Small-Signal Amplifiers: Configurations, Analysis & Design

Key Concepts and Configurations

Small-signal amplifiers amplify weak AC signals (e.g., audio, sensor outputs) while ignoring DC components. They are analyzed using small-signal models (hybrid-π for BJTs, simplified models for FETs) and classified by transistor configuration:

1. Amplifier Configurations

Three basic configurations determine input/output impedance, gain, and stability:

Why it matters: CE is most common for general-purpose amplification; CB excels at high frequencies; CC is used for impedance matching.

2. Hybrid-π Small-Signal Model for BJTs

The hybrid-π model simplifies BJT analysis by replacing the transistor with equivalent resistors and controlled sources. Key components:

• $r_{\pi }$: Input resistance ($r_{\pi }=\beta r_e$, where $r_e=\frac{V_T}{I_E}$)
• $g_m$: Transconductance ($g_m=\frac{I_C}{V_T}$)
• $r_o$: Output resistance (often ignored in low-frequency analysis)

graph LR
    A["V_in"] --> B["r_π"]
    B --> C["g_m"]
    C --> D["V_out"]
    B --> E["r_o"]
    E --> D

Worked Example: For a BJT with $\beta =100$, $I_C=1\text{ mA}$, and $V_T=25\text{ mV}$:

• $r_e=\frac{25\text{ mV}}{1\text{ mA}}=25\Omega$
• $r_{\pi }=100\times 25\Omega =2.5\text{ k}\Omega$
• $g_m=\frac{1\text{ mA}}{25\text{ mV}}=40\text{ mS}$

Frequency Response and Bandwidth

Amplifiers have limited bandwidth due to parasitic capacitances ($C_{\pi }$, $C_{\mu }$) and coupling capacitors. The frequency response is characterized by:

1. Lower cutoff frequency ($f_L$): Caused by coupling/emitter bypass capacitors.
2. Upper cutoff frequency ($f_H$): Caused by transistor junction capacitances.
3. Bandwidth ($BW$): $BW=f_H-f_L$.

Bode Plot for CE Amplifier

Key Idea: The gain rolls off at $-20\text{ dB/decade}$ below $f_L$ and above $f_H$.

Design Considerations

1. Gain Calculation

For a CE amplifier with $R_C$ and $R_E$: $$A_v=-g_m{R}_C^{\prime }$$ where $R_C^{\prime }=R_C\parallel R_L$.

Worked Example (Real-World Tie-In): In a Khalti payment app’s audio feedback system, a CE amplifier drives a small speaker. Given:

• $R_C=1\text{ k}\Omega$, $R_L=8\Omega$, $g_m=40\text{ mS}$.
• Calculate $A_v$: $R_C^{\prime }=1\text{ k}\Omega \parallel 8\Omega \approx 7.92\Omega$ $A_v=-40\text{ mS}\times 7.92\Omega \approx -0.3168$ (unity gain is achieved by adjusting $R_C$ or $g_m$).

2. Stability and Feedback

Negative feedback improves stability but reduces gain. Common feedback types:

• Voltage-series: Reduces output impedance, increases input impedance.
• Current-shunt: Increases output impedance, reduces input impedance.

In the Real World

1. Pathao’s Ride-Hailing App:

   • Uses CE amplifiers in its GPS signal conditioning circuits to boost weak antenna signals before processing. The high gain of CE stages ensures accurate location data for driver matching.

2. Ncell’s Base Station Transmitters:

   • CB amplifiers are employed in RF stages to amplify high-frequency signals (e.g., 4G/5G) with minimal distortion. Their low input impedance matches the antenna’s impedance for efficient power transfer.

3. eSewa’s Payment Terminals:

   • CC (Emitter Follower) amplifiers buffer signals between the payment terminal’s ADC and the display driver. Their unity gain and high input impedance prevent loading effects on the ADC.

Exam Tip

1. Memorize Hybrid-π Parameters:

   • $r_{\pi }=\beta r_e$, $g_m=\frac{I_C}{V_T}$, $r_e=\frac{V_T}{I_E}$. Always derive these from given data.

2. Configuration Trade-offs:

   • CE: High gain, inverting.
   • CB: High-frequency, non-inverting.
   • CC: Unity gain, buffer. Exam Question: "Which config would you choose for a 100 MHz amplifier?" → CB (high $f_T$).

3. Frequency Response:

   • Sketch the Bode plot for a given $f_L$ and $f_H$. Label the $-3\text{ dB}$ points and roll-off rates.

4. Feedback Analysis:

   • For a feedback amplifier, calculate the closed-loop gain using: $A_{cl}=\frac{A}{1+A\beta }$.
   • Identify if feedback is series/shunt and its effect on $Z_{in}$ and $Z_{out}$.

5. Real-World Scenarios:

   • Expect questions like: "Design a preamp for a Daraz delivery tracking sensor (output: 10 mV, $R_L=1\text{ k}\Omega$)." → Use CE with $R_C=10\text{ k}\Omega$, bias $I_C=1\text{ mA}$, and calculate $A_v$.

Labelled CE amplifier with $R_C$, $R_E$, and bias network. (Image: TedPavlic, CC BY-SA 3.0, via Wikimedia Commons) Hybrid-π model with $r_{\pi }$, $g_m$, and $r_o$. (Image: Michel Bakni, CC BY-SA 4.0, via Wikimedia Commons) Real OP-AMP chip (741) for comparison with discrete amplifiers. (Image: Inductiveload, Public domain, via Wikimedia Commons)