Monday, May 27, 2013

Maxwell’s Loop Current Method

This method is particularly well-suited to coupled circuit solutions employs of loop or mesh current instead of branch currents as in Kirchhoff’s laws. In this method the currents in different mesh are assigned continuous paths so that they do not split at a junction into branch currents. Basically, this method consists of writing loop voltage equation by Kirchhoff’s voltage law in terms of unknown loop currents.

 In fig above, there are two loops (or mesh).Let the current flowing in upper loop be I1 and the current flowing in lower loop is I2. Let the direction of flow is same (in clockwise direction) for both currents as shown in fig.2.4. Then applying KVL in upper loop, we get:
-10 -2I1 -1(I1 –I2) +12 = 0
Or      3I1 –I2 = 2 ….. (1)
Similarly, applying KVL in lower loop, we get:
-12 -1(I2 –I1) -5 I2 = 0
Or   I1 -6I2 = 12 …. (2)
Solving these equations, we get
I1 = 0 and I2 = 2 A.
            In this way in Maxwell’s Loop current method, loop current is calculated to find all the branch currents.

Some helpful points:
  • Assume that all the loop currents are flowing in same direction (say in clockwise).
  • When writing loop equation (KVL) on Nth loop, assume the current flowing in that loop i.e. In is largest current. See on above example: For loop (1) “-1(I1 –I2)“ and  for loop (2) “-1(I2 –I1)“ is taken because  Iis assumed as largest current in loop (1) and I2 Iis assumed as largest current in loop (2).

#Case: If a current source is present in the perimeter of the loop:
            If there is a current source in the perimeter of the Nth loop, then the current flowing through that loop (In) is  equal to the magnitude of the current of the current source.


            In this case, as shown in fig.2.5, there is a current source on the perimeter of loop (1). Hence I1 equals -2A because the direction of  I1 (assumed) is opposite to the direction of source current.

#Case: If a current source is present on the common branch of two loops:

            If a current source is present on the common branch of Mth and Nth loop, then sum or difference of currents flowing on those loops is equal to the magnitude of the current source present. And a new loop is considered to obtain the equation of loop which is known as super-loop. Super-loop is made by mixing those loops which has common current source (i.e. Mth and Nth loop) without considering that common branch.   


             For example, there is 3A current source on the common branch of loop (1) and loop (3).But in loop (2), there is no current source on perimeter as well as on common branch. So, for loop (2), we can write a simple KVL equation:

 for loop (2)…………(1)
For loop (1) and (3)
……………………. (2)
And from super loop:
……………. (3)
By solving these three equations we can get the unknown currents.

Cramer’s rule and matrix model

Cramer’s rule is used for the solution of linear equations by determinants. Let the system of linear equations in three unknown I1,I2 and I3 are:
May be written in matrix from as:


Where Rii = the total resistance of the ith loop with +ve sign.
            Rij = common resistance between ith loop and jth loop with –ve sign.
Then by Cramer’s rule, the solution of these simultaneous equations is given by:

            Cramer’s rule can be used for solving simultaneous equations if the numbers of unknown are more than two. This gives quick results.