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title: 'Lecture 20: Amplifier Parameters Gain, Bandwidth and Gain Bandwidth prod...'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Frequency Response of Transistor Amplifier
---

# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 20: Amplifier Parameters Gain, Bandwidth and Gain Bandwidth prod...
### Frequency Response of Transistor Amplifier

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Welcome to Lecture 20. Today we begin Unit 3: Frequency Response of Transistor Amplifiers. We will define amplifier gain parameters in linear and decibel scales, investigate low-frequency roll-off caused by coupling capacitors, analyze high-frequency roll-off driven by parasitic junction capacitances and the Miller Effect, and prove the fundamental Gain-Bandwidth trade-off.
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# Lecture Blueprint: Amplifier Parameters & Frequency Response

- Fundamental Amplifier Gain Definitions: Voltage, Current, Transconductance, Transresistance.
- Decibel ($dB$) Scale Representation & Logarithmic Bode Plot Curves.
- Low-Frequency Response ($f_L$): Coupling ($C_{C1}, C_{C2}$) & Emitter Bypass ($C_E$) Capacitor Poles.
- High-Frequency Response ($f_H$): Stray Parasitic & Transistor Junction Capacitances ($C_\pi, C_\mu$).
- Bandwidth ($BW$) & Half-Power (-3 dB) Corner Frequency Definitions.
- The Gain-Bandwidth Product ($GBP$) & High-Frequency Hybrid-\pi$ Model.
- Transition Frequency ($f_T$) Derivation & The Miller Effect Theorem.
- Feedback Effects on Gain-Bandwidth Trade-Offs & Worked Numerical Problems.

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We will examine gain definitions, decibel scaling, low-frequency poles, high-frequency parasitic limitations, Miller effect, $f_T$, and bandwidth extension via negative feedback.
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# Fundamental Amplifier Gain Parameters

- Voltage Gain ($A_v$): Ratio of AC output voltage to AC input voltage: $A_v = \frac{v_{out}}{v_{in}} = \frac{v_o}{v_i}$ (Dimensionless).
- Current Gain ($A_i$): Ratio of AC output current to AC input current: $A_i = \frac{i_{out}}{i_{in}} = \frac{i_o}{i_i}$ (Dimensionless).
- Transconductance Gain ($G_m$): Ratio of AC output current to AC input voltage: $G_m = \frac{i_{out}}{v_{in}}$ (Units: Siemens, $\text{S}$ or $\Omega^{-1}$).
- Transresistance Gain ($R_m$): Ratio of AC output voltage to AC input current: $R_m = \frac{v_{out}}{i_{in}}$ (Units: Ohms, $\Omega$).
- Power Gain ($A_p$): Ratio of AC output power to AC input power: $A_p = \frac{P_{out}}{P_{in}} = A_v \cdot A_i$.

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Amplifiers are categorized by their transfer functions: voltage gain $A_v$, current gain $A_i$, transconductance $G_m$, and transresistance $R_m$. Power gain is the product of voltage gain and current gain.
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# Decibel ($dB$) Scale Representation & Logarithmic Bode Curves

- Voltage Gain in Decibels: $A_{v,\text{dB}} = 20 \log_{10} |A_v| = 20 \log_{10} \left| \frac{v_o}{v_i} \right|$.
- Power Gain in Decibels: $A_{p,\text{dB}} = 10 \log_{10} A_p = 10 \log_{10} \left( \frac{P_o}{P_i} \right)$.
- Half-Power (-3 dB) Point Definition: Frequency at which output power drops to half its midband value ($P_{out} = 0.5 P_{mid}$), corresponding to voltage magnitude dropping to $\frac{1}{\sqrt{2}} \approx 0.707$ of midband voltage gain ($A_{mid}$).
- Decibel Loss at Cutoff: $20 \log_{10}(0.7071) = -3.0103\,\text{dB} \approx -3\,\text{dB}$.
- Bode Plot Slope Rates: Single-pole RC roll-off produces a attenuation slope of $-20\,\text{dB/decade}$ ($-6\,\text{dB/octave}$).

