How to do node analysis?

Question

So we are currently learning node analysis and was a given set of problems and I am stuck on problem 8 (top one in the attached pic). I can't figure out how to find the correct voltages at the nodes A and B (G is ground). Any help would be greatly appreciated. Also, the XX V

can be replaced with any quantity.

http://www.flickr.com/photos/29385059@N07/2744576925/

Answer 1

Ok, here's how you do it. First, you have to determine the number of nodes in the circuit. In this case, you have three nodes. Now that you know you have three nodes, you pick two up and use one as a reference, use the earth one as the reference and stay with nodes a and b to do your calculations.

For node a, this is the analysis.

I15+I10+I40+I=0

now, for each current, you have to write them using Va and Vb, like this

Va/15 + (Va-Vb)/10 + (Va-Vb-30)/40 + 3 = 0

making some math:

23Va/120-Vb/8 = -9/4

Now for node 2 you get that

I10 + I40+ I20 + I25 =0

so

(Va-Vb)/10 + (Va-Vb-30)/40 + Vb/20 + (Vb+xxV)/25 = 0

doing a little math

Va - 7Vb/200 + xxV/25 =3/4

and now, solving the equations simultaneously, you get:

Va = -30(2*xxV-69)/107 and Vb = -4(23*xxV-1275)/107

HOpe this helps.

Btw, when i write I15 it means that it's the current that passes through the 15 ohm element.

Answer 2

node analysis is basically an analysis of what goes in and what goes out

here's the analysis for A:

V(in) = I2 * (15  - 10) = 5 * I2

V(out) = I3 * 40 + 30 = (I2 - 3) * 40 + 30

I(node a) = I2 - I3 - 3 = 0 => I3 = I2 - 3

use the equations above to solve for V(in) and V(out) in terms of I2

VA = V(out) - V(in)  = 5 * I2 - (I2 - 3) * 40 - 30

simplify.

also do the same at B...  you should be able to find out what I2 is numerically. Plug into the above to get VA (etc)