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THE EXPANDED FREQUENCY RESPONSE CHEAT SHEET

To ensure 100% exam coverage, this expanded cheat sheet includes the core frequency cutoff rules plus the mandatory related topics: Decibels, Bode Plots, Transistor High-Frequency Parameters, and Square-Wave Testing.

1. The Miller Effect & Capacitance (Guaranteed Theory)

• Definition: In inverting amplifiers, the feedback capacitance ($C_f$) between the input and output is magnified by the amplifier’s midband voltage gain ($A_v$).
• The Formulas:
  • Input Miller: $C_{Mi}=(1-A_v)C_f$. (Since $A_v$ is negative, $C_f$ is heavily multiplied).
  • Output Miller: $C_{Mo}=(1-1/A_v)C_f\approx C_f$.
• The Effect on High-Cutoff: The total input capacitance is $C_i=C_{Wi}+C_{be}+C_{Mi}$. Because the Miller effect creates a massive $C_{Mi}$, the total $C_i$ becomes very large. Since high-cutoff frequency is inversely proportional to $C_i$ ($f_{Hi}=\frac{1}{2\pi R_{Thi}{C}_i}$), this massive capacitance forces the high-frequency cutoff to drop significantly, severely limiting the amplifier’s bandwidth.

2. Low-Frequency Response (The “Highest is Boss” Rule)

• Cause: Large coupling and bypass capacitors ($C_S,C_C,C_E$) blocking signals at low frequencies.
• The Rule: Calculate all three break frequencies. The HIGHEST calculated frequency dictates the overall lower cutoff frequency ($f_L$).

Capacitor	BJT Formula & Resistance	JFET Formula & Resistance
Input ($C_S/C_G$)	$f_{L_S}=\frac{1}{2\pi (R_s+R_i)C_S}$ $R_i=R_1\parallel R_2\parallel \beta r_e$	$f_{L_G}=\frac{1}{2\pi (R_{sig}+R_i)C_G}$ $R_i=R_G$
Output ($C_C$)	$f_{L_C}=\frac{1}{2\pi (R_o+R_L)C_C}$ $R_o=R_C\parallel r_o$	$f_{L_C}=\frac{1}{2\pi (R_o+R_L)C_C}$ $R_o=R_D\parallel r_d$
Bypass ($C_E/C_S$)	$f_{L_E}=\frac{1}{2\pi R_e{C}_E}$ $R_e=R_E\parallel \left(\frac{R_s\parallel R_1\parallel R_2}{\beta }+r_e\right)$	$f_{L_S}=\frac{1}{2\pi R_{eq}{C}_S}$ $R_{eq}=R_S\parallel \frac{1}{g_m}$ (if $r_d=\infty$)

3. High-Frequency Response (The “Lowest is Boss” Rule)

• Cause: Tiny internal parasitic capacitors ($C_{bc},C_{be},C_{gs},C_{gd}$) and wiring capacitors shorting the signal to ground at high frequencies.
• The Rule: Calculate the input cutoff and output cutoff. The LOWEST calculated frequency dictates the overall upper cutoff frequency ($f_H$).

Network	Cutoff Formula	BJT Capacitances	JFET Capacitances
Input	$f_{Hi}=\frac{1}{2\pi R_{Thi}{C}_i}$	C_i = C_{Wi} + C_{be} + C_{Mi}$$R_{Thi} = R_s \parallel R_1 \parallel R_2 \parallel R_i	C_i = C_{Wi} + C_{gs} + C_{Mi}$$R_{Thi} = R_{sig} \parallel R_G
Output	$f_{Ho}=\frac{1}{2\pi R_{Tho}{C}_o}$	C_o = C_{Wo} + C_{ce} + C_{Mo}$$R_{Tho} = R_C \parallel R_L \parallel r_o	C_o = C_{Wo} + C_{ds} + C_{Mo}$$R_{Tho} = R_D \parallel R_L \parallel r_d

4. Decibels, Phase, and Bode Plot Rules

• Decibel Gain: Total dB gain of cascaded stages is the sum of the individual dB gains: $G_{dBT}=G_{dB1}+G_{dB2}+...+G_{dBn}$.
• The -3 dB Point: The frequencies ($f_L$ and $f_H$) where the voltage gain drops to 0.707 of its maximum midband value. At this point, the output power is exactly half (Half-Power Frequencies).
• Drop-off Rates: A single RC stage rolls off at -20 dB/decade (which is equivalent to -6 dB/octave).
• Phase Shift: At the low-cutoff frequency ($f_L$), the signal experiences a 45° phase shift.

5. Multistage Bandwidth Shrinkage

• Concept: Cascading $n$ identical stages dramatically increases voltage gain, but the drop-off rate becomes steeper (e.g., -40 dB/decade for 2 stages, -60 dB/decade for 3 stages).
• The Effect: The overall lower cutoff frequency is pushed higher, and the upper cutoff frequency is pushed lower, causing the bandwidth to shrink.
• The Shrinkage Formulas:
  • New Lower Cutoff: $f_{L(n)}=\frac{f_1}{\sqrt{2^{1/n}-1}}$.
  • New Upper Cutoff: $f_{H(n)}=f_2\sqrt{2^{1/n}-1}$.

6. High-Frequency Transistor Parameters & Gain-Bandwidth Product (GBP)

• Beta Cutoff Frequency ($f_{\beta }$): The frequency at which the transistor’s current gain ($\beta$ or $h_{fe}$) drops by 3 dB from its midband value.
  • Formula: $f_{\beta }=\frac{1}{2\pi h_{f{e}_{mid}}{r}_e(C_{be}+C_{bc})}$.
• Unit Gain Frequency ($f_T$): The frequency at which the transistor’s current gain drops to exactly 1 (0 dB).
  • Formula: $f_T=h_{f{e}_{mid}}\times f_{\beta }$.
• Gain-Bandwidth Product (GBP): For any specific amplifier, the product of its midband gain and its bandwidth remains constant.
  • Formula: $GBP=A_{v_{mid}}\times BW$. (If you decrease the gain, you proportionally increase the bandwidth).

7. Square-Wave Testing (Time-Domain Response)

Applying a square-wave to an amplifier instantly reveals its frequency limitations without needing a full sweep.

• Tilt / Sag (Tests Low-Frequency): A poor low-frequency response causes the flat top of the square wave to tilt downwards.
  • Formula: $Tilt(P)=\frac{V_{peak}-V_{drop}}{V_{peak}}$.
  • Relation to Cutoff: $f_L=\frac{P}{\pi }{f}_s$ (where $f_s$ is the frequency of the square wave).
• Rise Time (Tests High-Frequency): A poor high-frequency response prevents the square wave from rising instantly, rounding off the leading edge.
  • Cascaded Rise Time: The total rise time of a cascaded amplifier is $t_r\approx \sqrt{t_{r1}^2+t_{r2}^2+...+t_{rn}^2}$.