32
ELECTRICAL CIRCUITS
3. INTRODUCTION TO TRANSFORMERS
Introduction
The purpose of this development is to present the basic theory and principles of
transformers. This handout was written assuming the reader has had the basic engineering
physics sequence and is at least presently taking a basic circuits course. We start with a
qualitative description from electromagnetic theory to enhance the concept of inductance
and present mutual inductance. We develop the methodology to analyze electrical circuits
containing mutual inductance. We use this as a basis to develop the concept of
transformers and we derive all the relationships for perfect transformers. We end the
development with a discussion of the limitations of transformers that cause the perfect to
become non-ideal.
Inductance and Mutual Inductance from EM Theory
As part of a basic circuits course the V-I characteristics of the passive elements were
defined as:
V
Resistance, R : V  IR or I 
(1)
R
Capacitance, C : V 
1
dV
Idt or I  C

C
dt
(2)
dI
1
or I   Vdt
dt
L
(3)
Inductance, L : V  L
Steady state AC circuit analysis used these relations with the following changes to
Equations 2 and 3:
d
 j ,
dt
 dt 
1
j
(4)
We focus on Equation 3. Figure 1 illustrates a perfect inductor, a coil of wire without
resistance. If that inductor is excited with a voltage source, then the following relations
can be obtained.
33
Figure 3 The perfect inductor
This voltage source applied to the coil will be opposed by a voltage drop across the coil
that is proportional to the changing magnetic flux within the coil and the number turns of
the coil. The current that flows from the source is inversely proportional to the number of
turns of the coil and proportional to the magnetic flux within the coil.
V  K1 N
I
d
dt
K2

N
(5)
(6)
From Equation 6 we obtain:
d N dI

dt K 2 dt
(7)
Plugging Equation 7 into Equation 5:
V
K1 2 dI
N
K2
dt
(8)
Now, from Equation 5 we obtain:

1
Vdt
K1 N 
(9)
Plugging Equation 9 into Equation 6:
I
K2
Vdt
K1 N 2 
(10)
Comparing Equations 8 and 10 with Equation 3 we observe that the inductance is given
by:
L
K1 2
N  K3 N 2
K2
(11)
34
In Equation 11 the ratio of constants is simply another constant. That constant is a
function of the coil geometry and the magnetic properties of the material within the core.
The important feature of this development is that the inductance is proportional to the
square of the turns. For a simple coil it can be shown that the inductance of a long tightly
wound coil is given by:
L
 AN 2
l
Where:  is the core magnetic permeability constant
A is the core cross sectional area
l is the core length
(12)
We do not derive Equation 12 as that is done in a junior EM course for EE’s.
Now, consider the case where we have 2 coils (both without resistance) where the first
one is excited with a voltage source and the second is just open circuit. Figure 2
illustrates the geometry.
Figure 2 geometry of 2 coupled coils
In Figure 2 if the first coil is excited by a source VIN current will flow and flux will be
generated as per Equations 3 and 6. Depending on the geometry of the second coil
relative to the first, some of the flux of the first coil could shine through the second coil
and as per Equation 5 that time changing flux will cause a potential to appear at the
terminals of the second coil. That potential of the second coil is proportional to the
number of turns on the second coil and has the same general relation as Equation 5 with
some caveats.
d
dt
The magnitude and sign of K are dependant on the following:
VCoil 2  KN 2
1
2
How well the flux from the primary shines through the secondary coil.
The “sense” of the turns.
(13)
35
1 impacts the magnitude of K while 2 impacts the sign of K . In regard to 2, the polarity
of VCoil 2 will always be such that if a current would flow driven by VCoil 2 , that current
would generate a flux in coil 2 that would oppose the direction of the change of flux
coming from coil 1. If this were not the case the principle of conservation of energy
would be violated (“There is no free cheese for the mouse.”). We modify Equation 7 to
become Equation 14 with I1 and N1 indicating the current and turns in coil 1:
d N1 dI1
(14)

dt K 2 dt
Plug Equation 14 into Equation 13:
VCoil 2 
dI
K
N1 N 2 1
K2
dt
(15)
Examining Equation 15 we identify the mutual inductance as given by:
K
(16)
N1 N 2  K 4 N1 N 2
K2
Again the ratio of constants is simply another constant, K 4 . Observe that the mutual
inductance is proportional to the product of the turns in coil 1 and coil 2. In an EM course
the complete derivation of the methodology of computing mutual inductance will be
given along with the determination of the polarity of the secondary coil voltage from the
sense of the turns. Here, we simply assume that the mutual inductance is given along with
the polarity of the second coil voltage via the convention as illustrated by Figure 3.
M12 
Figure 3 illustration of mutual inductance and the dot convention
That convention is that a positive potential at the dot of the first coil causes a positive
potential at the dot of the second coil. Thus Equation 15 becomes:
VCoil 2  M12
dI1
dt
Analyzing Circuits with Mutual Inductance
(17)
36
Circuits containing mutual inductance are best handled via KVL. We will illustrate the
concepts via some simple examples. The analysis will be steady state AC with s used in
place of j . Thus:
1
1
(18)
j L  sL, j M12  sM 12 ,

