Charles daEngineer

Minimum Stage Voltage Amp Bandwidth Calculations

Published on Tue Nov 04 2025

Introduction

This article will use the results of this post to create an n-stage voltage amp. Then I will determine the loop gain and the bandwidth.

Bandwidth for a Voltage Amp with a resistive feedback network

n-stage Voltage Amp with a resistive feedback network

The loop gain $L$ is equal to: $g_{m} {(\frac{- g m}{s C_{i s s}})}^{n - 1} \frac{d V_{1}}{d I_{g_{m}}}$ .

Circuit used to determine the loop gain, specifically

\frac{d V_{1}}{d I_{g_{m}}}

$\frac{d V_{1}}{d I_{g_{m}}} = - \frac{- R_{a} R_{l}}{s C_{i s s} (R_{s} R_{a} + R_{s} R_{b} + R_{s} R_{l} + R_{a} R_{l} + R_{a} R_{b}) + R_{a} + R_{b} + R_{l}}$

$∴ L = g_{m} {(\frac{- g m}{s C_{i s s}})}^{n - 1} \frac{R_{a} R_{l}}{s C_{i s s} (R_{s} R_{a} + R_{s} R_{b} + R_{s} R_{l} + R_{a} R_{l} + R_{a} R_{b}) + R_{a} + R_{b} + R_{l}}$

The loop gain is a low pass filter with $n - 1$ poles at the origin and one pole at $\frac{R_{a} + R_{b} + R_{l}}{C_{i s s} (R_{s} R_{a} + R_{s} R_{b} + R_{s} R_{l} + R_{a} R_{l} + R_{a} R_{b})}$ . The bandwidth of a butterworth filter is approximated using this equation.

$ω = \sqrt[n]{g_{m} {(\frac{g m}{C_{i s s}})}^{n - 1} \frac{R_{a} R_{l}}{C_{i s s} (R_{s} R_{a} + R_{s} R_{b} + R_{s} R_{l} + R_{a} R_{l} + R_{a} R_{b})}}$ $2 π B_{ω} = (\frac{g m}{C_{i s s}}) \sqrt[n]{\frac{R_{a} R_{l}}{(R_{s} R_{a} + R_{s} R_{b} + R_{s} R_{l} + R_{a} R_{l} + R_{a} R_{b})}}$

Now I can simplify further by noticing that a mosfet's transit frequency is: $f_{T} = \frac{g_{m}}{2 π C_{i s s}}$ .

$B_{ω} = f_{T} \sqrt[n]{\frac{R_{a} R_{l}}{(R_{s} R_{a} + R_{s} R_{b} + R_{s} R_{l} + R_{a} R_{l} + R_{a} R_{b})}}$ $f_{T} = B_{ω} \sqrt[n]{\frac{R_{s}}{R_{l}} (1 + \frac{R_{b}}{R_{a}}) + \frac{R_{s}}{R_{a}} + 1 + \frac{R_{b}}{R_{l}}}$