# finding open circuit voltage gain

below I have this schematic to find the open circuit voltage gain. This circuit basically utilizes three resistors from which R_f is supposed to be the internal feedback resistor. It also utilizes the voltage controlled current source. My job is to find A_vo = v_o/v_i. My work is illustrated below $$A_{vo}\:=\:\frac{v_o}{v_i};\:v_o\:=\:-g_mv_iR_2\: therefore \:A_{vo}=-g_mR_2$$ but the exact answer looks like this: $$\:A_{vo}=-g_mR_2\:\frac{1-\frac{1}{g_mR_f}}{1+\frac{R_2}{R_f}}$$ somehow am having trouble incorporating R_f into the equation. Any help please?

* My apologies, in the schematic, the dependant source is a voltage controlled current source*

• The source symbol is confusing. Also, is it supposed to be a current source? $g_m$ normally means transconductance. Actually, the whole diagram could do with re-drawing.
– Chu
May 25, 2018 at 9:29
• @Chu, thank you for pointing this out. I added a disclaimer to description. Unfortunately this is due to the limited resources available on LTspice, which is what I used to draw the schematic. Thank you. May 25, 2018 at 16:28

Part of the current of the transconductance amplifier will flow back to the input rather than through $R_2$. So the KCL law in the output is not

$$g_mv_i + \frac{v_o}{R_2} = 0$$

As this will lead to your incorrect expression. But the correct KCL law would be:

$$g_mv_i + \frac{v_o}{R_2} + \frac{v_o-v_i}{R_f} = 0$$

Solving this will result in the exact expression given.

• thank you so much. This actually worked. And I track your work and it actually makes a lot of sense. Thank you very much, now I understand how to solve the problem and found where my mistake was. May 30, 2018 at 14:06

The three resistors are excited by a current-source which is controlled by Vi. The output current of the source is i=gm*Vi
For calculöating the required ratio Vo/Vi you first need the voltage developped across the parallel combination of R2||(R1+Rf).

Hence, we have Vo=gm*Vi[R2||(R1+Rf)]

From this, it is a simple task to find the ratio Vo/Vi.

• I don't think this reflects the correct solution. Am trying to manipulate it such that it looks like the correct solution but it is not working. Also, the correct solution does not involve the use of R1 May 25, 2018 at 16:30
• I agree to your comment only if Vi is an ideal voltage source. This was not clear from the beginning. In this case, you must use the correct symbol for a voltage source!
– LvW
May 26, 2018 at 8:40

Your expression for Vo only holds true if there is no current flowing through Rf. If Vo and Vi are not the same then current will flow through Rf and eventually through R2. Apply kirchhoff's current law at Vo and you will find an expression with both Vi and Vo. Your expression should account for three currents in total and solve for Vo/Vi. You may find the answer in a different form that what's given, with some clever dividing by "one" you can derive the same form as what was given.

Hope this Helps.

• Can you provide a more mathematical based answer? Because I did nodal analysis and for some reason due to the dependence on v_i in the dependant current source things are cancelling and am not getting the same answer. May 26, 2018 at 14:16