1st order circuit analysis

Question

I'm having trouble with this exercise which I don't have a solution for, I'm used to dealing with simpler 1st order RL or RC circuits. Mostly I'm a little confused with the sinusoidal generator and the R2 resistance which doesn't allow me to treat it as the simpler RL circuits I've dealt with.

$$R_1=2 \Omega,\: R_2=5 \Omega ,\: R_3=20 \: \Omega,\: L=100 mH \: \; i_G(t)=10·cos (100·t)\: A$$

At t'=0, voltage is max at the current source and the switch is opened .

For t'>0 I have to find: $$\tau, u(0^+), u_{\infty}(t)$$

I found the current through the inductor branch (which cannot change abruptly) But I'm confused as to how to find tau and continue.

Answer 1

With the switch assumed closed for a long period, you must find the instantaneous current, \$\small i_L(t)\$, through \$\small L\$ when \$\small u(t)\$ is at a maximum value, i.e. when the current through \$\small R_3\$ is at a maximum. Also required is the value of \$\small i_G(t)\$ at the same instant. These define the initial conditions that will exist when the switch is opened.

The currents can be derived by solving the following differential equation via the IF method: $$\small -u(t)=L\frac{di_L}{dt}+R_1i_L=(i_G-i_L)R_3 $$

or $$\small\frac{di_L}{dt}+\frac{(R_1+R_3)}{L}i_L=\frac{R_3}{L}i_G $$

Now, the switch is opened, \$\small R_2\$ enters the arena, and a new clock starts ticking from \$\small t^{'}=0\$. A similar differential equation to the above may be derived, and solved to give the required unknowns.