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.



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