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The transconductance of an NMOS is

$$ g_m = \frac{\partial i_d}{\partial V_{gs}} $$

Is the transconductance of a PMOS the same?

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No, replace Vgs with Vsg. The models for a PMOS and NMOS transistor are the same, except for PMOS you use Vsg instead of Vgs, Vsd instead of Vds, and the absolute value of \$\lambda\$ and \$V_a\$.

For instance, in a velocity-limited model (\$I_d=V_{sat} C_{ox} W_g (V_{sg}-|V_t|) (1+|\lambda| V_{sd})\$), the transconductance would be \$V_{sat}C_{ox}W_g(1+|\lambda| V_{sg})\$. In a mobility limited model it would be \$ \mu C_{ox} \frac{W_g}{L_g} (V_{sg}-|V_t|) (1+|\lambda| V_{sd})\$.

These results makes sense, because in a PMOS transistor the current is proportional to Vsg (or negative Vgs). As you can see, in both these expressions, the transconductance is positively proportional to Vsg, and thus the current changes with Vsg.

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Formally, sciencectn is right, and the voltages for pMOS are inverted compared to nMOS.

But the principle for the transistors is the same, so they are different but similar. The idea is that the transconductance is a differential parameter that relates variations in the output current to variations in the input voltage (put some more stress on the fact that it's differential, so only for small signals, which is a linear approximation).

So, for both is how much the output current changes when you vary the input voltage, keeping the output voltage constant.

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