Superposition Theorem in Electrical Network

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SUPERPOSITION THEOREM 

The superposition theorem for electrical circuits states that for a linear system the response (voltage or current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedance 

To ascertain the contribution of each individual source, all of the other sources first must be “turned off” (set to zero) by-

 

(1) Replacing all other independent voltage source with a short circuit (thereby eliminating difference of potential i.e. V=0; internal impedance of ideal  voltage source is zero (short circuit).

(2) Replacing all other independent current source with an open circuit (thereby eliminating current i.e. I=0; internal impedance of ideal current source is infinite (open circuit).

ANALYSIS :

Suppose we are to analyse superposition theorem in given circuit , we are to find current in chart?cht=tx&chl=R %7B%7B2%7D%7D resistor with the help of superposition theorem 

 

superposition theorem

STEP (1) : Consider only one voltage source at a time and replace other voltage source with short circuit or with their internal resistance , and find the current  in resistor R2 due to this voltage source only 

superpostion theorem

 

STEP (2): Determine the a particular current ( suppose Ib=Ib1 in this case ) due to only one voltage source (due to voltage chart?cht=tx&chl=V %7B%7B1%7D%7D) in chart?cht=tx&chl=R %7B%7B2%7D%7D resistor

                                                       chart?cht=tx&chl=R %7B%7BT1%7D%7D   =    chart?cht=tx&chl=R %7B%7B1%7D%7D + chart?cht=tx&chl=R %7B%7B2%7D%7D%5Csetminus%5Csetminus%20R %7B%7B3%7D%7D

                                                      chart?cht=tx&chl=R %7B%7BT1%7D%7D    =   chart?cht=tx&chl=R %7B%7B1%7D%7Dchart?cht=tx&chl=%5Cfrac%7BR %7B2%7D%20%5Ctimes%20R %7B3%7D%7D%7BR %7B2%7D%2BR %7B3%7D%20%7D

                                                        chart?cht=tx&chl=I %7B%7BT1%7D%7D    =     chart?cht=tx&chl=%5Cfrac%7BV %7B1%7D%7D%7BI %7BT1%7D%20%7D

 

        So current in R2  ,            chart?cht=tx&chl=I %7B%7Bb1%7D%7D      =    chart?cht=tx&chl=I %7B%7BT1%7D%7D%5Ctimes chart?cht=tx&chl=%5Cfrac%7BR %7B3%7D%20%7D%7BR %7B2%7D%2BR %7B3%7D%20%7D  ( by current division rule )

 

 

STEP(3): Determine a particular current ( suppose Ib=Ib2 in this case ) due to only one voltage source (due to voltage chart?cht=tx&chl=V %7B%7B2%7D%7D ) in chart?cht=tx&chl=R %7B%7B2%7D%7D resistor 

 

superposition theorem 1

                                                             chart?cht=tx&chl=R %7B%7BT2%7D%7D  =  chart?cht=tx&chl=R %7B%7B3%7D%7D + chart?cht=tx&chl=R %7B%7B1%7D%7D%20%5Csetminus%5Csetminus%20R %7B%7B2%7D%7D

                                                                       =   chart?cht=tx&chl=R %7B%7B3%7D%7D%20%2B%20%5Cfrac%7B%20R %7B1%7D%20%5Ctimes%20R %7B2%7D%20%7D%7B%20R %7B1%7D%20%2B%20R %7B2%7D%7D

                                                             chart?cht=tx&chl=I %7B%7BT2%7D%7D    =    chart?cht=tx&chl=%5Cfrac%7BV %7B2%7D%20%7D%7BR %7BT2%7D%20%7D

      so current in R2     ,                   Ib2     =    chart?cht=tx&chl=I %7B%7BT2%7D%7D × chart?cht=tx&chl=%5Cfrac%7B%20R %7B1%7D%7D%7B%20R %7B1%7D%2BR %7B2%7D%20%7D (By current division rule )

 STEP (4): So total current in resistor chart?cht=tx&chl=R %7B%7B2%7D%7D will be (algebraic sum of two individual current ) :

                                                              Ib = Ib1+Ib2 

LIMITATION OF SUPERPOSITION THEOREM :

(1) It is valid for the direct calculation of only voltage or current , not valid for power calculation directly as power is proportional to square of current or voltage ( non linear nature )

(2) This theorem is valid  only for  linear and bilateral network 

(3) In this case, circuit element may be time variant or time invariant 

(4) It combines properties of homogeneity and superposition 

(5) It is used in a circuit when more than one active source is present 

(6) It is not applicable to the circuit consisting only dependent source