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Mutual Inductance measurement by Heydweiller Bridge
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Objective:  

To determine the mutual inductance of an unknown coil interms of standard capacitance.  

 

                                           Fig.1.  Heydweilling Bridge 

 This bridge is a modification of carey foster bridge. It is used to measure the mutual inductance interms of standard capacitance and is then known as Heydweilling bridge. 

The circuit diagram is shown in Fig. 1 under balance condition.

This bridge has an unusual feature, one of its arm is short circuited and therefore the potential drop across this arm is zero. In order to achieve balance, the potential drop across opposite arm should also be equal to zero and for this reason negative coupling is needed for the mutual inductance. 

At balance, 

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«msub»«mi»I«/mi»«mn»1«/mn»«/msub»«mo»(«/mo»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«mi»j«/mi»«mi»§#969;«/mi»«msub»«mi»L«/mi»«mn»1«/mn»«/msub»«mo»)«/mo»«mo»-«/mo»«mo»(«/mo»«msub»«mi»I«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«msub»«mi»I«/mi»«mn»2«/mn»«/msub»«mo»)«/mo»«mi»j«/mi»«mi»§#969;«/mi»«mi»M«/mi»«mo»=«/mo»«mn»0«/mn»«mo»§nbsp;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»1«/mn»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§nbsp;«/mo»«msub»«mi»I«/mi»«mn»1«/mn»«/msub»«mo»(«/mo»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«mo»+«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»j«/mi»«mi»§#969;«/mi»«msub»«mi»C«/mi»«mn»3«/mn»«/msub»«/mrow»«/mfrac»«mo»)«/mo»«mo»=«/mo»«msub»«mi»I«/mi»«mn»2«/mn»«/msub»«msub»«mi»R«/mi»«mn»4«/mn»«/msub»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»-«/mo»«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»«/mtable»«/math»

The solution of above two equation gives:

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»M«/mi»«mo»=«/mo»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«msub»«mi»R«/mi»«mn»4«/mn»«/msub»«msub»«mi»C«/mi»«mn»3«/mn»«/msub»«mo»§nbsp;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»3«/mn»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«msub»«mi»L«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mfrac»«mrow»«mi»M«/mi»«mo»(«/mo»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«mo»+«/mo»«msub»«mi»R«/mi»«mn»4«/mn»«/msub»«mo»)«/mo»«/mrow»«msub»«mi»R«/mi»«mn»4«/mn»«/msub»«/mfrac»«mo»=«/mo»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«msub»«mi»C«/mi»«mn»3«/mn»«/msub»«mfenced»«mrow»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«mo»+«/mo»«msub»«mi»R«/mi»«mn»4«/mn»«/msub»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»4«/mn»«mo»)«/mo»«/mtd»«/mtr»«/mtable»«/math»

If the above bridge is used to measurement of capacitance , equation (1) and (2) may be written as 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«msub»«mi»C«/mi»«mn»3«/mn»«/msub»«mo»=«/mo»«mfrac»«mi»M«/mi»«mrow»«msub»«mi»R«/mi»«mn»1«/mn»«/msub»«msub»«mi»R«/mi»«mn»4«/mn»«/msub»«/mrow»«/mfrac»«mo»§nbsp;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»5«/mn»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«msub»«mi»R«/mi»«mn»3«/mn»«/msub»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»R«/mi»«mn»4«/mn»«/msub»«mo»(«/mo»«msub»«mi»L«/mi»«mn»1«/mn»«/msub»«mo»-«/mo»«mi»M«/mi»«mo»)«/mo»«/mrow»«mi»M«/mi»«/mfrac»«mo»§nbsp;«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»6«/mn»«mo»)«/mo»«/mtd»«/mtr»«/mtable»«/math»

 In the measurement of mutual inductance with this bridge, R3 is a separate resistance while in measurement of capacitance C3 is not a separate unit but represents the equivalent series resistance of the capacitor and thus can be determined in terms of the elements of the bridge.

 

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