CHAPTER 1 SIGLE-PHASE TRASFORMER
1.0
DEFIITIO OF TRASFORMER
1.1
COSTRUCTIO
1.2
PRICIPLE OF OPERATIO
1.3
FARADAY’S LAW AD LEZ’S LAW 1.3.1
FARADAY’S FARADAY’ S LAW
1.3.2
LENZ’S LAW
1.4
EMF EQUATIO
1.5
TRASFORMER EQUIVALET CIRCUIT MODEL 1.5.1
IDEAL TRANSFORMER
1.5.2
PRACTICAL TRANSFORMER
1.5.3
IMPEDANCE TRANSFER
1.5.4
EXACT EQUIVALENT CIRCUIT
1.5.5
APPROXIMATE APPROXIMATE EQUIVALENT CIRCUIT
1.6
PHASOR DIAGRAMS
1.7
TRASFORMER TESTS 1.7.1
OPEN CIRCUIT TEST
1.7.2
SHORT CIRCUIT TEST
1.8
TYPES OF LOSSES
1.9
VOLTAGE REGULATIO
1.10
EFFICIECY AD MAXIMUM EFFICIECY
1.0
DEFIITIO OF TRASFORMER
•
A transformer is a device that transfers electrical energy from one circuit to another through a shared magnetic field. A changing current in the first circuit (the primary) creates a changing magnetic field; in turn, this magnetic field induces a voltage in the second circuit (the secondary).
•
It can raise (step-up) or lower (step-down) (step -down) the voltage in a circuit but b ut with a corresponding decrease or increase in current
1.1
COSTRUCTIO
A transformer is a static machine. Although it is not an energy conversion device, it is indispensable in many energy conversion systems. It is a simple device, having two or more electric circuits coupled by a common magnetic circuit.
A transformer essentially consists of two or more windings coupled by a mutual magnetic field. Ferromagnetic cores are used to provide tight magnetic coupling and high flux densities. Such transformers are known as iron core transformers. They are invariably used in high-power applications. Air core transformers have poor magnetic coupling and are sometimes used in low-power electronic circuits.
Two types of core constructions are normally used, as shown in Fig. 1.0. In the core type (Fig. 1.0a), the windings are wound around two legs of a magnetic core of rectangular shape. In the shell type (Fig. 1.0b), the windings are wound around the center leg of a three-legged magnetic core. To reduce core losses, the magnetic core is formed of a stack of thin laminations.
L-shaped laminations are used for core-type construction and E-shaped laminations are used for shell-type construction. To avoid a continuous air gap (which would require a large exciting current), laminations are stacked alternately as shown in Fig. l.0c and Fig. 1.0d.
Fig. 1.0: Transformer core construction. (a) Core-type (b) Shell-type (c) L-shaped lamination (d) E-shaped lamination
A schematic representation of a two-winding transformer is shown in Fig. 1.1. The two vertical bars are used to signify tight magnetic coupling between the windings. One winding is connected to an ac supply and is referred to as the primary winding. The other winding is connected to an electrical load and is referred to as the secondary winding.
The winding with the higher number of turns will have a high voltage and is called the high-voltage (HV) or high-tension (HT) winding. The winding with the lower number of turns is called the low-voltage (LV) or low-tension (LT) winding. To achieve tighter magnetic coupling between the windings, they may be formed of coils placed one on top of another (Fig. l.0a) or side by side (Fig. l.0b) in a “pancake” coil formation where primary and secondary coils are interleaved. Where the coils are placed one on top of another, the low-voltage winding is placed nearer the core and the high-voltage winding on top.
Fig. 1.1: Schematic representation of a two-winding transformer
Transformers have widespread use. Their primary function is to change voltage level. A transformer is a device for converting electric energy at one voltage level to electric energy at another voltage level through the action of a magnetic field. It plays an extremely important role in modern life by making possible the economical long distance transmission of electric power.
1.2
PRICIPLE OF OPERATIO
When a voltage is applied to the primary of a transformer, a flux is produced in the core as given by Faraday’s law. The changing flux in the core then induced a voltage in the secondary winding of the transformer. Because transformer cores have very high permeability, the net magnetomotive force required in the core to produce its flux is very small. Since the net magnetomotive force is very small, the primary circuit’s magnetomotive force must be approximately equal and opposite to the secondary circuit’s magnetomotive force. This fact yields the transformer current ratio. The transformer is based on two principles: •
First, that an electric current can produce a magnetic field (electromagnetism) and,
•
Second, that a changing magnetic field within a coil of wire induces a voltage across the ends of the coil (electromagnetic induction).
