1
Report of an experiment with a magnetic circuit .
A Report by Anirbit ( CMI , 1ST YEAR )
p.s There are some important hand written additions to this
report in the print out submitted.
1.
The Objectives :
Part A
Usually while writing the transformer equations we neglect the mutual
inductance of the coils and hence there is no dependence of the current in the primary to
the details of the secondary secondary circuit .
i) But it would be shown through the following experiment that the current
in the primary is non-trivially dependent on the resistance of the secondary and hence
the effect of the mutual inductance cannot be ignored.
It shall be shown theoretically that that the effect of the mutual inductance
manifests itself coupled with the self-inductance of the secondary coil as a factor named
as the “ reflected impedance “ .
Part B
It is to be noted that the differential equations in time that a three arm
transformer follows are extremely complex to solve though a steady state circuit
analysis is feasible.
i) In the following the aforementioned fact shall be exploited to get a
qualitative idea of the distribution of the flux of the primary in the left and the right
secondaries ( gauged by its manifestations through potential differences and currents )
in a 3 arm transformer by varying the loads in the 2 arms .
ii) Secondly we shall also show how the dependency of the potential across a
secondary coil on the current through it shows drastic changes depending on the load in
the other secondary.
iii) Thirdly we shall show that there exists a strong linear dependency of the
magnitude of impedance of the primary on the resistance of a secondary circuit
irrespective of what the load in the other secondary is.
2
Part A
a) The experimental set-up and the circuit details…
In this part of the experiment the following circuit is set up :
where
1. The left side is the primary circuit of the transformer and the right ride is the
secondary of the transformer .
2. The variables are as follows :
a)
b)
c)
d)
e)
f)
g)
h)
}
Ip = R.M.S value of the current in the primary circuit .
Is = R.M.S value of the current in the secondary circuit.
Rp = lumped resistance in the primary circuit .
Rl = lumped resistance in the secondary circuit.
Rs = the variable load in the secondary circuit.
Ls = the self inductance of the secondary .
Lp = the self inductancwe of the primary.
M = the mutual inductance of the secondary and the primary coils .
{ The currents are all in Ampere (A) units and the Voltages are in Volt (V) units
3. Ep is the sinusoidal potential source in the primary achived by a step down
18-0-18 V transformer ( from 230 V 50 Htz A.C mains supply ).
4. The coils in the transformer comprise of 300 turns of 23 SWG copper wires
over the laminated core.
5. 6 digital-multimeters are supplied for both the parts.
3
b) The observation table
The circuit is set up as as shown above and the R.M.S values of Ip is measured with
the DMM for various values of Rs and the following observation table is obtained :
Rs (  ) Ip ( A)
1
3.15
1.8
2.95
2.2
2.8
3.3
2.64
4.7
2.37
5.8
2.2
7
2.05
8.2
1.92
The graph of Ip vs Rs is :
Ip
3.2
3.0
Ip ( Amperes )
2.8
2.6
2.4
2.2
2.0
1.8
0
1
2
3
4
5
Rs ( Ohms )
6
7
8
9
4
c) The Inferences
The graph is non-linear , convex to the origin and downward sloping.
This allows us to make the qualitative inference that the current in the primary is inversely
proportional to the resistance in the secondary and the dependence although not very steep is
monotonically decreasing .
d) The Theoretical Analysis
Let Ip be the complex number whose real part is the approximately sinusoidal
steady state current in the primary and similarly Ep .
{ We note that the DMM s measure |
Ip
Ep
| and | | }
2
2
So in the complex representation of the AC circuit we have by Kirchoff’s Laws applied to
the steady state :
( Rp + i  Ls) Ip - i  M Is = Ep ( Primary Circuit )
( Rs + Rl +i  Lp) Is - i  M Ip = 0 ( Secondary circuit )
So eliminating Is we get :
Ip =
Ep
Rp  iLp 
 ^ 2M ^ 2
( Rs  Rl )  iLs
So we observe the following from the above expression :
1. If M is not accounted for then Ip is independent of Rs and by varying Rs we have shown a
strong dependence of | Ip | on M
2. Due to the complex representation of the above equation the dependence of Ip on Rs is
not apparent whereas the graph shows the inverse relation between them .
3. The term
 ^ 2M ^ 2
is called the “ Reflected Impedance “
Rs  Rl  iLs
5
4. If the secondary is open i.e (Rs + Rl) =  then the primary current is determined by the
parameters of the primary circuit alone ie
Ip =
Ep
Rp  iLs
5. If the secondary is shorted i.e ( Rs+ Rl)  0 then we shall have
Ip =
Ep
M ^2 

Rp  i  Lp 

Ls 

6 . As a further special case of the above if we consider that as in an ideal case M^2 =
K^2(LsLp) and that K = 1 then we have :
Ip =
Ep
Rp
6
e) Answers To The Questions Asked In The Handout
Ans 1 .)
