PMMC & Moving Iron

7/23/2019 PMMC & Moving Iron

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Classification measuring instrument:

The measuring instrument can broadly be classified into two groups as i)Absolute ii) Secondary

i) Absolute instrument:

This type of instrument measures the unknown quantity in an indirect way .Result of the measurement is computed based on quantity related to theunknown quantity and the constant of the instrument . No comparison withanother instrument is necessary .The accuracy of the measurement by thistype of instrument depends on the accuracy of the human obseration andtrue alue of the constant . !or e"ample# the tangent galanometer gies thealue of the current in terms of the tangent of the angle of deflection producedby the current # the radius and no. of turns of the galanometer coil # and

hori$ontal component of the earth%s magnetic field .

ii) Secondary instrument:

This type of instrument gies the result directly by the deflection of theinstrument and dial reading only if the latter has been calibrated bycomparison with either an absolute instrument or one which has already beencalibrated . The deflection obtained is meaningless until such a calibrationhas been made.

All secondary instrument can be classified according to the manner # the result orobseration is presented or displayed .The types are

i) Indicating instrument

These instruments indicate the magnitude of a quantity being measuredcontinuously. They generally make the use of dial and a pointer for thispurpose . Ammeter# &oltmeter #'attmeter etc belong these category.

ii) Recording instrument

This instrument gies continuous record of the quantity being measured oera specific period . The ariation of the quantity being measured are recordedby a pen(attached to the moing system of the instrument) giing permanent

mark oer the moing sheet of paper or chart at constant speed. "amplesare *+, plotter # strip chart recorder etc.

iii) Integrating instrument

This instrument totali$e eents oer a specific period of time . The summation# which they gie is the product of time and instantaneous quantity beingmeasured. "amples are Ampere+hour meter # 'att+hour or energy meter .

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the integration (summation alue) is generally gien by a register consisting ofa set of a pointer and dials.

The secondary instrument may be classified according to the principle ofoperation they utili$e . The effects utili$ed are:

i) -agneticii) Thermaliii) hemicali) lectrostatic) lectromagnetic /nduction

Magnetic effect:

'hen a current carrying conductor or coil is brought to near a bar magnet # itproduces 0. electromagnet . 0epending on the direction of flow of currentthrough coil # there will be force of attraction or repulsion . /f the bar magnet isstationary and the coil is mounted on a spindle # the coil e"perience a torqueabout its a"is of rotation and it moes to rotate . This effect is utili$ed in 1--instrument . /f there are two current carrying coils # one of which is fi"ed and theother is moable # there will be a motion of the moable coil .This effect is utili$edin the electrodynamoter type instrument.

Thermal effect:

The current or rate of change charge flow to be measured is passed through asmall element which heats it . The temperature rise is conerted to an e.m.f. bya thermocouple attached to the element. This thermo e.m.f. can be measured by

an indicating instrument or a ordinary galanometer.

Electrostatic effect:

'hen two plates are charged with opposite or same polarity # there is a force ofattraction or repulsion between the plates . This force is utili$ed to moe one ofthe plates and the moement is coupled to the driing mechanism of the pointergiing measurement of the quantity under measurement

Induction effect:

'hen an a.c. unknown quantity (e.g. current or oltage) produces alternatingmagnetic flu" which links another moing conducting part of the instrument # anemf is induced in the conducting part . /f a closed path is proided # the emfforces a current to flow in the conducting part . The force produced by theinteraction of alternating flu" and induced current will creates deflecting torquewhich makes the moing part of the instrument moe . The induction effect isutili$ed in only a.c. measuring instrument as alternating flu" is necessary .

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All secondary indicating type instrument are constructed with the subsystems asi) moing system ii) supporting mechanism ii) ontrol system to obtain steadydeflection . The constructional details are described

Operating forces of Electromechanical Indicating instrument:

Three types of forces are needed for the satisfactory operation . These are :i)0eflecting force ii) ontrolling force iii) 0amping force

Deflecting force:

The deflecting or operating force is required for moing the pointer from its $eroposition . the system producing the deflecting force is called 2 -oing system 2 .The deflecting force can be produced by utili$ing any of the aforesaid effects.Thus the moing system conerts the current into mechanical force inelectromechanical type instrument.

Controlling force:

This force is required for indicating instrument in order that the current producesa steady deflection which is proportional to the magnitude of the current flowing

the moing system. The function of the controlling force are:a) to produce a force equal and opposite to the deflecting force at the final

steady position of the pointer in order to make the deflection definite for aparticular magnitude of current . /n absence of a controlling system # thepointer will shoot beyond the final steady position for any magnitude ofcurrent and thus the deflection will be indefinite.

b) To bring the moing system back to $ero position when the force causingthe instrument moing system to deflect is remoed . /n the absence ofcontrolling system the pointer will not come back to $ero position whencurrent through the moing system ceases to flow .

Damping force:

'hen a deflecting force is applied to the moing system # it deflects and it shouldcome to rest at a position where the deflecting force is balanced by thecontrolling force . The deflecting and controlling forces are produced by thesystem which hae the inertia and therefore # the moing system can not

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immediately settle at its final position but oershoots or swings around the finalposition . This is illustrated in the following fig. Suppose 3 is the equilibrium orfinal steady position . 4ecause of the inertia of the moing system moes toposition 5a% . Now for any position 5a% beyond the equilibrium position thecontrolling force is more than the deflecting force and hence the moing system

swings back . 0ue to inertia it can not settle at 53% but swings to a position say 5b%behind the equilibrium position . At 5b% # the deflecting force is more than thecontrolling force and hence the moing system again swings . The pointer thusoscillates about its final steady position with decreasing amplitude till its kineticenergy is dissipated in friction and therefore # it will settle down at its final steadyposition . /f e"tra force are not proided to 2damp6 these oscillation # the moingsystem will take a considerable time to settle to the final position and hence timeconsumed in taking readings will be ery large. Therefore damping forces arenecessary so that the moing system comes to its equilibrium position rapidlyand smoothly without any oscillation.

Moving system:

The moing parts should be light and the frictional forces should be minimum

.These requirement should be fulfilled in order that power required by theinstrument for its operation is small. The power e"penditure is proportional to theweight of the moing parts and the frictional forces opposing the moement . Themoing part can be made light by using aluminum as far as possible . Thefrictional forces are reduced by using a spindle mounted between 7ewel bearingsand by carefully balancing the system.

Supporting system :

The force or torque deeloped by the moing element in electromechanicalindicating instrument is small in order that the power consumption be kept low sothat the introduction of the instrument into a circuit may cause minimum changein the e"isting circuit conditions . 4ecause of low power leel # the supporting themoing system is of great importance . 'ith operating forces being small # thefrictional forces must be kept to a minimum in order that the instrument readscorrectly and is not erratic in action and is reliable. Seeral types of supports areused # depending upon the sensitiity required and the operating conditions to bemet . Supports may be of the following types :

i) Suspension:

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/t consists of fine #ribbon shaped metal filament for the upper suspension and coilof fine wire for the lower part . The ribbon is made of a spring material likeberyllium copper or phosphor bron$e . This coiling of lower part of suspension indone is order to gie negligible restraint on the moing system . The type of

suspension requires careful leeling of the instrument # so that the moingsystem hangs in correct ertical position . This construction is # therefore # notsuited to field use and is employed only in those laboratory applications in whichery great sensitiity is required . /n order to preent shocks to the suspensionduring transit etc. a clamping arrangement is employed for supporting the moingsystem .

ii) Taut suspension:

Suspension type instruments can only be used in ertical position . The tautsuspension has a flat ribbon suspension both aboe and below the moingelement # with suspension kept under tension by a spring arrangement . Theadantage of this suspension is that e"act leeling is not required if the moingelement is properly balanced . Ribbon suspensions # in addition to supporting themoing element # e"ert a controlling torque when twisted . Taut suspensions arecustomarily used in instruments which require a low friction and high sensitiity .Thus the use of suspension results in elimination of piots # 7ewels and controlsprings and therefore piot less instruments are free from many defects.

iii) Piot ! "e#el $earings:

The moing system is mounted on a spindle made of hardened steel . The twoends of the spindle are made conical and then polished to form piots . Theseends fit conical holes in 7ewels located in the fi"ed parts of the instruments .These 7ewels # which are preferable made of natural or synthetic sapphire formthe bearings . The contact surface between the spindle and bearings giesminimum friction . /t has been found that frictional torque # for 7ewel bearings # isproportional to area of contact between the piot and 7ewel . the contact area