Transformer is only converting from AC signal into AC signal. The primary is connected to source of alternating (AC) voltage
By changing the current in the primary coil, one changes the strength of its magnetic field; since the secondary coil is wrapped around the same magnetic field, which it produces mutually-induced e.m.f (electromotive force)
If the secondary coil is closed, a current flows in it and so a voltage is induced across the secondary terminal. Therefore, electrical energy is transferred from primary to the secondary terminal.
Example Refer to Fig. 1.2
Fig. 1.2: Example of transformer structure
When the switch is closed: •
Current in primary coil increases
•
Creates increasing magnetic field in primary coil
•
Induces current in secondary coil
•
Lamp lights up
When current in primary coil becomes steady: • No more changes in the magnetic field in primary coil • No more current induced in secondary coil •
Lamp goes off
When switch is opened: •
Current in primary coil decreases to zero
•
Creates decreasing magnetic field in primary coil
•
Induces current in opposite direction in secondary coil
•
Lamp lights up again
1.3
FARADAY’S LAW AD LEZ’S LAW
1.3.1
FARADAY’S LAW
Faraday’s law states that if a flux passes through a turn of a coil of wire, a voltage will be induced in the turn of wire that is directly proportional to the rate of change in the flux with respect to time. In equation form, eind = −
d φ dt
………………………………… (1.0)
where eind is the voltage induced in the turn of the coil and φ is the flux passing through the turn. If a coil has turns and if the same flux passes through all of them, then the voltage induced across the whole coil is given by
eind = −
d φ dt
……………………………… (1.1)
where:
1.3.2
eind
=
voltage induced in the coil
=
number of turns of wire in coil
φ
=
flux passing through coil
LEZ’S LAW
The minus sign in the Faraday’s Law equations (Eqn. 1.1) above is an expression of Lenz’s law. Lenz’s law states that the direction of the voltage build up in the coil is such that if the coil ends were short circuited, it would produce current that would cause a flux
opposing the original flux change. Since the induced voltage opposes the change that causes it, a minus sign is included in Eqn. 1.1. To understand this concept clearly, examine Fig. 1.3. If the flux shown in the figure is increasing in strength, then the voltage built up in the coil will tend to establish a flux that will oppose the increase.
Fig.1.3: The meaning of Lenz’s law: (a) A coil enclosing an increasing magnetic flux; (b) Determining the resulting voltage polarity.
A current flowing as shown in Fig. 1.3(b) would produce a flux opposing the increase, so the voltage on the coil must be built up with the polarity required to drive that current through the external circuit. Therefore, the voltage must be built up with the polarity shown in the figure. Since the polarity of the resulting voltage can be determined from physical considerations, the minus sign in Eqn. 1.0 and Eqn. 1.1 is often left out. There is one major difficulty involved in using Eqn. 1.1 in practical problems. That equation assumes that exactly the same flux is present in each turn of the coil. Unfortunately, the flux leaking out of the core into the surrounding air prevents this from being true. If the windings are tightly coupled, so that the vast majority of the flux passing through one turn of the coil does indeed pass through all of them, then Eqn. 1.1 will give valid answers. But if leakage is quite high or if extreme accuracy is required, a different expression that does not make that assumption will be needed. The magnitude of the voltage in the i th turn of the coil is always given by
eind =
d (φ i ) dt
…………………………………. (1.2)
If there are turns in the coil of wire, the total voltage on the coil is eind =
∑e
1
i =1
( i) d φ
i =1
dt
=∑ =
d
∑ φ i ………………………….. (1.3) dt i =1
The term in parentheses in Eqn. 1.3 is called the flux linkage λ
of the coil, and
Faraday’s law can be rewritten in terms of flux linkage as eind =
d λ dt
………………………………… (1.4)
where λ =
∑φ i
…………………………….…… (1.5)
i =1
The units of flux linkage are weber-turns.
Faraday’s law is the fundamental property of magnetic fields involved in transformer operation. The effect of Lenz’s law in transformers is to predict the polarity of the voltages induced in transformer windings. Faraday’s law also explains the eddy current losses. A time-changing flux induces voltage within a ferromagnetic core in just the same manner as it would in a wire wrapped around that core. These voltages cause swirls of current to flow within the core. It is the shape of these currents that gives rise to the name eddy currents. These eddy currents are flowing in a resistive material (the iron of the core), so energy is dissipated by them. The lost energy goes into heating the iron core.
