Complex Circuit Analysis Simplification

[jonlg_uk]

I have the following circuit...

I have managed to derive an equation for U2:

Now I want to substitute U2 into U1, in order to derive equation for U1

I have defined U1 as:

Now I substitute U2 into the above equation. This is how far I get...I am stuck...


My circuit simulator is saying that U1 is equal to:

However I cannot simplify to get this equation. Can someone please help I have spent hours trying to get U1 to look like the U1 that the simulator produces.


I thank you in advance.

 

[Jony130]

Have you try to replace R1, R2 and E2 with his Thevenine equivalent circuit ?
Eth = R2/ ( R1 + R2 ) * E1 and Rth = R1||R2

V3 = I * R3

$$I = \frac{Eth}{\sqrt{(Rth + R3)^2 + Xc^2)}} = \frac{E1*\frac{R2}{R1+R2}}{\sqrt{(R1||R2 + R3)^2 + Xc^2)}}$$

And

$$V3 = \frac{E1*R2*R3}{(R1+R2)\sqrt{(R1||R2 + R3)^2 + Xc^2)}}$$

 

[jonlg_uk]

Have you try to replace R1, R2 and E2 with his Thevenine equivalent circuit ?
Eth = R2/ ( R1 + R2 ) * E1 and Rth = R1||R2

V3 = I * R3

$$I = \frac{Eth}{\sqrt{(Rth + R3)^2 + Xc^2)}} = \frac{E1*\frac{R2}{R1+R2}}{\sqrt{(R1||R2 + R3)^2 + Xc^2)}}$$

And

$$V3 = \frac{E1*R2*R3}{(R1+R2)\sqrt{(R1||R2 + R3)^2 + Xc^2)}}$$

No I haven't tried this before, but I did just try it and I still can't derive U1. Perhaps I am being stupid...

 

[Jony130]

A made a small error instead of V3 I should write U1

IMy circuit simulator is saying that U1 is equal to:

What is the name of your simulation program?

 

[jonlg_uk]

Thanks for your help Jony, the simulator is giving me this equation:

I haven't derived that myself^. When I expand out the equation you derived:


$$V3 = \frac{E1*R2*R3}{(R1+R2)\sqrt{(R1||R2 + R3)^2 + Xc^2)}}$$

I still not can get a the equation looking like the simulator one.


BTW the simulator is called Solve Elec, you can download it for free.

[URL="http://www.physicsbox.com/indexsolveelec2en.html[/URL]

 

[jonlg_uk]

Ah done it! you are right I was being stupid. I am upset with myself that I never thought to do a thevin equivalent :( thanks a lot for your help. I can sleep peacefully tonight.