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between piot and 7ewel should be small . The piot is ground to a cone and itstip is rounded to a hemispherical surface of small area. The 7ewel is ground to acone of somewhat larger angle . The piot may hae a radius at tips from 8.89;mm to as high as 8.8


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Tor%ue&'eig(t Ratio:

The frictional torque in indicating instrument depends upon the weight of themoing parts . /f the weight of the moing parts is large # the frictional torque willbe large . The frictional torque e"erts a considerable influence on theperformance of an indicating instrument . /f the frictional torque is large and iscomparable to a considerable fraction of the deflecting torque the deflection ofthe moing system will depend upon the frictional torque to an appreciable e"tent. Also the deflection will depend on the direction from which the equilibriumposition is approached and will be uncertain . 3n the other hand # if the fractionaltorque is ery small compared with the deflecting torque # its effect on deflectionis negligible . Thus the ratio of deflecting torque to fractional torque is a measure3f reliability of the instrument indication and of the inherent quality of the design .hence torque+weight ratio of the instrument is an inde" of its performance # thehigher the ratio # the better will be its performance . /f the deflecting torque ise"pressed in terms of a force which acting at a radius of 9 cm # produces fullscale deflection # the ratio of this torque to weight of the moing parts should notbe less than 8.9 as far as possible.

Control system:

The moing system is mounted on a pioted spindle . The quantity beingmeasured produces deflecting torque proportional to its magnitude . Thereshould a restraining or controlling torque acting in opposite to the deflectingtorque # which will bring the moing system to steady deflected position #otherwise the moing system will go on moing for indefinite time. Two types ofcontrol system are used in indicating instruments as i) =raity control ii) Sprin gcontrol .

raity control:

/n graity control instrument # a small weight is placed on a arm attached to themoing system in such a way that it produces a restoring or controlling torquewhen the system is deflected . The position of the weight is ad7ustable . 'henthe moing system is deflected by an angle # the controlling torque is gien by :

SinKWlSinT gc == # where l> distance from the a"is of rotation of moingsystem . gK > Wl>constant.

The controlling torque can be aried by simple ad7usting the position of controlweight upon the arm which carries it. The graity control instrument mustobiously be used in a ertical position in order that the control may operate. Theinstrument must be mounted in leel position otherwise there will be a eryserious $ero error .!or these reasons # graity control is not suited for portableindicating instruments .

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Spring control:

A flat spiral spring is attached to the moing system # produced controlling torque.

3ne end of the spring is attached to the spindle and the outer end carries aspigot which engages in a circular disc surrounding the 7ewel screw . This disccarries an arm which is slotted and splayed out at the end. The purpose ofslotted e"tension arm is to allow the spring to be coiled or uncoiled slightly # sothat the pointer may be set at $ero . the slotted arm is actuated by a set screwmounted at the front of instrument and # therefore # $ero setting of the instrumentcan be done without remoing the coer.The spring material should be i) non+magnetic ii) proof from mechanical fatigue .'here springs are used to lead the current into the moing system # they shouldhae a small resistance # their cross ?sectional area must be sufficient to carrythe current without temperature rise effecting their constant . they should also

hae a low resistance temperature coefficient. A no. of non magnetic materialslike silicon bron$e # hard rolled siler or copper # platinum siler # platinum ?iridium and =erman siler hae been used but hae not been found satisfactoryowing to some reason or the other . !or most applications phosphor ?bron$e hasbeen found to be most suitable material e"cept in instruments of low resistance () . /n this case some special bron$e alloys haing low resistance may be usedwith some sacrifice in mechanical quality .

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4y making the large no. of turns # the deformation per unit length is kept small onfull scale deflection . The controlling torque is thus made proportional to the angle

of deflection of the moing system . !or a flat spiral spring # the controlling torquedeeloped by deflection of the moing system is gien by :

cc Kl

EbtT == .

12

3

N+m.

where E> ,oung%s modulus of the spring material (N@m)b >width of spring (m)t>thickness of spring ( m)l>length of spring (m)>angular deflection (rad.)

cK > spring constant >l

Ebt

12

3

( N+m@rad.)

The spring should be stressed well below their elastic limit at ma"imum deflectionof the instrument in order that there is no permanent set or that no change indeflection (or $ero shift ) will occur from inelastic yield.

-a"imum fibre stress2max

6

bt

Tf c= (N@m)

Sometimes a parameter 2 length @ thickness6 ratio is used for design purpose .

Thus )2

(max

f

E

t

l=

ence in order that the material is not oer stressed the ration of length tothickness of the spring should not be much less that!atigue in spring may be aoided to a great e"tent by proper annealing andageing during manufacture . /n order to eliminate the effect of temperatureariations upon the length of the spring# two springs coiled in opposite directionsare used . 'hen the moing system deflects # one spring is e"tended while theother is compressed .

Comparison of spring and graity control instrument:

i) The graity control is cheap and is independent of temperatureariation

ii) =raity control instrument is independent of ageing and its

performance does not deteriorate with timeiii) The scale of graity control instrument is non+linear and is cramped at

lower end of the scale.i) The graity control instrument must be used in only ertically and must

be perfectly leeled.Damping System:

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The method for damping torque are :i) Air friction dampingii) !luid friction dampingiii) ddy current dampingi) lectromagnetic damping

Air friction damping:

/n the 9sttype of air friction damping a light aluminum piston is attached to themoing system . The piston moe with a ery small clearance in fi"ed airchamber closed at one end. ross section of the chamber may be either circularor rectangular . /f the piston is moing rapidly into the chamber # the air inside thechamber is compressed and thus the pressure inside the chamber opposes themotion of the piston i.e. of the moing system to which piston is attached. /f thepiston moing outside # pressure inside the chamber falls and therefore theoutside pressure becomes greater than that inside the chamber and thus the

motion of the piston or of the moing system is again opposed . /n this systemcare must be taken to ensure that the arm carrying piston is not bent and thepiston does not touch the sides of the chamber during its moement . when oncebent it is often difficult to straighten the piston arm so that it does not touch thesides of the chamber at any point during deflection . The solid friction occurringbetween piston and sides of air chamber may result in serious error in thedeflection . Such a system of damping is not faored now a days./n second type of air friction damping system # a ane mounted on the spindle ofthe moing system is utili$ed . The ane is thin aluminum sheet and moes in aclosed sector shaped bo" .

*luid friction damping:

/n this method of damping light ane or disc are attached to the spindle of themoing system and moe in a damping oil # which is iscous in nature Thedamping oil employed must be good insulator # non+eaporating # non+corrosieon the metal disc or ane . /t should net be sub7ected to change of property withtemperature. Two methods of fluid friction damping are aailable. /n the 9st

method # light anes are attached to the spindle of the moing system . the anesare dipped into a oil pot and are completely submerged by the oil . The motion ofthe moing system is always opposed by the friction of damping oil on the anes.

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The damping forces thus created always increases with the increase in elocityof anes . There is no damping force when the anes are stationary./n the second method # a disc attached to the spindle of the moing system isimmersed in damping oil . the frictional drag deeloped during the motion of thedisc always opposes the motion of the moing system and becomes $ero when

the disc is stationary . The suspension system of the disc should be cylindricaland of small diameter so that the surface tension effect is negligible .

Eddy current damping:

'hen a sheet of conducting but non+magnetic material like copper or aluminummoes in a magnetic field so as to cut through lines of force # an e.m.f. is inducedin the sheet # due to which a current flows through the short circuited path#encountered . This current is called eddy current which interacts with themagnetic field to produce an electromagnetic torque opposing the motion of theconducting sheet. The torque due to eddy current is proportional to themagnitude of eddy current and the magnetic field. The magnitude of current is

proportional to the elocity of the conductor . ence if the strength of themagnetic field is constant # the damping torque is $ero when the conductor is atrest # but increases when the conductor starts moing . There are two methods ofproiding eddy current damping .

/n the 9st method a thin disc or ane of conducting material like copper oraluminum is mounted on the spindle carrying the moing system and the pointer .The ane or disc is positioned so that its edge rotates between the poles of apermanent magnet . /n second method of proiding eddy current damping asused in 1-- instrument # the coil is wound on alight metal former in whicheddy current are produced due to moement of the coil in the field of apermanent magnet .

i) Eddy current damping tor%ue #it( metal disc:

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onsidering a metallic disc of thickness t # mean radius r and of resistiityrotating with angular elocity and cutting through the magnetic field ofpermanent magnet with uniform flu" density 4The instantaneous e.m.f. induced in this portion of disc in the inter polar gap #

BlrBlve ==

The resistance of the eddy current path # consisting only the portion of thedisc which is immediately under the pole #

wt

lR

=

The eddy current #

K

Bwrt

wt

lK

Blr

R

eIe ===

(A)

0amping force #

K

wlrtBlBIF e

2

== (N)

0amping torque# == rFT dd rK

wlrtB

2= (N+m)

3r==

K

twlrB

Td

22

K

AtrB 22

B Awl= # the area of the pole faceC

The actual resistance of the total eddy current path depends upon the radialposition of pole and let it be D times as high as that of section under the poles# D is always greater than unity .The magnitude of damping torque may be aried by ad7usting the radialposition of the poles. 0amping torque decreases with the distance of themagnet towards the edge of the disc and becomes $ero when the centre ofthe poles are at the edge of the disc .

ii) Eddy current damping tor%ue #it( a metal former:

/n 1-- instrument eddy current damping is obtained by employing ametallic former on which the coil iEs wound so that metal former acts as asingle turn short circuited coil rotating in the field of permanent magnet andthus producing eddy current and damping force on the moing system .Fet the metallic former of length l metre and diameter of d metre rotates inradial field of permanent magnet of uniform flu" density 4 'b@m with angular

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elocity w radians per second. Fet the thickness and width of the section ofmetallic former be t and w metre respectiely .

The e.m.f induced in each side of the former > Blv

Total induced e.m.f. in the former #

Bldd

BlBlve ===

2

22 (&) B

2

dv = C

Resistance of the path for eddy current #t

dlR

)(2 += (ohm)

'here is the resistiely of the metal of former in

ddy current

)(2)(2 dl

Bldwt

wt

dl

Bld

R

eI

e+

=

+

==

(A)

The damping force#)(2

22

dl

dwtlBlBIF e

+

==

(N)

0amping torque == FdTd)(2

222

dl

wtdlB

+

(N+m)

0amping constant ==

dd

TK

)(2

222

dl

wtdlB

+(N+m@rad)

ence the damping can be aried by arying the thickness of the metalformer .

Electromagnetic damping:

The moement of a coil in a magnetic field produces a current in the coil due toinduced e.m.f. #which interacts with the magnetic field to produce torque . Thistorque opposes the moement of the coil . The magnitude of the current andhence the damping torque is dependant on the resistance of the coil circuit .

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Magnet system:

Special alloys steel are used in instrument magnet construction . Tungsten steelhas been popular since the early days. obalt ?chrome steel has also been usedbecause of their higher coerciity and where economy of weight and space isimportant . 4ut recent years # Alnico magnets has been used to an increasinge"tent ./deally an instrument magnet should hae in its air gap high magnetic flu"density which may not change with time or temperature . The design ofinstrument magnets inoles consideration of weight and economy of space #e"pense of materials and manufacturing processes # and performance ofmagneti$ation both as well as the field strength desired in the air gap . /n most ofthe applications# the field strength may be e"pected to be between 8.8; and 8.;

T in air gap of 9.;+.; mm length # depending on the si$e and type of instrument.1ermanet magnets are made of hard materials # i.e. materials which hae abroad hysteresis loop (large co+ercie force) so that they are not sub7ect to selfdemagneti$ation . /n order that the olume of the permanent magnet is small #the (4) # of the material used should be large .The following table gies data formagnetic materials used for permanent magnets.

Material Remenance+'b&m,)

Co-ercieforce+A&m)

.alue of $for +$/)ma0+T)

.alue of /for +$/)ma0

+$/) ma0

98G arbon 8.E H#888 8.I #I88 9#;I8

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steel

IG Tungstensteel

9.8; ;#88 8.< J#


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magnitude of them in a closed circuit. 4y construction # a galanometer consistsof a moing coil and a permanent magnet producing 0.. magnetic field.

Torque equation of galvanometer:

The 0% Arsonal galanometer consists of a moing coil of rectangular geometryand permanent magnet haing the following parameters:l>length of ertical sided>width of the coilN >No. of turns of the coilB >!lu" density in the air gap at the coil positioni >steady current flowing through the coilS>torque constant

F >steady deflection

!orce on each conductor of the ertical side of the coil> BilSin

'here > angle between direction of magnetic field and the line passingthrough the side conductor. /f the 0.. magnetic field is radial # then >E8Then force on each side of the coil> NBil

0eflecting torque # dT > force * distance > NBild >NBiA (N+m)

where area of the coil face # ldA = (m )

Since for a galanometer # N #4 and A are constants # then GiTd = (N+m)

# where = is called the galanometer constant > NBld (N+m@A )

Dynamic ehaviour of galvanometer:

'hen the moing coil of the galanometer is energi$ed by the flow of steady

current # the moing system does not reach to its steady state deflectionimmediately but will take some time or period of transition during which thedeflection s changing . This is known as transient or dynamic behaior ofgalanometer . This behaiour will greatly depend on the design constants of thegalanometer # such as ) displacement constant /) inertia constant iii) dampingconstant ) control constant . The dynamic behaiour is e"pressed by theequation of motion of the moing system haing these design constants.

i) Displacement constant:

/t is the amount of deflecting torque produced due to flow unit current .e.

iTG d= (N+m@A)

ii) Inertia constant:

A retarding torque s produced owing to inertia of the moing systemThis torque s dependent on the moement of inertia of the moing system andangular acceleration .

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/nertia torque2

2

dt

dJTd

= # where L s called inertia torque constant and is

equal to the moment of inertia of the moing system. Thus the retarding torqueproduced per unit angular acceleration is called inertia constant (N+m@Rad+sec)

iii) Damping constant

0amping is proided by the friction due to motion of the moing system in thesurrounding air and also by induced electrical effects on the moing coil carryingcurrent . 0amping torque is proportional to the angular elocity of the moing

system . Thus damping torque is gien by the equation :

Ddt

dDTD == # where

D > damping coefficient ( N+m@Rad+sec)

i) Control constant:

A controlling torque is produced due to elasticity of the moing system which tries

to restore the system back to its equilibrium position . ontrolling torque STc =

#where S>control constant (N+m@Rad)

Dynamic e%uation of motion of t(e galanometer:

There are four torques acting on the moing system of the galanometer at anttime # since the energi$ation of the coil as

deflecting torque ( dT

) tends to accelerate the coil

/nertia torque ( iT ) tends to retard the moing system proportional to angular

acceleration

damping torque ( DT ) tends to retard the moing system in proportion to the

angular elocity

control torque ( cT ) tends to retard the moing system in proportion to angular

displacement

At any time # t within the transition period # the deflecting torque must be equalthe all retarding torque # Thus cDid TTTT ++=

3r GiSdt

dD

dt

dJ =++

2

2

'here # J> /nertia constantD >damping constantS>spring constant

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G >galanometer constant>deflection at time ti > current at time tThe aboe equation is a linear nddegree differential equation # whose solution sthe sum of .!. (complementary function representing the transient condition )

and 1/( particular integral representing the steady state condition)

Solution of .!.:

Assuming mdt

d=

# the au"iliary equation s : 02 =++ SDmJm

The roots are gien by:J

JSDDm

2

42

1

+= and

J

JSDDm

2

42

2

=

The solution : tmtm BeAe 21 += # where A #4 are constants # whose alues are to

be determined from the initial conditions.

