# Calculating input impedance of ICIS opamp circuit

My textbook just gives the formulas w/o derivations and I've been trying to prove them when possible. But I feel stuck while calculating the input impedance of below ICIS amplifier.  My work :

Since almost all the input current goes through $R_2$ , voltage at the meeting point of the 3 resistors is given by $-i_{in}R_2$ .

Then the output current is given by $$- i_{out} = i_{in} + \dfrac{i_{in} R_2}{R_1}$$

So the output voltage equals $$V_{out} = i_{out}R_L - i_{in}R_2$$

Also we have $V_{out} = -A_{VOL}V_{-}$

With some algebra to above 3 equation we get $$\dfrac{V_-}{i_{in}} = \dfrac{\left(1+\dfrac{R_2}{R_1}\right)R_L + R_2}{A_{VOL}} = \dfrac{\dfrac{\color{red}{R_L}}{B} + R_2}{A_{VOL}}$$

This is no where close to the given formula and I don't seem to simplify any further. Also I feel my work is wrong because $\color{red}{R_L}$ should not be there... Any help ?

• Possible duplicate of How do you determine the input impedance for an inverting amplifier? – Nick May 25 '18 at 14:02
• @Nick in that question input impedance is trivial. Its just $R_1$. Also that question is more about calculating compensating resistance, not input impedance.. – AgentS May 25 '18 at 14:05
• Since the voltage at V- is Vx + Iin*R2. So the V- voltage will depend on RL because Vx voltage depend on RL/ – G36 May 25 '18 at 16:25

I believe you have accidentally assumed that $V_- = 0V$ when you said

Since almost all the input current goes through $R_2$, voltage at the meeting point of the 3 resistors is given by $−i_{in}R_2$.

While it actually should be the voltage drop:

$$V_- - V_x = i_{in}R_2$$

You can still keep the equation

$$V_{out} = -A_{VOL}V_-$$

But the KCL at the crossing point ($V_x$) becomes:

$$i_{in} + \frac{V_{out} - V_x}{R_L} = \frac{V_x}{R_1}$$

This results in the expression

$$Z_{in} = \frac{V_-}{i_{in}} = \frac{(R_1 + R_2)R_L + R_1R_2}{R_L + (A_{VOL} + 1)R_1}$$

You can verify this formula by simulating the schematic below. I tried some combinations and I'm fairly certain that the input impedance is correct. 