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[Electrical engineering] Second order Parallel RLC Circuits.

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Homework Statement:

Source free Second order Parallel RLC Circuits. How to find functions for inductor current and capacitor voltage with respect to time after current source has been removed?

Relevant Equations:

α=1/2RC, w0=(1/lc)^1/2, v(t)=(A1+A2t)e^-αt
1571725964108.png
 

Answers and Replies

  • #2
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The inductor current curve and capacitor voltage curve should look like this after pulling the switch from which the circuit archived a steady state of 1 amp:
7PRPHDj.png


link to circuit: http://tinyurl.com/y4hq6c6u
 
  • #3
NascentOxygen
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The elements share a common voltage, u.

KCL says the sum of all currents = 0.

Form the second-order DE, then solve it.
 
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  • #4
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The elements share a common voltage, u.

KCL says the sum of all currents = 0.

Form the second-order DE, then solve it.
Can't I just use v(t)=(A1+A2t)e^-αt since the circuit is critically damped (α=w0) and solve for A1 and A2 based upon initial conditions?
 
  • #5
NascentOxygen
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Can't I just use v(t)=(A1+A2t)e^-αt since the circuit is critically damped (α=w0) and solve for A1 and A2 based upon initial conditions?
When it's critically damped, that is the way to solve it.
 
  • #6
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When it's critically damped, that is the way to solve it.
OK, but How do I find A1 and A2 based upon the initial conditions?
 
  • #7
NascentOxygen
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OK, but How do I find A1 and A2 based upon the initial conditions?
You set t=0 in the general solution, and substitute the known initial conditions.
 
  • #8
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You set t=0 in the general solution, and substitute the known initial conditions.
A2 is initial current or voltage
to get A1 you must differentiate the equation with respect to time
Find dv/dt or di/dt via KCL or KVL
the dv/dt or di/dt will come from the equation for the capacitor current or inductor voltage (basically ohms law for inductors and capacitors )
now you sub your dv/dt or di/dt into the equation you differentiated with respect to time

My functions for iL(t) and vc(t) have the exact same curves as the simulation:
e353fa5713.png
 
  • #9
NascentOxygen
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Your equation for v(x) agrees with what I arrived at.
 
  • #10
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Your equation for v(x) agrees with what I arrived at.
And what about the function for the current through the inductor iL(t)?
 

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