How to setup KVL equations with element laws?

[Laplace]

... BTW, there are actually three nodes in this circuit.

Then why is only one node equation necessary for a solution? Think about it.

 

[chopnhack]

I don't know the answer. All I can tell you is that 3 is the correct number of nodes per the good folks at MIT. Incidentally, these are the same folks that won't accept my answer for parallel resistance, so take it for what its worth.

 

[(*steve*)]

Laplace's method involves arbitrarily assigning one node as a ground now then calculating the voltage at each other now with respect to his ground node. In this case the node voltage at the other side of the voltage source is given be the characteristics of the voltage source itself. This leaves a single unknown node.

now the voltage across each element can be determined by the difference between the unknown node voltage and that of a known now voltage. Where those elements are resistors we can determine the current through them using ohms law. Finally you use these currents to create a kcl equation for the current at the unknown node. All of this is a single step! The equation is then simply solved.

This technique is covered by the course but you may not be up to it yet.

You come eventually to the same(or equivalent) equation using kvl and kcl, t it takes more steps.

So the key is the kcl now equation for the d with 4 elements connected to it, and then rewriting the currents in terms of the kvl equations and the resistances.

 

[(*steve*)]

Bloody phone wont let me fix words. Hope to makes some sense.

The end bit should add...

so you replace the currents with voltages from kvl divided by resistance and solve

 

[Harald Kapp]

This circuit has one node plus ground so a single node equation is required. But I never fuss with writing down node currents since the purpose of KCL is to find voltages, so I sum all the currents at the node in terms of the node voltage. To keep things simple I assume that all currents are positive (flowing away from the node) except for current sources flowing into the node which are negative. Then I can type the node equation directly from the schematic into MathCad. V3=12V. 2A flows through R3, 9A flows in reverse through the 6V source.

The notion of a current flowing into or out of a node is well supported by using arros as e.g. in post #25. Unfortunatley this is not very common in schematics based on the American way of thinking. ;(
If you don't know which way the current flows in reality, you can mark arbitrary directions. Once you have arrows in place, you can do the math and if it turns out that a current is negative, this only means that it flows against the direction of the arrow.

Although you can do the full math on the set of equations describing the complete original circuit (which is fine if you use a tool like e.g. mathematica), I find it very helpful to reduce the circuit to an equivalent circuit first, solving the equivalent circuit and using these solutions to refine the detail for the original circuit. In the above example I would replace R1 and R2 by an equivalent resistor R12 with a value of R12=R1*R2/(R1+R2). This reduces the circuit by one resistor and one current, thus makes it easier to solve. Once you have the solution for the reduced circuit, you get the voltage across R12 from V(R12)=I(R12)*R12. And since R1 and R2 are in parallel, V12=V(R1)=V(R2) and tehrefore I(R1=V(R12)/R1 and I(R2)=V(R12)/R2.
The more you can reduce the equivalent circuit, the easier it becomes to solve it. At the cost of a second step to refine the solution to the original circuit.

You may also look up Norton's theorem, Thevenin's theorem and the superposition theorem which help in simplifying a circuit for manual solution.

 

[Laplace]

... Although you can do the full math on the set of equations describing the complete original circuit (which is fine if you use a tool like e.g. mathematica) ...

I still remember when first learning circuit analysis that I would always use mesh analysis because dealing with loop currents seemed more intuitive than node voltages. But when I obtained Maple software for my Amiga computer to solve algebra problems it became most important to begin with a correct set of equations. So I evolved to a mechanical method of writing node equations that always gives the correct result without having to think about it much. Because in my experience the source of most errors lies in thinking about it and using simplifications with conceptual shortcuts to make the algebra easier.

 

[Harald Kapp]

Laplace, each man to his own.

I respect your proceedings as well as I prefer mine. I only wanted to give chopnhack another perspective. He'll have to find his own way around these matters.

Neither of both methods is wrong nor superior to the other. It is a matter of habit and personal preferences.