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title: 'Lecture 23: Frequency Response of Single Stage Amplifier'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Frequency Response of Transistor Amplifier
---

# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 23: Frequency Response of Single Stage Amplifier
### Frequency Response of Transistor Amplifier

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Welcome to Lecture 23. Today we unify low-frequency and high-frequency analysis into a complete single-stage frequency response framework. We will introduce the high-frequency hybrid-pi model, derive Miller's theorem, and construct full Bode plots for Common-Emitter amplifiers.
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# Lecture Agenda & Learning Roadmap

- Overview of the Three Frequency Response Regions (Low, Midband, High)
- Midband Voltage Gain Derivation using Small-Signal Hybrid-$\pi$ Model
- BJT High-Frequency Hybrid-$\pi$ Model & Junction Capacitances ($C_\pi, C_\mu$)
- Miller's Theorem Derivation and Equivalent Input/Output Capacitances
- High-Frequency Input and Output Cutoff Frequencies ($f_{H,in}, f_{H,out}$)
- Construction of Complete Bode Magnitude and Phase Response Curves
- Comprehensive Numerical Example: Full Frequency Response Calculation
- High-Frequency Performance Comparison Across Configurations (CE, CB, CC)
- MOSFET Common-Source (CS) High-Frequency Analysis
- Design Problem: Bandwidth Extension Techniques for CE Amplifiers
- Comprehensive Summary and Key Takeaways

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Here is our lecture roadmap. We start with midband gain, transition into internal transistor junction physics, apply Miller's theorem, calculate upper cutoffs, and compare CE, CB, and CC configurations.
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# Overview of the Complete Frequency Response Curve

- Low-Frequency Region ($f < f_L$): Gain attenuates at $+20\text{ dB/decade}$ per pole due to increasing reactance of coupling ($C_{C1}, C_{C2}$) and bypass ($C_E$) capacitors.
- Midband Frequency Region ($f_L \le f \le f_H$): Voltage gain magnitude is constant at maximum value $A_M$. External capacitors act as ideal AC short circuits, while internal parasitic capacitances act as open circuits.
- High-Frequency Region ($f > f_H$): Gain rolls off at $-20\text{ dB/decade}$ per pole due to decreasing reactance of internal junction capacitances $C_\pi$ (base-emitter) and $C_\mu$ (base-collector).
- Bandwidth & Half-Power Points: Operating bandwidth $BW = f_H - f_L$. Gain magnitude at $f_L$ and $f_H$ is $|A(f)| = \frac{A_M}{\sqrt{2}} = A_M - 3.01\text{ dB}$.

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The frequency response curve illustrates the three operating regimes of an analog amplifier. In the midband region, gain is flat. Below fL, external capacitors reduce gain. Above fH, internal device parasitics shunt signal current to ground.
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# Midband Small-Signal Voltage Gain Derivation

- Midband Hybrid-$\pi$ Circuit: External capacitors are replaced by AC short circuits. Transistor replaced by small-signal parameters $r_\pi$ and $g_m v_be$.
- Input Voltage Divider: Base-emitter signal voltage $V_{be} = V_s \left( \frac{R_{in}}{R_S + R_{in}} \right)$, where $R_{in} = R_1 \parallel R_2 \parallel r_\pi$.
- Output Voltage Expression: Output voltage $V_o = -g_m V_{be} (R_C \parallel R_L)$ (negative sign indicates $180^\circ$ phase inversion).
- Overall Midband System Gain ($A_{v,mid}$): $A_{v,mid} = \frac{V_o}{V_s} = -\left( \frac{R_1 \parallel R_2 \parallel r_\pi}{R_S + R_1 \parallel R_2 \parallel r_\pi} \right) g_m (R_C \parallel R_L)$, where $g_m = \frac{I_C}{V_T}$ and $r_\pi = \frac{\beta_0}{g_m}$.

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Midband gain represents the peak performance of the amplifier. The negative sign reflects the classic 180-degree phase reversal inherent to Common-Emitter amplifiers.
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# High-Frequency BJT Hybrid-$\pi$ Model & Parasitic Capacitances

- Base-Emitter Junction Capacitance ($C_\pi$): Sum of forward-bias depletion capacitance $C_{jE}$ and diffusion capacitance $C_{DE} = g_m \tau_F$: $C_\pi = C_{jE} + g_m \tau_F$ (typically $20 - 100\text{ pF}$).
- Base-Collector Junction Capacitance ($C_\mu$): Reverse-bias depletion capacitance $C_{jC} = \frac{C_{j0}}{(1 + V_{CB}/V_0)^m}$ (typically $1 - 5\text{ pF}$). Crosses the base and collector terminals.
- Collector-Substrate Capacitance ($C_{cs}$): Parasitic capacitance to substrate in integrated circuits (typically $1 - 3\text{ pF}$).
- Wiring & Layout Parasitics: Wiring capacitances $C_{w,in}$ and $C_{w,out}$ added in parallel with input and output nodes.

