ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 1 TC, 2009 Single-phase Transformers 1 Magnetic equivalent circuit We will consider a simple magnetic circuit comprising a coil wrapped around a magnetic core, as shown in Figure 1. The magnetic core has a cross-section area A. For simplicity, we will assume that magnetic flux is confined within the core (i.e. no flux leakage occurs). At the same time, the average length of the flux path in the core is l. We will also assume that the core material is homogenous, and therefore, that the value of magnetic field intensity, H, remains unchanged along the flux path. We can now apply the Ampere's Law to this circuit as follows: Hl dl H dl H NI i NI Hl (1) Figure 1: Simple magnetic circuit Using A B (2) T , Wb/m 2 0 H H B r (3) we obtain (4) The quantity Ni is called the magnetomotive force (mmf or F) and its unit is ampere-turn (At, or simply A). The quantity A l is called reluctance, R, of the magnetic circuit. The reciprocal of reluctance is called permeance, P. A l P R 1 (5) Therefore, RΦ F or RΦ mmf (6)
Transcript
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 1
TC, 2009
Single-phase Transformers
1 Magnetic equivalent circuit
We will consider a simple magnetic circuit comprising a coil wrapped around a magnetic
core, as shown in Figure 1. The magnetic core has a cross-section area A. For simplicity, we
will assume that magnetic flux is confined within the core (i.e. no flux leakage occurs). At the
same time, the average length of the flux path in the core is l. We will also assume that the
core material is homogenous, and therefore, that the value of magnetic field intensity, H,
remains unchanged along the flux path.
We can now apply the Ampere's Law to this circuit as follows:
HldlHdlH
NIi
NIHl
(1)
Figure 1: Simple magnetic circuit
Using
AB (2)
T ,Wb/m 2
0 HHB r (3)
we obtain
(4)
The quantity Ni is called the magnetomotive force (mmf or F) and its unit is ampere-turn (At,
or simply A). The quantity A
l
is called reluctance, R, of the magnetic circuit. The reciprocal
of reluctance is called permeance, P.
A
l
PR
1 (5)
Therefore,
RΦF or RΦmmf (6)
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 2
TC, 2009
Now we can see that
HlRΦF (7)
Equation (7) tells us that the amount of magnetomotive force, F, is equal to the amount of
magnetic potential drop, RΦ or Hl. This is similar to electric circuits in which, according to
the Kirchhof's Voltage Law, the source voltage is equal to the sum of voltage drops.
Following this similarity, we can see that reluctance in magnetic circuits has analogy with
resistance in electric circuits while magnetic flux, Φ, is a magnetic equivalent of electric
current. This leads us to a magnetic equivalent circuit diagram:
Figure 2: Simple magnetic equivalent circuit
Magnetic circuits can contain more than one coil, i.e. more than one source of mmf.
Additionally, magnetic flux can flow in a core whose sections are made of different magnetic
materials, including air gaps. Therefore, the general form of KVL applied to magnetic circuits
is written as follows:
HlNi (8)
Equation (6) suggests a straightforward, proportional relationship between the magnetic
excitation, F, and the magnetic flux Φ, in which reluctance R is the proportionality constant.
However, Figure 3 shows that most practical magnetic materials are highly non-linear. It is
easy to notice on the B-H curve that for the low values of excitation (i.e. low magnetic filed
intensity H - from the point of view of the magnetic circuit or low current i in the coil - from
the electric circuit view point) a small change of excitation can produce large variation of the
magnetic flux. At high values of excitation the magnetisation characteristic saturates. In this
region, any additional increase of the excitation does not produce a significant increase of the
flux. This flux saturation phenomenon has a significant impact on the performance of
electromagnetic devices. At the same time, to reflect the nonlinear relationship between the
flux and excitation, equation (6) should be re-written as follows:
ΦΦRF (9)
In other words, reluctance is a non-linear element in the magnetic equivalent circuit, while
resistance is usually a linear element in electric circuits.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 3
TC, 2009
Figure 3: Magnetization characteristics of various materials.
