8.3 Multistage Amplifiers, Frequency Response & Power Amplifiers

8.3 Multistage Amplifiers, Frequency Response & Power Amplifiers

1. Multistage Cascaded Amplifiers & Interstage Coupling

Single-stage amplifiers rarely provide sufficient voltage gain, input impedance, and output driving capability simultaneously. Multiple amplifier stages are cascaded in series, where the output of stage $n$ feeds the input of stage $n+1$.

Stage Loading Effects & Total Gain

When cascading stages, the input impedance of stage 2 ($Z_{in2}$) acts as a parallel AC load on stage 1 ($Z_{out1} \parallel R_{C1} \parallel Z_{in2}$).

• Total Voltage Gain:

$$A_{vT} = A_{v1} \times A_{v2} \times A_{v3} \times \dots \times A_{vn}$$

• Gain in Decibels ($dB$):

$$A_{vT,dB} = 20 \log_{10}|A_{vT}| = A_{v1,dB} + A_{v2,dB} + \dots + A_{vn,dB}$$

Interstage Coupling Methods

---

2. Frequency Response & Decibel Analysis

Decibel Fundamentals

• Voltage Gain in dB: $A_{v,dB} = 20 \log_{10}\left|\frac{V_o}{V_i}\right|$
• Power Gain in dB: $A_{p,dB} = 10 \log_{10}\left|\frac{P_o}{P_i}\right|$
• Half-Power (-3 dB) Cutoff Points: At frequencies $f_L$ and $f_H$, voltage gain drops to $\frac{1}{\sqrt{2}} \approx 0.707$ ($70.7\%$) of midband gain $A_{mid}$, and power drops to $50\%$.

Low-Frequency Response ($f_L$)

Determined by coupling capacitors ($C_S, C_C$) and emitter/source bypass capacitors ($C_E, C_S$). Each RC network introduces a lower cutoff frequency:

$$f_L = \frac{1}{2\pi R_{Th} C}$$

• The dominant lower cutoff frequency is the largest value among all lower cutoff frequencies ($f_{L,dom} = \max(f_{L1}, f_{L2}, f_{LE})$).

High-Frequency Response ($f_H$) & The Miller Effect

Determined by internal transistor parasitic junction capacitances ($C_{be}, C_{bc}$ in BJTs; $C_{gs}, C_{gd}$ in FETs) and stray wiring capacitance ($C_w$).

Miller Effect Theorem

For an inverting amplifier with voltage gain $-A_v$ and feedback capacitance $C_f$ ($C_{bc}$ or $C_{gd}$):

• Input Miller Capacitance:

$$C_{M1} = C_{bc} (1 - A_v) = C_{bc} (1 + |A_v|)$$

• Output Miller Capacitance:

$$C_{M2} = C_{bc} \left( 1 - \frac{1}{A_v} \right) \approx C_{bc}$$

• Total Input High-Frequency Capacitance:

$$C_{in,hi} = C_{wi} + C_{be} + C_{M1}$$

• Upper Cutoff Frequency ($f_H$):

$$f_H = \frac{1}{2\pi R_{Th,hi} C_{in,hi}}$$

• The dominant upper cutoff frequency is the smallest value ($f_{H,dom} = \min(f_{H1}, f_{H2})$).

Bandwidth & Gain-Bandwidth Product

• Bandwidth: $BW = f_H - f_L \approx f_H$ (since $f_H \gg f_L$).
• Gain-Bandwidth Product ($GBW$): Constant for a given transistor amplifier stage.

$$GBW = A_{v,mid} \times BW = f_T$$

Where $f_T$ is the transition frequency at which short-circuit current gain drops to unity ($0\text{ dB}$).

• Multistage Bandwidth Shrinkage (N identical non-interacting stages):

$$f_{L(overall)} = \frac{f_L}{\sqrt{2^{1/N} - 1}}, \quad f_{H(overall)} = f_H \sqrt{2^{1/N} - 1}$$

---

3. Large-Signal Power Amplifiers

Power amplifiers operate over large portions of the transistor's characteristic curves to deliver maximum AC power to a load (e.g., loudspeaker, antenna).

Summary of Power Amplifier Classes

Class B / AB Push-Pull Power Calculations

• Maximum AC Output Power:

$$P_{o,max} = \frac{V_{CC}^2}{2 R_L}$$

• DC Input Power:

$$P_{dc} = V_{CC} \cdot I_{dc} = V_{CC} \left( \frac{2 I_p}{\pi} \right) = \frac{2 V_{CC} V_p}{\pi R_L}$$

• Maximum Efficiency:

$$\eta_{max} = \frac{P_{o,max}}{P_{dc,max}} = \frac{\pi}{4} \approx 78.54\%$$

• Maximum Power Dissipation Per Transistor ($P_{D,max}$): Occurs at $V_p = \frac{2}{\pi} V_{CC} \approx 0.636 V_{CC}$:

$$P_{D,max} = \frac{V_{CC}^2}{\pi^2 R_L}$$

Harmonic Distortion & Thermal Management

• Total Harmonic Distortion (THD):

$$THD = \frac{\sqrt{V_2^2 + V_3^2 + V_4^2 + \dots + V_n^2}}{V_1} \times 100\%$$

• Thermal Resistance & Derating: Junction temperature $T_J$ must not exceed maximum limit ($150^\circ\text{C} - 200^\circ\text{C}$):

$$T_J - T_A = P_D (\theta_{JC} + \theta_{CS} + \theta_{SA})$$

Where $\theta_{JC}$ is junction-to-case, $\theta_{CS}$ is case-to-heatsink, and $\theta_{SA}$ is heatsink-to-ambient thermal resistance ($^\circ\text{C/W}$).