# Differential voltage to current converter

I have this circuit to solve:

with $R_2=R_3$ and $R_4=R_5$. I want to find the transfer function between $I_{\text{out}}$ and $V_{\text{in}}$ (differential). I'm trying to use superposition, but the math is very long. Do you have any other idea on how to solve it? Any hint?

I've divided $V_{\text{in}}$ in $V_1$ and $V_2$, and tried superposition. I've found the current on $R_2$ and $R_4$, but I'm having a problem finding out a relation between $V_{\text{x}}$ and $I_{\text{out}}$ in order to solve the system.

• Try to exploit the fact that the upper op-amp is a differential amplifier, while the lower one is a voltage follower. Between the two, lays a resistor... Oct 31 '16 at 9:36
• I don't get it. Can you tell me something more? Oct 31 '16 at 9:42
• As always, your opamps try everything they got to keep the voltage on both input pins equal and the current into the input pins are zero. If you have the current though all resistors, you have everything you need to calculate it. Oct 31 '16 at 9:47
• Consider the circuit neglecting R1 and the lower op-amp: that's a differential amplifier and it's characterized by a certain transfer function. Consider the remaining op-amp only in its configuration: that's a voltage follower. Now, the follower drives a branch of the differential amp and in turn it's driven by the differential amp through a resistor. Try to put together all the pieces with the knowledge on op-amps that you should have and you should be able to come out with a solution for your circuit. Oct 31 '16 at 9:50
• (Vx - V-)/R4 = Iout? Oct 31 '16 at 9:50

The following simplifies all the calculations:

simulate this circuit – Schematic created using CircuitLab

Steps:

• Opamp terminals draw zero current. So the current directions are as above.
• Write the equations for $i_n$ then solve for $V_k$ (1).
• Write the equations for $i_p$ then solve for $V_k$ (2).
• From (1) = (2), you'll find a nice relation between $(V_p - V_n)$ and $(V_x - V_o)$
• Finally, since $i_o=(V_x - V_o) / R_1$, you'll obtain the relationship between $i_o$ and $(V_p - V_n)$.

I have the result, but don't ask me the full calculations. All the info you need are above.

• Thank you. This was very helpfull. Thanks to everyone who answered my question. Oct 31 '16 at 10:05