Moving Coil Galvonometer

8/2/2019 Moving Coil Galvonometer

1/27

Galvanometer (Detecting presence of small currents (or) voltages.

Bridger Potentiometer

Moving Coil

Current carrying element Rectangular Circular

Use cylinder core Spherical core

The iron core is used to provide a flux path of low reluctance andtherefore to produce strong magnetic field for the coil to move in.

This creases the deflecting torque and hence the sensitivity of theGalvanometer.

RR

F

RF

1


2/27

dt

dC

emf

Damping is obtained by connecting a low resistance across thegalvanometer terminals.

Eddy current links with magnetic field Electromagnetic torque.

Moving Coil is supported

upper suspension lower suspension

Mirror spring controlling torque


3/27

Particular Integral

When we pass steady current through the galvanometer under steady stateconditioning

i

Fdt

d

dt

d

and;0;0

2

2

GiKdt

dD

dt

dJTTTT dCDj

2

2

(or) (i)

Putting above conditions in (i) we get (ii)K

GiF

Complete solution of D.E. is

F

tmtm

BeAePICF 21

(iii)

= Complementary function + particular integral

Transient Condition Steady State Condition


4/27

Complementary Function

Auxilary equation is

02 KDmJm

JKJDDm

JKJDDm

24;

24

2

2

2

1


5/27

Underdamped Motion of a Galvanometer

J

DKJjD

J

KJDD

mm 2

4

2

4

,

22

21

Real Imaginary

djmm 21,

tbtae t sincos

Fdd tbta ]sincos[eSolutionComplete t-

cos;sin FbFa

Fd

t

Fdd

t

tFe

tFtFe

][sin

]sincoscossin[


6/27

Fdt

J

D

tFe

sin2

where

KJDJDKJJ

DKJJ

radJ

DKJ

d

d

d

d

4444

42

sec/2

4

222

222

2

2

(1)

So, we have to remove and F.

Initial conditions At t = 0;=0

FF sin0

F

F sin (2) Here 2 unknown constant and F.

(6)


7/27

ddtJ

D

d

J

D

tJ

D

tFetFedtd

cossin 222

For finding maximum value for ,

Differentiate and equal if to zero.

At t= 0; [ derivative of constant value

att=0;

is constant ]

0

dt

d

cossin2

0 dFFJ

D (3)

DDKJ

DJ

DJTan dd

2

422

D

DKJTan

21 4 (4)


8/27

24

2

sin DKJ

KJF F

F

(5)

Substituting (4) and (5) in (1)

D

Jte

DKJ

KJ dd

tJ

D

F

2tansin

4

21 12

2

D

DKJ

J

tDKJe

DKJ

KJ tJD

F

21

22

2

4tan

2

4sin

4

21 (7)

From (6)


9/27

In underdamped case, d is the angular frequency

dd f 2

J

DKJf dd

2

4

2

1

2

2

(8)

Time period

24

22

21

DKJ

J

fT

dd

d

(9)


10/27

Undamped Motion of a Galvanometer:

(10)

From (5)

FF (11)

In this case, there are no damping forces i.e D = 0

But in practical cases, it is not possible.

J

K

J

KJdn

2

04

J

Kfn

2

1

(12)

Free period of oscillation is

K

J

fT

x

21

0 (13)


11/27

From (1)

Final deflection

Fdt

J

D

tFe

sin2

FxF te 90sin0

FxF t cos

txF cos1 (14)

0

1

1

90

tan

004tan

KJ


12/27

Critical Damped Motion of a Galvanometer (D2 = 4KJ)

zeroispartImaginary221

J

D

mm

Ftm

eBtA 1

Ft

J

D

eBtA

2 (15)

Where A and B are constants.

The value of A and B can be found by differentially eq. (15)

BeJ

DeBtA

dt

d tJ

DtJ

D

22

2

(16)


13/27

* Initial value condition at t = 0; = 0

From (15); 0 = A + F

A = -F

* Maximum value condition

From (16)

0;0 tdt

d

F

F

J

DB

BJ

D

BA

J

D

2

20

2

0

Note : The time taken by thecoil to reach its final steadyposition is smaller in the case ofcritical damping than that for

overdamping.


14/27

Solution ist

J

D

FFF etJ

D2

2

tJ

D

FF etJ

D2

21

tJ

De

tJ

D

F2

11 2 (17)

Now for critical damping KJDD C 2 (18)

Where DC = damping constant for critical damping.

Under critical damping conditions

J

K

J

KJ

J

Dx

2

2

2 (19)

For critical damped Galvanometer

tC xt

Fx

11 (20)


15/27

Overdamped Case: D2 >4KJ

In this case, the roots m1 and m2 are real and unequal.

1

2

21

2

2 22

12

1

12

11

tt

F

xx

ee

The above expression represents a decaying motion without oscillations(or) overshoot.

However, this motion is usually slow and is not desirable in indicatinginstruments.

The value of

CD

D


16/27

* Intrinsic constants = J, D and K.