Solution of 1./.:

ase a)

'hen JSD 42 < # The roots ( 21,mm ) are imaginary . Mnder these condition # the

motion is oscillatory

Now21

2

12

4

2jkk

J

DJSj

J

Dm +=

+=

And 212 jkkm =

F

tjkktjkk BeAe ++= + )()( 2121

or Ftjktjktk BeAee ++=

][ 221

or )]()([ 22221 tjSinktC!kBtjSinktC!kAe tk ++=

or Ftk BAtkSinBAtkC!e ++=

))](())(([22

1

orF

tk tSinkC!tC!kSinFe ++= ][

22

1

'here Fand are constants such that

22 )()( BABAF += and ])(

)([tan

1

BAj

BA

+=

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The aboe equation signifies that # when a steady current passes through thecoil # the angular deflection starts and will oscillate sinusoidally with e"ponentially

decreasing amplitude before reaching to a final steady state alue as f .The

frequency of oscillation is gien by :J

DJSf dd

2

41

2

2

==

And the time period is gien by :2

4

41

DJS

J

fT

d

d

==

The motion is called damped oscillatory and the galanometer is calledunderdamped. The plot of s tunder damped oscillation s shown below:

ase b)

'hen JSD 42 = # the roots of au"iliary equation are equal i.e.J

Dmmm

221 ===

The solution in this case is gien by : ][ BtAet

J

D

f ++=

}].2

{[]}{2

[ 22 BtBAJ

DeBeBtAe

J

D

dt

d tJ

Dt

J

Dt

J

D

++=++=

At 0=t # 0= and 0=dt

d Af += 0 BA

J

D+= .

20

fA = and fJ

DB .

2

=

The solution is : )]2

1(1[ 2 tJ

De

tJ

D

f +=

Now =D and cJ

S

J

D==

2

ence the solution is : )]1(1[ te ct

fc

+=

The aboe equation signifies that the moing system quickly moes to its finalsteady position without any oscillations. The motion is called critically dampedmotion and the galanometer is called critically damped . The plot of s . tunder critically damped condition s shown below:

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ase c)

'hen 0=D # the roots of the equation isJ

Sj

J

JSjm ==

21

AndJ

Sjm =

2 . Thus roots are imaginary with opposite sign

The solution is gien by: )]90(1[

+= tFSin nf

fF = and 90)(tan 1

== and

J

Sn =

where n > natural angular frequency of moing system

J

Sf nn

2

1

2== >natural frequency of sinusoidal oscillation

S

J

fT

n

n .21

== >Time period of oscillation .

)1( tC! nf =

The motion is called undamped under which the moing system of the

galanometer oscillates around the steady alue F sinusoidally with frequency

nf . the plot of s. t is shown below:

ase d)

'hen JSD 42 > # then the roots are real .

20


21/51

The solution is FtmtmBeAe ++= 21 # where

12

2

mm

mA F

=

and

12

1

mm

mB F

=

]1[ 21

12

1

12

2 tmtmF e

mmme

mmm

+

=

Now )1(422

4

2

2

02

22

1 =+=

+= J

S

J

D

J

D

J

J!D

J

Dm

And )1(2

4

2

2

0

2

2 ++=

= J

JSD

J

Dm

'herecD

D= # is called relatie damping factor

AndJ

S=0

2

0

222

0 444 JJ

SJJSD === or 2

0

2

024

DJ =

Now )(4)2

(4 22

0

2222

dd JJ

SJD ==

2

0

2

2

0

2

1

d

D

D= or # 2

0

22 1

d= #

2

0

1

=

d 2

0 1 = d

12)1()1( 202

0

2

012 =++= mm

ence the complete solution is :

]12

)1(

12

)1(1[

)1(

2

0

2

0)1(

2

0

2

0 2

02

0 tt

f ee

+

+

+=

+1[f

The aboe equation represents a decaying motion without oscillation . Thismotion is called oerdamped motion . This motion is usually slow and is notdesirable in indicating instrument . the plot s. under oerdamped case isshown below:

21


22/51

Damping of galanometer:

a) Mec(anical damping:

/n this type of damping # air friction is produced owing to the motion of moingcoil in the air surrounding it . So damping torque due to air friction is gien by :

-echanical damping:dt

dDT mm

=

# where mD >damping constant due to mechanical damping

! Electromagnetic damping:

lectromagnetic damping is produced by the induced effects when the coilmoes in the magnetic field and a closed path is proided by the currents toflow . The electromagnetic damping takes place due to ) eddy currents producedn the metal former ii) current circulated in the coil circuit by induced e.m.f.generated when coil rotates in the magnetic field.

22


23/51

i) Damping due to induced eddy current:

/f the coil is mounted on the metallic former # then eddy current flows through theformer due to induced eddy e.m.f.. 'ith the interaction of the magnetic field andeddy current # a damping torque s produced . The oltage induced n the metallicformer s gien by : e > no. of conductors * e.m.f. generated n each conductor >

)(2

22dt

dBld

dBlBlv

==

B metallic former has only two side conductors i.e. N>9C

/f fR be the resistance of the former # then damping torque produced by the

former is gien by: )()( 2

dt

d

R

BldT

f

f

=

B torque >conductors * force on each conductor * distance C

Now the displacement constant NBldG =

)()(2

2

dt

dD

dt

d

RN

GT frmer

f

f

==

where frmerD is the damping constant due to resistance of the former

f

frmerRN

GD

2

2

=

Eddy current damping tor%ue in metal former:

!or rectangular former haing dimensions as l #b and d #where lengt"l = of theformer# b >width of former (m) # d>breadth of former (m) and >resistiity offormer material # the resistance of path of eddy currents is gien by:

bt

dl

tb

dl

#re#

lengt"$re!i!tivitR

f

)(2)(2 +=

+=

=

ddy current :)(2/)(2 dl

Blbtd

btdl

Bld

R

EI

f

e

e+

=

+

==

0amping force :)(2)(2

22

dl

btdlBl

dl

BlbtdBlBIF eD

+

=

+

==

0amping torque :)(2)(2

222222

dl

btdlBd

dl

btdlBdFT DD

+

=

+

==

0amping constant)(2)(2

222222

dl

btdlB

dl

btdlBTK DD

+

=

+

==

(Nm@rad+sec)

23


24/51

Eddy current damping tor%ue in metal disc:

'hen a thin metal disc is attached to the moing system and when this discmoes in the field of a permanent magnet during rotation of the moing systemunder the action of deflection torque # then eddy current is induced in the metal

disc and an eddy current damping torque is produced in acting in oppositedirection to the deflecting torque.. The following mathematical analysis shows therelation between damping torque and arious parameters of metal disc.

!or a circular metal disc # if the poles of permanent magnet is placed near theperiphery of the disc at a distance R between the centre of the poles and centreof the disc # then the induced e.m.f. in the portion of the disc below the magnet isgien by:

BdRBlvEe ==

Resistance of the eddy current path under the poles >bt

d

Actual path of eddy current is not limited to the portion of the disc under themagnet but is greater than this . Therefore to take this factor into account the

actual resistance is taken as D timesbt

d

resistance of eddy current pathbt

dKRe

=

The alue of the constant D depends upon the position of poles.

ddy current

K

BRb

btdK

BdR

R

EI

e

ee ===

/

0amping force

K

RdbtB

dK

BRbt

BFD

2

==

0amping torque

K

dbtRBTD

22

=

0amping constant K

dbtRBTK DD

22

== ( Nm@rad+sec.)

24


25/51

/t is clear that the damping torque can be changed by changing the distance ofthe poles from the centre of the disc. (i.e. the radial distance of poles w.r.t. thecentre of the disc)

ii) Damping due to coil circuit resistance:

/n general # a series resistance of suitable alue is connected to moing coil ofthe galanometer for limiting the current .

Fet R > resistance of galanometer circuit # when closed

/f > cR resistance of coil and eR > e"ternal resistance # then ec RRR +=

The oltage induced in the coil due to its rotational motion is gien by:e > no. of conductor * e.m.f. induced due to relatie motion betweenmagnetic field and current flowing in the conductor

)()(2

22dt

dNBld

dt

ddNBlNBlve

===

where N > no. of turns in coil N2 >no. of conductors in coil l > length of the coil

d>width of the coil

v >tangential elocity of the conduction due to rotational speed >2

d

The current flowing in the coil due induced e.m.f. is gien by:

)(dt

d

R

NBld

R

ei

==

Torque produced owing to current flowing in the coil is gien by:

cilT >force acting on conductors * distance

> )()()(

)(22

dt

d

R

G

dt

d

R

NBld

dt

d

R

NBldNBldNBl%d

===

damping constant due to resistance of the coil circuitRGDcil

2=

Total electromagnetic damping torque s gien by :

dt

dD

dt

d

RN

G

R

GTTT e

f

fcile

=+=+= )(

2

22

where eD > damping constant due to electromagnetic effect >f

RN

G

R

G2

22

+

25


26/51

Total damping effect is made up of two components .e. damping due tomechanical effects and damping due to electromagnetic effects .

dt

dD

dt

dDD

dt

dD

dt

dDTTT emememD

=+=+=+= )(

where D > damping constant due to combined effects of mechanical and

electromagnetic type damping

Critical resistance for damping:

/t has been seen that electromagnetic damping occurs due to resistance ofmoing coil circuit and resistance of the eddy current effect on the metal former .

