single phase transformer

1. Electrical Power and
Machines
WEEK 8-9: SINGLE PHASE TRANSFORMERS

2. A transformer is an electrical device that transfers electrical energy between two or more circuits through electromagnetic induction.
The principal purpose of a transformer is to convert ac power at one voltage level to ac power of the same frequency at another
voltage level.
I. Construction
1. Transformer core
• The iron core is constructed from of a highly permeable material made from thin silicon steel laminations
assembled together to provide the required magnetic path with the minimum of losses. The laminations
are electrically insulated from each other by a very thin coating of insulating varnish or by the use of an
oxide layer on the surface.
Constructed on one of two types of cores.
◦ Core types transformer: Simple rectangular laminated piece of steel with the transformer windings wrapped around
two sides of the rectangle.
◦ Shell type transformer: three legged laminated core with the windings wrapped around the center leg.

3. 2. Transformer Winding
• The type of wire used in a transformer winding is either copper or aluminum.
• The primary and secondary windings in a physical transformer are wrapped one on top of the other concentrically with the low-
voltage winding innermost. Such an arrangement is usually used in the core-type transformer resulting in much less leakage flux
than would be the case if the two windings were separated by a distance on the core. Shell type transformer cores overcome this
leakage flux as both the primary and secondary windings are wound on the same central limb
• The insulation used to prevent the conductors shorting together in a transformer is usually a thin layer of varnish which is painted
onto the wire before it is wound around the core.
Wrapping windings on
top of the other
Reduces the leakage
flux+ simplifies
insulation problem for
the high voltage
winding

4. III. Ideal transformer
An ideal transformer is a theoretical linear transformer that is lossless and perfectly coupled; i.e. there are no electric
power losses and flux is completely confined within the magnetic core. Hence, 𝐸𝑃 = 𝑉𝑃 and 𝐸𝑆= 𝑉𝑆
By Faraday's law of induction
and
hence, , where a is the turns ratio
for step-down transformers, a > 1
for step-up transformers, a < 1
By law of Conservation of Energy, apparent, real and reactive power are each conserved in the input and output
By Ohm's Law and ideal transformer identity,
apparent load impedance Z'L (ZL referred to the primary)

5. II. Operation Principle
A varying current in the transformer's primary winding creates a varying magnetic flux in the core and a varying magnetic field
flows to the secondary winding. This varying magnetic field at the secondary induces a varying electromotive force (EMF) or
voltage in the secondary winding according to Faraday’s law. According to turns ratio between the primary and secondary
winding, the transformer is designed to efficiently change AC voltages from one voltage level to another within power networks.
Transformer universal EMF equation
If the flux in the core is purely sinusoidal,
∅ = ∅𝑚𝑥 sin 𝜔𝑡
The induced emf is;
𝐸 = −𝑁
𝑑∅
𝑑𝑡
= −𝑁∅𝑚𝑥𝜔 cos 𝜔𝑡
𝐸𝑟𝑚𝑠 =
𝑁∅𝑚𝑥𝜔
2
=
𝑁∅𝑚𝑥(2𝜋𝑓)
2
=
𝑁𝐴𝑐𝑜𝑟𝑒𝐵𝑚𝑥(2𝜋𝑓)
2
= 4.44 𝑓𝑁𝐴𝑐𝑜𝑟𝑒𝐵𝑚𝑥
Where Erms is rms induced voltage of the winding is volts, f is the supply frequency in Hz, N
is number of turns, Acore is core cross-sectional area in m2 and 𝐵𝑚𝑥 peak magnetic flux
density in Wb/m2 or T (tesla)
The universal emf equation of transformer;
𝑬𝑷 = 𝟒. 𝟒𝟒 𝒇𝑵𝑷𝑨𝒄𝒐𝒓𝒆𝑩𝒎𝒙 && 𝑬𝑺 = 𝟒. 𝟒𝟒 𝒇𝑵𝑺𝑨𝒄𝒐𝒓𝒆𝑩𝒎𝒙
Hence;
𝑬𝑷
𝑬𝑺
=
𝑵𝑷
𝑵𝑺