2 AC excitation in magnetic circuits
2.1 Hysteresis losses
Ferromagnetic materials are good conductors of magnetic flux. As such, they are commonly
used to build magnetic circuits. The process of magnetisation and demagnetisation of
ferromagnetic core is not completely reversible. Therefore, when the coil is supplied from an
AC source a hysteresis loop in the B-H curve is observed.
Figure 4: Magnetisation of ferromagnetic material.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 4
TC, 2009
Figure 5: Hysteresis loop.
The area on the B-H curve contained within the hysteresis loop represents magnetisation
energy losses. The magnetostatics theory tells us that the increment of energy density needed
to increment magnetic field is
w H B (area efag) (10)
From (10), the portion of energy delivered to the magnetic core (per unit volume) when H
increases from zero to its positive maximum is
a
e
B
BdBHw1 (area efag) (11)
The energy released when H decreases from its maximum to zero is
b
a
B
BdBHw2 (area agb) (12)
Note that Ba > Ba, therefore, w2 < 0, which indeed means that some energy is being released
from the magnetic circuit.
The remaining area abe represents the energy that is not returned to the source but instead it is
dissipated as heat in the magnetic core.
We have considered so far the change of excitation from zero to maximum and back to zero,
i.e. one half of the AC cycle. An identical process occurs during the second half of the cycle.
The energy dissipation over the entire AC cycle represents hysteresis losses.
The electrical energy flow from the source during magnetization is
dtieW (13)
At the same time,
dt
dΦNe - Faraday's law
BAΦ
HlNi - Ampere's circuital law
(14)
(15)
(16)
which allows us to write
g
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 5
TC, 2009
dBHAldBHlABAdHldΦHldtN
Hl
dt
dΦNW (17)
where Al = Vcore (volume of the magnetic material), therefore the hysteresis loss over one
cycle of AC becomes
dBHVW core [Joules] (18)
or specific hysteresis loss (per unit volume) is
dBHWh [J/m3] (19)
The amount of power lost in the core due to the hysteresis effect is
fWVP hcoreh [W] (20)
where f is frequency.
Because the amount of excitation can vary, it is convenient to have the relationship between
the magnetic flux density and the amount of power loss in the core. There is no analytical
expression for the highly non-linear B-H curve which prevents derivation of an analytical
formula for the power loss. Instead, an empirical relationship is frequently used in engineering
calculations.
fBkP n
hh max [W] (21)
where
kh = k·Vcore
n = 1.5 – 2.5
The material constants kh and n can be found from the magnetic material data sheets.
2.2 Eddy current losses
We are still considering AC magnetic fields. A time-varying magnetic field always induces
time-varying electric field in the medium. If the magnetic medium is at the same time an
electric conductor electric current will be generated as a consequence. The direction of such
current can be determined from the Lorenz's law:
The direction of the induced current is always such as to oppose the action of the magnetic
field that produces it.
Figure 6: AC currents induced in concentric conductors by an AC flux.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 6
TC, 2009
If instead of concentric conductors we have a solid iron core, the AC magnetic flux that flows
in the core will induce eddy currents. These currents will in turn cause electrical power losses
in the core (due to i2R).
Figure 7: Eddy currents in a solid (left) and laminated (right) iron core.
In order to reduce eddy current losses, we can use
(a) high-resistivity core material (silicon steel, ceramic ferrites) to increase R
(b) laminated core to break up the current path, and thus reduce i, (0.3 – 5 mm
laminations for 50 Hz, 0.01 – 0.5 mm for higher frequencies).
It has been determined empirically that
22
max fBkP ee [W] (22)
2.3 Total core losses
Both, hystereses losses and eddy current losses contribute to heat dissipation in the magnetic
core and reduce the overall efficiency of electromagnetic devices such as transformers and
rotating machines. The total core power loss is
ehc PPP [W] (23)
The presence of eddy currents has an effect on the overall flux (eddy currents generate their
own magnetic flux), which in turn effects the shape of the hysteresis loop (the hysteresis loop
"fattens"). From this point of view, the two effects cannot be separated from one another.
