annotate(f'GBW = A0 x f_p = {GBW/1e6:.1f} MHz',
             xy=(GBW, 0), xytext=(GBW/5, 15),
             arrowprops=dict(arrowstyle='->', color='#a78bfa'), color='#a78bfa', fontsize=9)
ax1.annotate(f'A0 = {A0_dB} dB', xy=(30, A0_dB), xytext=(200, A0_dB - 10),
             color='#34d399', fontsize=9)
ax1.set_xlabel('Frequency (Hz)', color='#9ca3af')
ax1.set_ylabel('Gain (dB)', color='#9ca3af')
ax1.set_title('Bode Plot — Single-Pole Amplifier Frequency Response', color='#6ee7b7', fontsize=12, pad=10)
ax1.legend(fontsize=9, framealpha=0.2)
ax1.set_ylim(-10, 70)
ax1.tick_params(colors='#9ca3af')
ax1.grid(True, color='#1f2937', which='both', alpha=0.6)
for spine in ax1.spines.values():
    spine.set_edgecolor('#374151')

# ---- Plot 2: Phase response ----
ax2 = fig.add_subplot(gs[1, 0])
ax2.set_facecolor('#0f0f0f')
ax2.semilogx(f, phase_deg, color='#60a5fa', linewidth=2)
ax2.axhline(-45, color='#fbbf24', linewidth=1, linestyle='--', label='-45 deg at f_p')
ax2.axvline(f_p, color='#f87171', linewidth=1, linestyle='--')
ax2.set_xlabel('Frequency (Hz)', color='#9ca3af')
ax2.set_ylabel('Phase (degrees)', color='#9ca3af')
ax2.set_title('Phase Response', color='#6ee7b7', fontsize=10)
ax2.legend(fontsize=8, framealpha=0.2)
ax2.tick_params(colors='#9ca3af')
ax2.grid(True, color='#1f2937', which='both', alpha=0.6)
for spine in ax2.spines.values():
    spine.set_edgecolor('#374151')

# ---- Plot 3: Gain vs feedback factor ----
ax3 = fig.add_subplot(gs[1, 1])
ax3.set_facecolor('#0f0f0f')
colors_fb = ['#34d399', '#fbbf24', '#f87171', '#a78bfa', '#60a5fa']
betas = [0, 0.001, 0.01, 0.1, 1.0]
labels_fb = ['beta=0 (open)', 'beta=0.001', 'beta=0.01', 'beta=0.1', 'beta=1.0']
for beta, col, lbl in zip(betas, colors_fb, labels_fb):
    if beta == 0:
        Hf = A0 / (1 + 1j * f / f_p)
    else:
        # Closed-loop: Af = A / (1 + A*beta)
        A_ol = A0 / (1 + 1j * f / f_p)
        Hf = A_ol / (1 + A_ol * beta)
    ax3.semilogx(f, 20 * np.log10(np.abs(Hf)), color=col, linewidth=1.5, label=lbl)

ax3.set_xlabel('Frequency (Hz)', color='#9ca3af')
ax3.set_ylabel('Gain (dB)', color='#9ca3af')
ax3.set_title('Closed-Loop: Effect of Feedback', color='#6ee7b7', fontsize=10)
ax3.legend(fontsize=7, framealpha=0.2)
ax3.tick_params(colors='#9ca3af')
ax3.grid(True, color='#1f2937', which='both', alpha=0.6)
for spine in ax3.spines.values():
    spine.set_edgecolor('#374151')

plt.suptitle('Amplifier Frequency Response & Gain-Bandwidth Product', color='#6ee7b7', fontsize=13, y=1.01)
plt.savefig('output.png', dpi=130, bbox_inches='tight', facecolor='#0a0a0a')
print('Saved output.png')
print(f'DC Gain A0         = {A0_dB} dB = {A0:.0f} V/V')
print(f'Pole frequency     = {f_p:.0f} Hz')
print(f'Gain-Bandwidth     = {GBW/1e6:.2f} MHz')
print(f'-3 dB bandwidth    = {f_3dB:.0f} Hz (open loop)')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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