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The decibel scale compresses huge dynamic ranges. A 3 dB drop in voltage corresponds to a 50% drop in power, defining the cutoff frequencies $f_L$ and $f_H$.
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# Low-Frequency Cutoff ($f_L$) Dynamics: Coupling & Bypass Capacitors

- Low-Frequency Attenuation Mechanism: At low frequencies ($f \to 0$), capacitive reactances $X_C = \frac{1}{2\pi f C} \to \infty$, acting as open circuits that attenuate signals.
- Input Coupling Capacitor Pole ($C_{C1}$): Forms high-pass filter with total input resistance $R_{in}' = R_{sig} + (R_1 \parallel R_2 \parallel r_p)$: $f_{L1} = \frac{1}{2\pi (R_{sig} + R_{in}) C_{C1}}$.
- Output Coupling Capacitor Pole ($C_{C2}$): Forms high-pass filter with load resistance: $f_{L2} = \frac{1}{2\pi (R_C + R_L) C_{C2}}$.
- Emitter Bypass Capacitor Pole ($C_E$): Bypasses emitter resistor $R_E$. Resistance looking into emitter is $R_e' = R_E \parallel \left[ r_e + \frac{R_{TH} \parallel R_{sig}}{1+\beta} \right]$: $f_{LE} = \frac{1}{2\pi R_e' C_E}$.
- Overall Lower Cutoff Frequency: Dominant pole approximation: $f_L \approx \sqrt{f_{L1}^2 + f_{L2}^2 + f_{LE}^2}$ or $f_L \approx f_{L1} + f_{L2} + f_{LE}$.

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At low frequencies, coupling capacitors $C_{C1}, C_{C2}$ and bypass capacitor $C_E$ introduce high-pass RC filter poles. The highest pole frequency acts as the dominant lower cutoff $f_L$.
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# High-Frequency Cutoff ($f_H$) Dynamics: Parasitic Junction Capacitances

- High-Frequency Attenuation Mechanism: At high frequencies ($f \to \infty$), parasitic internal junction reactances $X_C = \frac{1}{2\pi f C} \to 0$, shorting AC signals to ground.
- Base-Emitter Diffusion & Transition Capacitance ($C_\pi$ or $C_{be}$): Forward-biased base-emitter junction capacitance ($10\,\text{pF} - 100\,\text{pF}$).
- Base-Collector Depletion Capacitance ($C_\mu$ or $C_{bc}$): Reverse-biased base-collector junction capacitance ($1\,\text{pF} - 5\,\text{pF}$).
- Stray Wiring Capacitance ($C_{w,in}, C_{w,out}$): Parasitic PCB trace capacitances ($2\,\text{pF} - 10\,\text{pF}$).
- Overall Upper Cutoff Frequency ($f_H$): Set by internal low-pass RC network poles: $f_H = \frac{1}{2\pi R_{in,eq} C_{in,total}}$.

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At high frequencies, external capacitors are short circuits, but internal semiconductor junction capacitances $C_\pi$ and $C_\mu$ take over, shunting high-frequency signals to ground.
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# Bandwidth ($BW$) & Midband Operational Range

- Bandwidth Definition: The continuous range of signal frequencies over which amplifier gain remains within $3\,\text{dB}$ of its peak midband gain $A_{mid}$.
- Mathematical Formula: $BW = f_H - f_L$.
- Wideband Approximation ($f_H \gg f_L$): For standard audio and video amplifiers where $f_H \ge 100 f_L$, $BW \approx f_H$.
- Midband Region Characteristics: Frequency range where external capacitors ($C_{C1}, C_{C2}, C_E$) act as perfect AC short circuits AND internal capacitors ($C_\pi, C_\mu$) act as perfect AC open circuits, yielding constant gain $A_{mid}$ and zero phase shift distortion.

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Bandwidth is defined as $f_H - f_L$. Because $f_H$ is typically orders of magnitude larger than $f_L$, bandwidth is approximately equal to $f_H$.
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# The Gain-Bandwidth Product ($GBP / f_T$): Invariance Principle

- Gain-Bandwidth Product Definition: $GBP \equiv |A_{mid}| \times BW \approx |A_{mid}| \times f_H$.
- Fundamental Invariance Principle: For a single-pole amplifier system, the product of midband voltage gain and bandwidth is a CONSTANT parameter dictated by internal transistor physics.
- Gain vs. Bandwidth Trade-Off: Increasing midband gain $|A_{mid}|$ proportionally decreases bandwidth $BW$; trading gain extends bandwidth.
- Transition Frequency ($f_T$): Frequency at which short-circuit common-emitter current gain drops to unity ($|h_{fe}(f_T)| = 1$). $f_T$ represents the absolute maximum theoretical $GBP$ of the device.