Thusf

eRN

G

R

GD

2

22

+=

4ut damping due to metal former is usually small and can be neglected .So

R

GDe

2

= !or critical damping caseR

GKJDD

c

2

2 ===

Thus for critical damping # the total resistance of the galanometer circuit should

be : KJG

R2

2

= .

!or a gien galanometer are constants . So critical damping condition can beobtained by connecting an e"ternal resistance in series with the moing coil .

"ternal series resistance required for critical damping : gge RKJG

RRR ==2

2

# where is resistance of the galanometer coil .

This e"ternal resistance is called critical damping resistance e"ternal (0R*)

Sensitiity of galanometer:

There are three different sensitiity parameters for galanometer as i) urrentsensitiity ii) &oltage sensitiity iii) -eg+ohm sensitiity

i) urrent sensitiity:

/t is defined as the full scale deflection in mm. 1roduced by unit current and isgien by:

iS Fi /= # but KGF /= # KGiSi /= Rad@A

ii) &oltage sensitiity:

The oltage sensitiity is defined as the deflection in scale diisions per unitoltage

26


27/51

Thus 610=

g

viR

dS scale diision@u&

'here gR >resistance of moing coil

> current through galanometer coil

iii) -eg+ohm sensitiity:

The meg+ohm sensitiity is the resistance of the coil circuit so that thedeflection will be 9 scale diision with 9 olt impressed to the moing coil .

Thus =S resistance in meg ohm to gie a deflection of one scale diision

with 9 olt >6

10

i

d -+ohm @ scale diision

"ermanent Magnet Moving Coil #"MMC!Instrument:

The 1-- instrument is commonly used for d.c. measurement . /t utili$es themagnetic effect produced by current carrying conductor and the working principleis similar to that of 0% Arsonal galanometer # the difference being that a directreading instrument is proided with pointer and scale arrangement. This type ofinstrument consists of the following fundamental parts:

a) Moing coil

The moing coil is wound with many turns of enameled or silk coeredcopper wire . The coil is mounted on a rectangular aluminum former which ispioted on 7eweled bearings The coil moes freely in the field of a permanentmagnet .

27


28/51

b) Magnet system

The permanent magnet consisting of long M shape haing soft iron polepieces is used in 1-- type meter. The material used for magnet is

Alcoma" # Alnico # which hae a high coercie force .The flu" density ariesfrom 8.9 'b@m to 9 'b@ m. Since coerciely is high # it is possible to use asmall coil haing small no. of turns and hence a reduction in olume isachieed . Alternatiely in instruments haing a large scale length it ispossible to increase the air gap length to accommodate large no. of turns.

c) Control mec(anism

'hen the coil is supported between two 7ewel bearings # the control torque isproided by two phosphor bron$e hair springs . These springs also seres asleads to the moing coil .

d) Damping mec(anism

0amping is proided by either air friction mechanism or fluid frictionmechanism . Air friction mechanism consists of a light aluminum piston whichis attached to the moing system . This piston moes in a fi"ed air chamber

which is closed at one end . The clearance between piston and chamberwalls is uniform throughout and is ery small. The cross section n of thechamber is either circular or rectangular . 'hen the piston moes into thechamber # the air inside is compressed and the pressure of air # thus built up #opposes the motion of piston and hence whole of the moing system . 'henthe piston moes out of the chamber # pressure in the closed chamber falls #and the pressure on the open side of the piston is greater than on the closedside . Thus there is again an opposition to motion. /n an alternatie

28


29/51

arrangement # an aluminum ane is attached to the spindle and moes in asectored shaped air chamber . This chamber is coered at the top andbottom by plates. The aluminum piston shold not touch the wall otherwiseerror may occur. /n fluid friction damping mechanism # a light disc is attachedto the spindle and dips into an oil pot and is completely submerged in iscous

oil . 'hen the spindle rotates # the disc also rotates and a frictional dragforce acting on the both surfaces of the oil is produced . This force alwaysopposes the motion . /f the iscosity of the oil is greater # more drag force canbe produced . /ncreased damping is obtained by attaching a no. of anes tothe spindle .

e) Pointer and scale:

The pointer is carried by the spindle and moes oer a graduated scale . Thepointer is of light+weight construction and # apart from those used in someine"pensie instruments has the section oer the scale ?twisted to form afine blade . This helps to reduce paralla" errors in the reading of the scale .The pointer blade usually moes oer mirror ad7acent to the scale so thatparalla" can be aoided by its reflection while noting the reading from thescale . The weight of the instrument is normally counter balanced by weightrigidly attached at the end the #diametrically opposite

Torque equation:

Fet N > No. of turns of the moing coilB >flu" density of 0.. magnetic field produced by permanent magnetl>>length of the coild>width of coilI >the current flowing through moing coil

then the torque equation is gien by: GINBIdlTd ==

where NBldG = > a constant!or spring control 1-- meter # steady deflection is produced whendeflecting torque becomes equal to control torque

29


30/51

Thus # cd TT = or KGI = # where > steady deflection and > spring

constantIKG )/(= or )/( GKI =

As the deflection is directly proportional to the current passing through themeter # an uniform scale can be made .

"MMC $mmeter and %oltmeter:

3wing to low current carrying capacity of the moing coil instrument # theseinstrument can be used directly for measurement of current and oltage . /nmicro+ammeter and low range mili+ammeters about 8mA # the entirecurrent to be measured can be forced to pass through moing coil ia springTheir current carrying capacity limits the current which can be safely carriedaround to about 8mA. !or higher currents # the 1-- instruments arelargely used in con7unction with shunts # when used as ammeters and with ahigh series resistance when used as oltmeters. 0.. ammeters are

normally designed to hae a oltage drop of nearly ;8m& to 988m& for fullscale deflection . &oltmeter hae a range of 8+;8m& or 8+988m& .

$mmeter shunt:

The basic moement of a 0.. ammeter is a 1-- type 0%Arsonalgalanometer . The moing coil is small and light and can carry ery smallcurrents# since the construction of d.c. ammeter for measuring current more than988mA is not feasible owing to the bulk and weight of the coil. 'hen heaycurrents are to be measured # the ma7or part of the current should be bypassed

through a low resistance parallel path # called% shunt. -anganin usually used forshunt as it gies low alue of thermal e.m.f. with copper although it is liable tocorrosion and is difficult to solder . onstantan is a useful material for a.c. circuitsince its comparatiely high thermal e.m.f. # being unidirectional # is ineffectie onthese circuits . The construction of shunt is the same as that of low resistancestandards .Shunts for low currents are enclosed in the meter casing but forcurrents aboe 88A # they are mounted separately (so that heat produced canbe effectiely dissipated.) .The general requirement for the ammeter shunt is

i) the temperature coefficients of the material for shunt and moing coilmust be low and should be nearly as possibly the same

ii) the shunt should carry the full scale current without e"cessietemperature riseiii) The shunt should hae a low thermal e.m.f. with temperaturei) The resistance of the shunt should remain constant with time

The resistance of the shunt can be calculated using conentional circuit.

Fet mR >internal resistance of the moing coil

!"R >resistance of shunt

30


31/51

mI >full scale deflection current

I > current to be measuredSince the shunt resistance is in parallel with the moing coil # the oltage dropsacross shunt and coil must be same .

