04 Multistage Bandwidth Effects

Here is your strictly PYQ-focused study note for the final conceptual topic in the Frequency Response chapter.

Topic 4: Multistage Bandwidth Effects (Bandwidth Shrinkage)

Exam-Focused Concept Note

This section is entirely theoretical. The examiners do not ask for heavy calculations here; instead, they want you to explain physically and mathematically what happens to the bandwidth when you chain (cascade) multiple amplifier stages together, and they strictly require a graphical sketch to prove it.

1. Core Concept: The “Shrinkage” Phenomenon

(Prerequisite Theory) The primary reason we cascade amplifier stages is to achieve a massive overall voltage gain (since $A_{t o t a l} = A_{1} \times A_{2} \times ... \times A_{n}$ ). However, there is a severe trade-off: Gain comes at the cost of bandwidth.

When you cascade multiple stages, the overall frequency response degrades. Specifically:

The overall low-cutoff frequency ( $f_{L}$ ) is pushed higher. It is dictated primarily by the single stage that has the highest low-cutoff frequency.
The overall high-cutoff frequency ( $f_{H}$ ) is pushed lower. It is dictated primarily by the single stage that has the lowest high-cutoff frequency.
The Result: Because the low-end limit moves up and the high-end limit moves down, the overall bandwidth of the system drastically shrinks.

2. The Mathematical Proof for Identical Stages

If you cascade $n$ identical amplifier stages, the drop-off at the edges of the frequency band becomes much steeper.

A single stage drops off at a rate of 20 dB/decade (or 6 dB/octave) outside the midband.
Two stages drop off at 40 dB/decade (12 dB/octave).
Three stages drop off at 60 dB/decade (18 dB/octave), and so on.

Because the edges drop off so much faster, the frequency at which the total gain drops by 3 dB (the new cutoff point) moves inward. If the original cutoff frequencies for a single stage are $f_{1}$ (lower) and $f_{2}$ (upper), the new overall system cutoff frequencies for $n$ stages are calculated using this specific factor: $2^{1/ n} - 1$ .

Memorize these two formulas for the exam:

Parameter	Formula for $n$ Cascaded Stages	Effect on Bandwidth
New Lower Cutoff ( $f_{L (n)}$ )	$f_{L (n)} = \frac{f _{1}}{2 ^{1/ n} - 1}$	The denominator is less than 1, so the new frequency becomes higher.
New Upper Cutoff ( $f_{H (n)}$ )	$f_{H (n)} = f_{2} 2^{1/ n} - 1$	The multiplier is less than 1, so the new frequency becomes lower.

(Example: For 2 identical stages ( $n = 2$ ), the factor becomes 0.64. The new upper cutoff is $0.64 \times f_{2}$ , meaning it dropped to 64% of its original value!).

3. The Graphical Illustration (Mandatory for Full Marks)

When the PYQ asks you to “illustrate graphically,” you must draw the Bode plot showing the shrinkage.

How to draw it: Draw a standard Bode plot curve (Gain vs. Frequency on a log scale). Draw a wide, flat “dome” and label it $n = 1$ . Then, inside that dome, draw a narrower dome with steeper sides and label it $n = 2$ . Finally, draw an even narrower dome inside that and label it $n = 3$ .
Draw a horizontal dotted line near the top representing the -3 dB drop level. Show how the intersection points ( $f_{L}$ and $f_{H}$ ) move closer and closer together as $n$ increases, visually proving that the bandwidth ( $f_{H} - f_{L}$ ) is shrinking.

Exact PYQs to Master for this Topic:

“How is the frequency response varied with number of stages in multistage amplifier? Illustrate graphically.” (Asked in: 2019, 2018, 2016)

That officially completes all the high-yield topics, numerical blueprints, and core theory for the Frequency Response chapter!

Based on your ECE 1209 curriculum shift analysis, the last major section to secure easy, predictable marks is Optoelectronics & Displays (LEDs, LCDs, Solar Cells), which is mostly straightforward theory.

Would you like to start tackling the optoelectronics notes, or is there anything else you’d like to review from Frequency Response first?

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