j C
sC
Example 1
Obtain the loop equations for the circuit of Figure 4.
Figure 4 Example 1 circuit for mutual inductance
We obtain the following loop equations.
Loop 1:
VIN  I1 ( R1  sL1 )  VL1 fromL2
The term VL1 fromL2 is the additional voltage developed in L1 that will be positive at the dot
is caused by the current flowing into the dot at L2 and is expressed in terms of mutual
inductance as: VL1 fromL2   I 2 sM12 (the current into the dot at L2 is  I 2 ). Thus the equation
for Loop 1 becomes:
VIN  I1 ( R1  sL1 )  I 2 sM12
(19)
Now the equation for loop 2 is defined as:
Loop 2:
0  I 2 ( RL  sL2 )  VL2 fromL1
The term VL2 fromL1 is the additional voltage developed in L2 that will be positive at the dot
is caused by the current flowing into the dot at L1 and is expressed in terms of mutual
inductance as: VL2 fromL1   I1sM12 (it’s negative because it’s a voltage rise relative to the
direction of loop current I 2 and not a voltage drop). Thus the equation for Loop 1
becomes:
37
0  I 2 ( RL  sL2 )  I1sM12
0   I1sM12  I 2 ( RL  sL2 )
(20)
Example 2
Obtain the loop equations for the circuit of Figure 5.
Figure 5 Example 2 circuit for mutual inductance
We obtain the following loop equations using the technique of Example 1.
Loop 1:
VIN  I1 ( R1  sL1  sL2 )  I 2 (sL2 )  VL1 fromL2  VL2 fromL1
The term VL1 fromL2 is the additional voltage developed in L1 that will be positive at the dot
and is caused by the current flowing into the dot at L2 . It is expressed in terms of mutual
inductance as: VL1 fromL2  ( I1  I 2 )sM12 (the current into the dot at L2 is I1  I 2 ).
The term VL2 fromL1 is the additional voltage developed in L2 that will be positive at the dot
and is caused by the current flowing into the dot at L1 . It expressed in terms of mutual
inductance as: VL2 fromL1  I1sM12 (the current into the dot at L1 is I1 ).
Thus the equation for Loop 1 becomes:
VIN  I1 ( R1  sL1  sL2 )  I 2 (sL2 )  ( I1  I 2 )sM12  I1sM12
VIN  I1 ( R1  sL1  sL2  2sM12 )  I 2 (sL2  sM12 )
Loop 2:
(21)
0  I 2 ( RL  sL2 )  I1 (sL2 )  VL2 fromL1
The term VL2 fromL1 is the additional voltage developed in L2 that will be positive at the dot
and is caused by the current flowing into the dot at L1 . It expressed in terms of mutual
inductance as: VL2 fromL1  I1sM12 (the current into the dot at L1 is I1 ). However, the polarity
38
of VL2 fromL1 is that of a voltage rise relative to the direction of loop current I 2 and thus must
appear in the Loop 2 equation with a minus sign.
The equation for Loop 2 becomes:
0  I 2 ( RL  sL2 )  I1 (sL2 )  I1sM12
0   I1 (sL2  I1sM12 )  I 2 ( RL  sL2 )
(22)
Derivation of the Transformer Relations from Mutual Inductance
Transformer relationships are essentially a special case of mutual inductance whereby the
design of the mutual coupled inductors is such that some powerful simplifying
assumptions can be made. Referring to Figure 2 where flux generated in the first coil
partly shines through the second coil and this is what causes the mutual inductance effect.
Now, if all of the flux from the first coil were to pass through the second coil without any
leakage and if the inductances were very large, then these powerful assumptions can be
made. If the core material for the coils has an infinite permeability constant  relative to
air then the conditions to enable these assumptions are present. In fact,  for transformer
cores is about 50,000 times larger than that of air. Figure 4 for Example 1 illustrates the
circuit that could be a simple transformer. Under these conditions of almost non-existent
flux leakage, lots of turns on the coils and very high core  the inductances and mutual
inductance, based upon Equation 12, can be approximated by:
L1 
 AN12
l
, L2 
 AN 22
l
A
M12  K L1L2  K
N1 N 2
l
(23)
(24)
Thus we can represent these relations as:
L1  KG N12 , L2  KG N 22
(25)
M12  KKG N1 N2
Where: K G is a geometry material constant
K is a unit less constant of the fraction of flux that doesn’t leak
N1 is the turns on inductor L1
N 2 is the turns on inductor L2
(26)
We are now ready to derive the transformer relations. Consider the circuit of Figure 6 a
simple transformer circuit that will be analyzed with loop equations.
39
Figure 6 basic transformer circuit
We obtain the following loop equations using the mutual inductance relations:
VIN  I1sL1  I 2 sM12
0  I 2 (sL2  RL )  I1sM12
(27)
(28)
Solving the system of equations yields:
I1 
sL2  RL
VIN
s ( L1L2  M122 )  sL1RL
(29)
I2 
sM12
VIN
s ( L1L2  M122 )  sL1RL
(30)
2
2
Now plug in Equations 25 and 26 into Equations 29 and 30:
I1 
sKG N 22  RL
VIN
s 2 ( KG2 N12 N 22  K 2 KG2 N12 N 22 )  sKG N12 RL
(31)
I2 
sKKG N1 N 2
VIN
s ( K N N  K 2 KG2 N12 N 22 )  sKG N12 RL
(32)
2
2
G
2
1
2
2
Now let the flux non-leakage constant K =1, Equation 31 becomes:
sKG N 22  RL
VIN N 22
VIN
I1 
VIN  2 
2
sKG N1 RL
N1 RL sKG N12
If we rearrange these terms and plug L1 back in:
I1 
VIN
VIN

