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title: 'Lecture 21: Amplifier Parameters Gain, Bandwidth and Gain Bandwidth prod...'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Frequency Response of Transistor Amplifier
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# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 21: Amplifier Parameters Gain, Bandwidth and Gain Bandwidth prod...
### Frequency Response of Transistor Amplifier

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Welcome to Lecture 21. Today we will establish a rigorous quantitative framework for assessing transistor amplifier performance across frequency domains. We will analyze how gain, cutoff frequencies, and gain-bandwidth products interconnect, and how feedback can be leveraged to tune bandwidth at the expense of voltage gain.
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# Lecture Agenda & Learning Roadmap

- Voltage, Current, and Power Gain Definitions and Decibel Conversions
- Amplifier Frequency Response: Half-Power (-3 dB) Cutoff Frequencies $f_L$ and $f_H$
- Bandwidth ($BW$) and Transfer Function Representation of Single-Pole Amplifiers
- Gain-Bandwidth Product (GBW) and Unity-Gain Frequency ($f_T$)
- Gain-Bandwidth Trade-Off via Negative Feedback Configuration
- BJT High-Frequency Figures of Merit: Cutoff Frequencies $f_\alpha$, $f_\beta$, and $f_T$
- Physical Factors Influencing GBW: Transconductance $g_m$ and Parasitic Capacitances
- Op-Amp Closed-Loop Bandwidth Determination and Noise Gain Considerations
- Gain-Bandwidth Shrinkage in Multistage Cascaded Amplifiers
- Step-by-Step Numerical Examples and Comprehensive Summary

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Here is our roadmap for today's lecture. We begin with core parameter definitions in decibels, transition into frequency response transfer functions, explore the GBW invariant property under feedback, and solve practical engineering examples.
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# Amplifier Gain Metrics: Voltage, Current, and Power Gains in Decibels

- Voltage Gain Definition: Linear ratio $A_v = \frac{V_o}{V_i}$ and decibel representation $A_v(\text{dB}) = 20 \log_{10} \left| \frac{V_o}{V_i} \right|$, quantifying signal voltage amplification.
- Current Gain Definition: Linear ratio $A_i = \frac{I_o}{I_i}$ and decibel representation $A_i(\text{dB}) = 20 \log_{10} \left| \frac{I_o}{I_i} \right|$, measuring current multiplication across low-impedance nodes.
- Power Gain Relationship: Overall power amplification $A_p = \frac{P_o}{P_i} = A_v \cdot A_i$. In decibels, $A_p(\text{dB}) = 10 \log_{10} A_p = A_v(\text{dB}) + A_i(\text{dB})$ when input and output load resistances are equal.
- Source-to-Load System Gain ($A_{vs}$): Incorporates signal source internal resistance $R_S$ and load resistance $R_L$: $A_{vs} = \frac{V_o}{V_s} = \left(\frac{R_{in}}{R_S + R_{in}}\right) A_v \left(\frac{R_L}{R_{out} + R_L}\right)$.

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When evaluating amplifier gain, we must distinguish between internal voltage gain Av and overall system voltage gain Avs. Notice that power gain uses a factor of 10 in decibel conversion, whereas voltage and current use a factor of 20 because power scales with the square of voltage.
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# Frequency Response Characteristics & Bandwidth ($BW$)

- Midband Voltage Gain ($A_M$): The maximum, frequency-independent voltage gain in the intermediate frequency range where coupling/bypass capacitors act as AC short circuits and parasitic capacitances act as open circuits.
- Half-Power (-3 dB) Cutoff Frequencies: Lower cutoff $f_L$ and upper cutoff $f_H$ represent frequencies where the magnitude drops to $|A(f)| = \frac{A_M}{\sqrt{2}} \approx 0.707 A_M$.
- Power Reduction at Cutoff: At $f_L$ and $f_H$, output power drops to half of midband power ($P_{out} = 0.5 P_{mid}$), corresponding to a $-3.01\text{ dB}$ change: $20 \log_{10}(0.707) = -3.01\text{ dB}$.
- Bandwidth Definition: $BW = f_H - f_L$. For broadband amplifiers where $f_H \gg f_L$ (e.g., $f_H = 1\text{ MHz}$ and $f_L = 10\text{ Hz}$), the bandwidth is approximated as $BW \approx f_H$.

