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OpAmp circuit analysis

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The Electrician

Gold Member
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150
##\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}#
How about trying this:
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}##
 
How about trying this:
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
1,235
150
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

Mentor
9,159
1,021
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

Mentor
9,159
1,021
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.
 
If you use the D operator technique, I think you will improve your ability to solve the system:

http://www.codecogs.com/library/maths/calculus/differential/linear-simultaneous-equations.php

http://www.solitaryroad.com/c658.html
Even with d operators, how would I get rid of V2? There is only one equation with V2
If I combine NodeV2 and NodeV1, I get rid of the derivatives.
If I derive the combined equation and then solve NodeV1 for V_2' and plug in to get rid of V_2', would that work?
 

NascentOxygen

Mentor
9,159
1,021
If I derive the combined equation and then solve NodeV1 for V_2' and plug in to get rid of V_2', would that work?
That's what I've been anticipating.
 

rude man

Homework Helper
Insights Author
Gold Member
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642
You seem to want to 'get rid of derivatives'. You can't. In fact the problem asks for the differential equation for Vo. That will include time derivatives of Vo and/or Vin.

Had you been asked to find Vo itself that would mean solving the diff. eq. But you weren't given an expression for Vin so you can't solve for Vo.
 

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