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At high frequencies, we must include internal physical capacitances. C_pi is relatively large because the base-emitter junction is forward biased, whereas C_mu is smaller but acts as a feedback element across base and collector.
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# Miller's Theorem & High-Frequency Input/Output Capacitance Derivation

- Miller's Theorem Concept: An impedance $Z$ connected between input node (1) and output node (2) with voltage gain $K = \frac{V_2}{V_1}$ can be replaced by two equivalent shunting impedances: $Z_1 = \frac{Z}{1 - K}$ and $Z_2 = \frac{Z}{1 - 1/K}$.
- Miller Input Capacitance ($C_{M,in}$): For bridging feedback capacitance $C_\mu$ with stage inverting gain $K = A_{v,stage} = -g_m (R_C \parallel R_L)$: $C_{M,in} = C_\mu (1 - A_{v,stage}) = C_\mu (1 + |A_{v,stage}|)$.
- Miller Output Capacitance ($C_{M,out}$): $C_{M,out} = C_\mu \left(1 - \frac{1}{A_{v,stage}}\right) \approx C_\mu$ since $|A_{v,stage}| \gg 1$.
- Total High-Frequency Input Capacitance ($C_{in}'$): $C_{in}' = C_\pi + C_{M,in} + C_{w,in} = C_\pi + C_\mu (1 + |A_{v,stage}|) + C_{w,in}$.
- Impact of Miller Multiplication: Even a small $C_\mu = 4\text{ pF}$ with gain $A_{v,stage} = -100$ creates $C_{M,in} = 4\text{ pF} \times 101 = 404\text{ pF}$, completely dominating input capacitance!

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Miller's theorem is one of the most critical concepts in high-frequency analog electronics. Because the collector moves in opposition to the base, the AC voltage across C_mu is (1 + |Av|) times larger than the input signal, multiplying the effective input capacitance by (1 + |Av|).
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# High-Frequency Cutoff Calculation ($f_{H,in}, f_{H,out}$ and Overall $f_H$)

- Input High-Frequency Pole ($f_{H,in}$): Equivalent resistance $R_{in}' = R_S \parallel R_1 \parallel R_2 \parallel r_\pi$. Input cutoff: $f_{H,in} = \frac{1}{2\pi R_{in}' C_{in}'} = \frac{1}{2\pi (R_S \parallel R_1 \parallel R_2 \parallel r_\pi) \left[ C_\pi + C_\mu(1+|A_{v,stage}|) + C_{w,in} \right]}$.
- Output High-Frequency Pole ($f_{H,out}$): Equivalent resistance $R_{out}' = R_C \parallel R_L \parallel r_o \approx R_C \parallel R_L$. Output cutoff: $f_{H,out} = \frac{1}{2\pi R_{out}' C_{out}'} = \frac{1}{2\pi (R_C \parallel R_L) \left[ C_\mu + C_{cs} + C_{w,out} \right]}$.
- Dominant High-Frequency Pole: Because $C_{in}' \gg C_{out}'$ due to Miller multiplication, $f_{H,in}$ is substantially smaller than $f_{H,out}$, making $f_{H,in}$ the dominant upper pole.
- Overall Upper 3-dB Cutoff Frequency ($f_H$): $f_H \approx f_{H,in}$ or using root-square combination: $f_H = \frac{1}{\sqrt{(1/f_{H,in})^2 + (1/f_{H,out})^2}}$.

