Lecture 4
Single-phase Transformers
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 made of thin silicon steel laminations. Eddy current losses are reduced by making the laminations very thin. 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.
• There are two types of cores;
1. One type consists of a simple rectangular laminated piece of steel with the
transformer windings wrapped around two sides of the rectangle. This type of
construction is known as core form.
2. The other type consists of a three-legged laminated core with the windings
wrapped around the center leg. This type of construction is known as shell form.
2. Transformer Winding
• Either copper or aluminium winding are used. Although Aluminium wire is lighter and generally less expensive than copper wire, a larger cross
sectional area of conductor must be used to carry the same amount of current as with copper so it is used mainly in larger power transformer
applications.
• 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.
Φ
I1
NP/2
I1
NP/2
I1
Φ/2
NS/2
NS/2
Φ
NP
NS
Shell form
Core form
Φ/2
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 ration 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
𝐸𝑟𝑚𝑠 =
𝑁∅𝑚𝑥 𝜔
2
=
𝑑∅
= −𝑁∅𝑚𝑥 𝜔 cos 𝜔𝑡
𝑑𝑡
𝑁∅𝑚𝑥 (2𝜋𝑓)
2
=
𝑁𝐴𝑐𝑜𝑟𝑒 𝐵𝑚𝑥 (2𝜋𝑓)
2
= 4.44 𝑓𝑁𝐴𝑐𝑜𝑟𝑒 𝐵𝑚𝑥
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 𝑓𝑁𝑃 𝐴𝑐𝑜𝑟𝑒 𝐵𝑚𝑥 && 𝐸𝑆 = 4.44 𝑓𝑁𝑆 𝐴𝑐𝑜𝑟𝑒 𝐵𝑚𝑥
𝐸𝑃
𝑁
Hence;
= 𝑃
𝐸𝑆
𝑁𝑆
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)
A dot convention is often used in transformer circuit diagrams, nameplates or
terminal markings to define the relative polarity of transformer windings.
Positively increasing instantaneous current entering the primary winding's dot
end induces positive polarity voltage at the secondary winding's dot end
IV. Real transformer Model
eP(t)
eS(t)
Real transformer model
∅𝑃 = ∅𝑀 +∅𝐿𝑃
∅𝑃 total average 1ry flux
∅𝑀 mutual linkage flux
∅𝐿𝑃 leakage 1ry flux,
𝑣𝑃 (t)=𝑁𝑃
Hence,
𝑑∅𝑃
𝑑𝑡
𝑑∅
Similarly;
∅𝑆 = ∅𝑀 +∅𝐿𝑆
∅𝑆 total average 2ry flux
∅𝑀 mutual linkage flux
∅𝐿𝑆 leakage 2ry flux,
𝑑∅
=𝑁𝑃 𝑀+𝑁𝑃 𝐿𝑃
𝑑𝑡
𝑑𝑡
=𝑒𝑃 (t) + 𝑒𝐿𝑃 (t)
𝑑∅𝑀 𝑒𝑃 (t) 𝑒𝑆 (t)
=
=
𝑑𝑡
𝑁𝑃
𝑁𝑆
AND
𝑣𝑆 (t)=𝑁𝑆
𝑒𝑃 (t)
𝑒𝑆 (t)
=
𝑑∅𝑆
𝑑𝑡
𝑁𝑃
𝑁𝑆
𝑑∅
𝑑∅
=𝑁𝑆 𝑀+𝑁𝑆 𝐿𝑆
𝑑𝑡
𝑑𝑡
=𝑒𝑆 (t) + 𝑒𝐿𝑆 (t)
eP(t)
eS(t)
Real transformer model
Hysteresis transformer curve
𝐓𝐡𝐞 𝐧𝐨𝐧 − 𝐢𝐝𝐞𝐚𝐥 𝐫𝐞𝐚𝐥 𝐭𝐫𝐚𝐧𝐬𝐟𝐨𝐫𝐦𝐞𝐫 𝐦𝐨𝐝𝐞𝐥 𝐝𝐢𝐟𝐟𝐞𝐫𝐬 𝐟𝐫𝐨𝐦 𝐭𝐡𝐞 𝐢𝐝𝐞𝐚𝐥 𝐭𝐫𝐚𝐧𝐬𝐟𝐨𝐫𝐦𝐞𝐫 𝐦𝐨𝐝𝐞𝐥 𝐢𝐧 𝐭𝐡𝐞 𝐟𝐨𝐥𝐥𝐨𝐰𝐢𝐧𝐠 𝐩𝐨𝐢𝐧𝐭𝐬;
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
(XP and XS )
𝑁 𝑃 𝑖𝑃
•
𝑑 ℜ
𝑑∅𝐿𝑃
𝑒𝐿𝑃 (t)=𝑁𝑃 𝑑𝑡 =𝑁𝑃 𝑑𝑡
=
2
𝑁𝑃
𝑑𝑖𝑃
ℜ 𝑑𝑡
=
𝑑𝑖
𝐿𝑃 𝑑𝑡𝑃,
Similarly,
𝑁𝑆2 𝑑𝑖𝑆
𝑒𝐿𝑆 (t)= ℜ 𝑑𝑡
= 𝐿𝑆
𝑑𝑖𝑆
𝑑𝑡
II. Transformer Core
•
•
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.