If a bulb is connected to the primary of the circuit keeping the secondary open
then due to the high value of self inductance of the primary coil , flux linkage through it is
very high and hence the major part of the supplied voltage is dropped across it and hence the
potential difference across the bulb is not sufficient to make it glow .
Ans 2.)
Due to the reasons already specified above it is evident that the maximum part
of the supplied potential is being dropped across the primary coil.
Ans 3.)
For the safety of the apparatus the current in the primary should be kept low
i.e within 2 A ( further too low a current will make it difficult to be able to get the
qualitative analysis ) and when it is near 2 A it should not be allowed to pass for a long
time .
It should be so arranged that the maximum potential is dropped across the
primary coil.
7
3. Part B
a)The experimental set-up and the circuit details…
The circuit diagram is as follows :
where
1. E0 = the potential across the terminals of the step down transformer which
steps
down from 220 V (approx.) A.C mains to 7.4 V (approx.)
2. Ip , Il , Ir = the RM.S values of the steady state currents in the primary circuit
and left and the right secondary respectively.
3. Rp , Rl , Rr = the lumped resistances of the primary circuit and the left and the
right secondary respectively.
4. Lp , Ll , Lr = the self inductances of the primary circuit and the left and the
right secondary respectively.
5. Ml , Mr = the mutual inductances between the left coil and the primary coil
and
between the right coil and the primary coil respectively .
6. Ep , El , Er = the potential difference across the coils in the primary , left and
the
right secondary respectively.
8
b) Certain Observations About The Procedure Of This Experiment
1. We neglect any mutual inductance between the coils of the left and the right
secondaries.
2. Because of the very geometry of the transformer we find it difficult to measure the
flux linkage through any coil.
3. Due to the above mentioned constraint we measure/observe 3 other parameters of
sny of the arms as and when which is found to be more easy or useful. i.e
a) the current in that arm of the circuit .
b) potential drop across a coil or a resistance .
c) or qualitatively estimate by observing the glow of the bulb in the left arm.
4. We necessarily observe that in this experiment the number of parameters is > 2 ( the
number of independent variables is already = 2 i.e Rl and Rs ) hence the dependencies are
very high dimensional . So to be able to demonstrate the dependencies well we have 2
alternatives :
a) We plot 3 dimensional plots to show the dependencies on the 2
independent variables.
b) We take our independent variables in such combinations (by keeping
either of them constant) that we can pick out pairs of interdependent
parameters to plot and hence effectively take projections in our very high
dimensional parameter space.
2. We assume the 2 sides of the primary coil to be symmetrical and the 2 coils (each of 300
turns of 23 SWG wires ) of the left an the right secondaries to be identical . We assume
this symmetry in the circuit and assume that if the circuit conditions in the left and the
right interchanged then the results would be identical . So we create out parameters on
either side depending on when which is found convenient.
9
c) The Observation Tables
We vary the load resistances in the left and the right secondary and measure the
parameters Ep ( in the sense as specified in the brackets above ) ,El , Er , Ip , Il
and Ir
For each of the configurations.
