differential output amplifier formula derivation

i was reading Art of Electronics book and then I counter this circuit which is differential output amplifier the author just give the formula without derivation , (i try to derivate it my self and it took me many hours and i fail )can somebody prove this formula with a circuit analysis equations thanks [![enter image description here][1]][1]

• show your working and then maybe we can spot your error, most on here don't have the time to do all the work for you. – Solar Mike Oct 14 '17 at 9:05
• @Solar Mike I added my work I hope the picture is clear to read – user156489 Oct 14 '17 at 9:43
• @isam you want $V_{01}$ and $V_{02}$ ? – Rohit Oct 14 '17 at 9:54
• @Rohit the final output is what I want it is the difference between Vo1 and Vo2 , Voutput=Vo1-Vo2 – user156489 Oct 14 '17 at 10:16

Apply nodal at the non inverting and inverting termial of upper Op amp

at non inveting terminal

$\frac{V_x-v_{in}}{Rg}+\frac{V_x-V_{out2}}{R_f}=0----------------(1)$

at inverting terminal

$\frac{V_x-v_{out2}}{R_4}+\frac{V_x-V_{out1}}{R_1}=0---- -----------(2)$

Also nodal at the lower op amp inverting terminal

$\frac{0-v_{out2}}{R_2}+\frac{0-V_{out1}}{R_3}=0---------------(3)$

As given $R_1=R_2=R_3=R_4=R$

So from equation $(3)$

$V_{out2}=-V_{out1}$

and from equation $(2)$,$V_x=\frac{V_{out1}+V_{out2}}{2}$ from this $V_x=0$

Now from equation $(1)$

$V_{out2}=-\frac{V_{in}R_f}{R_g}$ also $V_{out1}=\frac{V_{in}R_f}{R_g}$

Now the desired output $G=\frac{V_{out1}-V_{out1}}{Vin}=2\frac{R_f}{R_g}$

• MathJAX tip: if you use the  full line syntax you can add a \tag 1, etc., to autogenerate the right-aligned line references without the need for '------'. – Transistor Oct 14 '17 at 10:50
• @isam let me know if you get it from here or not if not I will update it for the desired output you wanted that is $2R_f/R_g$ – Rohit Oct 14 '17 at 12:23
• @Rohit thanks I solve it but in your first equation there must be Vout2 not vout1 and I am not sure if your third equation is true or not please edit this mistakes – user156489 Oct 14 '17 at 12:34
• @isam Yes you are right,I edited it.actually I got that result so just wanted to update it...before that i asked you. – Rohit Oct 14 '17 at 12:58

Prove at first the following useful formula:

Use this formula to express the input voltages of the opamps. If the circuit is stable, then both inputs of an opamp have the same voltage. This gives two equations to you. One (after eliminating the equal resistors in your original drawing before any edits of the question) tells that Vout2=-Vout1 and the other says the relation between Vin and Vout1. The gain is (Vout1-Vout2)/Vin ie. 2Vout/Vin

The whole calculation starting from the formula for Vz takes less than 2 minutes.