Engineering [Electrical engineering] Second order Parallel RLC Circuits.

[Muskyboi]

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

 

[Muskyboi]

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:

link to circuit: http://tinyurl.com/y4hq6c6u

 

[NascentOxygen]

The elements share a common voltage, u.

KCL says the sum of all currents = 0.

Form the second-order DE, then solve it.

 

[Muskyboi]

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?

 

[NascentOxygen]

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.

 

[Muskyboi]

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?

 

[NascentOxygen]

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.

 

[Muskyboi]

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:

 

[NascentOxygen]

Your equation for v(x) agrees with what I arrived at.

 

[Muskyboi]

Your equation for v(x) agrees with what I arrived at.

And what about the function for the current through the inductor iL(t)?