i) Both The Secondaries Have Finite Resistances (With One Constant)
Rr
Rl
El
(Ohms) (Ohms) ( Volts)
Ep
(Volts)
Er
Il
Ip
Ir
(Volts) (Amperes) (Amperes) (Amperes)
1.3
1.2
2.28
6.17
2.27
2.000
2.000
1.98
1.3
2.1
1.96
6.30
3.3
1.486
1.474
1.455
1.3
2.4
1.844
6.34
3.55
1.38
1.374
1.35
1.3
3.5
1.545
6.50
4.2
1.164
1.155
1.127
1.3
5.1
1.250
6.67
4.82
0.943
0.938
0.905
1.3
6.0
1.117
6.73
5.10
0.845
0.838
0.806
1.3
7.3
0.974
6.79
5.37
0.735
0.733
0.696
1.3
8.0
0.824
6.79
5.59
0.608
0.602
0.577
1.3
8.9
0.779
6.80
5.66
0.574
0.58
0.528
1.3
10.6
0.670
6.90
5.9
0.500
0.500
0.475
1.3
15.6
0.515
7.01
6.23
0.381
0.39
0.341
1.3
18.4
0.448
7.02
6.33
0.337
0.344
0.291
1.3
22.6
0.364
6.93
6.3
0.36
0.272
0.234
1.3
27.0
0.317
7.09
6.62
0.244
0.244
0.144
10
ii) The Left Secondary is shorted ( Rl  0 )
Rr (Ohms)
Rl (Ohms)
El ( Volts)
Ep (Volts)
Er (Volts)
Il (Amperes)
Ip (Amperes)
Ir (Amperes)
0
1.2
0.1864
5.93
3.17
2.92
2.86
2.84
0
2.1
0.143
6.12
4.22
2.2
2.21
2.17
0
2.4
0.437
5.98
4.29
1.725
1.705
1.662
0
3.5
0.333
6.02
4.76
1.347
1.327
1.291
0
5.1
0.273
6.51
5.55
1.104
1.09
1.054
0
6.0
0.242
6.61
5.77
0.976
0.962
0.924
0
7.3
0.209
6.68
5.96
0.839
0.83
0.79
0
8.0
0.191
6.66
5.98
0.772
0.764
0.725
0
8.9
0.178
6.73
6.09
0.715
0.708
0.668
0
10.6
0.154
6.77
6.1
0.623
0.617
0.575
0
15.6
0.114
6.93
6.51
0.454
0.45
0.407
0
18.4
0.0979
6.97
6.54
0.396
0.395
0.35
0
22.6
0.0833
6.99
6.63
0.338
0.338
0.291
0
27.0
0.0721
7.04
6.63
0.293
0.294
0.245
0
50.7
0.1974
6.97
6.73
0.1737
0.1804
0.1283
0
99.5
0.1334
7.02
6.83
0.1174
0.1249
0.0672
11
iii) The Left Secondary Is Open ( Rl   )
Rr
(Ohms)
Rl (Ohms)
El (
Volts)
Ep
(Volts)
Er
(Volts)
Il
(Amperes)
Ip
(Amperes)
Ir
(Amperes)
infinity
1.2
6.92
7.15
0.1389
0
0.0715
0.0628
infinity
2.1
6.85
7.11
0.1884
0
0.0712
0.0619
infinity
2.4
6.83
7.12
0.214
0
0.0709
0.0615
infinity
3.5
6.77
7.10
0.276
0
0.0702
0.0604
infinity
5.1
6.70
7.1
0.362
0
0.0695
0.0590
infinity
6.0
6.64
7.09
0.413
0
0.0689
0.0579
infinity
7.3
6.54
7.04
0.479
0
0.0685
0.0568
infinity
8.0
6.54
7.06
0.511
0
0.0683
0.0563
infinity
8.9
6.51
7.05
0.558
0
0.0675
0.0553
infinity
10.6
6.45
7.07
0.631
0
0.0669
0.0541
infinity
15.6
6.28
7.09
0.846
0
0.0651
0.0502
infinity
18.4
6.2
7.12
0.955
0
0.0640
0.0483
infinity
22.6
6.05
7.08
1.09
0
0.0629
0.0461
infinity
27.0
5.93
7.11
1.229
0
0.0614
0.0435
infinity
50.7
5.4
7.1
1.765
0
0.0569
0.0338
12
d) Analysis Of The Graphs To Determine The Qualitative Behaviour.
I) Both Secondaries Have Finite Non Zero Resistances
i) Behaviour of the currents when both the secondaries have finite
resistances
a) The graph of Il vs Rr :
Il
2.2
2.0
1.8
1.6
Il (Amperes)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
Rr (Ohms)
We can easily infer from the above graph that Il is(non-linear) inversely
proportional to Rr
and Il undergoes a slight rise and fall after 15 Ohms
13
b) The graph of Ip vs Rr
Ip
2.2
2.0
1.8
Ip (Amperes)
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
Rr (Ohms)
We can infer from the above graph that Ip is (non-linear) inversely proportional to
Rr
c) The graph of Ir vs Rr
Ir
2.0
Ir (Amperes)
1.5
1.0
0.5
0.0
0
5
10
15
20
25
30
Rr (Ohms)
We can infer from the above graph that Ir is (non-linear)inversely proportional to Rr
14
ii) Behaviour of the potential differences when both the secondaries have
finite resistances
a) The graph of El vs Rr
El
2.5
2.0
E l (Ohms)
1.5
1.0
0.5
0.0
0
5
10
15
20
25
30
R r (Ohms)
We can infer from the graph that El is (non-linear) inversely proportional to Rr
b) The graph of Ep vs Rr
Ep
7.2
7.0
Ep (Volt)
6.8
6.6
6.4
6.2
6.0
0
5
10
15
20
25
30
R r (Ohms)
We can infer from the graph that Ep is (non-linear) directly proportional to Rr.
15
and it undergoes a slight fall and rise after around 20 ohms.
c) The graph of Er vs Rr
Er
7
6
E r (Volts)
5
4
3
2
0
5
10
15
20
25
30
R r (Volts)
We can infer from this graph that Er is (non-linear)inversely proportional to Rr
and it undergoes a slight fall and rise from around 20 ohms.
d) The graph of Er vs Ir
Er
7
6
Er (Volts)
5
4
3
2
0
5
10
15
Ir (Amperes)
20
25
30
16
We can infer from the above graph that Er is (non-linear) directly proportional to Ir
and undergoes a slight fall and rise from around 20 ohms.
II) The Left Secondary Is Shorted
i) Behaviour of the currents when the left secondary is shorted .
a) The graph of Il vs Rr
Il
3.0
2.5
Il (Amperes)
2.0
1.5
1.0
0.5
0.0
0
20
40
60
80
100
Rr (Ohms)
We can infer from the above graph that Il is (non-linear) inversely proportional to
Rr
17
b) The graph of Ip vs Rr
Ip
3.0
2.5
I p (Amperes)
2.0
1.5
1.0
0.5
0.0
0
20
40
60
80
100
Rr ( Ohms )
We can infer from the above graph that Ip is (non-linear) inversely proportional to
Rr
c) The graph of Ir vs Rr
Ir
3.0
2.5
Ir (Amperes)
2.0
1.5
1.0
0.5
0.0
0
20
40
60
80
100
R r (Ohms)
We camn infer from the above graph that Ir is (non-linear) inversely proportional to
Rr
18
ii) Behaviour of the potentials when the left secondary is shorted.
a) The graph of El vs Rr
El
0.45
0.40
0.35
E l ( Volts)
0.30
0.25
0.20
0.15
0.10
0.05
0
20
40
60
80
100
Rr ( Ohms)
We can infer from the above graph that El is ( non-linear ) inversely proportional to
Rr and it undergoes a rise and fall in the approximate ranges 1 to 2.5 ohms and after
around 27 ohms.
b) The graph of Ep vs Rr
Ep
7.2
7.0
Ep (Volts)
6.8
6.6
6.4
6.2
6.0
5.8
0
20
40
60
R r (Ohms)
80
100
19
We can infer from the above graph that Ep is (non –linear) directly proportional to
Rr and it increases with a decreasing rate and thus showing an asymptotic behaviour.
c) The graph of Er vs Rr
Er
7.0
6.5
6.0
E r ( Volts)
5.5
5.0
4.5
4.0
3.5
3.0
0
20
40
60
80
100
Rr (Ohms)
We can infer from the above graph that Er is (non –linear) directly proportional to
Rr and it increases with a decreasing rate and thus showing an asymptotic behaviour.
20
III) The Left Secondary Is Open
i) Behaviour of the currents when the left secondary is open .
a) The graph of Ip vs Rr
Ip
0.072
0.070
Ip (Amperes)
0.068
0.066
0.064
0.062
0.060
0.058
0.056
0
10
20
30
40
50
Rr (Ohms)
We can infer from the above graph that Ip is inversely proportional to Rr
And the dependency is almost linear
b) The graph of Ir vs Rr
Ir
0.065
0.060
Ir (Amperes)
0.055
0.050
0.045
0.040
0.035
0.030
0
10
20
30
Rr ( Ohms)
40
50
21
We can infer from the above graph that Ir is inversely proportional to Rr
and the dependency is almost linear.
ii) Behaviour of the potentials when the left secondary is open.
a) The graph of El vs Rr
El
7.0
6.8
6.6
E l (Volts)
6.4
6.2
6.0
5.8
5.6
5.4
5.2
0
10
20
30
40
50
R r (Ohms)
We can infer from the above graph that El is inversely proportional to Rr
and the dependency is almost linear.
22
a) The graph of Ep vs Rr
Ep
7.4
7.3
7.2
Ep (Volts)
7.1
7.0
6.9
6.8
6.7
6.6
0
10
20
30
40
50
R r ( Ohms)
We can infer from the above graph that Ep is almost constant with respect to Rr but
slightly oscillating around a mean value of 7.1 V which is slightly lower than the
potential across the transformer.
a) The graph of Er vs Rr
Er
1.8
1.6
1.4
Er (Volts)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
10
20
30
40
50
Rr (Ohms)
We can infer from the above graph that Er is directly proportional to Rr
And the dependency is almost linear.
23
IV) Some Special Dependencies That Were Observed
i) Dependency of Er on Ir when both secondary resistances are
finite(non-zero)
Er
7
6
Er (Volts)
5
4
3
2
0
5
10
15
20
25
30
Ir (Amperes)
We observe that Er is non-linearly and directly proportional to Ir and though it
shows a slight rise and fall at around 20 – 30 ohms we might expect it to have an
asymptotic behaviour.
This also indicates from its slope that the magnitude of the impedance of the right
secondary is also increasing sharply with increase of its current.
24
ii) Dependency of Er on Ir when the left secondary is open.
Er
1.8
1.6
1.4
Er (Ohms)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.030
0.035
0.040
0.045
0.050
0.055
0.060
0.065
Ir (Amperes)
We observe from the above graph that Er is inversely related to Ir when the left
secondary is open ( exactly reverse behaviour from previous case when the left
secondary had a finite non zero resistance ).
Secondly this dependency is strongly linear whereas the previous case was highly
non-linear.
Thirdly from the slope of the graph we have the unavoidable conclusion that it
shows a negative value of the magnitude of impedance (!)
iii) Dependency of Impedance of the primary on the resistances of either
of the secondaries.
We see that the value of
Ep
is equal to the effective A.C resistance or Impedance of
Ip
the primary which we denote as Zp. In the following we observe some interesting
dependencies of Zp on the values of the load resistances in the secondaries.
25
a) The graph of Zp vs Rr when both the secondary resistances are finite
EpIpZp
30
Ep/Ip = Zp (Ohms)
25
20
15
10
5
0
0
5
10
15
20
25
30
Rr (Ohms)
We observe that Zp is directly proportional to Rr and the dependency is strongly
linear.
b) The graph of Zp vs Rr when the left secondary is shorted.
EpIpZp
60
Ep/Ip = Zp (Ohms)
50
40
30
20
10
0
0
20
40
60
80
100
Rr (Ohms)
We observe that Zp is directly proportional to Rr and the dependency is strongly
linear.
26
c) The graph of Zp vs Rr when the left secondary is open.
EpIpZp
125
Ep/Ip = Zp (Ohms)
120
115
110
105
100
0
10
20
30
40
50
Rr (Ohms)
We observe that Zp is directly proportional to Rr and the dependency is strongly
linear.
27
28
d) The Theoretical Analysis
In order to avoid the complex differential equations that a dynamic analysis of
time evolution of the circuit will involve we look at the approximate steady state
analysis
of the flux distribution of the primary coil among the 2 secondary coils.
We assume for theoretical ease that the magnetic permeability of the
laminated core of the conductor is sufficiently high to contain the total flux produced
and there is no loss of flux out of the transformer .
Let  p be the flux through the primary coil of the transformer and
let  l and  r be the flux through the left and the right secondary coils. Therefore
from the above assumption we get  p =  l +  r.
We note that that  p ,  l,  r are all due to the self inductance of the
respective coils and mutual inductances ( ignoring mutual inductance between the 2
secondaries ).
We also observe that the potential differences and the current
measured are a reflection of the time variation of the  ’s . But as the time intervals
considered are the same for all ultimately the measurements indicate how is  p
distributed among  l and  r .
We consider the following special cases to get a qualitative feel of
the situation :
29
i) Both Secondaries Are Open ( Rr = Rl   )
Here we have Rr = Rl   and we expect the current in the
primary to behave as if there are no secopndaries and hence determined by the parameters of
the primary alone. Let Np , Nl and Nr be the number of turns in the primary and the left and
the right secondaries . Here we note that there will exist a El and an Er but no Il or Ir .So
we have the following equations :
Ep = - Np
dp
dt
,
d l
dp
d r
=
+
dt
dt
dt
By symmetry we have  l =  r and Np = Nl = Nr and hence
d l
Ep
d r
= - Nr
=
dt
2
dt
Ep
Er = El =
2
Er = El = - Nl
i.e
ii) Both Secondaries Are Shorted ( Rr = Rl  0 )
Since the geometries of the 3 arms of the transformer are equal we
expect their inductances to follow the following equations :
Ll = CNl2 , Lr = CNr2 and Mr = k LpLr = kCNpNr and similarly Ml = kCNpNl
Where C is a constant which depends on the inherent geometries of the coils and hence
equal for both the left and the right coils . k is the coupling constant between the primary
and the secondary and by symmetry is the same for both .
Now invoking the fact that Np = Nl = Nr we get Mp = Mr = Ll = Lr and let Mp
= Mr be = M and Ll = Lr = L .
So when the above equality conditions are invoked in the steady-state’s maximal current
equations along with the fact that Rr = Rl  0 we get :
i  MIp = i  LIl and i  MIp = i  LIr
and hence the conclusion that Il = Ir = Ip .
Further due to the above equations we also get that there will be ideally no net flux
intercepted through either of the secondaries and hence no potential differences across them
and by the equation  p =  l +  r we get that there will be no flux through the primary
as well .As a result the effective impedance of the primary is greatly reduced and hence the
primary current is very high .
30
iii) Extreme Asymmetric Loading ( Rl   and Rr  0 )
Here the Ir is not limited by any load resistance and it will be ideally of such value
so that the magnetic flux it produces almost completely negates the flux produced by mutual
induction from the primary i.e ideally
MIp = LIr
So  r = 0 and hence Er = - Nr
 l.
d r
= 0 . Further  p =  l +  r and hence  p =
dt
Therefore taking the time derivative and since Np = Nl we get Ep = El .and we already
have
Er = 0 .
And since M = L we get Ip = Ir .
iv) Both Secondaries Have Equal Resistances
We note that the three legged core of the transformer is
symmetrically fabricated ( with respect to the number of turns of the wire , the wire
type and dimensions ) hence the values of Lp , Ll , Lr are equal and and also Ml = Mr
for similar reasons . Hence with symmetric loading the circuit is expected not to
differentiate between the two sides and we have  p =  l +  r , and  l =  r and
hence  l =  r =  p/2 and hence we expect their time derivatives to be also equal and
hence El = Er =
Ep
and Il = Ir .
2
V) Any Combination Of Values Of Rr And Rl
From a theoretical standpoint this is the most important case as
the behaviour of the circuit is characterized by its response to this range of finite values .
Here the theoretical analysis is very complex for the dynamic time evolution of the circuit
but the steady state values can be estimated by the solutions of the 3 variables Ip , Il , Ir for a
given value of Rr , , Rl and Eo of the following approximate equations in the complex
number representation :
Ip Rp + i  ( M( Il + Ir ) - L Ip ) = Eo
Rl Il + i  (M Ip - L Il ) = 0
Rr Ir + i  ( M Ip - L Ir ) = 0
31
{
This equations other than being approximate in neglecting the very small transient
component of the steady state current are also assuming resistances of the wires to be = 0
and hence we should ideally have the potential across the transformer to be equal to the
potential across the primary but in practice this assumption is not found to be true always as
shown in the readings of section (e)
}
Hence to overcome these above cited difficulties we resort to experimental results to be able
to atleast qualitatively gauge the mode of dependency of the currents and the potential drops
on the Rr and Rs from the graphs .
Here we more clearly see the need to do the measurements by varying only one of Rr or Rl
at a time so that the individual dependencies of the complex solutions can be gauged.
32
e) Conclusions About Some Distinctive Observations Asked To Be
Confirmed
1.
Let a bulb be attached to the left secondary.We observe that the
bulb glows when the right secondary is complete and not when the right
secondary is open although there is no electrical connection between
them.
As explained in Part B section d (iii) when one of the secondaries is shorted the
entire flux of the primary links to the other secondary whereas when both of the
secondaries have resistances ( may be infinite when open as here ) the flux distributes itself
in some proportion and hence the flux linkage is less than when it is shorted .
And potential drop across the bulb is proportional to the flux linkage in the coil to
which it is connected . Hence the above observation is explained.
2.
Confirmation of theory when both the secondaries are open.
We make the following observations in this case :
Ep = 7.25 V
El = 3.58 V
Er = 3.52 V
Ip = 0.02 A
Il = 0.00 A
Ir = 0.00 A
We note the following agreements with the theory of d (i)
1. El  Er  Ep/2
2. Potential created by the transformer across its terminals is = 7.36 V and it is
found to be very close to Ep = 7.25 .
33
3.
Confirmation of theory when both the secondaries are shorted .
We observe the following
El = 0.10 V
Er = 0.14 V
Ep = 6.50 V
We observe the following correspondences with theory:
i) El and Er are almost equal as predicted by theory .
ii) Potential across the transformer is not equal to Ep
4.
Confirmation of theory in the case of asymmetric loading
We set Rl   and Rr  0 and we make the following observations
Ep = 7.25 V
El = 7.20 V
Er = 0.00 V
Il = 0.01 A
Ir = 0.05 A
Ip = 0.05 A
We observe the following correlations with the theory of d (iii)
1. Ep and El are almost equal as predicted in theory (deviation of 0.69 %)
2. Ip and Ir are equal as predicted by theory .
3. Er is equal to 0 as predicted .
4. Ep and ( El + Er ) are almost equal as predicted by theory (deviation of
0.69%)
34
5.
Confirmation of theory when both the secondaries have the same
resistance.
We fix Rr = Rl = 3.3 Ohms and obtain the following observation table.
Ep = 6.92 V
El = 3.25 V
Er = 3.05 V
Ip = 0.92 A
Il = 0.92 A
Ir = 0.91 A
We see the following correspondences with the theory of d (iv)
1. El and Er are very close ( deviation of 6.56 % ) but not exactly equal as
predicted.
2. El and Er are close to Ep/2 but not exactly equal as predicted (deviation of
about 6.46%)
3. Potential across the transformer is not equal to Ep .
We note from the comparison of values of Ep and Eo that only when
the secondaries are in extreme asymmetric loading or both open state
the resistance of the wires of the primary circuit can be neglected and
otherwise not.
35
36
f) Answers To The Questions Asked In The Handout
Ans 4
The maxwell’s law about the curl of the electric field states that the curl of the
electric field in an infinitesimal region is equal to the negative of the parial derivative of
the magnetic field at that region at that instant. This law when surface integrated and
surface integral of the curl of the electric field converted to line integral of the field along
the circuit loop involved in the secondary coil tell us that that a potential difference will
be generated across the secondary as the flux of the magnetic field through it is changing
with time since the current passing in the primary i.e the current causing the magnetic
field in the secondary coil is AC .
Since the number of turns in the primary and the secondary coils is the same
the flux through them is the same and hence (assuming the coupling to be near ideal ) the
induced potential is almost equal to the potential in the primary.
Ans 5
If the secondary circuit is completed then the resistance in it will be very low
and it will allow a high current to pass through it and its maximal value will increase till
the net flux through through it becomes equal to 0.
The current flowing in the secondary will cause a time variant flux to be
induced in the primary (mutual induction ) and then the current in the primary will also
increase to nullify this change . Hence a high current will flow in the primary making the
attached bulb glow brighter .
The process will equilibriate (i.e the maximal values of the sinusoidal variations
will stabilize ) when the flux linkage in the secondary becomes 0 and the flux linkage is
restored to its original values . (final maxima of the currents will be solutuions of 2
simultaneous equations )
Ans 6
The phenomenon of mutual inductance makes the current in the primary
dependent on the current in the secondary and hence by shorting the secondary or
making any change in the secondary changes the impedance ( or the effective resistance in
an A.C circuit ) .The dependence is strong if the coupling constant is near to 1 as it is in
this experiment.
37
Ans 7
Increasing the number of turns in the secondary will increase the flux linked
through it and the potential induced across it proportionately ( if current is not allowed to
flow through the secondary ) if it is done without affecting the relative geometries
between the coils i.e the coupling constants remain remain near to 1 even after the change
.
A higher potential induced in the secondary will trigger the same response as
explained in answer 5 and will cause a huge current to flow in the primary and this may
cause burning or damaging of the components in the primary if it goes beyong its
tolerance limit.
Ans 8
The above explanation and answer 5 together indicate that we must not allow
high current to flow through the secondary for the safety of components connected t0 the
primary.
But if flux linkage through the secondary increases (either due to changes in the
primary or due to effects as in ans 7 or effects in ans 6) nature will take its own course to
increase the flux linkage of the opposite sign to negate the effect . So that nature does it
not at the expense of passing more current we must use the coils of high inductance so
that current in the secondary is kept low . This will be able to prevent the sudden rise of
current in the primary since that can increase only due to mutual induction which here
will only depend on the secondary current .
Overall we must avoid shorting the secondary.
38
4. Analysis Of This Experiment On The Magnetic Circuit
a) The Technical Aspects
i) The experimenter’s observations regarding the procedure of
the experiment have been already put down in section 3)Part B (b) .
ii) The various types of precautions that need to be taken while
doing the experiment have been explained through the answers to the questions 5 ,
6 , 7 & 8 in the section 3) Part B (f) .
iii) The Sources Of Errors And Their Remedies
suggested/Adopted
i) The DMM shows arbitary fluctuations and at times the amplitude is about 1 V
or 1 A . To glean the truth out of such situations we have adopted the following
measures :
a) Instead of the DMM probes banana clips were used so that the fluctuation caused
due to the movement of the contact point of the pin head probe and the large
socket is reduced .
b) Attempt was made to reduce disturbances in the circuit due to movement of the
wires as minimum as possible.
c) The rust on the sockets could also have caused the readings to fluctuate
frantically and so it is suggested to use sockets of stainless steel
d) The mean value of the fluctuations has been taken as the data
ii)
The resistors heat up very fast and causing thus making the current drop
at times by around 1.5 A per 7 min . To counter this the following things
were done :
a) 6 DMM’s were used in the second try of the experiment to be able to take the
measurements of the 6 parameters simultaneously ( initially due to insufficient
DMM’s error due to continual change of circuit was huge) . Further this reduces
time for which current has to continually pass through the circuit and hence
prevents heating up.
b) Power was turned off in between circuit rearrangements.
c) Measurements were tried to be noted down within 1 min to reduce the time for
which current passes.