6. I. Transformer 1ry and 2ndry Windings
• Copper losses (I2R) i.e. the resistive heating losses in the 1ry and 2ndry windings (RP and RS)
• Leakage flux i.e. ∅𝐿𝑃 and ∅𝐿𝑆 which escape the core and each pass through one of the transformer windings. Hence,
leakage inductance (LP and LS )in the 1ry and 2ndry windings are produced that should be compensated for (XP and XS )
𝐓𝐡𝐞 𝐧𝐨𝐧 − 𝐢𝐝𝐞𝐚𝐥 𝐫𝐞𝐚𝐥 𝐭𝐫𝐚𝐧𝐬𝐟𝐨𝐫𝐦𝐞𝐫 𝐦𝐨𝐝𝐞𝐥 𝐝𝐢𝐟𝐟𝐞𝐫𝐬 𝐟𝐫𝐨𝐦 𝐭𝐡𝐞 𝐢𝐝𝐞𝐚𝐥 𝐭𝐫𝐚𝐧𝐬𝐟𝐨𝐫𝐦𝐞𝐫 𝐦𝐨𝐝𝐞𝐥 𝐢𝐧 𝐭𝐡𝐞 𝐟𝐨𝐥𝐥𝐨𝐰𝐢𝐧𝐠 𝐩𝐨𝐢𝐧𝐭𝐬;
Real transformer model
eP(t) eS(t)

7. II. Transformer Core
𝐓𝐡𝐞 𝐜𝐨𝐫𝐞 𝐞𝐱𝐜𝐢𝐭𝐚𝐭𝐢𝐨𝐧 𝐛𝐫𝐚𝐧𝐜𝐡 𝐢𝐬 𝐩𝐥𝐚𝐜𝐞𝐝 𝐚𝐭 𝐭𝐡𝐞 𝟏𝐫𝐲 𝐬𝐢𝐝𝐞 𝐛𝐞𝐜𝐚𝐮𝐬𝐞 𝐭𝐡𝐞 𝐯𝐨𝐥𝐭𝐚𝐠𝐞 𝐚𝐩𝐩𝐥𝐢𝐞𝐝 𝐭𝐨 𝐭𝐡𝐞 𝐜𝐨𝐫𝐞 𝐢𝐬 𝐩𝐫𝐨𝐝𝐮𝐜𝐞𝐝 𝐟𝐫𝐨𝐦 𝑽𝑷
𝐼𝑒𝑥𝑐𝑖𝑡𝑎𝑡𝑖𝑜𝑛 = 𝐼𝐶+𝐼𝑀, Iex should be much smaller than the full load current in a well designed transformer
Real transformer model
eP(t) eS(t)
• Linkage flux (∅𝑀) in the core and the magnetization current (IM) to produce the flux in the core produce magnetization
inductance (LM )in core ; (XM)
• Core losses (Pcore) and core loss current (IC) required to make up for the hysteresis and eddy current losses in the core (Rc)
 Eddy current losses are resistive heating losses in the core .
 Hysteresis losses are associated with the rearrangement of the magnetic domains in the core during each half cycle.
Hysteresis transformer curve

8. V. Equivalent circuit of a real transformer
Real transformer model
Exact Equivalent circuit
referred to primary side
Approximate Equivalent circuit
referred to primary side

9. V. Voltage Regulation
Approximate Equivalent circuit
referred to primary side
where
𝑉𝑆𝑛𝑙 : RMS value of No-load secondary voltage of the transformer
𝑉𝑆𝑓𝑙 : RMS value of Full-load secondary voltage of the transformer
Voltage regulation (VR):𝑉𝑅% =
𝑉𝑆𝑛𝑙−𝑉𝑆𝑓𝑙
𝑉𝑆𝑓𝑙
× 100
𝑉𝑅% =
𝑉𝑃−𝑎𝑉𝑆𝑓𝑙
𝑎𝑉𝑆𝑓𝑙
× 100

10. VI. Power flow of a real transformer
Approximate Equivalent circuit
referred to primary side
𝑃𝑜𝑢𝑡 = 𝑎𝑉
𝑠
𝐼𝑆
𝑎
𝑝𝑓𝑠 = 𝑉
𝑠 𝐼𝑠cos(𝜃𝑉𝑆
− 𝜃𝐼𝑆
)
𝑃𝑖𝑛 = 𝑉𝑃𝐼𝑃𝑝𝑓𝑃 = 𝑉𝑃 𝐼𝑃cos(𝜃𝑉𝑃
− 𝜃𝐼𝑃
)
𝑃𝑐𝑜𝑝𝑝𝑒𝑟 =
𝐼𝑆
𝑎
2
𝑅𝑒𝑞𝑃 , 𝑅𝑒𝑞𝑃=𝑅𝑃+ 𝑎2
𝑅𝑠
𝑃𝑐𝑜𝑟𝑒 =
𝑉𝑃
2
𝑅𝐶

11. No information what so ever about the equivalent circuit
parameters
VI. Determining the Values of Components in the Transformer Model
Single phase transformer

