1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Nov 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the differential equation for VC
    Circuit #1
    upload_2016-11-29_1-49-13.png
    Circuit #2
    upload_2016-11-29_2-6-8.png
    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?
     

    Attached Files:

  2. jcsd
  3. Dec 4, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted