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Q meter Experiment
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Objective :  To determine accurate Quality Factor of an unknown coil. 

The determination of the storage factor Q is one of the most widely used means in the laboratory for testing radio frequency coils, inductors and capacitors. The storage factor is equal to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»Q«/mi»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»§#969;«/mi»«mn»0«/mn»«/msub»«mi»L«/mi»«/mrow»«mi»R«/mi»«/mfrac»«/math» , where w0 is the resonant frequency, L is the inductance and R is the effective resistance of the a coil. The effective resistance, R, is never determined directly since its value depends upon the value of frequency. 

Principle of Working: 

                Fig.1. The practical circuit of unknown coil 

The principle of working of this useful laboratory instrument is based upon the well-known characteristics of a resonant series R-L-C circuit.  

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»A«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»r«/mi»«mi»e«/mi»«mi»s«/mi»«mi»o«/mi»«mi»n«/mi»«mi»a«/mi»«mi»n«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»f«/mi»«mi»r«/mi»«mi»e«/mi»«mi»q«/mi»«mi»u«/mi»«mi»e«/mi»«mi»n«/mi»«mi»c«/mi»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»f«/mi»«mn»0«/mn»«/msub»«mo»§nbsp;«/mo»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»w«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»h«/mi»«mi»a«/mi»«mi»v«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«msub»«mi»X«/mi»«mi»C«/mi»«/msub»«mo»=«/mo»«msub»«mi»X«/mi»«mi»L«/mi»«/msub»«/mtd»«/mtr»«mtr»«mtd»«mi»w«/mi»«mi»h«/mi»«mi»e«/mi»«mi»r«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«mi»a«/mi»«mi»p«/mi»«mi»a«/mi»«mi»c«/mi»«mi»i«/mi»«mi»t«/mi»«mi»i«/mi»«mi»v«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»r«/mi»«mi»e«/mi»«mi»a«/mi»«mi»c«/mi»«mi mathvariant=¨normal¨»tan«/mi»«mi»c«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«msub»«mi»X«/mi»«mi»C«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mo»*«/mo»«mi»§#960;«/mi»«mo»*«/mo»«msub»«mi»f«/mi»«mn»0«/mn»«/msub»«mo»*«/mo»«mi»C«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§nbsp;«/mo»«mi»i«/mi»«mi»n«/mi»«mi»d«/mi»«mi»u«/mi»«mi»c«/mi»«mi»t«/mi»«mi»i«/mi»«mi»v«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»r«/mi»«mi»e«/mi»«mi»a«/mi»«mi»c«/mi»«mi mathvariant=¨normal¨»tan«/mi»«mi»c«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«msub»«mi»X«/mi»«mi»L«/mi»«/msub»«mo»=«/mo»«mn»2«/mn»«mo»*«/mo»«mi»§#960;«/mi»«mo»*«/mo»«msub»«mi»f«/mi»«mn»0«/mn»«/msub»«mo»*«/mo»«mi»L«/mi»«/mtd»«/mtr»«/mtable»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»r«/mi»«mi»e«/mi»«mi»s«/mi»«mi»o«/mi»«mi»n«/mi»«mi»a«/mi»«mi»n«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»f«/mi»«mi»r«/mi»«mi»e«/mi»«mi»q«/mi»«mi»u«/mi»«mi»e«/mi»«mi»n«/mi»«mi»c«/mi»«mi»y«/mi»«mo»§nbsp;«/mo»«msub»«mi»f«/mi»«mn»0«/mn»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mi»§#960;«/mi»«msqrt»«mrow»«mi»L«/mi»«mi»C«/mi»«/mrow»«/msqrt»«/mrow»«/mfrac»«mo»§nbsp;«/mo»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«mi»u«/mi»«mi»r«/mi»«mi»r«/mi»«mi»e«/mi»«mi»n«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»a«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»r«/mi»«mi»e«/mi»«mi»s«/mi»«mi»o«/mi»«mi»n«/mi»«mi»a«/mi»«mi»n«/mi»«mi»c«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«msub»«mi»I«/mi»«mn»0«/mn»«/msub»«mo»=«/mo»«mfrac»«mi»E«/mi»«mi»R«/mi»«/mfrac»«mo»§nbsp;«/mo»«mo».«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»T«/mi»«mi»h«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»v«/mi»«mi»o«/mi»«mi»l«/mi»«mi»t«/mi»«mi»a«/mi»«mi»g«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»a«/mi»«mi»c«/mi»«mi»r«/mi»«mi»o«/mi»«mi»s«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«mi»a«/mi»«mi»p«/mi»«mi»a«/mi»«mi»c«/mi»«mi»i«/mi»«mi»t«/mi»«mi»o«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«msub»«mi»E«/mi»«mi»C«/mi»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mn»0«/mn»«/msub»«mo»*«/mo»«msub»«mi»X«/mi»«mi»C«/mi»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mn»0«/mn»«/msub»«mo»*«/mo»«msub»«mi»X«/mi»«mi»L«/mi»«/msub»«mo»=«/mo»«msub»«mi»I«/mi»«mn»0«/mn»«/msub»«mo»*«/mo»«msub»«mi»w«/mi»«mn»0«/mn»«/msub»«mi»L«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»I«/mi»«mi»n«/mi»«mi»p«/mi»«mi»u«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»V«/mi»«mi»o«/mi»«mi»l«/mi»«mi»t«/mi»«mi»a«/mi»«mi»g«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»E«/mi»«mo»=«/mo»«mo»§nbsp;«/mo»«msub»«mi»I«/mi»«mn»0«/mn»«/msub»«mo»*«/mo»«mi»R«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»t«/mi»«mi»h«/mi»«mi»e«/mi»«mi»n«/mi»«mo»§nbsp;«/mo»«mfrac»«msub»«mi»E«/mi»«mi»C«/mi»«/msub»«mi»E«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»w«/mi»«mn»0«/mn»«/msub»«mi»L«/mi»«/mrow»«mi»R«/mi»«/mfrac»«mo»=«/mo»«mi»Q«/mi»«/mtd»«/mtr»«mtr»«mtd»«msub»«mi»E«/mi»«mi»C«/mi»«/msub»«mo»=«/mo»«mi»Q«/mi»«mi»E«/mi»«mo».«/mo»«/mtd»«/mtr»«/mtable»«/math»

 If the input voltage is kept constant the voltage across capacitor is Q times E and a voltmeter connected across the capacitor can be calibrated to read the value of Q directly. 

Practical Circuit: 

 The practical circuit is shown in Fig. 1. It consists of self contained variable frequency RF oscillator. This oscillator delivers current to a low value shunt resistance Rsh: value may be 0.02 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#937;«/mi»«/math». The small value of input voltage E is injected into circuit that would be measured by thermocouple voltmeter.  An electronic voltmeter is connected across this capacitor. The coil under test is connected to terminals T1 and T2

Measurement of Q : 

The circuit for measurement of Q  shown in Fig.1. The oscillator is set to the desired frequency and then the tuning capacitor is adjusted for maximum value E0. The input voltage E is kept constant then the voltage across capacitor is calibrated  to read the value of Q directly. The measured value of Q is defined whole circuit not of the coil. There are errors caused due to shunt resistance and distributed capacitance of the circuit. 

 Correction for shunt resistance :

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«msub»«mi»Q«/mi»«mrow»«mi»m«/mi»«mi»e«/mi»«mi»a«/mi»«mi»s«/mi»«/mrow»«/msub»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»w«/mi»«mn»0«/mn»«/msub»«mi»L«/mi»«/mrow»«mrow»«mi»R«/mi»«mo»+«/mo»«msub»«mi»R«/mi»«mrow»«mi»s«/mi»«mi»h«/mi»«/mrow»«/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»«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»t«/mi»«mi»r«/mi»«mi»u«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»v«/mi»«mi»a«/mi»«mi»l«/mi»«mi»u«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«msub»«mi»Q«/mi»«mrow»«mi»t«/mi»«mi»r«/mi»«mi»u«/mi»«mi»e«/mi»«/mrow»«/msub»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»w«/mi»«mn»0«/mn»«/msub»«mi»L«/mi»«/mrow»«mi»R«/mi»«/mfrac»«mo»=«/mo»«msub»«mi»Q«/mi»«mrow»«mi»m«/mi»«mi»e«/mi»«mi»a«/mi»«mi»s«/mi»«/mrow»«/msub»«mo»(«/mo»«mn»1«/mn»«mo»+«/mo»«mfrac»«msub»«mi»R«/mi»«mrow»«mi»s«/mi»«mi»h«/mi»«/mrow»«/msub»«mi»R«/mi»«/mfrac»«mo»)«/mo»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mfenced close=¨§nbsp;¨ open=¨§nbsp;¨»«mrow»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«/math»

 

Correction for distributed capacitance :

 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«msub»«mi»Q«/mi»«mrow»«mi»t«/mi»«mi»r«/mi»«mi»u«/mi»«mi»e«/mi»«/mrow»«/msub»«mo»=«/mo»«msub»«mi»Q«/mi»«mrow»«mi»m«/mi»«mi»e«/mi»«mi»a«/mi»«mi»s«/mi»«/mrow»«/msub»«mo»(«/mo»«mn»1«/mn»«mo»+«/mo»«mfrac»«msub»«mi»C«/mi»«mi»d«/mi»«/msub»«mi»C«/mi»«/mfrac»«mo»)«/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»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mi»w«/mi»«mi»h«/mi»«mi»e«/mi»«mi»r«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«msub»«mi»C«/mi»«mi»d«/mi»«/msub»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»d«/mi»«mi»i«/mi»«mi»s«/mi»«mi»t«/mi»«mi»r«/mi»«mi»i«/mi»«mi»b«/mi»«mi»u«/mi»«mi»t«/mi»«mi»e«/mi»«mi»d«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«mi»a«/mi»«mi»p«/mi»«mi»a«/mi»«mi»c«/mi»«mi»i«/mi»«mi»t«/mi»«mi»o«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§nbsp;«/mo»«mi»C«/mi»«mo»=«/mo»«mi»t«/mi»«mi»u«/mi»«mi»n«/mi»«mi»i«/mi»«mi»n«/mi»«mi»g«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«mi»a«/mi»«mi»p«/mi»«mi»a«/mi»«mi»c«/mi»«mi»i«/mi»«mi mathvariant=¨normal¨»tan«/mi»«mi»c«/mi»«mi»e«/mi»«mo».«/mo»«mo»§nbsp;«/mo»«/mtd»«/mtr»«/mtable»«/math»

 Measurement of self Capacitance: 

The value of inductance is given by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»L«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»4«/mn»«msup»«mi»§#960;«/mi»«mn»2«/mn»«/msup»«msup»«msub»«mi»f«/mi»«mn»0«/mn»«/msub»«mn»2«/mn»«/msup»«mi»C«/mi»«/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»«mo»-«/mo»«mo»(«/mo»«mn»3«/mn»«mo»)«/mo»«/math»

 the values of f0 and C are known and therefore the value of inductance may be calculated. 

 

Measurement of Effective Resistance: 

The value of effective resistance may be computed from the relation «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»R«/mi»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»§#969;«/mi»«mn»0«/mn»«/msub»«mi»L«/mi»«/mrow»«msub»«mi»Q«/mi»«mrow»«mi»t«/mi»«mi»r«/mi»«mi»u«/mi»«mi»e«/mi»«/mrow»«/msub»«/mfrac»«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»-«/mo»«mo»-«/mo»«mfenced»«mn»4«/mn»«/mfenced»«/math»

 

Measurement of Self-Capacitance: 

 The self capacitance is measured by making two measurements at different frequencies. The capacitor is set to a high value and the circuit is resonated by adjustment of the oscillator frequency. Resonance is indicated by the circuit Q meter. Let the values of tuning capacitor be C1 and that of  frequency be f1 under these condition. Therefore,

                                           «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»f«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mi»§#960;«/mi»«msqrt»«mrow»«mi»L«/mi»«mo»(«/mo»«msub»«mi»C«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«msub»«mi»C«/mi»«mi»d«/mi»«/msub»«mo»)«/mo»«/mrow»«/msqrt»«/mrow»«/mfrac»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mfenced»«mn»5«/mn»«/mfenced»«/math»

 The frequency is now increased to twice its initial value and the circuit is resonated again this time  with the help of the tuning capacitor. Let the values of tuning capacitor be C2 and that of  frequency be f2 under these condition. Therefore,

                                         «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»f«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mi»§#960;«/mi»«msqrt»«mrow»«mi»L«/mi»«mo»(«/mo»«msub»«mi»C«/mi»«mn»2«/mn»«/msub»«mo»+«/mo»«msub»«mi»C«/mi»«mi»d«/mi»«/msub»«mo»)«/mo»«/mrow»«/msqrt»«/mrow»«/mfrac»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»6«/mn»«mo»)«/mo»«/math»

                                       Now f2=2*f1

Then distributed capacitance, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»C«/mi»«mi»d«/mi»«/msub»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»C«/mi»«mn»1«/mn»«/msub»«mo»-«/mo»«mn»4«/mn»«msub»«mi»C«/mi»«mn»2«/mn»«/msub»«/mrow»«mn»3«/mn»«/mfrac»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»-«/mo»«mo»(«/mo»«mn»7«/mn»«mo»)«/mo»«/math»

 

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