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eehelp150
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Homework Statement
Find the differential equation for Vo
Homework Equations
KCL
The Attempt at a Solution
At node v1:
##\frac{V_1-V_{in}}{R_1}+\frac{V_1-V_p}{R_2}+C_2(\dot{V_1}-\dot{V_2})=0##
At node vp:
##C_1\dot{V_P}+\frac{V_P-V_1}{R_2}=0##
At node vn:
##\frac{V_N}{R_3}+\frac{V_N-V_o}{R_4}=0##
At node v2:
##\frac{V_2}{R_6}+\frac{V_2-V_o}{R_5}+C_2(\dot{V_2}-\dot{V_1})=0##
At node Vo:
##\frac{V_o-V_N}{R_4}+\frac{V_o-V_2}{R_5}=0##This is my attempt at solving for Vo:
Take equation of node VP and solve for V1
##C_1\dot{V_P}+\frac{V_P-V_1}{R_2}=0##
##V_1=R_2C_1\dot{V_P}+V_P##
derivative
##\dot{V_1}=R_2C_1\ddot{V_P}+\dot{V_P}##
Take equation of node Vo and solve for V2
##\frac{V_o-V_N}{R_4}+\frac{V_o-V_2}{R_5}=0##
##V_2 = \frac{R_5}{R_4}(V_o-V_{N})+V_o##
derivative
##\dot{V_2} = \frac{R_5}{R_4}(\dot{V_o}-\dot{V_{N}})+\dot{V_o}##
Plug these four (V1, dV1, V2, dV2) equations into node V1 equation
##\frac{V_1-V_{in}}{R_1}+\frac{V_1-V_p}{R_2}+C_2(\dot{V_1}-\dot{V_2})=0##
##\frac{R_2C_1\dot{V_P}+V_P-V_{in}}{R_1}+\frac{R_2C_1\dot{V_P}+V_P-V_P}{R_2}+C_2(R_2C_1\ddot{V_P}+\dot{V_P}-\frac{R_5}{R_4}(\dot{V_o}-\dot{V_{N}})+\dot{V_o})=0##
Simplify
##\frac{R_2C_1\dot{V_P}+V_P-V_{in}}{R_1}+C_1\dot{V_P}+C_2(R_2C_1\ddot{V_P}+\dot{V_P}-\frac{R_5}{R_4}(\dot{V_o}-\dot{V_{N}})+\dot{V_o}=0##
By property of opamps: VN = VP
##\frac{R_2C_1\dot{V_P}+V_P-V_{in}}{R_1}+C_1\dot{V_P}+C_2(R_2C_1\ddot{V_P}+\dot{V_P}-\frac{R_5}{R_4}(\dot{V_o}-\dot{V_{P}})+\dot{V_o}=0##
This is where I am stuck... Are my original equations correct? Can someone give me a hint as to how to get rid of Vp?