The following points highlight the two theorems used for determining the current flowing in a circuit. The theorems are: 1. Maxwell Circulating Current Theorem 2. Node Voltage Theorem.

#### 1. Maxwell Circulating Current Theorem:

If a network with several sources has more than two nodes the current in it may be determined by Maxwell circulating current theorem. This is one of the most universal methods for solving networks.

In a number of cases, a network may be considered as consisting of a set of adjoining loops, each of which forms a polygon made up of several branches of the network (without any diagonals). Some branches of the network are common to two adjacent loops, while others form an external circuit where each branch occurs in one loop only.

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This theorem involves representing a current that is assumed to circulate around a closed loop by a curved arrow and labelling the arrow with its identifying current symbol I with a subscript.

By this theorem the current flowing through the branch common to two meshes will be equal to the algebraic sum of the two loop currents flowing through it. The direction of any loop current may be taken either as clockwise or counter-clockwise but for systematic solution the directions of all loop currents are assumed to be the same (say clockwise).

Then Kirchhoff’s second law is applied to each mesh and algebraic equations are obtained. The total number of independent equations is equal to the number of meshes (i.e. there are fewer equations than in a purely Kirchhoffian solution). Therefore, they can be solved as simultaneous equations to give the circulating currents and then the branch currents. Thus, this method eliminates a great deal of tedious calculation work involved in the branch current method.

Application of Maxwell circulating current theorem will be more clear from the following illustrations:

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Example 1:

Solve the network shown in Fig. 2.65 by mesh current method.

Solution:

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The network is redrawn, as illustrated in Fig. 2.66. There are two independent loops. The loop currents have been taken clockwise, as marked in the circuit diagram. The individual branch currents along with their directions of flow are also shown in the circuit diagram.

Applying Kirchhoff’s voltage law to meshes I and II we have:

#### 2. Node Voltage Theorem:

For application of node voltage theorem one of the node is taken as reference or zero potential or datum node and the potential difference between each of the other nodes and the reference node is expressed in terms of an unknown voltage (symbolized as V1, V2 or VA, VB or Vx, Vy etc.) and at every node Kirchhoff’s first (or current) law is applied assuming the possible directions of branch currents.

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This assumption does not change the statement of problem, since the branch currents are determined by the potential difference between respective nodes and not by absolute values of node potentials.

Like Maxwell’s circulating current theorem, node-voltage theorem reduces the number of equations to be solved to determine the unknown quantities. If there are n number of nodes, there shall be (n – 1) number of nodal equations in terms of (n – 1) number of unknown variables of nodal voltages. By solving these equations, nodal voltages are known to compute the branch currents.

When the number of nodes minus one is less than the number of independent meshes in the network, it is, in fact more advantageous. Moreover, it is particularly suited for networks having many parallel circuits with common ground connected node such as in electronic circuits.

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Example 2:

Consider, for example, a two node network, as illustrated in Fig. 2.72.

Node C has been taken as reference node. Let respectively with respect to node C. Let the current distribution be as shown on the circuit diagram (Fig. 2.72) arbitrarily. Now let us get independent equations for these two nodes.

Node A is the junction of resistors R1, R2, and R4. So current equation for node A is:

Node B is the junction of resistors R2, R3, and R5. So current equation for node B is:

The Eqs. (2.34) and (2.35) can now be solved to get the values of VA and VB and then the values of currents I1, I2, I3, I4 and I5 can be computed easily.

Example 3:

Two batteries A and B are connected in parallel to a load of 10 ohm. Battery A has an emf of 12 V and an internal resistance of 2 ohm and battery B has an emf of 10 V and internal resistance of 1 ohm. Using nodal analysis, determine the currents supplied by each battery and load current.

Solution:

Taking node C as reference node and the potentials of nodes A and B be VA and VB respectively and current distribution as shown in Fig. 2.73 arbitrarily.

From circuit diagram shown in Fig. 2.73.