12. Short-circuit test (Computing Req & Xeq)
LV
HV
• In the short-circuit test, the low-voltage terminals of the transformer are short
circuited, and the high-voltage terminals are connected to a variable voltage source
(voltage applied is a fraction of the rated voltage. Working at the high-voltage side of
the transformer is easier, since currents will be lower on that side.
• The short circuit voltage, current, and power applied to the transformer are measured i.e.
(VSC, ISC, and PSC)
• Since the input voltage is so low during the short-circuit test, negligible current flows
through the excitation branch. If the excitation current is ignored, then all the voltage
drop in the transformer can be attributed to the series elements in the circuit
PSC
ISC
VSC
𝑍𝑒𝑞𝑃 =
𝑉𝑆𝐶
𝐼𝑆𝐶
& 𝑅𝑒𝑞𝑃 =
𝑃𝑆𝐶
𝐼𝑆𝐶
2
𝑋𝑒𝑞𝑃 = 𝑍𝑒𝑞𝑃
2
− 𝑅𝑒𝑞𝑃
2
𝑅𝑒𝑞𝑃=𝑅𝑃+ 𝑎2𝑅𝑠, for simplicity
𝑅𝑃= 𝑎2
𝑅𝑠=0.5𝑅𝑒𝑞𝑃
𝑋𝑒𝑞𝑃=𝑋𝑃+ 𝑎2𝑋𝑠, for simplicity
𝑋𝑃= 𝑎2
𝑋𝑠=0.5𝑋𝑒𝑞𝑃
Measured values Symbol
short circuit voltage 𝑉
𝑠𝑐
short circuit current 𝐼𝑠𝑐
short circuit power 𝑃𝑠𝑐
Calculated values Symbol
winding resistances 𝑅𝑒𝑞
leakage reactances 𝑋𝑒𝑞

13. VI. Determining the Values of Components in the Transformer Model
Open-circuit test (Computing RC & XM)
• In the open-circuit test, one transformer winding is open-circuited, and the other
winding is connected to full rated line voltage. The latter is normally done on the
low voltage side of the transformer, since lower voltages are easier to work with.
• The open circuit voltage, current, and power applied to the transformer are measured
i.e. (VOC, IOC, and POC)
• The series elements, RP and XP are too small in comparison to Rc and XM to cause a
significant voltage drop, so essentially all the input voltage is dropped across the
excitation branch.
POC
IOC
VOC
LV HV
Measured values Symbol
Open circuit voltage 𝑉
𝑜𝑐
Open circuit current 𝐼𝑜𝑐
Open circuit power 𝑃
𝑜𝑐
Calculated values Symbol
Core loss resistance 𝑅𝑐
Magnetizing reactance 𝑋𝑀

14. Circuit Parameters: Open-Circuit Test
oc
oc
oc
oc I
V
P 
cos











 
oc
oc
oc
oc
oc
oc
oc
oc
oc
I
V
P
or
I
V
P
PF 1
cos
,
cos 

oc
oc
M
oc
oc
c I
I
I
I 
 sin
,
cos 
 

The open circuit power (no load power) is given by:
The open-circuit power factor and power factor angle can be determined:
Open-circuit test (no load test)
Since the no load current 𝐼𝑜𝑐 will correspond to the magnetic branch current which constitutes two current components, 𝐼𝑀
and 𝐼𝑐
Then one can easily determine 𝑋𝑀 and 𝑅𝑐:
c
oc
c
M
oc
M
I
V
R
I
V
X 


 ,
Io
Ic
IM
E1

qo

15. • This test is carried out to measure the resistance of each winding, thus the test can be carried on one side and the other
side is concluded with the help of the short circuit test
• To determine the primary winding resistance, the secondary winding is open circuited and the primary winding is
excited with dc source. The dc voltage is varied till the primary current reaches rated current. Each time the voltage is
increased, the corresponding primary current reading is recorded.
• The resistance of each winding is obtained from the Ohms Law:
dc
dc
p
dc
I
V
R
R




𝑉 𝑑𝑐
𝐼 𝑑𝑐
Measured values Symbol
Dc voltage 𝑉𝑑𝑐
Dc current 𝐼𝑑𝑐
DC test (Computing Rs & Rp)

16. Questions
• Derive the EMF equation of the transformer
• Discuss the construction of single phase transformer and mention the types of transformer connections
and applications
• Explain the principle of operation of single phase transformer
• Explain the differences between ideal and non ideal transformer and draw the equivalent circuit for each
• Discuss briefly the type of tests used to determine transformer parameters