Input Impedance of a diode connected MOSFET

Question

I was trying to derive the small-signal input impedance of a diode-connected MOSFET (figure included), and came up with this expression.

Is my method correct? I have been reading about methods of deriving the input and output impedances using a test voltage, and that is the method I have followed myself.

Some validation on the approach and the answer would help.

This is my solution.

Answer 1

[edit] To avoid confusion, from the comments you state that you want to find the impedance looking into the source. In other words, this situation:

^{simulate this circuit – Schematic created using CircuitLab}

I could find a few mistakes:

• In the third step, you missed an \$R_D\$:

$$ v_D = \frac{g_mR_D}{1+g_mR_D}V_T $$

• You also made an additional mistake in the second-last step. But it is irrelevant because of the first mistake.

The expression should resolve to:

$$ I_D = g_m\left(\frac{g_mR_D}{1+g_mR_D}-1\right)V_T $$

• Then another mistake is that by your voltage/current convention, the impedance will be

$$ Z = -\frac{V_T}{I_D} $$

Don't forget the minus sign.

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The result with these corrections will be:

$$Z = \frac{1}{g_m} + R_D$$

As expected. A diode-connected transistor has an approximate impedance of \$1/g_m\$ which is in series with \$R_D\$.