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The bandwidth of an amplifier defines its effective operating frequency range. The -3 dB points represent the boundaries where signal power drops by half. In broadband communication circuits, lower cutoff fL is virtually negligible compared to fH.
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# Single-Pole Transfer Function Analysis & High-Frequency Roll-off

- Single-Pole High-Frequency Transfer Function: $A(s) = \frac{A_M}{1 + s/\omega_H}$, where $\omega_H = 2\pi f_H$ is the high-frequency pole.
- Frequency Response Vector Expression: Substituting $s = j\omega$ yields $A(j\omega) = \frac{A_M}{1 + j(\omega/\omega_H)} = \frac{A_M}{\sqrt{1 + (f/f_H)^2}} \angle -\arctan(f/f_H)$.
- Asymptotic High-Frequency Roll-Off: For frequencies well above cutoff ($f \gg f_H$), the gain magnitude simplifies to $|A(f)| \approx A_M \cdot \frac{f_H}{f}$, giving a roll-off rate of $-20\text{ dB/decade}$ (or $-6\text{ dB/octave}$).
- Phase Response Shift: Phase lag at midband ($f \ll f_H$) is $0^\circ$, at upper cutoff ($f = f_H$) is $-45^\circ$, and approaches $-90^\circ$ as $f \to \infty$.

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The single-pole transfer function is fundamental to amplifier modeling. Above fH, the product of gain magnitude and frequency becomes constant, which leads directly to the concept of Gain-Bandwidth Product.
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# Gain-Bandwidth Product (GBW) and Unity-Gain Frequency ($f_T$)

- Gain-Bandwidth Product Definition: $GBW = A_M \cdot f_H$, representing the constant product of midband voltage gain magnitude $A_M$ and upper 3-dB bandwidth $f_H$ for a single-pole system.
- Unity-Gain Frequency ($f_T$): The frequency at which the short-circuit/open-loop voltage gain drops to unity ($|A(f_T)| = 1 = 0\text{ dB}$).
- Equivalence of GBW and $f_T$: Setting $|A(f_T)| = \frac{A_M}{\sqrt{1 + (f_T/f_H)^2}} = 1$ yields $f_T \approx A_M \cdot f_H = GBW$ when $A_M \gg 1$.
- Figure of Merit Significance: GBW remains constant across different closed-loop gain configurations, serving as an intrinsic device parameter determined by internal transistor dynamics.

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GBW is a crucial figure of merit. It tells an engineer the maximum bandwidth attainable for a given required gain, or conversely, the maximum gain available for a given operating bandwidth.
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# Gain vs. Bandwidth Trade-off via Negative Feedback

- Closed-Loop Gain Derivation: Applying negative feedback with feedback factor $\beta$ reduces low-frequency gain: $A_{cl}(s) = \frac{A(s)}{1 + \beta A(s)}$.
- Closed-Loop Transfer Function Expansion: Substituting $A(s) = \frac{A_0}{1 + s/\omega_H}$ yields $A_{cl}(s) = \frac{A_0 / (1 + A_0 \beta)}{1 + \frac{s}{\omega_H (1 + A_0 \beta)}}$.
- Closed-Loop Bandwidth Extension: The new midband gain is $A_{cl,0} = \frac{A_0}{1 + A_0 \beta}$ and new upper cutoff is $f_{H,cl} = f_H (1 + A_0 \beta)$.
- Invariance of Gain-Bandwidth Product: Multiplying closed-loop gain by closed-loop bandwidth gives $GBW_{cl} = A_{cl,0} \cdot f_{H,cl} = \left(\frac{A_0}{1 + A_0 \beta}\right) \cdot f_H (1 + A_0 \beta) = A_0 f_H = GBW$.

