# OpAmp circuit analysis

• Engineering
The Electrician
Gold Member
##\frac{V_N}{R_3}+\frac{V_N-V_O}{R_4}=0##
##V_N=V_P##
##\frac{V_P}{R_3}+\frac{V_P-V_O}{R_4}=0##
##V_P(1+\frac{R_4}{R_3})-V_O=0##

Solve eq1 for Vp
##V_P = \frac{R_2}{R_1}(V_1-V_{in})+V_1+C_2R_2(\dot{V_1}-\dot{V_2})##
Plug into earlier equation
##V_P(1+\frac{R_4}{R_3})-V_O=0##
##(\frac{R_2}{R_1}(V_1-V_{in})+V_1+C_2R_2(\dot{V_1}-\dot{V_2}))(1+\frac{R_4}{R_3})-V_O=0##
Expand
##\frac{R_2}{R_1}(V_1-V_{in})+V_1+C_2R_2(\dot{V_1}-\dot{V_2})+\frac{R_4R_2}{R_3R_1}(V_1-V_{in})+\frac{R_4V_1}{R_3}+\frac{R_4C_2R_2(\dot{V_1}-\dot{V_2})}{R_3}-V_O=0##

solve eq2 for V_1
##C_1\dot{V_P}+\frac{V_P-V_1}{R_2}=0##
##V_1=V_P+R_2C_1\dot{V_P}##
derive
##\dot{V_1}=\dot{V_P}+R_2C_1\ddot{V_P}##
Am I on the right track? I still don't know how to get rid of ##V_2## and ##\dot{V_2}#
Solve this: ##\frac{V_P}{R_3}+\frac{V_P-V_O}{R_4}=0##
for Vp, then substitute that in: ##V_1=V_P+R_2C_1\dot{V_P}##

Solve this: ##\frac{V_P}{R_3}+\frac{V_P-V_O}{R_4}=0##
for Vp, then substitute that in: ##V_1=V_P+R_2C_1\dot{V_P}##
##V_P=\frac{R_3V_O}{R_3+R_4}##
##\dot{V_P}=\frac{R_3\dot{V_O}}{R_3+R_4}##
##V_1=\frac{R_3V_O}{R_3+R_4}+R_2C_1(\frac{R_3\dot{V_O}}{R_3+R_4})##

The Electrician
Gold Member
OK. Now you have V1 in terms of Vo. Substitute that in some of the earlier equations. Get Vin in terms of V1. Slog away.

OK. Now you have V1 in terms of Vo. Substitute that in some of the earlier equations. Get Vin in terms of V1. Slog away.
I need Vin, so Equation 1 has to be used right? I still can't see any way to get rid of V2

NascentOxygen
Staff Emeritus
If you combine your first and fourth equations, the derivatives can disappear and you will be left with V2 in terms of things you know.

If you combine your first and fourth equations, the derivatives can disappear and you will be left with V2 in terms of things you know.
##\frac{V_1-V_{in}}{R_1}+\frac{V_1-V_p}{R_2}+\frac{V_2}{R_6}+\frac{V_2-V_o}{R_5}=0##

becomes

##\frac{\frac{R_3V_O}{R_3+R_4}+R_2C_1(\frac{R_3\dot{V_O}}{R_3+R_4})-V_{in}}{R_1}+\frac{R_2C_1(\frac{R_3\dot{V_O}}{R_3+R_4})}{R_2}+\frac{V_2}{R_6}+\frac{V_2-V_o}{R_5}=0##
How would I get rid of V2?

NascentOxygen
Staff Emeritus
Isolate V2. Determine ##\dot V_2##.

Now you can substitute for all necessary terms so as to leave you with an equation relating Vo to Vin.

Isolate V2. Determine ##\dot V_2##.

Now you can substitute for all necessary terms so as to leave you with an equation relating Vo to Vin.
I'm not really following. There is no other equation with V2.

NascentOxygen
Staff Emeritus