39
d) The process of taking the mean of the readings was used to get quality data.
iii) In the attempt to reduce the time for which current had to be passed
continually there could have been some minor errors in not allowing the circuit to reach
the steady state after the transient components become negligible.
iv) The magnetic fields of the transformer in the 3 arm case are not well
contained and hence was a fairly good amount of time variant magnetic field around it
producing a potential difference of about .3 to .4 V in the atmosphere near it . Presence of
this magnetic field was causing unwanted induction effects in the adjoining parts of the
circuits . (especially the crocodile clips ) . This induction effect introduced caused a spatial
dependence of the resistance of the circuit elements . To minimize this attempt was made to
keep the movement of the circuit linkages as low as possible and positions as constant as
possible during the process of reading the data. 2 remedies can be suggested here :
a) The transformer can be made of better layering by reduction of gaps and by using
material of higher magnetic permeability.
b) To create some magnetic shield around the transformer.
40
b) The educational aspects
i) Fundamental Knowledge
We observe that this experiment involves the knowledge of the following broadly
classified areas of physics and hence the student is motivated to read in details about
these topics :
a) The theory of A.C circuits especially analysis of steady-state currents
in L-R circuits with an A.C source using complex number representation.
b) The theory of electromagnetic induction involving the practical
applications and the manifestations of the maxwell’s law about the curl of the electric
field. i.e in inductors and transformers.
c) The general theory of electrical circuits.
ii) Connecting Theory And Experiment Through The Questions
The questions in the handout make the student think and try to understand
the following two areas:
a) The questions involve understanding the maxwell’s law about the curl
of electric field especially the questions 1 to 4 .
b) The questions from 5 to 8 make the student think about the various
possible accidents that can occur while working with transformer circuits as
manifestations of the maxwll’s equation mentioned .
Starting from the basic fundamentals of electrodynamics the student
is motivated to think out how each of the accidents can be prevented and hence
he/she tries to devise his/her own precautionary methods.
iii) Laboratory Skills
This experiment develops in the student the following laboratory skills :
a) Use of sophisticated measuring instruments like the DMM and he/she
gets acquainted with the various errors that are possible with electronic gadgets that
he/she might be unaccustomed to having worked with mostly mechanical apparatus
previously.
41
Analysis of the various types of errors ( like fluctuations) and how the
student can counter them have been explained in the earlier section.
b) The student gets familiar to using electrical circuits involving high
magnitudes of voltages and currents and hence he/she learns to work with them
judiciously . And as has been explained in the answers 5 to 8 in the process the
students fundamentals in electrodynamics gets strengthened as he/she analyses the
causes of the risk prone situations from the depth and also hence understands how
the precautionary methods evolve out of the fundamentals.
c) The various combinations of circuit involved in this experiment gives the
student practice in connecting various types of circuits and being able to judge when
which type of connection should be used among the available choices of banana clips ,
crocodile clips etc.
iv) Analytical Skills & Data Representation
This experiment’s data has the property of having very complex
interdependencies and the number of parameters very large . Hence the student gets
the scope to train himself in the following skills :
a) The ability to make the right choice of parameters to bring out the
dependencies clearly i.e how and which parameters should be kept constant when to
be able to see the relationship clearly.
b) Since the value of the various parameters in the various situations are
varying over various ranges the student is exposed to the art of being able to choose
i) the origin of the graph
ii) and the scale along the axes
so that the dependencies are visually apparent on the graph paper.
c) The experiment hones the students analytical skills in being able to to the
following :
i) interprete the dependencies among the parameters
ii) and to understand the correlations and discrepancies between the
theoretical predictions and the practical realities.
42
v) Oppurtunity To Innovate
Most importantly the experiment in its utter complexity of parameters is actually
a gold mine of opportunities to explore fascinating relationships and innovate
different combinations of circuits and hence encourages the student to think
independently and thus create new experiments with the given apparatus to explore
beyond the asked question.
.