𝐼𝑒𝑥𝑐𝑖𝑡𝑎𝑡𝑖𝑜𝑛 = 𝐼𝐶 +𝐼𝑀 , Iex should be much smaller than the full load current in a well designed transformer
𝐓𝐡𝐞 𝐜𝐨𝐫𝐞 𝐞𝐱𝐜𝐢𝐭𝐚𝐭𝐢𝐨𝐧 𝐛𝐫𝐚𝐧𝐜𝐡 𝐢𝐬 𝐩𝐥𝐚𝐜𝐞𝐝 𝐚𝐭 𝐭𝐡𝐞 𝟏𝐫𝐲 𝐬𝐢𝐝𝐞 𝐛𝐞𝐜𝐚𝐮𝐬𝐞 𝐭𝐡𝐞 𝐯𝐨𝐥𝐭𝐚𝐠𝐞 𝐚𝐩𝐩𝐥𝐢𝐞𝐝 𝐭𝐨 𝐭𝐡𝐞 𝐜𝐨𝐫𝐞 𝐢𝐬 𝐩𝐫𝐨𝐝𝐮𝐜𝐞𝐝 𝐟𝐫𝐨𝐦 𝑽𝑷
V. Equivalent circuit of a real transformer
Real transformer model
Exact Equivalent circuit
referred to primary side
Approximate Equivalent circuit
referred to primary side
V. Voltage Regulation
Voltage regulation (VR):
𝑉𝑅% =
𝑉𝑆𝑛𝑙 −𝑉𝑆𝑓𝑙
𝑉𝑆𝑓𝑙
× 100
where
𝑉𝑆𝑛𝑙 : RMS value of No-load secondary voltage of the transformer
𝑉𝑆𝑓𝑙 : RMS value of Full-load secondary voltage of the transformer
Approximate Equivalent circuit
referred to primary side
𝑉𝑅% =
𝑉𝑃 −𝑎𝑉𝑆𝑓𝑙
𝑎𝑉𝑆 𝑓𝑙
× 100
VI. Power flow of a real transformer
Approximate Equivalent circuit
referred to primary side
𝑃𝑜𝑢𝑡 = 𝑎𝑉𝑠
𝐼𝑆
𝑝𝑓 = 𝑉𝑠 𝐼𝑠 cos(𝜃𝑉𝑆 − 𝜃𝐼𝑆 )
𝑎 𝑠
𝑃𝑖𝑛 = 𝑉𝑃 𝐼𝑃 𝑝𝑓𝑃 = 𝑉𝑃 𝐼𝑃 cos(𝜃𝑉𝑃 − 𝜃𝐼𝑃 )
𝑃𝑐𝑜𝑝𝑝𝑒𝑟 =
𝐼𝑆 2
𝑅𝑒𝑞𝑃
𝑎
𝑃𝑐𝑜𝑟𝑒
, 𝑅𝑒𝑞𝑃 =𝑅𝑃 + 𝑎2 𝑅𝑠
𝑉𝑃2
=
𝑅𝐶
VI. Determining the Values of Components in the Transformer Model
Approximate values of the inductances and resistances in the transformer model can be experimentally determined
with only two tests, the open-circuit test and the short-circuit test
Open-circuit test (Computing RC & XM)
IOC
POC
VOC
•
•
•
HV
LV
In the open-circuit test, one transformer winding is open-circuited,
and the other winding is connected to full rated line voltage where
measurements occur. 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.
𝑆𝑂𝐶 = 𝑉𝑂𝐶 𝐼𝑂𝐶
𝑄𝑂𝐶 =
2
2
𝑆𝑂𝐶
− 𝑃𝑂𝐶
2
2
𝑉𝑂𝐶
𝑉𝑂𝐶
𝑅𝐶 =
𝑎𝑛𝑑 𝑋𝑀 =
𝑃𝑂𝐶
𝑄𝑂𝐶
Short-circuit test
Short-circuit test (Computing Req & Xeq)
ISC
PSC
VSC
HV
LV
• 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 where measurements occur. 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
𝑍𝑒𝑞𝑃
𝑉𝑆𝐶
𝑃𝑆𝐶
=
& 𝑅𝑒𝑞𝑃 = 2
𝐼𝑆𝐶
𝐼𝑆𝐶
𝑋𝑒𝑞𝑃 =
2
2
𝑍𝑒𝑞𝑃
− 𝑅𝑒𝑞𝑃
𝑅𝑒𝑞𝑃 =𝑅𝑃 + 𝑎 2 𝑅𝑠 , for simplicity
𝑅𝑃 = 𝑎2 𝑅𝑠 =0.5𝑅𝑒𝑞𝑃
𝑋𝑒𝑞𝑃 =𝑋𝑃 + 𝑎2 𝑋𝑠 , for simplicity
𝑋𝑃 = 𝑎2 𝑋𝑠 =0.5𝑋𝑒𝑞𝑃
• Discuss construction of single-phase transformer
• Discuss the theory of operation of single-phase transformer
• What is ideal transformer?
• What’s the need of dot notification?
• Draw the equivalent circuit of ;
Real transformer model - Exact equivalent circuit referred to 1ry - Approximate equivalent circuit referred to 1ry
• Draw power flow diagram of single-phase transformer
• Discuss single-phase transformer tests (Open-circuit – short circuit)