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The Gain-Bandwidth Product is constant for a given transistor architecture. You can trade gain for bandwidth, but their product $GBP = A_{mid} \times BW$ remains invariant.
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# High-Frequency Hybrid-\pi Transistor Model Parameters

- Hybrid-\pi Model Components: Replaces low-frequency $h$-parameter model for high-frequency dynamic AC circuit analysis.
- Base Spreading Resistance ($r_bb'$ or $r_x$): Ohmic resistance of bulk base semiconductor material between base terminal and active junction ($20\,\Omega - 100\,\Omega$).
- Internal Base-Emitter Resistance ($r_b'e$ or $r_\pi$): Dynamic resistance of forward-biased base-emitter junction: $r_\pi = \frac{\beta_0}{g_m} = \frac{\beta_0 V_T}{I_{CQ}}$.
- Transconductance ($g_m$): $g_m = \frac{I_{CQ}}{V_T} = \frac{I_{CQ}}{26\,\text{mV}}$ at room temperature ($25^\circ\text{C}$).
- Junction Capacitances: $C_{b'e} = C_\pi$ (diffusion + depletion) and $C_{b'c} = C_\mu$ (collector overlap depletion).

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At high frequencies, we use the Hybrid-\pi model. $g_m = I_C/V_T$ models small-signal current generation, while $C_\pi$ and $C_\mu$ capture physical charge storage.
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# Transition Frequency ($f_T$) Derivation & Hybrid-\pi Equation

- Short-Circuit Current Gain Formulation: Output short-circuited ($R_L = 0 \implies v_{ce} = 0$). Output current $i_o = -g_m v_{b'e}$.
- Input Current Equation: $i_i = v_{b'e} \cdot \left[ \frac{1}{r_\pi} + j 2\pi f (C_\pi + C_\mu) \right]$.
- Short-Circuit Current Gain Expression: $A_{i,sc}(f) = \frac{i_o}{i_i} = \frac{-g_m}{\frac{1}{r_\pi} + j 2\pi f (C_\pi + C_\mu)} = \frac{-h_{fe0}}{1 + j \frac{f}{f_\beta}}$, where $f_\beta = \frac{1}{2\pi r_\pi (C_\pi + C_\mu)}$ is the $\beta$-cutoff frequency.
- Setting $|A_{i,sc}(f_T)| = 1$: At $f = f_T \gg f_\beta$, $|A_{i,sc}| \approx \frac{g_m}{2\pi f_T (C_\pi + C_\mu)} = 1$.
- Transition Frequency Formula: $f_T = \frac{g_m}{2\pi (C_\pi + C_\mu)}$.

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Here we derive the expression for unity-gain transition frequency $f_T$. Setting short-circuit current gain to unity yields $f_T = g_m / [2\pi (C_\pi + C_\mu)]$.
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# Miller Effect Theorem: Impact of Feedback Capacitance $C_\mu$

- Miller Effect Origin: In an inverting amplifier ($A_v = -|A_v|$), feedback capacitance $C_\mu$ connected between base (input) and collector (output) experiences an amplified AC voltage swing across its terminals.
- Input Miller Capacitance Derivation: $C_{M,in} = C_\mu \left( 1 - A_v \right) = C_\mu \left( 1 + |A_v| \right)$.
- Output Miller Capacitance Derivation: $C_{M,out} = C_\mu \left( 1 - \frac{1}{A_v} \right) \approx C_\mu$.
- Total High-Frequency Input Capacitance: $C_{in,total} = C_\pi + C_{M,in} = C_\pi + C_\mu (1 + |A_v|)$.
- Bandwidth Impact: The massive equivalent input capacitance $C_{in,total}$ creates a low-frequency dominant pole at the input node, severely reducing high-frequency cutoff $f_H$.