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This slide proves mathematically why negative feedback preserves GBW. By reducing gain by a factor of (1 + A0*beta), the upper cutoff frequency is expanded by the exact same factor, maintaining a constant gain-bandwidth product.
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# Numerical Example 1: BJT Amplifier Bandwidth and GBW Calculation

- Problem Statement: A single-stage BJT amplifier has an open-loop midband voltage gain $A_0 = 80\text{ dB}$ ($10,000\text{ V/V}$) and a unity-gain frequency $f_T = 50\text{ MHz}$. (1) Calculate open-loop bandwidth $f_H$. (2) If negative feedback reduces closed-loop gain to $A_{cl} = 40\text{ dB}$ ($100\text{ V/V}$), find new bandwidth $f_{H,cl}$.
- Step 1 - Open-Loop Bandwidth $f_H$: Using $GBW = A_0 \cdot f_H = f_T$, we get $f_H = \frac{f_T}{A_0} = \frac{50 \times 10^6\text{ Hz}}{10,000} = 5,000\text{ Hz} = 5\text{ kHz}$.
- Step 2 - Closed-Loop Bandwidth $f_{H,cl}$: Using constant GBW, $f_{H,cl} = \frac{GBW}{A_{cl}} = \frac{50 \times 10^6\text{ Hz}}{100} = 500,000\text{ Hz} = 500\text{ kHz}$.
- Verification of Feedback Factor $\beta$: $1 + A_0 \beta = \frac{A_0}{A_{cl}} = \frac{10000}{100} = 100 \implies \beta = \frac{99}{10000} = 0.0099$. Bandwidth expanded 100-fold from $5\text{ kHz}$ to $500\text{ kHz}$.

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Walk through this calculation step by step. Notice how trading 40 dB of gain (from 80 dB to 40 dB, a factor of 100 drop) increases the bandwidth from 5 kHz all the way up to 500 kHz!
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# Transistor High-Frequency Limits: Cutoff Frequencies $f_\alpha$, $f_\beta$, and $f_T$

- Common-Base Short-Circuit Cutoff ($f_\alpha$): Frequency where common-base short-circuit current gain $\alpha$ drops by 3 dB from low-frequency value $\alpha_0$: $f_\alpha = \frac{1}{2\pi \tau_F}$, where $\tau_F$ is base transit time.
- Common-Emitter Short-Circuit Cutoff ($f_\beta$): Frequency where common-emitter current gain $\beta(f)$ drops by 3 dB: $f_\beta = \frac{1}{2\pi r_\pi (C_\pi + C_\mu)} = \frac{g_m}{2\pi \beta_0 (C_\pi + C_\mu)}$.
- Relationship Between $f_T$ and $f_\beta$: Short-circuit current gain relation $\beta(f) = \frac{\beta_0}{1 + j(f/f_\beta)}$. Unity gain frequency occurs when $|\beta(f_T)| = 1 \implies f_T = \beta_0 f_\beta = \frac{g_m}{2\pi (C_\pi + C_\mu)}$.
- Alpha-Beta Cutoff Relation: Since $\beta_0 \gg 1$, $f_\beta = \frac{f_T}{\beta_0}$ is substantially lower than $f_T$ and $f_\alpha \approx 1.2 f_T$.

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It is vital to distinguish between f_beta (the 3dB bandwidth of short-circuit current gain beta) and f_T (the unity gain frequency). f_T is beta_0 times f_beta, demonstrating another manifestation of the GBW product principle.
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# Physical Parameters Governing GBW: $g_m$, $C_\pi$, and $C_\mu$

- Transconductance Dependence: $g_m = \frac{I_C}{V_T}$, where $I_C$ is DC collector current and $V_T \approx 26\text{ mV}$ at $300\text{ K}$. Higher bias current $I_C$ elevates $g_m$ and increases $f_T$.
- Diffusion Capacitance $C_{DE}$: Proportional to collector current: $C_{DE} = g_m \tau_F = \left(\frac{I_C}{V_T}\right) \tau_F$. Dominates base-emitter junction capacitance $C_\pi = C_{jE} + C_{DE}$ at higher bias currents.
- Depletion Capacitances $C_{jE}, C_\mu$: Voltage-dependent junction capacitances: $C_j(V_R) = \frac{C_{j0}}{(1 + V_R / V_0)^m}$. Collector-base capacitance $C_\mu$ is minimized by higher reverse-bias collector-base voltage $V_{CB}$.
- Optimum Bias Current Region: At very high $I_C$, high-level injection and Kirk effect increase base transit time $\tau_F$, causing $f_T$ and GBW to roll off after reaching a peak value.