The ratio of total current to the coil current is called the multiplying factor of shunt

!"

m

m R

R

I

Im +== 1

resistance of shunt > )1/( = mRR m!"

ffect of temperature changes in Ammeter:

Effect of temperature change in $mmeters:

The temperature error can be eliminated when the shunt and the moing coil aremade of the same material and kept at the same temperature . This method #howeer # is not satisfactory in practice as the temperature of the two parts arenot likely to change at the same rate . Additional disadantage of using coppershunt is that they are bulky as the resistiity of copper is small . Alternatiely thetemperature effect can be eliminated by adding a series resistance made of

magnanin to the moing coil in addition to manganin shunt . The manganin hasresistiity 8 to J8 times higher than copper and also it has negligibletemperature co+efficient . Since copper forms a small fraction of the seriescombination # the proportion in which the current would diide between the shuntand coil would not change appreciably with the change in temperature . Theseries resistance is called in this arrangement 2s#amping resistance6

31


32/51

Multi&range $mmeters:

The current range of 0.. ammeter may be further e"tended by adding multipleshunts # selected by a range switch . /f # if there are n No. of multiplying factors as# there should be n No. of shunt resistances as . At a time only one shuntresistance is to be selected by a selector switch as shown in the fig. /n analternatie arrangement # the Mniersal shunt (or Ayrton shunt) may be used formulti+range ammeter. /t uses a single shunt resistor with multiple tapping point sothat different alues for shunt resistor can be selected (shown in fig.) by selector

switch. 4y setting the selector position to 9st#nd # O nth # different multiplyingfactors (as ) can be set. This is illustrated in the following way :

onsidering that the meter ranges hae to be e"tended to nIIII ...,, 321 . # the

range selector switch position is to be set at 9# #O nth respectiely. !or switchposition 9#

11 )( RIIRI mmm =

1

11 1

R

R

I

Im m

m

+== or #

Thus the alues of different sections of resistance i.e. . The adantage of anAyrton shunt is that it eliminates the possibilities of the meter being in the circuitwithout a shunt .

32


33/51

%oltmeter multiplier:

A 1-- meter can be conerted into 0.. oltmeter by connecting a seriesresistance with it . This series resistance is called as multiplier . The multiplierlimits the current through the meter so that it does not e"ceed the alue for fullscale deflection and thus preents the moing coil being damaged .

The resistance of the multiplier can be calculated as follows:

Fet mR > internal resistance of the moing coil

!R >multiplier resistancev >oltage across the moing coil for current& >!ull scale oltage of the 1-- type oltmeter

Since the multiplier resistance is connected in series # then mmRIv =

And )( !mm RRI& +=

m

mm

mm

! R

I

&

I

RI&R =

=

So the oltage multiplying factor ism

!

mm

!mm

R

R

RI

RRI

v

&

m +=+

== 1

)(

resistance of multiplier m! RmR )1( =

ence for the measurement of oltage m times the oltage range of theinstrument # the series resistance should be (m+9) times the meter resistance .

The essential requirement of the multiplier are:

33


34/51

i) The multiplier resistance must be non+inductie typeii) The change in multiplier resistance with temperature should be small

or the temperature co+efficient of multiplier material must be ery smalliii) They should be non+inductiely wound for a.c. meters.iv) The resistance materials used for multipliers are manganin and

constantan

Multirange D'C' %oltmeter:

/n a multirange oltmeter # different full scale oltage ranges may be coered by

i) use of indiidual multiplier resistance ii) by a potential diider arrangement.

a) $y multiplier resistance:

/n this method different oltage ranges are obtained by connecting differentalues of multiplier resistors in series with the meter . The number of theseresistors is equal to the number of ranges required. !or e"ample # if there are n

no. of oltage ranges used as nmmmm ....,, 321 # then the no of different alues

resistors will be as !n!!! RRRR .......,, 321 etc. such that their alues will be equal

to :

m!

RmR )1(11

=

m! RmR )1( 22 =

m! RmR )1( 33 =

OOOOOO

mn!n RmR )1( =

where mR >meter resistance

34


35/51

b) $y Potential diider:

-ultiple resistances are connected in series with the 1-- instrument suchthat connection to unknown oltage source is made to any one 7unctions of

these series resistances nRRRR ....,, 321 for selection of proper oltage range

as . The series resistances for the oltage ranges n&&& ,...., 21 can be

computed as follows:

mmmm

m

m

m

RmRRmRRv&R

I&R )1(

/ 11

111 ===

mmmmmm

m

m

m

RmmRmRRmRmRRv

&RR

I

&R )()1()1(

/ 12121

2

1

2

2 ====

Similarly

mm

m

RmmRRRI

&R )(

2321

3

3 ==

This method has the adantage that all multipliers e"cept the first hae

standard resistance alues and can be obtained commercially in precisiontolerances . The range multiplier # 1R # is the only special resistor which must

be manufactured to meet specific circuit requirement .

35


36/51

T(e sources of error in PMMC instrument:

i) The weakening of permanent magnets due to aging at temperatureeffects

ii) 'eakening of springs due to aging and temperature effects:The weakening of springs with time can be reduced by careful use ofmaterial and presaging during manufacture . oweer # the effect ofweakening of springs on the performance of the instrument is opposite tothat of aging of magnet. The weakening of magnet tends to decrease thedeflection for a particular alue of current # while the weakening of springs

tends to increase the deflection. /n 1-- instrument # a 9 increase oftemperature reduces the strength of spring by about 8.8H G and reducesflu" density in the air gap of the magnet by about 8.8 G . Thus the neteffect # on the aerage # is to increase the deflection by about 8.8 G per.

iii) hange of resistance of the moing coil with temperature. The moingcoil is usually wound with copper wire haing a temperature co+efficient of8.88H@ . when the instrument is used as a micro+ammeter or milli+ammeter and the moing coil is directly connected to the output terminalsof the instrument # the indication of the instrument for a constant current

would decrease by 8.8H G per rise in temperature. /n case the moingcoil instrument is used as a oltmeter a large series resistance ofnegligible temperature co+efficient is used. . This eliminates the error dueto temperature . This is because the copper coil forms a ery smallfraction of the total resistance of the instrument circuit and thus anychange in its resistance has a little effect on the total resistance.

36


37/51

$dvantages and disadvantages of "MMC instruments:

i) The scale is uniformly diided .ii) The power consumption is ery low as ;u' to 88u'iii) The torque ?weight ratio is high which gies a high accuracy of about

G of full scale deflection.i) Since the operating force is large on account of large flu" densitieswhich may be as high as 8.; 'b@m # the errors due to stray magneticfields are small.

) These instrument c an only be used in 0.. circuit # but not in A..circuit because if the direction of current flow through moing coilreerses periodically # the directions of deflecting torque also reerses .0ue to inertia of the moing system # it can not follow the rapid reersal. Thus the pointer attached to the moing system and occupies themean position ( which is $ero )oer the scale .

i) The cost of these instrument is comparatiely higher than other type of

instruments.

The chief disadantages are:

These instruments are useful only for 0. measurement . The torquereerses if the current reerses . /f the instrument is connected to a.c. # thepointer can not follow the rapid reersal and the deflection correspondingto mean torque # which is $ero . ence these instruments can not be usedfor a.c. The cost of these instruments is higher than that of moing ironinstruments.

37


38/51

Moving Iron Instrument:

/n 1-- type instrument # the deflection is directly proportional to the d.c.quantity being measured . /f this instrument is used to measure the a.c. quantity #the torque acting on the moing system is alternating .This means that themoing system tends to oscillate about its $ero position . /f the frequency ofalternation of a.c. quantity is high # the moing system of the instrument can not #in general # follow these alternations # and so no deflection will be obsered . So1-- instrument can not measure the a.c. quantity .

/n moing iron type instrument # there is no moing coil # instead of which a plateor ane of soft iron with high permeability forms the moing element of thesystem . The iron ane is so situated that it can moe in a magnetic fieldproduced by a stationary coil . The coil is e"cited by the the current or oltageunder measurement . when the coil is e"cited # it becomes an electromagnet andthe iron ane moes in such a way so as to increase the flu" of theelectromagnet by occupying a position of minimum reluctance . The deflectingforce or torque produced is always in such direction so as to increase theinductance of coil .The moing iron instrument is classified into two types according to theconstruction as i! attraction type ii! repulsion type '

/n attraction type construction # the moing iron # consisting a flat disc made ofsoft iron is eccentrically mounted or pioted near the end of coil . 'hen thecurrent flows through the coil # a magnetic field is produced # which attracts thedisc . The disc tend to moe from the weaker magnetic field outside the coil intothe stronger field inside it . 'hateer the direction of the current in the coil # themagneti$ation of the moing iron is always such that attraction takes place.Thus # the direction of deflecting torque will not depend on the direction of

38


39/51

current flowing through the coil and hence the pointer deflects to only onedirection . The controlling torque is proided by springs but graity control can beused for panel type instrument which are ertically mounted. 0amping isproided by air friction with the help of a light aluminum piston # which moes in afi"ed chamber closed at one end or with the help of a ane attached to the

moing system.