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The Miller Effect multiplies bridging capacitance $C_\mu$ by $(1 + |A_v|)$. For a stage with voltage gain of -100, a 2 pF $C_\mu$ acts like a huge 202 pF capacitor at the input node!
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# Feedback Effects on Gain-Bandwidth Product: Desensitivity & Extension

- Closed-Loop Gain with Negative Feedback: $A_f = \frac{A_{mid}}{1 + \beta_{fb} A_{mid}}$, where $\beta_{fb}$ is the feedback factor.
- Closed-Loop Upper Cutoff Frequency: $f_{Hf} = f_H \left( 1 + \beta_{fb} A_{mid} \right)$.
- Closed-Loop Lower Cutoff Frequency: $f_{Lf} = \frac{f_L}{1 + \beta_{fb} A_{mid}}$.
- Closed-Loop Bandwidth Extension: $BW_f = f_{Hf} - f_{Lf} \approx f_H \left( 1 + \beta_{fb} A_{mid} \right)$.
- Conservation of Gain-Bandwidth Product: $GBP_f = A_f \times BW_f = \left( \frac{A_{mid}}{1 + \beta_{fb} A_{mid}} \right) \times \left[ f_H (1 + \beta_{fb} A_{mid}) \right] = A_{mid} \times f_H = GBP$.

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Applying negative feedback reduces gain by factor $(1 + \beta_{fb} A)$, but extends bandwidth by the exact same factor $(1 + \beta_{fb} A)$. Their product remains perfectly constant!
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# Numerical Example: High-Frequency Hybrid-\pi & $f_T$ Calculation

- Given Transistor Parameters: Operating quiescent current $I_{CQ} = 2.6\,\text{mA}$, $\beta_0 = 100$, $V_T = 26\,\text{mV}$. Parasitic junction capacitances $C_\pi = 30\,\text{pF}$, $C_\mu = 2\,\text{pF}$. Inverting voltage gain $|A_v| = 50$. Source resistance $R_{sig} = 600\,\Omega$, $R_{TH} = 10\,\text{k}\Omega$.
- Step 1: Calculate Transconductance $g_m$: $g_m = \frac{I_{CQ}}{V_T} = \frac{2.6\,\text{mA}}{26\,\text{mV}} = 0.1\,\text{S} = 100\,\text{mS}$.
- Step 2: Calculate Input Resistance $r_\pi$: $r_\pi = \frac{\beta_0}{g_m} = \frac{100}{0.1} = 1000\,\Omega = 1.0\,\text{k}\Omega$.
- Step 3: Calculate Transition Frequency $f_T$: $f_T = \frac{g_m}{2\pi (C_\pi + C_\mu)} = \frac{0.1}{2\pi (30\text{pF} + 2\text{pF})} = \frac{0.1}{2\pi \times 32 \times 10^{-12}} = 497.35\,\text{MHz}$.
- Step 4: Calculate Miller Input Capacitance: $C_{M,in} = C_\mu (1 + |A_v|) = 2\,\text{pF} \times (1 + 50) = 102\,\text{pF}$.
- Step 5: Total Input Capacitance: $C_{in,total} = C_\pi + C_{M,in} = 30\,\text{pF} + 102\,\text{pF} = 132\,\text{pF}$.

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Let's perform this hybrid-$\pi$ calculation. Notice how $g_m = 100\,\text{mS}$, giving a transition frequency $f_T = 497.35\,\text{MHz}$. Miller capacitance expands $C_\mu$ from 2 pF to 102 pF!
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# Summary & Core Amplifier Frequency Response Principles

- Frequency Response Regions: Low-frequency roll-off ($f < f_L$) set by coupling/bypass capacitors; High-frequency roll-off ($f > f_H$) set by junction capacitors ($C_\pi, C_\mu$).
- Half-Power Cutoff: $-3\,\text{dB}$ point where gain drops to $0.707 A_{mid}$ and power drops to $50\%$.
- Miller Effect: Multiplies feedback capacitance by $(1 + |A_v|)$, creating the dominant high-frequency input pole.
- Gain-Bandwidth Invariance: $GBP = |A_{mid}| \times BW = f_T = \frac{g_m}{2\pi(C_\pi+C_\mu)}$.
- Feedback Extension: Negative feedback trades gain for bandwidth, extending $BW_f = f_H (1 + \beta_{fb} A)$ while conserving overall $GBP$.

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To summarize: low frequency is limited by external capacitors, high frequency is limited by junction capacitors and Miller multiplication. The Gain-Bandwidth Product remains constant.
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