Circuit Theory/1Source Excitement/Example 7
Prior Work calculating Steady State/Particular Solution
Have already found the steady state/Particular solution:
Calculating the transient/Homogeneous Solution
Need to find the transient/Homogeneous Solution to:
There is no VS ... this makes the homogeneous solution easy!
Finding the time constant:
Now find see if it works:
- divide through by A, cancel L's
It works, therefore it must be the solution:
Now must find the initial conditions.
Determining the Constants
There are two constants. and come from any homogeneous solution to a non-homogeneous differential equation equation. These were not ignored in the steady state phasor solutions earlier, the fact that they were not being computed was pointed out.
There are two initial conditions that have to be true:
- initial source voltage has a value at t=0:
- initial current through the inductor has at t=0 has to be 0, thus the current throughout the entire series circuit is 0 at t=0
Finding two initial conditions
Two equations are necessary to find A and C.
Initially the current through the inductor and the entire circuit is going to be zero:
- , thus .
This means that setting t=0, have one equation:
Evaluating this at t=0:
The second equation comes from the loop:
Substituting for i(t)S and V(t)S, taking the differential and then evaluating at t=0, get:
So solving get:
This agrees with the Laplace solution and simulation.