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To find the upper 3-dB cutoff, we compute the input and output high-frequency poles. Because C_in' is hugely multiplied by Miller effect, the input pole fH,in is much lower than fH,out and sets the upper bandwidth limit fH.
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# Bode Magnitude and Phase Plots of Single-Stage Amplifier

- Complete Transfer Function: $A(s) = A_{v,mid} \cdot \left(\frac{s}{s + \omega_L}\right) \cdot \left(\frac{1}{1 + s/\omega_H}\right)$.
- Bode Magnitude Asymptotes: Flat response at $A_{v,mid}(\text{dB})$ in midband ($f_L < f < f_H$). $+20\text{ dB/decade}$ roll-off for $f < f_L$, $-20\text{ dB/decade}$ roll-off for $f > f_H$.
- Corner Frequencies: At $f = f_L$ and $f = f_H$, actual gain magnitude is $-3\text{ dB}$ below asymptotic lines.
- Bode Phase Plot Progression: At $f \ll f_L$, phase is $+270^\circ$ ($+90^\circ$ lead over midband $180^\circ$). At $f = f_L$, phase is $+225^\circ$. At midband, phase is $180^\circ$. At $f = f_H$, phase is $135^\circ$ ($45^\circ$ lag). At $f \gg f_H$, phase approaches $+90^\circ$ ($90^\circ$ lag).

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The complete Bode plot synthesizes low, midband, and high frequency dynamics. Note that at midband, the phase shift is 180 degrees due to CE inversion. At fH, phase lag drops the shift to 135 degrees.
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# Comprehensive Numerical Example: Complete Frequency Response of CE Amplifier

- Given Circuit & Transistor Parameters: $V_{CC}=15\text{ V}, R_1=33\text{ k}\Omega, R_2=10\text{ k}\Omega, R_C=3.3\text{ k}\Omega, R_E=1\text{ k}\Omega, R_S=500\ \Omega, R_L=10\text{ k}\Omega, \beta_0=150, I_C=1.5\text{ mA}$. Parasitics: $C_\pi=30\text{ pF}, C_\mu=4\text{ pF}, C_{w,in}=5\text{ pF}, C_{w,out}=3\text{ pF}$. External: $C_{C1}=10\ \mu\text{F}, C_{C2}=10\ \mu\text{F}, C_E=100\ \mu\text{F}$. Small-signal parameters: $V_T=26\text{ mV} \implies r_e = 17.33\ \Omega, g_m = 57.69\text{ mS}, r_\pi = 2.6\text{ k}\Omega$.
- Step 1 - Midband Gain ($A_{v,mid}$): $R_{in} = 33\text{ k} \parallel 10\text{ k} \parallel 2.6\text{ k} = 1.954\text{ k}\Omega$. Loaded collector resistance $R_L' = R_C \parallel R_L = 3.3\text{ k} \parallel 10\text{ k} = 2.481\text{ k}\Omega$. Internal stage gain $A_{v,stage} = -g_m R_L' = -57.69\text{ mS} \times 2481 = -143.13\text{ V/V}$. Overall gain $A_{v,mid} = -\left(\frac{1954}{500+1954}\right) \times 143.13 = -113.97\text{ V/V}$ ($41.14\text{ dB}$).
- Step 2 - Miller Input Capacitance ($C_{M,in}$): $C_{M,in} = C_\mu (1 + |A_{v,stage}|) = 4\text{ pF} \times (1 + 143.13) = 576.52\text{ pF}$. Total input capacitance $C_{in}' = C_\pi + C_{M,in} + C_{w,in} = 30 + 576.52 + 5 = 611.52\text{ pF}$.
- Step 3 - High-Frequency Input Pole ($f_{H,in}$): Equivalent resistance $R_{in}' = R_S \parallel R_{in} = 500 \parallel 1954 = 398.1\ \Omega$. $f_{H,in} = \frac{1}{2\pi \times 398.1\ \Omega \times 611.52\text{ pF}} = 653.8\text{ kHz}$.
- Step 4 - High-Frequency Output Pole ($f_{H,out}$): $C_{out}' = C_\mu + C_{w,out} = 4 + 3 = 7\text{ pF}$. $R_{out}' = 2.481\text{ k}\Omega$. $f_{H,out} = \frac{1}{2\pi \times 2481 \times 7\text{ pF}} = 9.16\text{ MHz}$. Upper cutoff $f_H \approx f_{H,in} = 653.8\text{ kHz}$.

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Work through this numerical example with precision. Notice how Miller multiplication turns 4 pF of C_mu into 576.5 pF of input capacitance, pulling the upper cutoff frequency down to 653.8 kHz, while the output pole is up at 9.16 MHz.
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# High-Frequency Response Comparison: CE, CB, and CC Configurations

- Common-Emitter (CE): Provides high voltage and current gain, but suffers from strong Miller effect $C_{M,in} = C_\mu(1+|A_v|)$, resulting in moderate upper cutoff frequency $f_H$.
- Common-Base (CB): Low input impedance ($R_{in} \approx r_e$), unit current gain, but ZERO Miller effect because collector is AC grounded relative to input. Achieves extremely high upper cutoff $f_H \approx f_T$, making it ideal for RF amplifiers.
- Common-Collector (CC / Emitter Follower): Voltage gain $A_v \approx 1 \implies C_{M,in} = C_\mu(1 - 1) = 0$. Low Miller multiplication, high input impedance, wide bandwidth, ideal as high-speed output buffers.
- Summary Comparison: CB configuration provides the widest bandwidth for voltage amplification due to elimination of Miller effect.

