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.
- 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.
Part A
a)The experimental set-up and the circuit details…
In this part of the experiment the following circuit is set up :
where
- The left side is the primary circuit of the transformer and the right ride is the secondary of the transformer .
- The variables are as follows :
a)Ip = R.M.S value of the current in the primary circuit .
b)Is = R.M.S value of the current in the secondary circuit.
c)Rp = lumped resistance in the primary circuit .
d)Rl = lumped resistance in the secondary circuit.
e)Rs = the variable load in the secondary circuit.
f)Ls = the self inductance of the secondary .
g)Lp = the self inductancwe of the primary.
h)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 }
- 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 ).
- The coils in the transformer comprise of 300 turns of 23 SWG copper wires over the laminated core.
- 6 digital-multimeters are supplied for both the parts.
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 :
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
Letbe the complex number whose real part is the approximately sinusoidal steady state current in the primary and similarly .
{ We note that the DMM s measure || and || }
So in the complex representation of the AC circuit we have by Kirchoff’s Laws applied to the steady state :
( Rp + iLs)- iM= ( Primary Circuit )
( Rs + Rl +iLp)- iM= 0 ( Secondary circuit )
So eliminating we get :
=
So we observe the following from the above expression :
- If M is not accounted for then is independent of Rs and by varying Rs we have shown a strong dependence of || on M
- Due to the complex representation of the above equation the dependence of on Rs is not apparent whereas the graph shows the inverse relation between them .
- The term is called the “ Reflected Impedance “
- If the secondary is open i.e (Rs + Rl) = then the primary current is determined by the parameters of the primary circuit alone ie
=
5. If the secondary is shorted i.e ( Rs+ Rl) 0 then we shall have
=
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 :
=
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.
3. Part B
a)The experimental set-up and the circuit details…
The circuit diagram is as follows :
where
- 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.)
- Ip , Il , Ir = the RM.S values of the steady state currents in the primary circuit and left and the right secondary respectively.
- Rp , Rl , Rr = the lumped resistances of the primary circuit and the left and the right secondary respectively.
- Lp , Ll , Lr = the self inductances of the primary circuit and the left and the right secondary respectively.
- Ml , Mr = the mutual inductances between the left coil and the primary coil and between the right coil and the primary coil respectively .
- Ep , El , Er = the potential difference across the coils in the primary , left and the right secondary respectively.
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.
- 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.
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 (Ohms) / Rl (Ohms) / El ( Volts) / Ep (Volts) / Er (Volts) / Il (Amperes) / Ip (Amperes) / Ir (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
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
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
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 :
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
b) The graph of Ip vs Rr
We can infer from the above graph that Ip is (non-linear) inversely proportional to Rr
c) The graph of Ir vs Rr
We can infer from the above graph that Ir is (non-linear)inversely proportional to Rr
ii) Behaviour of the potential differences when both the secondaries have finite resistances
a) The graph of El vs Rr
We can infer from the graph that El is (non-linear) inversely proportional to Rr
b) The graph of Ep vs Rr
We can infer from the graph that Ep is (non-linear) directly proportional to Rr.
and it undergoes a slight fall and rise after around 20 ohms.
c) The graph of Er vs Rr
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
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
We can infer from the above graph that Il is (non-linear) inversely proportional toRr
b) The graph of Ip vs Rr
We can infer from the above graph that Ip is (non-linear) inversely proportional to Rr
c) The graph of Ir vs Rr
We camn infer from the above graph that Ir is (non-linear) inversely proportional to Rr
ii) Behaviour of the potentials when the left secondary is shorted.
a) The graph of El vs Rr
We can infer from the above graph that El is ( non-linear ) inversely proportional toRr 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
We can infer from the above graph that Ep is (non –linear) directly proportional toRr and it increases with a decreasing rate and thus showing an asymptotic behaviour.
c) The graph of Er vs Rr
We can infer from the above graph that Er is (non –linear) directly proportional toRr and it increases with a decreasing rate and thus showing an asymptotic behaviour.
III) The Left Secondary Is Open
i) Behaviour of the currents when the left secondary is open .
a) The graph of Ip vs Rr
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
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
We can infer from the above graph that El is inversely proportional to Rr
and the dependency is almost linear.
a) The graph of Ep vs Rr
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
We can infer from the above graph that Er is directly proportional to Rr
And the dependency is almost linear.
IV) Some Special Dependencies That Were Observed
i) Dependency of Er on Ir when both secondary resistances are finite(non-zero)
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.
ii) Dependency of Er on Ir when the left secondary is open.
We observe from the above graph that Er is inversely related to Ir when the leftsecondary 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 is equal to the effective A.C resistance or Impedance of 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.
a) The graph of Zp vs Rr when both the secondary resistances are finite
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.
We observe that Zp is directly proportional to Rr and the dependency is strongly linear.
c) The graph of Zp vs Rr when the left secondary is open.
We observe that Zp is directly proportional to Rr and the dependency is strongly linear.
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 :
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 , = +
By symmetry we have l = r and Np = Nl = Nr and hence
Er = El = - Nl = - Nr=
i.e Er = El =
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 = 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 :
iMIp = iLIl and iMIp = iLIr
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 .
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 = 0 . Further p = l + r and hence p = l.
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 = and Il = Ir .
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 :
Rp + i( M( + ) - L ) =