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 Superposition Theorem.

This Theorem is applied when we are to determine that current in one particular branch of a network containing several voltage source or current source or both voltage sources and current sources. This scheme is to determine how much current each of the individual sources contributes to the branch in question, and then add algebraically these components currents.

The supervision Theorem can be started as below :
In a linear resistance network containing two or more voltage source the current through any element ( resistance or source )
May be determined by adding together algebraically the current Produce by each  source are acting alone when all other voltage source had no internal resistance, the terminal to which it was connected are joined together. If there are current sources present They are removed and the network terminals to which they were connected are left open.

Maxwell circulating Current Theorem

If a network with several sources has more than two nodes the current in it may be determined by this Theorem. This Theorem one of the most universal methods for solving networks.
This Theorem involved representing a current that is assumed to circulate around a closed loop by curved arrow and labeling the arrow with it's identifying current symbols I with a subscript.
By this Theorem the current flowing through the branch common to the meshes will be equal to the algebraic sum of the two loop current flowing through it. The direction of any loop current may be taken either as clock Wise or counter clock Wise but for systemic solutions the directions of all loops Current are assumed to be the same. 
The kirchhoff's Second law is applied to each mesh and algebraic Equation are obtained. The total number of Independent equation equal to the number of mashes [ i.e. there are fewer Equations than in a purly kirchhoffian solution].
There are they can be solved  simultaneous Equations to give the circulating Currents and then the branch Current. 
Thus this method eliminates a great deal of tedious circulation work involved in the branch Current method.

Substitution Theorem.

According to the substitution Theorem, any branch in a network may be substitution by a different branch *{ having a Voltage source or a current sources or an impedance along with a Voltage source }*  without disturbing the voltage and current in the entire Circuits 
Provided the new branch has the same set of terminal Voltage and current as the original Branch. 
Some of the following equivalent Networks can be substitution in place of the other.

A+      4A             15 Ω               +100v-      -B

+A      4A                           + 40v-          - B

+ A                              40v                     -B

The substitution Theorem is applicable to any Circuit and can be applied to a branch which is not couple to other branches of the Circuit.
This Theorem cannot applied to the 
               I) Branches which are coupled to other branches of the Circuit.
              II) Branches of the Circuit unless the voltages and currents of the those branches are known.

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