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Verification of Maximum Power Transfer Theorem.
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 Objective:  Verification of Maximum power transfer theorem.

Maximum power is transferred from a source of given voltage and an internal impedence to the load impedence ZL in a circuit (Fig.-1) under three conditions.

 

                                                    (A) When only XL is adjustable:

Under this condition the power consumed by the load (I2*RL) is maximum, when I is maximum, since RL is constant.

                                                       

                                                              «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»I«/mi»«mo»=«/mo»«mfrac»«msub»«mi»V«/mi»«mi»s«/mi»«/msub»«mrow»«msub»«mi»R«/mi»«mi»i«/mi»«/msub»«mo»+«/mo»«mi»j«/mi»«msub»«mi»X«/mi»«mi»i«/mi»«/msub»«mo»+«/mo»«msub»«mi»R«/mi»«mi»L«/mi»«/msub»«mo»+«/mo»«mi»j«/mi»«msub»«mi»X«/mi»«mi»L«/mi»«/msub»«/mrow»«/mfrac»«mo»;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mfenced»«mn»1«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«msub»«mfenced close=¨|¨ open=¨|¨»«mi»I«/mi»«/mfenced»«mrow»«mi»m«/mi»«mi»a«/mi»«mi»x«/mi»«/mrow»«/msub»«mo»=«/mo»«mfrac»«msub»«mi»V«/mi»«mi»s«/mi»«/msub»«mrow»«msub»«mi»R«/mi»«mi»i«/mi»«/msub»«mo»+«/mo»«msub»«mi»R«/mi»«mi»L«/mi»«/msub»«/mrow»«/mfrac»«mo»;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»w«/mi»«mi»h«/mi»«mi»e«/mi»«mi»n«/mi»«mo»,«/mo»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi»X«/mi»«mi»L«/mi»«/msub»«mo»=«/mo»«mo»-«/mo»«msub»«mi»X«/mi»«mi»i«/mi»«/msub»«/mtd»«/mtr»«/mtable»«/math»

  

 This means that if the load reactance(XLis made equal magnitude and opposite in sign to the internal reactance(Xi), the power transferred is maximum.

                                             (B) When only RL is adjustable:

 

From equation (1) in section (a), one may write,

 

                                                                      «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»P«/mi»«mo»=«/mo»«mfenced close=¨|¨ open=¨|¨»«msup»«mi»I«/mi»«mn»2«/mn»«/msup»«/mfenced»«mo»*«/mo»«msub»«mi»R«/mi»«mi»L«/mi»«/msub»«mo»;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mfrac»«msup»«msub»«mi»V«/mi»«mi»s«/mi»«/msub»«mn»2«/mn»«/msup»«mrow»«mo»(«/mo»«msub»«mi»R«/mi»«mi»i«/mi»«/msub»«mo»+«/mo»«msub»«mi»R«/mi»«mi»L«/mi»«/msub»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«mo»+«/mo»«mo»(«/mo»«msub»«mi»X«/mi»«mi»i«/mi»«/msub»«mo»+«/mo»«msub»«mi»X«/mi»«mi»L«/mi»«/msub»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»*«/mo»«msub»«mi»R«/mi»«mi»L«/mi»«/msub»«mo»:«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mfenced»«mn»3«/mn»«/mfenced»«/mtd»«/mtr»«/mtable»«/math»

Differentiating the equation ( 3 ) w.r.t  RL and equating to zero, one obtains.  

 

                                                                «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»R«/mi»«mi»L«/mi»«/msub»«mo»=«/mo»«msqrt»«mrow»«msup»«msub»«mi»R«/mi»«mi»i«/mi»«/msub»«mn»2«/mn»«/msup»«mo»+«/mo»«mo»(«/mo»«msub»«mi»X«/mi»«mi»i«/mi»«/msub»«mo»+«/mo»«msub»«mi»X«/mi»«mi»L«/mi»«/msub»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/mrow»«/msqrt»«mo»:«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mfenced»«mn»4«/mn»«/mfenced»«/math»

                                                       

                                         (C) When both RL and XL are adjustable:

 

Under this condition both equation (2) are valid simultaneously and one obtains.

                                                              «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»R«/mi»«mi»L«/mi»«/msub»«mo»=«/mo»«msub»«mi»R«/mi»«mi»i«/mi»«/msub»«mo»,«/mo»«mo»§nbsp;«/mo»«msub»«mi»X«/mi»«mi»L«/mi»«/msub»«mo»=«/mo»«mo»-«/mo»«msub»«mi»X«/mi»«mrow»«mi»i«/mi»«mo»§nbsp;«/mo»«/mrow»«/msub»«/math»

                         

  

                                   

  

                                                                                                             Fig.-1.

 

 

 

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