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APPENDIX 1
DERIVATION OF TRANSFORMER EQUATIONS
The necessary equations for the formation of optimum design
problem of a transformer is derived below.
OUTPUT / CONSTRAINT EQUATION
Single Phase Transformer
The voltage induced in a transformer winding with T turns and
excited by a source having a frequency f Hz is given by
Voltage per turn Et
E
4.44 f m
T
(A1.1)
The window in a single phase transformer contains one primary and
one secondary winding.
Total copper area in window
Ac
= Copper area of primary winding + Copper area of
secondary winding
= primary turns x area of primary conductors +
secondary turns x area of secondary conductors
= T p a p Ts a s
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Taking the current density
secondary windings, a p I p /
and a s
to be same in both primary and
Is /
Ip
Total conductor area in window, Ac T p
I
Ts s
(T p I p Ts I s ) /
2 AT
as T p I p Ts I s
(A1.2)
AT if magnetizing mmf is neglected.
The window space factor K w is defined as the ratio of copper area
in window to total area of window,
or
Kw
Ac
Aw
Conductor area in window, Ac K w Aw
From Equations (A1.2) and (A1.3),
or
AT
2 AT
(A1.3)
K w Aw
Kw Aw
2
(A1.4)
Rating of a single phase transformer in kVA,
Q Vp I p 10 3 E p I p 10 3 (asVp E p )
Et T p I p 10 3 Et AT 10 3
K A
K A
Et w w 10 3 4.44 f m w w 10 3
2
2
2.22 f m Kw Aw
10 3
(A1.5)
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But m Bm A
i
Q 2.22 f Bm K w Aw Ai 10 3 kVA
(A1.6)
Three Phase Transformer
In this case, each window contains two primary and two secondary
windings.
Proceeding as in the case of single phase transformer,
Total conductor area in each window,
Ac = 2(a pTp +a sTs )
= 2(IpTp + IsTs ) /
4 AT
(A1.7)
Total conductor area is also equal to K w Aw
4 AT
or
AT
K w Aw
Kw Aw
4
(A1.8)
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Rating of a three phase transformer in kVA
Q 3Vp I p 10 3 3 E p I p 10 3
3 Et T p I p 10 3
3 Et AT 10 3
K A
3 4.44 f m w w
4
3.33 f m Kw Aw
10 3
10 3
(A1.9)
3.33 f Bm Kw Aw Ai 10 3 kVA (A1.10)
As the transformer is expected to deliver its rated kVA,
Equations (A1.6) and (A1.10) are taken as the constraint equations for single
phase and three phase transformers respectively.
CONDITIONS FOR OPTIMUM DESIGN
Minimum Cost / Volume / Weight
Consider a single phase transformer. Its kVA output is
Q = 2.22fBm K w A A x10-3
w i
Assuming that the flux and current densities are constant, the
product Ac A is constant for a transformer of given rating. Let this product
i
Ac Ai
M2
(A1.11)
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The optimum design problem is, therefore, that of determining the
minimum value of total cost.
Now, r
where
Ai
Ac
Bm Ai and AT
r
2 Bm
2
(A1.12)
is a function of r only, as Bm and
are constant.
Thus from Equations (A1.11) and (A1.12),
Ai
Ac
2Bm Ai
Ac
r
or
m AT and m
M
and Ac
M
Let CT be the total cost,
CT
C Cc
i
ci Gi cc Gc
ci gi Al
i i cc g c Ac Lmt
Substituting for A and Ac ,
i
CT c g l M
i ii
cc g c Lmt
M
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Differentiating CT w.r.t ,
dCT
d
1
2 cc g c Lmt M
1
c glM
2 i ii
For minimum cost,
dCT
d
ci gi li cc g c Lmt
or
3
2
0
1
ci gi li Ai cc g c Lmt Ac
or
c G
or
Ci
i i
cc Gc
Cc
(A1.13)
Hence, for minimum total cost, the cost of iron must be equal to the
cost of copper or conductor.
Similar conditions apply to other quantities, namely, for minimum
volume of transformer,
Volume of iron = Volume of conductor
i.e.
Ui Uc
(A1.14)
and for minimum weight of transformer,
Weight of iron = Weight of conductor
i.e.
Gi Gc
(A1.15)
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The above condition also holds good for a three phase transformer.
Minimum Full Load Loss
The main component of full load loss is
i)
Full load iron loss, P and
i
ii)
Full load copper loss, Pc .
The efficiency will be maximum for minimum full load loss.
Total full load loss, PT
Pi Pc
(A1.16)
At any fraction x of full load, the total losses are P
i
x2 Pc .
If Q is the output at full load, the output at fraction x of full load
is xQ .
Efficiency at output, xQ ,
x
xQ
xQ P x2 Pc
i
This efficiency is maximum when
d x
0
dx
2
d x ( xQ Pi x Pc ) Q xQ (Q 2 x Pc )
dx
( xQ Pi x2 Pc )2
For maximum efficiency,
( xQ Pi x2 Pc ) x (Q 2 x Pc )
xQ Pi x2 Pc xQ 2 x2 Pc
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or
Pi
x2 Pc
For full load (i.e. x =1),
Pi Pc
Hence for minimum full load loss, the condition is
Full load iron loss = Full load copper loss
i.e.
Pi Pc
(A1.17)
The above condition is applicable to both single and three phase
transformers.