* Operational constants = sensitivity, critical damping resistance and thetime period.

relative damping = free period = T0sensitivity = F

(1) Damping ration

(2)

(3)

CD

D

KJ

D

2

xJ

K

KJ

D

J

D

22

2

22

2

22

1

tan

111

11

24tan

D

KJ

D

DKJ


17/27

(4)

(5)

(6) Substituting all these values in

cos;1sin 2

2

222

2

22

1

42

4

xd

xxdJ

D

J

K

J

DKJ

2

1

2

2

1tansin1

sin4

21

te

teDKJ

KJ

d

t

d

xF

d

tJD

F

x

224

2

4

2

DKJ

KJ

DKJ

J

J

K

d

x

21

21sinsin

1

11

te d

t

Fx

From (5)

21

1

d

x


18/27

(7)

(8)

(9)

All the above equations are derived in terms of , T0, F.

20

20

0

1

1

2

2

d

x

d

d

x

d

x

d

x

d

d

T

TT

T

f

f

f

f

T

T

00

22

TTx

212

0

2

21sin1

2sin

1

11 0

tT

et

T

F

dd

tTd

FT

Tt

Te

T

T 012

0

sin2

sin1 0

2

0

2

12

1

Td

xd


19/27

0dt

d

Logarithmic Decrement

The time required for the deflection to reach a maximum value may be obtained by putting

x

t

ddd

t

d

xF

xx ettedt

d

sincos0Differentiating

221

2

11tantan

1tan

1tan

sincos

sincos0

t

t

t

tt

tte

d

d

x

dd

dxdd

dxdd

t

d

xF

x

The above equation is the only when where N is an integer. Ntd


20/27

When N = even integer, we get minimum value of deflection, since the initial value t = 0represents minimum, when N=odd, t corresponds to a maximum value.

The first maximum value of deflection occurs at t1 for N=1.

We have dt1 =

21

1

xd

t

Substituting this value in

* In place of = 1 and t=t1 and dt1= .

211

21 1sinsin

1

11 1

te d

t

Fx

(I)

211

21 1sin.1sin

1

11

2

x

x

eF

21

1 1

eF

21

1

eFF (1)

This equation gives the first overshoot.

2


21/27

* The first minimum deflection occurs at t = t2 where

d

t

22

211

2

22 1sin2sin

1

11 1

2

eF

21

2

21

1

11

2

eF

2

2

1

2

2

1

2

2 1

e

e

F

F

(2)

From (1) and (2); we get

22

1

1

2

1

1

log

2

F

Fe

F

F e


22/27

Logarithmic decrement is defined as the Naperian logarithm of the ratio of successiveswings.

Logarithmic decrement2

1

2

2

1

1

loglog

F

Fe

F

Fe

20 1 dT

TBut

00

T

T

T

T

d

d

From (I)

1

1

2

0

01

2

0

sinsin1

sin2

sin1

sin2

sin1 0

te

T

te

T

T

T

T

T

te

T

T

dd

x

F

d

tTd

F

dd

tTd

F

d

d


23/27

Damping due to coil circuit resistance:

The damping effect produced by current flowing in the coil is only present when circuit ofthe coil is closed.

R = resistance of galvanometer circuit when closed.

= Rg + Re

Rg = resistance of galvanometer coil,

Re = external resistance required for damping.

The voltage induced in the coil by its motion.i

currenteddy

currenteddy

dt

dGe

dt

dBlNd

diameter

r

d

BlN

rBlN

BlvN

ForceNe

222

2

2

2


24/27

The current flowing in the circuit due to emf e is

dt

d

R

GT

dt

d

R

GG

iBlNd

BilliBFdBilN

T

coil

coil

2

0 fieldradialin90

sin

distanceconductoreachonforceconductors

dt

d

R

G

R

ei

Torque produced owing to current flowing in the coil

(1)

But (2)dt

dDT circuitcoil

From (1) and (2)

R

GD

circuit

2

eg RRRwhereR

GD

2


25/27

* For critical damping

KJ

G

R

R

GKJ

RGD

KJeDDc

2

2

2

2

2

* External series resistance required for critical damping is

gge R

KJ

GRRR

2

2

This external resistance required for critical damping is called CDRXor ECDR, critical damping resistance external.


26/27

When ac is passed through the moving coil

Moving coil vibrates upto the frequency of current passing through coil a.c.

This wire passes over a small pulley at the top and is pulled tight by a spring

attached to the pulley

The tension of the spring can be adjusted by twining a milled head attached tothe spring

The loop wire is stretched over two ivory bridge pieces, the distance betweenthese pieces is adjustable


27/27

Moving coil vibration Galvanometer

The moving coil consists of a fine platinum silver wire superded b/w the piecesof a permanent magnet

This wire passes over a small pulley at the top and is pulled tight by a spring

attached to the pulley

The tension of the spring can be adjusted by twining a milled head attached tothe spring

The loop wire is stretched over two ivory bridge pieces, the distance betweenthese pieces is adjustable

Moving Coil Galvonometer

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