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Comparing transistor configurations highlights why Common-Base is preferred in RF applications. Because the base is AC grounded in CB, there is no inverting voltage gain between input and output, completely eliminating Miller multiplication of C_mu.
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# MOSFET Common-Source (CS) High-Frequency Response

- MOSFET High-Frequency Parasitics: Gate-source capacitance $C_{gs}$, gate-drain capacitance $C_{gd}$, and drain-source capacitance $C_{ds}$.
- Midband Gain: $A_{v,mid} = -g_m (R_D \parallel R_L)$, where $g_m = 2 \sqrt{K_n I_D} = \sqrt{2 \mu_n C_{ox} \frac{W}{L} I_D}$.
- Miller Gate-Drain Capacitance: Gate-drain capacitance $C_{gd}$ bridges input and output: $C_{M,in} = C_{gd} (1 + g_m (R_D \parallel R_L))$.
- CS High-Frequency Pole ($f_{H,in}$): $f_{H,in} = \frac{1}{2\pi (R_S \parallel R_G) \left[ C_{gs} + C_{gd}(1 + g_m (R_D \parallel R_L)) \right]}$. Very high input resistance $R_G$ makes CS amplifiers sensitive to signal source resistance $R_S$.

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MOSFET Common-Source amplifiers mirror BJT Common-Emitter behavior. Cgd undergoes Miller multiplication, generating a large input capacitance Cgs + Cgd(1 + |Av|) that limits high-frequency operation.
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# Design Problem: Bandwidth Extension of a CE Stage

- Problem Challenge: An engineer needs to extend the upper 3-dB bandwidth $f_H$ of a Common-Emitter stage from $500\text{ kHz}$ to $2\text{ MHz}$. Evaluate design options.
- Option 1 - Emitter Degeneration ($R_e$): Adding an unbypassed emitter resistor $R_e$ reduces stage gain to $A_{v,stage}' \approx -\frac{R_C \parallel R_L}{r_e + R_e}$. Lower gain reduces $C_{M,in} = C_\mu(1+|A_{v,stage}'|)$, extending upper cutoff $f_H$ proportionally.
- Option 2 - Source Impedance Reduction ($R_S$): Lowering signal source resistance $R_S$ using an upstream Emitter Follower buffer reduces $R_{in}' = R_S \parallel R_{in}$, boosting $f_{H,in} = \frac{1}{2\pi R_{in}' C_{in}'}$.
- Option 3 - Device Selection: Replacing BJT with a high-$f_T$ microwave transistor having smaller $C_\mu$ ($0.5\text{ pF}$ vs $4\text{ pF}$).
- Quantitative Trade-off: Adding $R_e$ to trade gain for bandwidth preserves the constant Gain-Bandwidth Product (GBW).

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When extending bandwidth in practical designs, adding a small unbypassed emitter resistor Re is the most effective circuit technique. It trades midband gain for reduced Miller capacitance, successfully quadrupling the upper cutoff frequency.
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# Summary of Single-Stage Amplifier Frequency Characteristics

- Complete Transfer Function: $A(s) = A_{v,mid} \left(\frac{s}{s + \omega_L}\right) \left(\frac{1}{1 + s/\omega_H}\right)$.
- Midband Gain: $A_{v,mid} = -\left(\frac{R_{in}}{R_S + R_{in}}\right) g_m (R_C \parallel R_L)$.
- Low-Frequency Dominant Pole: Set by external capacitors $C_{C1}, C_{C2}, C_E$, dominated by $C_E$ due to small $R_{eq,E}$.
- High-Frequency Dominant Pole: Set by internal parasitics $C_\pi, C_\mu$, dominated by Miller input capacitance $C_{M,in} = C_\mu (1 + |A_{v,stage}|)$.
- Configuration Comparison: Common-Base provides superior high-frequency performance by eliminating Miller effect.

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To summarize Lecture 23, single-stage frequency response is bounded at low frequencies by external coupling/bypass capacitors and bounded at high frequencies by internal junction capacitances expanded via Miller effect.
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