Such an enlarged hysteresis loop is called the dynamic hysteresis loop and this is the one that
can be observed experimentally.
The total core losses can be easily measured using a wattmeter, a procedure commonly
performed on transformers and rotating machines as a no-load (open circuit) test.
2.4 Magnetising current
In most practical applications electromagnetic devices are energised from a voltage source. A
power transformer connected to a distribution circuit is one example of such a device.
According to Faraday's law, equation (14),
dt
dΦNe
If we assume that the core flux is sinusoidal
tΦΦ max sin (24)
then the voltage induced in the coil will be
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 7
TC, 2009
tEtNΦdt
dΦNte max coscos max (25)
Equations (24) and (25) show that if flux is sinusoidal the induced voltage is also sinusoidal.
The converse is also true, i.e. that sinusoidal voltage applied to the coil will result with
sinusoidal flux. The Faraday's law enforces this phenomenon. An implication of this fact is
that magnetic flux in most practical electromagnetic devices will be sinusoidal. Because
magnetic core is non-linear the resulting magnetic field intensity, H, will become a distorted
sinusoid. Because Hl = Ni, the exciting current supplied to the coil will be also a distorted
sinusoid. The relationship between voltage and current in a magnetic coil is illustrated in
Figure 8 and Figure 9 for magnetisation without and with the hysteresis respectively.
Equation (25) allows also calculation of AC voltage induced in a coil as a function of flux
maxmaxmax
rms NfΦNΦE
E 44.422
(26)
Equation (26) is frequently used for calculating the amount of magnetic flux in electrical
machines energised from an AC voltage source, such as power grid.
Figure 8: Exciting current with no hysteresis in the magnetic circuit1.
Figure 9: Exciting current with hysteresis in the magnetic circuit1.
1 From P.C. Sen, "Principles of Electric Machines and Power Electronics", 2
nd edition, Wiley, 1997
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 8
TC, 2009
3 Transformer theory
3.1 Magnetising and leakage fluxes
The purpose of this section is a brief revision of transformer theory. We begin with a model of
a tightly coupled transformer, i.e. a transformer in which magnetic losses do occur but one
magnetic flux links all the windings. In other words, no flux leakage occurs in the
transformer.
Figure 10: Diagram of a tightly coupled two-winding transformer.
Figure 11: Equivalent circuit of a tightly coupled transformer.
The mutual magnetic flux, m, is in fact composed of two mutual fluxes, m1 and m2, driven
by the two mmf's, N1I1and N2I2 and it is confined to the magnetic core. In a practical
transformer, magnetic coupling is imperfect and some of the flux produced by the mmf's
"leaks" outside of the magnetic core.
Figure 12: A practical transformer.
The mutual fluxes m1 and m2 induce emf's E1 and E2 in the two windings. The leakage flux
is only linked with its own source winding. As such, it does not contribute to the emf induced
in the other winding. This applies to both sides of the transformer. However, leakage flux
N1
I1 Φm
N2
I2
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 9
TC, 2009
induces some emf in "its own" winding. This effect can be accounted for by introducing an
inductance, called leakage inductance, to the transformer model. Flux leakage occurs in each
winding, therefore, there should be leakage inductance attached to each winding of the
transformer. Additionally, each winding has some resistance that also must be included in the
model of a practical transformer. All these elements are added to the model of the tightly
coupled transformer and the final result in the form of the equivalent circuit of a practical
transformer is shown in Figure 12.
3.2 Equivalent circuit
Figure 13: Complete equivalent circuit of a loaded transformer.
Principal relationships between voltages and currents
aN
N
E
E
2
1
2
1 (27)
where a is the turns ratio.
aN
N
I
I 1
1
2
2
1 (28)
3.3 Losses
Copper losses:
2
Cu p eqP I R (29)
At full load Ip = 1 pu, which gives
PUCu eqP R (30)
Core (iron) losses:
m
p
cR
EP
2
(31)
Normally, Ep E's = 1 pu, thus
m
cR
P1
PU (32)
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 10
TC, 2009
3.4 Efficiency
2
2
2 2
cos
cos
out out s
in out Cu c s eq c
P P E I Θ
P P P P E I Θ I R P
(33)
Figure 14: Transformer efficiency as a function of load current.
