Example 1: Find the three unknown currents ( ) and three unknown voltages ( ) in the circuit below:
Note: The direction of a current and the polarity of a voltage can be assumed arbitrarily. To determine the actual direction and polarity, the sign of the values also should be considered. For example, a current labeled in left-to-right direction with a negative value is actually flowing right-to-left.
All voltages and currents in the circuit can be found by either of the following two methods, based on KVL or KCL respectively.
Find currents from a to b, from c to b, and from b to d.
In the same circuit considered previously, there are only 2 nodes and ( and are not nodes). We assume node is the ground, and consider just voltage at node as the only unknown in the problem. Apply KCL to node , we have
We could also apply KCL to node d, but the resulting equation is exactly the same as simply because this node d is not independent.
As special case of the node-voltage method with only two nodes, we have the following theorem:
If there are multiple parallel branches between two nodes and , such as the circuit below (left), then the voltage at node can be found as shown below if the other node is treated as the reference point.
Assume there are three types of branches:
Applying KCL to node , we have:
The dual form of the Millman's theorem can be derived based on the loop circuit on the right. Applying KVL to the loop, we have:
Example 2: Solve the following circuit:
Example 3: Solve the following circuit with , , , , , . This circuit has 3 independent loops and 3 independent nodes.
Assume three loop currents (left), (right), (top) all in clock-wise direction. We take advantage of the fact that the current source is in loop 1 only, with loop current , and get the following two (instead of three) loop equations with 2 unknown loop currents and :
Assume the three node voltages with respect to the bottom node treated as ground to be (left), (middle), (right). We take advantage of the fact that one side of the voltage source is treated as ground, and get the note voltage . Then we have only two (instead of three) node equations with 2 unknown node voltages and :
In other words, to simplify the analysis, it is preferable to
Example 4: In the circuit below, , , , , , .
Find all node voltages with respect to the top-left corner treated as the ground. Then do the same when the middle node where all three resistors , , and join is treated as ground is treated as the ground.
Example 5: The two circuits shown below are equivalent, but you may want to choose wisely in terms of which is easier to analyze. Solve this circuit using both node voltage and loop current methods. Assume , , , , , and .