Loop Analysis of Electric Circuits

In this method, we set up and solve a system of equations in which the unknowns are loop currents. The currents in the various branches of the circuit are then easily determined from the loop currents. (Click here for a tutorial on loop currents vs. branch currents.)

The steps in the loop current method are: Example 1: Find the current flowing in each branch of this circuit.
Solution:
• The number of loop currents required is 3. • We will choose the loop currents shown to the right. In fact these loop currents are mesh currents.

• Write down Kirchoff's Voltage Law for each loop. The result is the following system of equations: Collecting terms this becomes: This form for the system of equations could have been gotten immediately by using the inspection method.

• Solving the system of equations using Gaussian elimination or some other method gives the following currents, all measured in amperes:
I1=0.245, I2=0.111 and I3=0.117 • Reconstructing the branch currents from the loop currents gives the results shown in the picture to the right. Example 2: Find the current flowing in each branch of this circuit.

Solution:
• The number of loop currents required is 3. • This time we will choose the loop currents shown to the right.

• Write down Kirchoff's Voltage Law for each loop. The result is the following system of equations: Collecting terms this becomes: This form for the system of equations could have been gotten immediately by using the inspection method.

• Solving the system of equations using Gaussian elimination or some other method gives the following currents, all measured in amperes:
I1 = - 4.57, I2 = 13.7 and I3 = - 1.05 • Reconstructing the branch currents from the loop currents gives the results shown in the picture to the right.