/n repulsion type construction# there are two anes inside the coil # one is fi"edand the other is moable. Two different types of design are followed as radialane ii) coa"ial type ane . /n radial ane type # two rectangular soft iron stripsare used # one is fi"ed at inside surface of the coil and the other is attached to thespindle radially .'hen the coil is energi$ed by the current # the anes aresimilarly magneti$ed and a force of repulsion e"ists between them . This forcemoes the moable ane away from the fi"ed ane. /n coa"ial type design #thefi"ed iron may consists of tongue shaped piece of sheet iron bent into acylindrical form # the moing iron being another piece of sheet iron bent andmounted so as to moe parallel to the fi"ed iron and towards its narrower end.

The controlling torque is proided by springs The damping torque is proided byair friction piston moing in enclosed chamber. The operating magnetic field inmoing iron instrument is ery weak due air core electromagnet and thereforeeddy current damping is neer used . /t is clear that whateer may be thedirection of current in the coil # the iron anes are so magneti$ed that there isalways a force of attraction or repulsion . The moing iron is unpolarised #i.e. theyare independent of the direction in which the current passes . therefore # theseinstrument can be used with both a.c. and d.c.

39


40/51

Torque equation :

An e"pression of torque of a moing iron instrument may be deried byconsidering the energy relations when there is a small increment in currentsupplied to the instrument . when this happens there will be small deflection and

some mechanical work will be done .Fet dT be deflecting torque

-echanical work done > dTdSuppose the initial current is I # and the inductance of the coil is '/f the current increases by dI then the deflection changes by d and the/nductances changes by d'

/f e be the applied oltage # thendt

dI'

dt

d'I'I

dt

de +== )(

The electrical energy supplied is I'dId'IeIdt += 2

The stored energy changes from 'I2

2

1 to )()(

2

1 2 d''dII ++

The change in stored energy is 'Id''dIIdII 222

2

1))(2(

2

1+++

Neglecting nd and higher order terms #

The change in stored energy is d'II'dI 2

2

1+

!rom the conseration of energy# lectrical energy supplied >increase in storedenergyP mechanical work done

Thus Tdd'II'dII'dId'I ++=+ 22

2

1

d'ITd 2

2

1=

or deflecting torqued

d'ITd

2

2

1=

The moing system is proided with spring control and it comes to steadyposition when the deflecting torque is balanced by the controlling torque.

Thus cd TT = or

d

d'IK

2

2

1= or #

d

d'

K

I2

2

1=

ence the deflection is proportional to square of the rms alue of the operatingcurrent . The deflecting torque is # therefore # unidirectional whateer may be thepolarity of the current. As the deflection is proportional to square of current # it iseident that the scale of such an instrument is non uniform ./f there is no

saturation # the change of inductance with angle of deflection is uniform (i.e.d

d'

> a constant ) . Therefore the instrument e"hibits a pure square law response .

40


41/51

/n actual instrumentsd

d' is not constant and is usually a function of angular

position of the moing iron and thus the scale is distorted from the square law ina manner dependant upon the way in which inductance aries with angle ofdeflection . This ariation can be controlled by suitable design i.e. by choosing

proper dimensions # shape and position of iron anes . Thus it is possible todesign and construct an instrument with a scale which is ery nearly uniform oer

a considerable part of its length . The necessary condition relating tod

d'

against for lineari$ation may be obtained when it is assumed as CI =

Also 2

2

C

K

d

d'= or

2

2

C

K

d

d'=

>a constant . Thus for a linear scale # the product (

d

d') should be a constant . This is not possible as it requires to be infinite at

. /n practice the scale is made linear from the ma"imum deflection down to about

101 th of the ma"imum deflection . The plot of

dd' against oer the range is

a rectangular hyperbola . /t is possible to design the instrument in which a smallportion of the range # which is of particular interest or importance # is e"pandedoer a large part while the remainder of the scale is compressed into a relatielysmall space.

S(unts for moing iron instruments:

-oing iron instrument can be used for a range of ;8A since in these instrument #moing part does not carry ant current . ence shunts are not necessary # e"ceptfor large currents . oweer # range of the instrument can further be e"tendedusing shunt. !or a.c. operation multiplying factor of the shunt depends on thefrequency and inductance ?resistance ratios of coil and shunt as illustrated in thefollowing way:

Since the shunt is connected in parallel to the coil# the currents in coil and shuntare in inerse ratio of their impedances.

41


42/51

or2

2

22

22

)(1

)(1

)(

)(

!"

!"m

m

m!"

!"!"

mm

m

!"

R

'R

R

'R

'R

'R

I

I

+

+

=

+

+=

/n order that the diision of currents through coil and shunt shall remain same forall frequencies # the ratio of time constants of two branches must be same or in

other words #m

m

!"

!"

R

'

R

'= .

Multiplier for moing iron instruments:

The oltage range of moing iron instrument can be e"tended by the use of aseries resistance called multiplier with the fi"ed coil .!or operation on a.c. withdifferent frequencies # it is necessary that the total impedance of the oltmetercircuit should remain substantially constant oer a large frequency range . Sincethe series resistance forms a ma7or portion of the total impedance of theoltmeter circuit # it is desirable that this resistor should be either of non+inductietype or hae as small as inductance as possible .Fet R > resistance of meter' > inductance of meter

mI >meter current for full scale deflection>angular frequency&oltage drop across the meter for full scale deflection : 222 'RIv m +=

!R >resistance of multiplier& >oltage to be measured

Total resistance of meter circuit > !RR +

Total impedance of circuit > 222)( 'RR ! ++

The meter current 222)( 'RR

&I

!

m

++=

ence # oltage multiplying factor222

222)(

'R

'RR

v

&m

!

+

++

==

/t is eident that the multiplying factor will change with frequency .

Comparison bet#een Attraction and Repulsion type moing ironinstrument:

42


43/51

Sources of errors in moving iron instruments:

There are two types of errors which occur in moing iron instruments ?errorswhich occur with both a.c. and d.c. and the other which occur only with a.c. only .

Error (ith oth D'C' and $'C' :i) /ysteresis error:

This error occurs as the alue of flu" density is different for the samecurrent for ascending and descending order . The alue of flu" densityis higher for ascending alues of current and therefore # the instrumenttends to read higher for descending alues of current than forascending alues. This error can be minimi$ed by making the ironparts small so that they demagneti$e themseles quickly . Anothermethod is to operate the iron parts at low alues of flu" density so that

the hysteresis effects are small. 'ith the use of nickel iron alloys withnarrow hysteresis (4+ loop) # the error may be brought down to lessthan 8.8; G

ii) Temperature error :

The effect of temperature changes on moing iron on moing ironinstruments arises chiefly from the temperature coefficient of spring .The error may be 8.8 G per change in temperature . /n oltmeter #errors are caused due to self heating of coil and series resistance . Thetemperature of the coil may increase by 98 to 8 for a power

consumption of 9' . Therefore # the resistance increased by about H toK G # causing a decrease in current for a gien oltage . This producesa decreased deflection . Therefore# the series resistance should bemade of material like -aganin which has a small temperature co+efficient . the alue of series resistance should be ery large ascompared with the coil resistance in order to minimi$e errors due toself heating .

iii) Stray magnetic field error:

The errors due to stray magnetic field may be appreciable as theoperating magnetic field is weak ( about 8.88I to 8.88


44/51

hanges in frequency may cause errors due to changes of reactance of theworking coil and also due to changes of magnitude of eddy currents set up inthe metal parts of instrument. The change of impedance of the coil # ofresistance R and series resistance r is only of importance in case of oltmeter.s. /f F is the inductance of the coil # the current for an applied oltage & will be

gien by: 222)( 'RR&I! ++

=

The deflection of the moing ?iron oltmeter depends upon the currentthrough the coil . Therefore # the deflection for a gien oltage will be less athigh frequencies than at low frequencies . To some e"tent # compensation tothis type of error is possible by connecting a capacitor across the series

resistance !R

This shunting capacitor will make the circuit behae like a pure resistance sothat the frequency changes hae no effect on the readings of the instrument .Thus the compensated instrument will hae a power factor nearly equal tounity .

Now the impedance of the circuit222

2

11 !

!!

!

!

