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Verification of Superposition Theorem
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 Objective:   To Verify Superposition Theorem.

 

If a number of voltage or current source are acting simultanously in a linear network, the resultant current in any branch is the algebraic sum of the currents that would be produced in it, when each source acts alone replacing all other independent sources by their internal resistances.

 

 Circuit Diagram:

  

  

 

 

  In a given figure apply superposition theorem , let us first take the sources V1 alone at first replacing V2 by short circuit.

                                         

                                                        

    
 
 
                                                                      
 
 
  
Here,
                                               «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd/»«/mtr»«mtr»«mtd»«msub»«mi»I«/mi»«mrow»«mn»1«/mn»«mo»§apos;«/mo»«/mrow»«/msub»«mo»=«/mo»«mfrac»«msub»«mi»V«/mi»«mn»1«/mn»«/msub»«mrow»«mfrac»«mrow»«msub»«mi»R«/mi»«mn»2«/mn»«/msub»«mo»*«/mo»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«/mrow»«mrow»«msub»«mi»R«/mi»«mn»2«/mn»«/msub»«mo»+«/mo»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«/mrow»«/mfrac»«mo»+«/mo»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«msub»«mi»I«/mi»«mrow»«mn»2«/mn»«mo»§apos;«/mo»«mo»§nbsp;«/mo»«/mrow»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mrow»«mn»1«/mn»«mo»§apos;«/mo»«/mrow»«/msub»«mo»*«/mo»«mfrac»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«mrow»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«mo»+«/mo»«msub»«mi»R«/mi»«mn»2«/mn»«/msub»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«msub»«mi»I«/mi»«mrow»«mn»3«/mn»«mo»§apos;«/mo»«/mrow»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mrow»«mn»1«/mn»«mo»§apos;«/mo»«/mrow»«/msub»«mo»-«/mo»«msub»«mi»I«/mi»«mrow»«mn»2«/mn»«mo»§apos;«/mo»«mo»§nbsp;«/mo»«/mrow»«/msub»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd/»«/mtr»«/mtable»«/math»
 Next, removing V1 by short circuit, let the circuit be energized by V2 only

                 

 

  
 
 
                                                                
 
 
            
           Here,
                                                                                                    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd/»«/mtr»«mtr»«mtd»«msub»«mi»I«/mi»«mrow»«mn»2«/mn»«mo»§apos;«/mo»«mo»§apos;«/mo»«/mrow»«/msub»«mo»=«/mo»«mfrac»«msub»«mi»V«/mi»«mn»2«/mn»«/msub»«mrow»«mfrac»«mrow»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«mo»*«/mo»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«/mrow»«mrow»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«/mrow»«/mfrac»«mo»+«/mo»«msub»«mi»R«/mi»«mn»2«/mn»«/msub»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«msub»«mi»I«/mi»«mrow»«mn»1«/mn»«mo»§apos;«/mo»«mo»§apos;«/mo»«/mrow»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mrow»«mn»2«/mn»«mo»§apos;«/mo»«mo»§apos;«/mo»«/mrow»«/msub»«mfrac»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«mrow»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«mo»+«/mo»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«msub»«mi»I«/mi»«mrow»«mn»3«/mn»«mo»§apos;«/mo»«mo»§apos;«/mo»«/mrow»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mrow»«mn»2«/mn»«mo»§apos;«/mo»«mo»§apos;«/mo»«/mrow»«/msub»«mo»-«/mo»«msub»«mi»I«/mi»«mrow»«mn»1«/mn»«mo»§apos;«/mo»«mo»§apos;«/mo»«/mrow»«/msub»«/mtd»«/mtr»«/mtable»«/math»
 
 
 As per superposition theorem,

 
                        «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi»I«/mi»«mn»3«/mn»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mrow»«mn»3«/mn»«mo»§apos;«/mo»«/mrow»«/msub»«mo»+«/mo»«msub»«mi»I«/mi»«mrow»«mn»3«/mn»«mo»§apos;«/mo»«mo»§apos;«/mo»«/mrow»«/msub»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi»I«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mrow»«mn»2«/mn»«mo»§apos;«/mo»«/mrow»«/msub»«mo»-«/mo»«msub»«mi»I«/mi»«mrow»«mn»2«/mn»«mo»§apos;«/mo»«mo»§apos;«/mo»«/mrow»«/msub»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi»I«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mrow»«mn»1«/mn»«mo»§apos;«/mo»«/mrow»«/msub»«mo»-«/mo»«msub»«mi»I«/mi»«mrow»«mn»1«/mn»«mo»§apos;«/mo»«mo»§apos;«/mo»«/mrow»«/msub»«/mtd»«/mtr»«/mtable»«/math»
 
 

 

 

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