# Voltage gain of a differential amplifier with tail resistor

Suppose the following differential amplifier:

simulate this circuit – Schematic created using CircuitLab

To calculate the differential gain, I take advantage of node P being an AC ground and I draw the half circuit small signal model:

simulate this circuit

I know the gain to be:

Is the gain the same if I replace the ideal current source with a resistor? What changes in my analysis?

simulate this circuit

• Do you only need the differential gain? That stays the same by symmetry. If you want the common mode gain as well then you have more work to do. Most people would want at least an estimate of that as inputs might not be exactly in antiphase. Jan 11, 2022 at 9:46
• For the common mode gain I probably need to calculate the transconductance of the whole circuit, right? Jan 11, 2022 at 9:50
• Right, and you'll certainly need the impedance at P. Jan 11, 2022 at 9:54
• So I guess a general question I have is when to just draw the half circuit and apply my knowledge of transistor topologies and when to use Av = - Gm*Rout where Gm is the general transconductance of a sub-circuit. Jan 11, 2022 at 10:10
• It depends whether you can use the symmetry or not. Pure diff in to pure diff out, use symmetry. Anything else, so single ended to diff, or diff to single ended, or when the two sides don't use equal components (and that's always, because tolerances) then do a full analysis. Jan 11, 2022 at 11:33

## 1 Answer

If I replace the ideal current source with a resistor Is the gain the same? What changes in my analysis?

The differential gain remains the same because the voltage at node P still behaves like 0 volts whether the resistor is placed there or not. It is forced to be 0 volts due to the antiphase nature of the two differential inputs having equal magnitude (that's how you measure differential gain).