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SITUATION:

Nodal Analysis is nothing more than applying KCL and is very powerful. I'm working on developing a more clearer perspective of two approaches in Nodal Analysis "GENERAL & FORMAT". Mesh Analysis is also quite powerful but my questions below are pertaining to Nodal Analysis.

The "General Approach" seems to be more extensive when it comes to factoring and simplifying equations into a form to later solve using simultaneous calculation. After studying the "Format Approach" I started to see a more clearer picture of what my equations were actually representing after the factoring and simplification steps. It was showing that the voltage at the node I was applying KCL too, was simply being multiplied by the sum of the conductance attached to that node.

QUESTION #1: Among the many methods used in solving for unknown currents and unknown voltages would I be theoretically correct in saying that the "Format Approach" is effective in most cases? If not could you explain?

EXAMPLE #1:

I'm including an example here from my textbook to show the "Format Approach". circuit

"Format" Equation

QUESTION #2: How do I write the equation, "in the same format as the first example", that is used to solve the circuit below? Since the circuit below has 2 components that are shared, "12V Source & 10 ohm Resistor", I just can't figure out how to write the equation using the same approach as above.

enter image description here

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Do you mean "formal" approach? Could you give us a link to a description of this approach/method? It's not something that very many people will know off the top of their heads. –  The Photon Feb 26 '13 at 20:26
    
Also, when your circuit has voltage sources in it, you have to use the "modified nodal analysis" (see Wikipedia) instead of the "nodal analysis". So I'm not sure it's possible to answer question #2. –  The Photon Feb 26 '13 at 20:28
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3 Answers 3

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You can solve this circuit by more or less the same method you've given in the question; however you need to plug in one more equation (\$V_1=V_2+12\$) into the system and introduce an unknown current variable. So I'm not sure if we can call it a 'pure' nodal analysis.

This is what you've got to do:

  • Write KCL on the left node (\$I_s\$ is the current through the voltage source): $$6A=\frac { { V }_{ 1 }-{ V }_{ 2 } }{ { R }_{ 3 } } +\frac { { V }_{ 1 } }{ { R }_{ 3 } } +{ I }_{ s }$$

  • Do it again on the node on the right side: $$4A={ I }_{ s }-\frac { { V }_{ 2 } }{ { R }_{ 2 } } +\frac { { V }_{ 1 }-{ V }_{ 2 } }{ { R }_{ 3 } } $$

  • So far, we're in line with the method described in the question. As a last step, write down this one: $$V_1=V_2+12$$

Now we're left with 3 equations in three unknowns, which you can easily solve to get \$V_1, V_2 and I_s\$

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That was exactly what I was missing. I was not implementing the last step (V1=V2+12). So how do I dictate when and when not to enter a 3rd equation into a system of equations to solve? –  Shane Yost Feb 27 '13 at 16:28
    
An observation: introducing an unknown current variable for the current through the voltage source takes one out of the realm of node voltage analysis. Use of the supernode is the accepted practice. For example: enjoy-electrical.blogspot.com/2012/05/… –  Alfred Centauri Feb 27 '13 at 22:56
    
@AlfredCentauri: I complete agree with your first point; this is not a strict nodal analysis. But then, it gets the job done and the concept is rather simpler; and I guess that was what the OP was looking for. Anyway, thanks for pointing it out :) –  nav Feb 28 '13 at 6:45
    
@ShaneYost: A short answer would be whenever the formal methods fail. To get a better insight, you can try to analyse this circuit: (voltage_source) parallel to (resistor) parallel to (current_source + resistor in series). The straightforward mesh and nodal analyses fail here, but you can survive with the rather simple ohm's law, KCL and KVL :) –  nav Feb 28 '13 at 6:57
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I just can't figure out how to write the equation using the same approach as above.

You won't be able to because it's impossible.

You need two equations for the two unknown node voltage variables but you can only write one independent KCL equation in the two unknowns. This is done by treating the floating voltage source as a supernode.

\$6A - I_{R1} - I_{R2} - 4A = 0 = 2A - \dfrac{V_1}{4} - \dfrac{V_2}{2}\$

(Note that current through R3 both enters and exits the supernode and so cancels out of the KCL equation).

To solve this equation, you must use the KVL equation: \$V_1 - V_2 = 12V\$

You now have two independent equations for the two unknown node voltage variables.

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Format approach to nodal analysis only works for networks containing only independent current sources. No mixed voltage and current sources.

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