Solving Circuits with Kirchoff Laws

Example 1: Find the three unknown currents ( $I_1,I_2,I_3$) and three unknown voltages ( $V_{ab}, V_{bd}, V_{cb}$) 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.

Example 2: Solve the following circuit:


We see that either of the loop-current and node-voltage methods requires to solve a linear system of 3 equations with 3 unknowns.

Example 3: Solve the following circuit with $I=0.5 A$, $V=6 V$, $R_1=3\Omega$, $R_2=8\Omega$, $R_3=6\Omega$, $R_4=4\Omega$. This circuit has 3 independent loops and 3 independent nodes.


In summary, here we have taken advantage of either the given current source by the following: By either of these methods, the number of unknown loop currents or node voltages is reduced by one.

In other words, to simplify the analysis, it is preferable to

Example 4: In the circuit below, $V_1=12 V$, $V_2=6 V$, $R_1=3\Omega$, $R_2=8\Omega$, $R_3=6\Omega$, $R_4=4\Omega$.


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 $R_1$, $R_2$, and $R_3$ 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 $R_1=100\Omega$, $R_2=5\Omega$, $R_3=200\Omega$, $R_4=50\Omega$, $V=50V$, and $I=0.2A$.

problembase1.gif problembase1a.gif