Solving Impedance/Admittance for V_s w/ 2 Caps, 2 Inductors, 2 Res.

[VinnyCee]

Impedance and Admittance: Find Vs given Io in a circuit with 2 caps,2 inductors,2res

Homework Statement

Find $V_s$ if $I_0\,=\,2\angle0\deg$ A.

http://img237.imageshack.us/img237/6291/problem954zz0.jpg

Homework Equations

KCL, KVL

The Attempt at a Solution

I made a new diagram:

http://img201.imageshack.us/img201/9359/problem954part2gz3.jpg

But how do I combine the left hand equivalent impedance, so that the final circuit to work on would be this:

http://img201.imageshack.us/img201/1690/problem954part3fj1.jpg $$\frac{1}{Z_3}\,=\,\frac{1}{Z_1}\,+\,\frac{1}{j4\Omega}\,=\,\frac{1}{2\,+\,2j}\,+\,\frac{1}{4j}$$

Figuring $V_o$:

$$V_o\,=\,I_o\,Z_2\,=\,\left(2\angle0\right)\left(2\angle45\right)\,=\,4\angle45$$

$$I_L\,=\,\frac{V_0}{j2}\,=\,\frac{4\angle45}{\sqrt{2}\angle63.43}\,=\,\frac{4}{\sqrt{2}}\angle-18.43$$

$$I_1\,=\,-\left(I_L\,+\,I_0\right)\,=-\left[\,\left(\frac{4}{\sqrt{2}}\angle-18.43\right)\,+\,\left(2\angle0\right)\right]$$

$$V_1\,=\,I_1\,Z_1$$

I need this to find $V_s$.

$$V_s\,=\,V_1\,-\,V_0$$

How do I finish? Please help!

 

[VinnyCee]

I did the calculations and conversions of complex rectangular to polar and got this as a final answer:

$$V_S\,=\,9.581\,cos\left(t\,+\,29.70)$$

Does that look right?