Transformer Design

1. 1
Chapter # 5
Transformer Design Trade-Offs
Introduction: -
 Size and weight of the transformer have significant effect on the overall efficiency,
voltage regulation and cost.
 Because of the interdependence and interaction of these parameters, judicious trade-
offs are necessary to achieve design optimization.
Design Constraints: -
 Constraints generally faced by the design engineer in transformer design are:
 The secondary winding of the transformer must be capable of
delivering the load within the specified voltage regulation limits
(within %5 ).
 It must be designed to give maximum efficiency at rated load.
 Temperature rise should be within permissible limit.
 One of the basic steps in transformer design is the selection of proper core material.
Magnetic materials used to design low and high frequency transformers are shown in
Table 5-1.
 Each one of these material has its own optimum point in the cost, size, frequency and
efficiency spectrum. The designer should be aware of the cost difference between
silicon-iron, nickel-iron, amorphous and ferrite materials.
 Other constraints relate to volume occupied by the transformer and, particularly in
aerospace applications, weight minimization is an important goal.
 Finally, cost effectiveness is always an important consideration.
 Depending upon the application, some of these constraints will dominate. Parameters
affecting others may then be traded off, as necessary, to achieve the most desirable
design.
 It is not possible to optimize all parameters in a single design because of their
interaction and interdependence. For example, if volume and weight are of great

2. 2
significance, reductions in both can often be affected, by operating the transformer at
a higher frequency, but with the penalty being in efficiency.
 When, the frequency cannot be increased, reduction in weight and volume may still
be possible by selecting a more efficient core material, but with the penalty of
increased cost.
 Thus, judicious trade-offs must be affected to achieve the design goals.
Table 5.1 Magnetic materials and their characteristics.
Power Handling Ability: -
 For years, manufacturers have assigned numeric codes to their cores to indicate their
power-handling ability. This method assigns to each core a number called the area
product (Ap).
 Area product Ap is the product of window area (Wa) and the core cross-sectional area
(Ac). The core suppliers use these numbers to summarize dimensional and electrical
properties in their catalogs.
Relationship, Ap, to Transformer Power Handling Capability: -
Total power handling capability of the transformer,
0PP
PowerOutputPowerInputP
in
t


(5.1)

3. 3
Primary voltage,
pcac
pp
NABf
NfV
)10(44.4
44.4
4

 
(5.2)
Where, ′𝑓′ is the supply frequency in Hz, ′𝐵𝑎𝑐′ is the maximum flux density in the
transformer core, ′𝐴 𝑐′ is the core area in cm2
and 𝑁𝑝 is the number of primary turns.
From equation 5.2, number of primary turns can be expressed as,
cac
p
p
AfB
V
N
44.4
104

 (5.3)
Winding area of a transformer is fully utilized when
wsswppau ANANWK  (5.4)
Where, ′𝐾 𝑢
′
is the window utilization factor, ′𝑊𝑎′ is the window area, ′𝑁𝑝′ is the
primary number of turns, ′𝐴 𝑤𝑝′ is the cross-sectional area of primary winding conductor,
′𝑁𝑠′ is the secondary number of turns and ′𝐴 𝑤𝑠′ is the cross-sectional area of secondary
winding conductor.
l
h
Low voltage winding
(Awp or Aws)
High voltage winding
(Aws or Awp)
Insulation between windings
Bobbin
Bobbin
Figure 5.1 Cut-section of a shell type transformer.
Conductor cross-sectional area,
J
I
Aw  (5.5)
Where, ′𝐼′ is the current through the conductor and ′𝐽′ is the current density.
Substituting equation (5.5) in (5.4)
J
I
N
J
I
NWK s
s
p
pau  (5.6)
Putting the values of ′𝑁𝑝′ and ′𝑁𝑠′ in equation (5.6) referring equation (5.3)

4. 4
J
I
AfBK
V
J
I
AfBK
V
J
I
N
J
I
NWK s
cacf
sp
cacf
ps
s
p
pau 




44
1010
JKfBK
IVIV
AW
uacf
sspp
ca
4
10)( 

So area product,
ufac
t
p
KfJKB
P
A
4
10
 (5.7)
Where, ′𝑃𝑡′ is the total power handling capability of the transformer (= power
handling capability of primary winding + power handling capability of secondary winding).
′𝐾𝑓′ is waveform coefficient (= 4.44 for sine wave and 4.0 for square wave).
Core geometry ′𝑲 𝒈′ and its relationship with voltage regulation and power
handling capability: -
Power handling capability,
egt KKP 2 . (5.8)
Where, ′𝐾𝑔′ is the core geometry and is dependent on core area, window area and
mean length turn. The constant ′𝐾𝑒′ is determined by the magnetic and electrical loading
conditions.
Therefore,
 MLTWAfK acg ,, and
 me BfgK ,
Voltage regulation,
100
sincos 20222022



sV
XIRI 
 (5.9)
+ve sign is taken for lagging and -ve sign for leading power factor.
At unity power factor, voltage regulation is
22
2
2'
2022
2002001002
100100100100
)/(
100100100
)(
100
p
p
p
ppp
p
pp
s
ss
p
pp
s
ss
p
pp
s
ss
s
ps
s
sp
s
V
RVA
V
RIV
V
RI
V
RI
V
RI
V
RI
kV
RkkI
V
RI
V
RkI
V
RRI
V
RI







5. 5
2
200
p
p
V
RVAinratingrTransforme 
 (5.10)
Primary winding resistance,
p
up
a
p
p
p
p
N
K
W
NMLT
a
l
R


  (5.11)
Where, ′𝐾 𝑢𝑝′ is the primary window utilization factor and ′𝑊𝑎′ is the window area.
upa
p
p
KW
NMLT
R



2
 (5.12)
Generally,
2
u
usup
K
KK  (5.13)
Primary winding voltage,
4
10
 cmpfp ABfNKV (Core area ′𝐴 𝑐′ in 𝑐𝑚2
) (5.14)
From equation (5.10), rating of the transformer can be written as

p
p
R
V
VA
200
2
(5.15)
Putting the value of ′𝑉𝑝′ from equation (5.14) in equation (5.15)
 
















MLT
KWABfK
KW
NMLT
ABfNK
VA
upacmf
upa
p
cmpf
2
10
200
10
102222
2
24
(5.16)
Taking, resistivity of copper as, m 6
10724.1 ,




MLT
KWABfK
VA upacmf
42222
1029.0
(5.17)
Let primary magnetic and electric loading condition,
4222
1029.0 
 mfe BfKK (5.18)
Primary core geometry,
MLT
KAW
K upca
g
2
 (5.19)
Putting the values of ′𝐾𝑒′ & ′𝐾𝑔′ form equations (5.18) & (5.19)
 ge KKVA (5.20)

6. 6
Taking
2
u
usup
K
KK  and putting this in equation (5.17)










MLT
KWABfK
MLT
K
WABfK
VA
uacmf
u
acmf
42222
42222
10149.0
10
2
29.0
(5.21)
From equation (5.21),
Magnetic and electric loading condition,
4222
10145.0 
 mfe BfKK (5.22)
MLT
KAW
K uca
g
2
 (5.22)
Therefore equation (5.21) can be written as power handling capability of primary
winding is,
gep KKVA  (5.23)
Total power handling capability,


ge
geget
KK
ondaryforKKprimaryforKKP
2
)sec()(


(5.24)