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To maximize GBW in high-frequency design, engineers select transistors with low parasitic capacitances C_pi and C_mu, and bias the device at an optimal collector current IC that maximizes transconductance gm without triggering Kirk effect degradation.
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# Numerical Example 2: Operational Amplifier Closed-Loop Bandwidth

- Problem Statement: An operational amplifier has a specified Gain-Bandwidth Product $GBW = 12\text{ MHz}$. It is configured as a non-inverting voltage amplifier with feedback resistors $R_F = 18\text{ k}\Omega$ and $R_1 = 2\text{ k}\Omega$. Determine: (1) Closed-loop midband voltage gain $A_{cl}$. (2) Closed-loop 3-dB bandwidth $f_{H,cl}$. (3) Gain at an operating signal frequency of $f = 2.4\text{ MHz}$.
- Step 1 - Closed-Loop Gain $A_{cl}$: $A_{cl} = 1 + \frac{R_F}{R_1} = 1 + \frac{18\text{ k}\Omega}{2\text{ k}\Omega} = 1 + 9 = 10\text{ V/V}$ ($20\text{ dB}$).
- Step 2 - Closed-Loop Bandwidth $f_{H,cl}$: $f_{H,cl} = \frac{GBW}{A_{cl}} = \frac{12 \times 10^6\text{ Hz}}{10} = 1.2\text{ MHz}$.
- Step 3 - Gain Magnitude at $f = 2.4\text{ MHz}$: $|A(f)| = \frac{A_{cl}}{\sqrt{1 + (f / f_{H,cl})^2}} = \frac{10}{\sqrt{1 + (2.4 / 1.2)^2}} = \frac{10}{\sqrt{1 + 4}} = \frac{10}{\sqrt{5}} \approx 4.47\text{ V/V}$ ($13.01\text{ dB}$).

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This example demonstrates op-amp circuit performance calculation. With an op-amp GBW of 12 MHz and a target closed-loop gain of 10, the amplifier provides a flat response up to 1.2 MHz. Beyond 1.2 MHz, gain rolls off according to the single-pole transfer equation.
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# Gain-Bandwidth Considerations in Multistage Amplifiers

- Multistage Bandwidth Shrinkage Effect: Cascading $N$ identical non-interacting single-pole stages increases overall low-frequency gain to $A_{M,sys} = (A_M)^N$, but shrinks upper 3-dB bandwidth.
- Upper Cutoff Formula for $N$ Cascaded Stages: $f_{H,sys} = f_H \cdot \sqrt{2^{1/N} - 1}$, where $f_H$ is the upper cutoff of an individual stage.
- Bandwidth Reduction Factors: For $N=2$, $f_{H,sys} = 0.643 f_H$. For $N=3$, $f_{H,sys} = 0.510 f_H$. For $N=4$, $f_{H,sys} = 0.435 f_H$.
- Overall GBW Non-Constancy: Because overall gain grows exponentially $(A_M)^N$ while bandwidth shrinks moderately, the total multi-stage GBW is not equal to single-stage $GBW$, requiring careful stage-by-stage optimization.

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A common design trap is assuming multi-stage amplifiers maintain single-stage GBW rules. Cascading identical stages causes bandwidth shrinkage because each stage contributes -20 dB/decade roll-off, compounding the total gain reduction at frequencies near fH.
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# Summary & Key Engineering Formulas for Amplifier Parameters

- Gain Relationships: Voltage Gain $A_v(\text{dB}) = 20 \log_{10}|A_v|$, Power Gain $A_p(\text{dB}) = 10 \log_{10} A_p = A_v(\text{dB}) + A_i(\text{dB})$.
- Bandwidth & Cutoff Limits: Half-power (-3 dB) frequencies $f_L, f_H$; Bandwidth $BW = f_H - f_L \approx f_H$ for broadband systems.
- Gain-Bandwidth Invariance: $GBW = A_M \cdot f_H = f_T$. Negative feedback scales gain down by $(1 + A_0 \beta)$ and expands upper bandwidth by $(1 + A_0 \beta)$, keeping GBW constant.
- Transistor Frequency Parameters: Unity-gain frequency $f_T = \beta_0 f_\beta = \frac{g_m}{2\pi (C_\pi + C_\mu)}$, driven by bias current $I_C$ and internal capacitances.

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To conclude Lecture 21, remember that Gain-Bandwidth Product is the fundamental currency in analog circuit design. Every gain boost costs bandwidth, and feedback provides the exact knob to balance this trade-off deterministically.
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