The amount of energy lost to eddy currents is proportional to the size of the paths they follow within the core. For this reason, it is customary to break up any ferromagnetic core that may be subject to alternating fluxes into many small strips, or laminations, and to build the core up out of these strips. An insulating oxide or resin is used between the
strips, so that the current paths for eddy currents are limited to very small areas. Because the insulating layers are extremely thin, this action reduces eddy current losses with very little effect on the core’s magnetic properties. Actual eddy current losses are proportional to the square of the lamination thickness, so there is a strong incentive to make the laminations as thin as economically possible.
Example Figure below shows a coil of wire wrapped around an iron core. If the flux in the core is given by the equation φ = 0.05 sin 377 t Wb If there are 100 turns on the core, what voltage is produced at the terminals of the coil? Of what polarity is the voltage during the time when flux is increasing in the reference direction shown in the figure? Assume that all the magnetic flux stays within the core (i.e., assume that the flux leakage is zero).
Solution The direction of the voltage while the flux is increasing in the reference direction must be positive to negative, as shown in figure. The magnitude of the voltage is given by eind =
d φ dt
= (100turns )
d dt
(0.05 sin 377 t )
= 1885 cos 377 t @ 1885 sin(377 t + 90°)V
1.4
EMF EQUATIO
Let →
1
= No. of turns in primary
2
= No. of turns in secondary
φ m
= Maximum flux in core in Webers = Bm * A
f
= Frequency of AC input in Hz
The flux increases from its zero value to maximum value from in one quarter of the cycle i.e. in 1/4 f second
Rate of change of flux
=
Average e.m.f/ turns
=
φ m 1 / 4 f
Unit: wb/s or volt
4 f φ m
If flux (φ m ) varies sinusoidally, then:
R.M.S value/turn
=
1.11* average value
=
1.11* 4 f φ m
=
4.44 f φ m volts
E.M.F induced in primary
=
(r.m.s / turn) * (No. of primary turns)
=
4.44 f φ m 1
=
4.44 fBm A 1 ………………………………… (a)
E.M.F induced in secondary =
(r.m.s / turn) * (No. of secondary turns)
=
4.44 f φ m 2
=
4.44 fBm A 2 ………………………………… (b)
From (a) and (b): E 1
= 4.44 f φ m
1 E 2
= 4.44 f φ m
2
E.M.F per turns is same in both windings
From equation (a) and (b), we get: E 1 E 2
=
1 2
=a
Where a = voltage transformation ratio
If a <1, then the transformer is called step- up transformer If a >1, then the transformer is called step- down transformer
1.5
TRASFORMER EQUIVALET CIRCUIT MODEL
1.5.1
IDEAL TRASFORMER
V p = E p V s = E s
Fig. 1.4: Basic Transformer
Fig. 1.5: Ideal Transformer Equivalent Circuit
Consider a transformer with two windings, a primary winding of 1 turns and a secondary winding of 2 turns, as shown schematically in Fig.1.6. In a schematic diagram it is a common practice to show the two windings in the two legs of the core, although in an actual transformer the windings are interleaved. Let us consider an ideal transformer that has the following properties:
Fig. 1.6(a): Ideal transformer
1.
The winding resistances are negligible.
2.
All fluxes are confined to the core and link both windings; that is, no leakage fluxes are present. Core losses are assumed to be negligible.
3.
Permeability of the core is infinite (i.e.�→ ∞ ). Therefore, the exciting current required to establish flux in the core is negligible; that is, the net mmf required to establish a flux in the core is zero.
When the primary winding is connected to a time-varying voltage v1 , a time-varying flux φ is established in the core. A voltage e1 will be induced in the winding and will equal the applied voltage if resistance of the winding is neglected: v1 = e1 = 1
d φ dt
……………….……..….. (1.6)
The core flux also links the secondary winding and induces a voltage e2 , which is the same as the terminal voltage v 2 : v 2 = e2 = 2
d φ dt
……………….………... (1.7)
From Eqn. 1.6 and Eqn. 1.7, v1 v2
=
1 2
= a …………………………….. (1.8)
where a is the turns ratio. Eqn. 1.8 indicates that the voltages in the windings of an ideal transformer are directly proportional to the turns of the windings.
