CIRCUIT ANALYSIS: 3 Resistors, 3 Voltage Src, 1 Current Src - Find v1, v2, v3

[VinnyCee]

Homework Statement

Find the node voltages $V_1$ through $V_3$ in the circuit below.

http://img413.imageshack.us/img413/2240/chapter3problem328an.jpg

Homework Equations

KCL, KVL, V = i R, super-node

The Attempt at a Solution

I added 5 currents and 2 KVL loops.

http://img101.imageshack.us/img101/3699/chapter3problem32part24yt.jpg

$$V_2\,=\,42\,V$$ <----- Right?

Now I do KVL loop 1

$$(-12\,V)\,+\,(20\,V)\,+\,V_3\,=\,0\,\,\longrightarrow\,\,V_3\,=\,-8\,V$$

And KVL at loop 2

$$(-20\,V)\,+\,(10\,V)\,+\,(V_1\,-\,V_3)\,=\,0\,\,\longrightarrow\,\,V_1\,-\,V_3\,=\,10\,V$$

$$V_1\,-\,(-8\,V)\,=\,10\,\,\longrightarrow\,\,V_1\,=\,2\,V$$

Does that seem right?

 

[mjsd]

i think you meant V2=12V. otherwise correct

 

[VinnyCee]

What about the 10 V and the 20 V that are also connected to node $V_2$?

 

[mjsd]

your statement represents a common mistake/misconception about circuit theory... when we talk about V2 we actually mean V2 with respect to (w.r.t.) the ground we are chosen (that's why I say this all the time), because if you choose a different ref node, value of V2 can change. For example, if you choose V3 to be your ref node (ie. ground), then V2 will have the value of 20V.
you get V2=12V for this daigram because, you see that the potential difference between V2 and ref node is V2-0 = 12 so V2 =12.
Now, you could also say
V2-V3=20 => V2 = 20+V3 (with V3 still unknown as this stage)
or
V2-V1=10 => V2 = 10+V1 (V1 too is unknown)

note that they never quite "add together", in fact if V1 and V3 are connected directed to ground, you have an inconsistent circuit...because all sources in circuit theory are assumed to be ideal.

anyway, if you sub V1, V3 you calculated into those equations above, you will find that all gives V2=12