# RLC Circuit, Find Diff Eq for VC - Can someone check my work?

1. Nov 29, 2016

### eehelp150

1. The problem statement, all variables and given/known data
Find the differential equation for VC
Circuit #1

Circuit #2

2. Relevant equations
KCL

3. The attempt at a solution
Circuit #1:
Node VC:
$\frac{V_C-V_{in}}{R_1}+\frac{1}{L_1}\int_{0}^{t}(V_C-V_2)+C_1\dot{V_C}=0$
Node V2:
$\frac{1}{L_1}\int_{0}^{t}(V_2-V_C)+\frac{V_2}{R_2}=0$

Derive NodeVC and solve for V2
$\frac{\dot{V_C}}{R_1}-\frac{\dot{V_{in}}}{R_1}+\frac{V_C}{L_1}-\frac{V_2}{L_1}+C_1\ddot{V_C}=0$
$\frac{\dot{V_C}}{R_1}-\frac{\dot{V_{in}}}{R_1}+\frac{V_C}{L_1}+C_1\ddot{V_C}=\frac{V_2}{L_1}$
$\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}=V_2$

Plug that into node V2 equation
$\frac{1}{L_1}\int_{0}^{t}((\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C})-V_C)+\frac{\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}}=0$

Simplify
$\frac{1}{L_1}\int_{0}^{t}(\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+C_1L_1\ddot{V_C})+\frac{\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}}{R_2}=0$
get rid of integral
$\frac{1}{L_1}(\frac{L_1{V_C}}{R_1}-\frac{L_1{V_{in}}}{R_1}+C_1L_1\dot{V_C})+\frac{\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}}{R_2}=0$
Simplify
$(\frac{{V_C}}{R_1}-\frac{{V_{in}}}{R_1}+C_1\dot{V_C})+\frac{\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}}{R_2}=0$

Multiply everything by R2
$R_2(\frac{{V_C}}{R_1}-\frac{{V_{in}}}{R_1}+C_1\dot{V_C})+\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}=0$

I end up with:
$\ddot{V_C}+\frac{\dot{V_C}}{R_1C_1}+\frac{\dot{V_C}}{L_1}+\frac{V_CR_2}{R_1C_1L_1}+\frac{V_C}{C_1L_1}=\frac{V_{in}}{R_1C_1}+\frac{V_{in}}{R_1C_1L_1}$

Circuit #2 (V2=VC)
Node V1
$\frac{V_1-V_{in}}{R_1}+\frac{1}{L}\int_{0}^{t}(V_1)+\frac{V_1-V_2}{R_2}=0$
Node V2
$\frac{V_2-V_1}{R_2}+C\dot{V_2}=0$
Solve NodeV2 for V1
$V_2-V_1+R_2C\dot{V_2}=0$
$V_1=V_2+R_2C\dot{V_2}$
plug into NodeV1
$\frac{V_2+R_2C\dot{V_2}-V_{in}}{R_1}+\frac{1}{L}\int_{0}^{t}(V_2+R_2C\dot{V_2})+\frac{V_2+R_2C\dot{V_2}-V_2}{R_2}=0$

Simplify
$\frac{V_2}{R_1}+\frac{R_2C\dot{V_2}}{R_1}-\frac{V_{in}}{R_1}+\frac{1}{L}\int_{0}^{t}(V_2+R_2C\dot{V_2})+C\dot{V_2}=0$
Derive everything to get rid of integral
$\frac{\dot{V_2}}{R_1}+\frac{R_2C\ddot{V_2}}{R_1}-\frac{\dot{V_{in}}}{R_1}+\frac{V_2}{L}+\frac{R_2C\dot{V_2}}{L}+C\ddot{V_2}=0$

Combine like terms
$C\ddot{V_2}+\frac{R_2C\ddot{V_2}}{R_1}+\frac{\dot{V_2}}{R_1}+\frac{R_2C\dot{V_2}}{L}+\frac{V_2}{L}=\frac{\dot{V_{in}}}{R_1}$
Divide everything by C
$\ddot{V_2}+\frac{R_2\ddot{V_2}}{R_1}+\frac{\dot{V_2}}{CR_1}+\frac{R_2\dot{V_2}}{L}+\frac{V_2}{LC}=\frac{\dot{V_{in}}}{CR_1}$
Simplify
$\ddot{V_2}(1+\frac{R_2}{R_1})+\dot{V_2}(\frac{1}{CR_1}+\frac{R_2}{L})+\frac{V_2}{LC}=\frac{\dot{V_in}}{CR_1}$

Am I doing these two problems right?

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2. Dec 4, 2016