Let us now connect a load (by closing the switch in Fig. 1.6(a)) to the secondary winding. A current i2 will flow in the secondary winding, and the secondary winding will provide an mmf 2 i2 for the core. This will immediately make a primary winding current i1 flow so that a counter mmf 1i1 can oppose 2 i2 . Otherwise 2 i2 would make the core flux change drastically and the balance between v1 and e1 would be disturbed. Note that in Fig. 1.6 that the current directions are shown such that their mmf’s oppose each other. Because the net mmf required to establish a flux in the ideal core is zero, 1 i1 − 2 i2 = net mmf = 0
1i1 = 2 i2 i1 i2
=
2 1
=
………..……………………. (1.9) 1 a
……………………...……. (1.10)
The currents in the windings are inversely proportional to the turns of the windings. Also note that if more current is drawn by the load, more current will flow from the supply. It is this mmf-balancing requirement (Eqn. 1.9) that makes the primary know of the presence of current in the secondary. From Eqn. 1.8 and Eqn. 1.10 v1i1 = v2 i2 ……………………………… (1.11) That is, the instantaneous power input to the transformer equals the instantaneous power output from the transformer. This is expected, because all power losses are neglected in an ideal transformer. Note that although there is no physical connection between load and supply, as soon as power is consumed by the load, the same power is drawn from the supply. The transformer, therefore, provides a physical isolation between load and supply while maintaining electrical continuity. If the supply voltage v1 is sinusoidal, then Eqn. 1.8, 1.10, and 1.11 can be written in terms of rms values: V 1 V 2 I 1 I 2
= =
1 2 2 1
=a =
V 1 I 1 = V 2 I 2
1 a ……………………………. (1.12)
1.5.2
PRACTICAL TRASFORMER
Fig. 1.6(b): Equivalent circuit of practical transformer
V p
=
Primary terminal voltage (input)
V s
=
Secondary terminal voltage (output)
R p , R s
=
Leakage resistance on the primary and secondary respectively
X p , X s
=
Leakage reactance on the primary and secondary respectively
Rc
=
Core resistance
X m
=
Magnetizing reactance
I p
=
Primary current
I s
=
Secondary current
p
=
Primary winding turns
s
=
Secondary winding turns
1.5.3
IMPEDACE TRASFER
Consider the case of a sinusoidal applied voltage and secondary impedance Z 2 , as shown in Fig. 1.7a.
Fig. 1.7: Impedance transfer across an ideal tra nsformer.
Z 2 =
V 2 I 2
The input impedance is Z 1 =
V 1 I 1
=
aV 2 I 2 / a
= a2
V 2 I 2
= a 2 Z 2 So Z 1 = a 2 Z 2 = Z ' 2
Impedance Z 2 connected in the secondary will appear as impedance Z ' 2 looking from the primary. The circuit in Fig. 8a is therefore equivalent to the circuit in Fig. 1.7b. Impedance can be transferred from secondary to primary if its value is multiplied by the square of the turns ratio. Impedance from the primary side can also be transferred to the secondary side, and in that case its value has to be divided by the square of the turns ratio: Z ' 1 =
1 a
2
Z 1
This impedance transfer is very useful because it eliminates a coupled circuit in an electrical circuit and thereby simplifies the c ircuit.
1.5.4
EXACT EQUIVALET CIRCUIT
Fig. 1.8: The model of the real transformer
•
The figure above shows the accurate model of the transformer but it is quite difficult to analyze practical circuits containing transformers
•
Thus the equivalent circuit normally has converted to the equivalent circuit at a single voltage level.
•
Therefore, the equivalent circuit must be referred either to its primary side or to its secondary side in problem solutions.
Fig. 1.9: Exact equivalent circuit (referred to the primary side)
Fig. 1.10: Exact equivalent circuit (referred to the secondar y side)
1.5.5
APPROXIMATE EQUIVALET CIRCUIT
Fig. 1.11: The model of the real transformer
•
The excitation branch has a very small current compared to the load current of the transformers.
•
Moreover, it is so small that under normal circumstances it causes a completely negligible voltage drop in R p and X p . Thus, it can be simplified into approximate equivalent circuits.
Fig. 1.12: Approximate equivalent circuit (referred to the primary side)
Fig. 1.13: Approximate equivalent circuit (referred to the secondary side)
1.6
PHASOR DIAGRAMS
•
It is easy to determine the effect of the impedances and the current phase on the transformer voltage regulation by drawing the phasor diagram.
