Engineering Solve V_A in Circuit with Nodal Analysis

[VinnyCee]

Homework Statement

Use nodal analysis to find V_A in the circuit below.

http://img180.imageshack.us/img180/7004/chapter3problem236vy.jpg

Homework Equations

KVL, KCL, v = i R, super-node?

The Attempt at a Solution

I made a few currents and showed the obvious KVL loop.

http://img201.imageshack.us/img201/4716/chapter3problem23part22nf.jpg

V_1\,=\,V_A <---- Right?

I_1\,=\,\frac{V_1\,-\,0}{2\Omega}

I_2\,=\,\frac{V_3\,-\,0}{16\Omega}

I_3\,=\,\frac{V_1\,-\,V_2}{4\Omega} <------Right?I know KCL for V_1:

\frac{30\,-\,V_1}{1\Omega}\,=\,\frac{V_1\,-\,0}{2\Omega}\,+\,\frac{V_1\,-\,V_2}{4\Omega}

7\,V_1\,-\,V_2\,=\,120I know KCL for V_3:

\frac{V_1\,-\,V_2}{4\Omega}\,+\,3\,=\,\frac{V_3\,-\,0}{16\Omega}

4\,V_1\,-\,4\,V_2\,-\,V_3\,=\,-48For KVL1:

-V_1\,+\,4\,I_3\,+2\,V_A\,+\,V_3\,=\,0

2\,V_1\,-\,V_2\,+\,V_3\,=\,0Now, putting those three EQs into a matrix and rref:

\left[\begin{array}{cccc}7&amp;-1&amp;0&amp;120\\4&amp;-4&amp;-1&amp;-48\\2&amp;-1&amp;1&amp;0\end{array}\right]\,\,\longrightarrow\,\,\left[\begin{array}{cccc}1&amp;0&amp;0&amp;\frac{648}{29}\\0&amp;1&amp;0&amp;\frac{1056}{29}\\0&amp;0&amp;1&amp;-\frac{240}{29}\end{array}\right]

This gives me V_1\,=\,V_A equal to 22.34 V.

Thanks for the help mjsd!

 

[mjsd]

all your equations look sound to me..

 

[mjsd]

oh..hang on... 2nd line for KCL for V3 is wrong where is the 48 gone?

 

[VinnyCee]

Corrected, thanks much!