Transformer Model

where r = diag [r1 r2], a diagonal matrix, and

The resistances r1 and r2 and the flux linkages l1 and l2 are related to coils 1 and 2, respectively. Because it is assumed that 1 links the equivalent turns of coil 1 and 2 links the equivalent turns of coil 2, the flux linkages may be written as

Voltage Equation of a transformer in matrix form is:

Where

Linear Magnetic System

Reluctance is impossible to measure

accurately, could be determined using:

1 1 1 1 2 21

1

2 2 2 2 1 12

2

l m m

l m m

l

A

N i N i N i

N i N i N i

m

f

f

 =

= + +Â Â Â

= + +Â Â Â

Flux Linkages2 21 1 1 2

1 1 1 21

2 22 2 1 2

2 2 2 12

l m m

l m m

N N N Ni i i

N N N Ni i i

l

l

= + +Â Â Â

= + +Â Â Â

The coefficients of the 1st two terms on the right–hand side depend upon the turns of coil 1 and the reluctance of the magnetic system; i.e. independent of coil 2. Similar situation exist in equation for 2

Self Inductances

From the previous equations one can define Self-Inductances:

2 21 1

11 1 11

2 22 2

22 2 22

l ml m

l ml m

N NL L L

N NL L L

= + = +Â Â

= + = +Â Â

Where

Ll1 and Ll2 are leakage inductances of coil 1 and 2 respectively.

Lm1 and Lm2 are the magnetizing inductances of coils 1 and 2 respectively.

Magnetizing Inductances

The two magnetizing Inductances are related as:

2 12 22 1

m mL L

N N=

Where

m the Magnetizing Reluctance being common for both coils.

The mutual Inductances are defined: 1 2

12

2 121

m

m

N NL

N NL

=Â

=Â

Mutual Inductances

Mutual Reluctance being common for both

Circuits; Mutual Inductances are

related to Magnetizing Inductances too:

2 112 21 1 2

1 2m m

N NL L L L

N N= = =

Flux Linkages

Flux Linkages may be written as:

= Li

Where1 1 2

111 1211

221 222 12 2

l mm

mm

L L NLL L

NL NLL L

N L L

é ù+é ù ê úê ú ê ú= =ê ú ê úë û ê ú+ë û

Flux Linkages

The Flux Linkage may also be derived based on self and mutual inductances:

21 1 1 1 1 2

1

12 12 2 2 1 2

2

( )

( )

l m

m

NL i L i i

N

NL i L i i

N

l

l

= + +

= + +

Example 1A

It is instructive to illustrate the method of deriving an

equivalent T circuit from open- and short-circuit measurements. For

this purpose let us assume that when coil 2 of the two-winding

transformer shown in Fig. is open-circuited, the power input to coil 1

is 12 W with an applied voltage is 100 V (rms) at 60 Hz and the

current is 1 A (rms). When coil 2 is short-circuited, the current

flowing in coil 1 is 1 A when the applied voltage is 30 V at 60 Hz.

The power during this test is 22 W. If we assume Ll1 = L’l2, an approximate equivalent T circuit can be determined from these measurements with coil 1 selected as the reference coil.

Equivalent T Circuit of Transformer

Where i’2 =(N2/N1)i2

Example 1…..

1 1 1 cosP V I f® ®

=

Vand Ir r

The Power may be expressed:

Where are phasor and is the phase angle between them.

Solving for during the open circuit test, we have:

1 1 01 12cos cos 83.7

110 1

P

xV If - -= = =r r

Example 1……

Vr

0

0

110 012 109.3

1 83.7

VZ j

I

Ð= = = + W

Ð-

r

r

as a reference phasor and in an inductive circuit of the transformer I phasor would lag behind by the angle of =83.70

Z, the impedance may therefore be determined by:

That suggests that Xl1+Xm1 = 109.3, while r1 =12

V

I

Example 1…….

For short circuit test i1= -I’2 because transformers are

designed so that Xm1>> |r’2+jX’12|. Hence using phase

angle equation: 1 022cos 42.8

30 1xf -= =

In this case input Impedance is (r1+r’2)+j(Xl1+X’l2) and that is determined by:

0

0

30 022 20.4

1 42.8Z j

Ð= = + W

Ð-

That means r’2 = 10 and Xl1=X’l2 both are 10.2

Example 1….

That leads to conclusion that:

Xm1= 109.3 -10.2 =99.1

Hence other parameters are:

r1 = 12 Lm1 = 262.9mH r’2 = 10

Ll1 = 27.1mH L’l2 = 27.1mH

V1

I1V’2

E1

Phasor Diagram

T circuit ref. to Primary

Equivalent Circuit ref. to Primary

Active & Reactive Power

Magnetic Laws

Flux Linkage of a Coil

Fig. 1 shows a coil of N turns. All these N turns link flux lines of Weber resulting in the N flux linkages.In such a case:

Where

N is number of turns in a coil;

e is emf induced, and

is flux linking to each coil

N

de N

dt

y f

f

=

=

Design of Transformer

Let's try to proportion a transformer for 120 V, 60 Hz supply, with a full-load current of 10 A. The core material is to be silicon-steel laminations with a maximum operating flux density Bmax = 12,000 gauss. This is comfortably less than the saturation flux density, Bsat. The first requirement is to ensure that we have sufficient ampere-turns to magnetize the core to this level with a permissible magnetizing current I0A.

Design of Transformer….

Let's choose the magnetizing current to be 1% of the full-load current, or 0.1 A. The exact value is not sacred; this might be thought of as an upper limit. Here, we will assume a simple, uniform magnetic circuit for simplicity. If l is the length of the magnetic circuit, H is 0.4πN(√2I0)/l, and the magnetization curve for the core iron gives the H required for the chosen Bmax. From this, we can find the number of turns, N, required for the primary.

Design of Transformer

We could also estimate the ampere-turns required by using an assumed permeability μ. Experience will furnish a satisfactory value. It is not taken from the magnetization curve, but from the hysteresis loop. Let's take μ = 1000. Then, N = Bmaxl / 0.4π√2 μI0. If we estimate l = 20 cm, the number of primary turns required is N = 1350. The rms voltage induced per turn is determined from Faraday's Law:

√2 e = (2πf)BmaxA x 10-8. Now, e must be 120 / 1350 = 0.126 V/turn, f is 60, and Bmax = 12,000 gauss. We know everything but A, the cross-sectional area of the core. We find A = 2.8 cm2.

Design of Transformer

Powdered iron and ferrite cores have low Bsat and permeability values. A type 43 ferrite has Bsat = 2750 gauss, but a maximum permeability of 3000, and is recommended for frequencies from 10 kHz to 1 MHz. Silicon iron is much better magnetically, but cannot be used at these frequencies. The approximate dimensions of an FT-114 ferrite core (of any desired material) are OD 28 mm, ID 19 mm, thickness 7.5 mm. The magnetic dimensions are l = 74.17 mm, A = 37.49 mm2, and volume 2778 mm3. Similar information is available for a wide range of cores. There are tables showing how much wire can be wound on them, and even the inductance as a function of the number of turns.

Design Parameters

l average length

A:X-section Area

N number of turns

B = /A Laminated Core

Laminations

Flux Density & MMF

Bmax

Hysteresis loop

Hysteresis Loop