RC

CRjR'j

CRj

R'j(

+

+=

+

+=

Since 1 2222222

)()21( !!! CR'RCR + B 1444


45/51

Since the deflection of the instrument depends upon the current through thecoil # the deflection for a gien oltage will be less at high frequency than atlow frequency . The error due to this problem can be eliminated somehow byconnecting a capacitor in parallel to series resistance r .The idea of shuntingthe series resistor is to make the circuit behae like a pure resistor so that the

frequency changes hae no effect on the readings of the instrument . Thusthe compensated instrument will hae a power factor nearly equal to unity .

The changes in frequency will cause the change in eddy current induced inmetal parts of the instrument. Fet the mutual inductance between the coil andthe iron parts be ) . The induced oltage due to current / in the instrumentcoil lags the current / by E8 . As a result of this induced oltage an eddy

current flows and its magnitude is gien by: 222ee

e

'R

)II

+

= # where

A component of this eddy current #which will oppose the instrument current

is gien by:)()90(

'

eeeee SinIC!II ==

This current will crease opposing field thus reducing the torque on themoing system . !rom the phasor diagram # it is seen that

IR

)'

'R

)ISinII

e

e

ee

eee 2

2

222

' )(

+

== # when ee 'R >> # i.e. when

is small

and ee '

)II

'

# when ee R' >>

# i.e. when is large.Thus at low frequency # the eddy current effect increases with square of thefrequency while at high frequency # the effect is practically constant . !orthese reason moing iron instruments are unsuitable for frequencies aboe9; $.

45


46/51

%iration galvanometer:

The instrument is used generally as a.c. bridge null detector . n construction #the galanometer is similar to 0%Arsonal type # haing a moing coilsuspended between the poles of strong parmanent magnet . 4ut since theinstrument s used with a.c. circuit # the moing coil is energsed by a.c. . 'henan alternating current is passed through the coil # an torque is produced #which makes the coil ibrate with a frequency equal to the frequency of thea.c. . 3n account of inertia of the moing parts # the amplitude of ibrationbecomes small . oweer when the natural frequency of oscillation of themoing part is made to equal to the frequency of a.c. # the mechanicalresonance is obtained and the moing system ibrate about its a"is of rotationwith large amplitude . The damping is ery small in these galanometer norder to obtain sharp resonance characteristic.

46


47/51

onstruction:The moing coil consists of a few turns of wire made of fine bron$e orplatinum siler . This wires passes oer a small pulley at the top and s pulledtight by a spring attached to the pulley . The tension of the spring can bead7usted by tuning a head attached to the spring . the loop of wire s stretched

oer two iory bridge pieces # the distance between the pieces s ad7ustable .The moing coil carries a small mirror #upon which a focused light beam isincident and reflected . when the moing coil ibrate due to passage ofcurrent # the reflected beam from the mirror throws a band of light on thegraduated scale . the effectie resistance of the wire loop is about ;8 ohm .The current sensitiity s ;8 mm @uA

1erformance analysis :

Fet the alternating current flows through the moing coil # producing

instantaneous deflection from the equilibrium position .The equation of motion is gien by :

dcD TTTT =++

where

dT > deflection torque > )( tSinGIm

iT > /nertia torque>2

2

dt

dJ

DT >0amping torque>dt

dD

cT >ontrolling torque> S

G >0isplacement constant

)(2

tSinGISdt

dD

dt

dJ m

=++

The deflection can be determined by soling the aboe differential equationconsisting of .!. and 1./. # which represents the transient and steady statebehaiour of the moing system .

i.e. ftt +=)(

where

t > solution of au"iliary equation

i.e. 02

2

=++ SdtdD

dtdJ

and f >solution of particular integral

i.e. )(2

2

tSinGISdt

dD

dt

dJ m

=++

The roots of the au"iliary equations are :

47


48/51

J

JSDDm

2

421

= andJ

JSDDm

2

422

+

=

The solution of .!. is :tmtm

t BeAe 21

+=

Since the damping of the ibration galanometer is made ery small ( i.e.undamped condition ) # the roots and are comple"

The defection under steady state condition is found by soling the particular

integral : )(2

2

tSinGISdt

dD

dt

dJ m

=++

The 1.. is of the form : )(max = tSinf

)(max

= tC!dt

d f and )(2max2

2

= tSindt

d f

Substituting these alues in equation of motion # we get #

)()()(max

2

max

tSinGItC!DtSinJm

=+

or # )()(]).[( max2

maxmax tSinGItC!DJStSin m =+

Now when # at =t # then )(max SinGID m=

'hen2

)( =t # then )(max

2

max C!GISJ m=+

Squaring and and adding them # we get #22222

max

222

max )( mIGJSD =+

or # 222max)()(

JSD

GIm

+

=

Also from # 2tanJSD

= or # )(tan2

1

JS

D

=

)()()(

222

+

= tSinJSD

GImf

ence the complete solution is gien by:

)()()(

)2

4(..)(

222

2

2

++

=

tSinJSD

GIt

J

DJSSinFet m

tJ

D

Since the 9stterm of the aboe equation represent transient behaiour #whichusually affects only the 9stfew ibrations after switching on # and diminishes to$ero # the steady state deflection id gien by:

)()()(

)(222

+

= tSinJSD

GIt m

48


49/51

Thus under steady state condition # the deflection of the ibrationgalanometer shows sinusoidal ibration with amplitude as gien by :

222max

)()(

JSD

GIm

+

=

And the frequency of ibration s gien by :J

Sf

2

1= > undamped natural

frequency of the moing coil ./n the design of ibration galanometer # the damping s made e"tremely small( i.e. 0D ) # so that transient behaiour anishes ery quickly

Tuning of &ibration galanometer:

/t is the method by which the natural frequency of the moing system isad7usted to alue such that ma"imum amplitude of ibration will occurs for apassage of alternating current through the coil . /n tuning of the galanometer# the ob7ectie is to make the amplitude as large as possible for a giencurrent . The amplitude of ibration can be made ma"imum by ) increasingthe alue of galanometer constant (.e. =) ) by decreasing the alue of

222)()( JSD + . Since NBAG = # it may be made large by using a coil of

large area A and with large no. of turns N . Also = can be increased byincreasing the air gap flu" density 4 . 4ut increasing the alue of N and A will

increase also the alue of inertia constant L and the term 222 )()( JSD + .

therefore by increasing the alue of 4 by using a powerful magnet is moreeffectie for obtaining large amplitude of ibration and deflection sensitiity . f

the alue of the term 222 )()( JSD + can be decreased # the amplitude of

ibration will increase . 4ut for a gien designed galanometer # the alues ofconstants L and 0 are fi"ed and can not be changed . the control springcoefficients S can only be changed by arying the distance between the iorybridge pieces . This will ary the length of the wire loop and thereby ad7ustingthe tension of the suspension of the moing system .

N order that the amplitude be ma"imum # the term 222 )()( JSD + should

be minimum . This requires to fulfill the condition :

02= JS or # 0 ==

J

S

So when the frequency of the a.c. signal # applied s equal to the naturalfrequency of the moing system # the amplitude of ibration reaches the

ma"imum alue and is equal toS

J

D

GIm . . The condition is known as

mechanical resonance .The deflection of the moing system at arious frequencies of a.c. signal and

at constant amplitude can be calculated from which a plot of f s. canbe generated # which is a bell shaped cure . /t is found that for agalanometer haing a sharply peaked resonance cure a slightly deiation

49


50/51

from the resonance frequency results in large reduction in deflection.Therefore # the galanometer responds greatly to only fundamentalfrequency of e"citation # but less to the harmonic frequencies.&ibration galanometer can be used for frequencies between 988 to 9K88$ . the deflection sensitiity w. r. t. frequency is obtained by taking deriatie

of the deflection

w.r.t. .Thus # )]2)(.(2.2[

)()(2

1 2222

JJSDJSD

GI

d

dmf

+

=

>2

3

222

2

])()[(

)](.[2.

2

1

JSD

JSJDGIm

+

AtJ

S= # the sensitiity is gien by :

SDJGI

J

SD

J

SDGI

J

SD

GIdd m

mm

J

S

f

..

).(

..

).(

2

332

3

2

===

=

3r # )1

(.

.

.

.2222

D

GI

SD

JGI

SD

JGI

d

dmmm

J

S

f===

=

.

The ma"imum deflection occurs at frequency # when 0=

d

d f

i.e. 0)( 2 = JSJD or # )( 2JS

J

D= or#

2

2

J

D

J

S= or #

2maxJD

JS=

50


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Documents

PMMC & Moving Iron