•
V s is assumed to he at an angle of 0° and all other voltages and currents are compared to that references.
•
A transformer phasor diagram is presented by applying Kirchhoff’s \/voltage the transformer equivalent circuit and an equation will be as follows. V p a
= V s ∠0° + Req I s ∠θ ° + jX eq I s ∠θ
Lagging Power Factor
V p a
= V s ∠0° + Req I s ∠ − θ ° + jX eq I s ∠ − θ °
Fig. 1.14: Lagging power factor
Unity Power Factor
V p a
= V s ∠0° + Req I s ∠0° + jX eq I s ∠0°
Fig. 1.15: Unity power factor
Leading Power Factor
V p a
= V s ∠0° + Req I s ∠ + θ ° + jX eq I s ∠ + θ °
Fig. 1.16: Leading power factor
1.7
TRASFORMER TEST
1.7.1
OPE CIRCUIT TEST
•
Must be conducted at the rated terminal voltage
•
The values of Rc (core resistance) and X m (magnetizing reactance) can be determined by opening the output line at the secondary side of transformer as in figure below
θ OC = cos X M =
−1
P oc
Fig. 1.17: Open-circuit test
V oc I oc
V OC I M
I C = I OC cos θ
Rc =
V OC I C
I M = I OC sin θ
CHECK →
If V p = V oc
→
Therefore Rc and X m (Primary side)
→
Therefore Rc and X m (Secondary side)
Else → V s = V oc
1.7.2
SHORT CIRCUIT TEST
•
Always conducted at the rated winding current
•
Current flowing in excitation branch is neglect since the input voltage is so low during the short-circuit test.
•
Thus the entire voltage drop in the transformer can be attributed to the series elements in the circuit:
•
The determined values of Req (winding resistance) and X eq (winding reactance)
Fig. 1.18: Short-circuit test
θ SC = cos −1
Z eq =
P SC V sc I sc
V sc ∠0° I sc ∠ − θ sc
Z eq = Req + jX eq
(Note: minus sign → inductance only)
CHECK →
If I p = I sc Therefore Z eq = Z p = R p + jX p Else If I s = I sc Therefore Z eq = Z s = R s + jX s I p =
1.8
S rated V p
I s =
S rated V s
TYPES OF LOSSES
• Copper ( I 2 R ) Losses :
resistive heating losses in the secondary and primary windings
• Eddy current losses
:
resistive heating losses in the core of the transformer
• Hysteresis losses
:
a heat loss caused by the magnetic properties of the armature
• Leakage flux
:
in turn causes losses due to frictional heating in susceptible cores. Magnetic flux lines produc ed by the primary winding that do not link the turns of the secondary winding
1.9
VOLTAGE REGULATIO
Defined as the difference between the voltage magnitude at the load terminals of the transformer at full load and at no load in percent of full load voltage.
a)
Equivalent circuit referred to primary side
Fig. 1.19(a): Approximate equivalent circuit (referred to primary)
Voltage regulation =
V s L − V s FL V s FL
× 100%
Since at no load, V s ( L ) = V p the voltage regulation can also be expressed as Voltage regulation =
V p − V s ( FL ) V s ( FL )
× 100%
b)
Equivalent circuit referred to secondary side
Fig. 1.19(b): Approximate equivalent circuit (referred to secondary)
Voltage regulation = Since at no load, V s ( L ) = V p / a
V s L − V s FL V s FL
× 100%
the voltage regulation can also be expressed as
Voltage regulation =
V p / a − V s ( FL ) V s ( FL )
× 100%
1.10
EFFICIECY AD MAXIMUM EFFICIECY
To measure the efficient of the transformer a)
Equivalent circuit referred to primary side
Fig. 1.20(a): Approximate equivalent circuit (referred to primary side)
P core =
V p
2
R c
P cu = I p Req 1 2
η =
η =
η =
P out P out + P loss
x100%
V s I s cos θ V s I s cos θ + P Cu + P core
x100%
nV s I s cos θ nV s I s cos θ + n 2 P Cu + P core
x100%
b)
Equivalent circuit referred to secondary side
Fig. 1.20(b): Approximate equivalent circuit (referred to secondary side)
(V / a )
2
P core =
p
R c
P cu = I s Req 2 2
η =
η =
P out P out + P loss
x100%
nV s I s cos θ nV s I s cos θ + n 2 P Cu + P core
When maximum efficiency occurs:
n =
P c P cu
and cos θ = 1
x100%