 N 2  sL1
RL  12 
 N2 
(33)
40
Now, do the same for Equation 32:

sKG N1 N 2
N2
N  1 
VIN 
VIN   VIN 2  
2
sKG N1 RL
N1RL
N1  RL 


N  1 
I 2   VIN 2  
N1  RL 

I2 
(34)
Equations 33 and 34 result in some very powerful observations that are the basis of the
transformer relations:
1. In Equation 34 we see that the voltage applied to L1 , the primary, reflects to the
N
secondary as the ratio of the turns, 2 .
N1
Destination
2. Also observe that this ratio followed the direction of the reflection:
Orgin
3. In Equation 33 we see that the load resistance has reflected to the primary as the
N 
square of the ratio of the turns:  1 
 N2 
2
4. Again this ratio followed the direction of the reflection:
Destination
Orgin
VIN
, this is referred to as magnetizing
sL1
current and if the primary impedance, sL1 is large, it is neglected.
5. Observe the second term in Equation 33:
Thus far we have derived the voltage and load impedance relationships for the
transformer. We need the current relationship. Equation 34 gives the current in the
secondary. If we neglect the magnetizing current in the primary, the current in the
primary becomes:
I1 
VIN
 N2 
RL  12 
 N2 
Now, we plug Equation 34 into 35:
N 
VIN  2 
VIN
 N1   N 2   I  N 2 
I1 


 2

2
RL
 N1 
 N1 
 N1 
RL 

 N2 
(35)
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N 
I1  I 2  2 
 N1 
(36)
Equation 36 provides the final transformer relationship, current reflects from one side of
Orgin
the transformer to the other side as the inverse of the turns,
. This makes
Destination
sense because it is keeping with the “No free cheese for the mouse” concept. Consider a
perfect transformer. Power at the primary is:
PPRIM  VIN I IN
(37)
And power at the secondary is:


N
N
PSEC  VSEC I SEC   VPRIM SEC   I PRIM PRIM
N PRIM 
N SEC


  PPRIM

(38)
The following summarizes the transformer relationships:
1.
2.
3.
The dot is the convention for winding to winding polarity.
Destination Turns
Voltage reflects across as the ratio of turns with:
Orgin Turns
Impedance reflects across as the square of the ratio with:
 Destination Turns 


 Orgin Turns 
2
Orgin Turns
Destination Turns
As a final note, observe the common connection of grounds between the primary circuit
and the secondary circuit in Figure 6. This connection is in fact optional as all this
development is perfectly valid with or without that connection. This feature is a very
important attribute of transformers, the ability to realize ground isolation between
primary and secondary circuits.
4.
Current reflects across as the inverse ratio with:
The Non-Ideal Transformer
We have seen that the perfect transformer assumptions allow for analysis simplification,
the reflection of voltage, current and impedance back and forth across the windings with
the appropriate turns ratio factor. The non-ideal transformer deviates from this, forcing
the analysis to become more complicated. The first departure is with flux leakage and
non-negligible magnetizing current. Additional complication comes with non-zero
winding resistance, interwinding capacitance, core loss and less than infinite primary and
secondary inductance. Inclusion of a few of these effects will very quickly, generate a
complicated model that will require computer simulations to replace simple hand
analysis. Additionally, the issue of measuring or estimating these non-ideal parameters
42
can be very difficult. Manufactures data sheets may provide some information for typical
parameters or data to allow estimated for typical parameters. Figure 7 illustrates the nonideal transformer equivalent circuit.
Figure 7 non-ideal transformer equivalent circuit
The parameters in Figure 7 are:
RWP : primary winding resistance
LE : leakage inductance
LP : primary inductance
N P : primary turns
N S : secondary turns
LS : secondary inductance
RWS : secondary winding resistance
CIW : interwinding capacitance
RCL : resistor to accommodate core loss
A few final comments on some of these degrading parameters are of interest. A DC
measurement of the winding resistance may yield values that are much smaller than
operational values because of the Skin Effect. The core loss resistance will be a
complicated function of frequency because of Magnetic Hysteresis loss and Skin Effect.
Interwinding capacitance in addition primary to secondary that is shown, also exists
shunting both primary and secondary windings. The capacitance shown will couple noise
from primary to secondary. The capacitance not shown will cause a self resonance effect.