Maximum efficiency condition:
cCu PP (34)
3.5 Voltage regulation
=
Figure 15: Equivalent circuit for voltage regulation.
%100
FL
FLNL
E
EEVR (35)
Normally, we regulate to have EFL = 1 pu, which gives
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 11
TC, 2009
1PU NLEVR (36)
4 Transformer basic tests
In this section, we describe tests that are carried our in order to determine that winding
polarities are correct and to determine equivalent circuit parameters of the transformer. The
tests described here are part of factory tests performed on all newly manufactured power
transformers.
4.1 Polarity tests
Winding polarity tests are necessary to assure correct phasing of the transformer windings.
This is important to know specially when two transformers are to be connected for parallel
operation.
Figure 16: AC polarity test.
Method:
1. Connect a jumper lead between to adjacent HV and LV terminals.
2. Connect AC source across HV winding.
3. Measure voltage Ex between the other two adjacent HV and LV terminals and voltage
Ep applied to the HV winding.
4. If Ex < Ep, the HV and LV terminals are in phase.
Figure 17: DC polarity test.
Method:
1. Connect a DC source in series with a switch to the LV terminals.
2. Connect a DC voltmeter to HV terminals such that the positive side of the source and
the positive side of the voltameter are connected to adjacent terminals.
3. Turn on the switch. If the voltmeter deflects in the positive direction the two sides of
the transformer are in phase.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 12
TC, 2009
4.2 Open-circuit test
This test determines the transformer turns ratio, a, core losses Pc as well as resistance Rm
representing core losses and magnetising reactance Xm.
The test is conducted at full rated voltage. For convenience, the low voltage winding of the
transformer is usually energised during this test.
Measured quantities: Ep, Es, Io, POC.
Calculations:
aE
E
s
p
OC
(37)
The value of Pc = POC is obtained directly from the measurement while Rm and Xm are
calculated as follows:
m
m
fm
m
f
m
I
EX
III
R
EI
P
ER
OC
22
0
OC
OC
2
OC
(38)
4.3 Short-circuit test
This test determines copper losses and the equivalent winding impedance. The impedance
contains resistance of the windings, Rp, and the leakage reactance, Xp.
This test is conducted at full rated current. This is achieved by short-circuiting the low voltage
winding and supplying the high voltage winding with a voltage that will result with rated
current circulating in the windings.
Measured quantities: ESC, ISC, PSC.
Calculations:
22
2
SC
SC
SC
SC
ppp
p
p
RZX
I
PR
I
EZ
(39)
Note: PSC = PCu FL
The heat from copper losses occurring during the short-circuit test is also utilised to perform
the heat-run test. This is a factory test, carried out on newly built power transformers to
confirm thermal design parameters of the transformer.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 13
TC, 2009
5 Special single-phase transformers
5.1 Autotransformer
Smaller in size then an equivalent two-winding transformer, this is an advantage. No isolation
between the primary and secondary circuits may be a disadvantage.
Figure 18: Autotransformer winding configuration.
The principal relationships between voltages and currents are maintained in autotransformers,
namely
aN
N
E
E
2
1
2
1 (40)
aN
N
I
I 1
1
2
2
1 (41)
Figure 19: Current flow in an autotransformer.
Equation (41) results from the mmf balance equation that can be derived directly from the
diagram shown in Figure 19.
212211 NIINNI (42)
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 14
TC, 2009
Figure 20: Variable autotransformer ('variac').
Equivalent circuity of an autotransformer has identical structure with the two winding
transformer. In comparison with a two-winding transformer of the same power rating, an
autotransformer has lower leakage reactances, lower losses and exciting current. The leower
leakage reactance can be an advantage (better voltage regulation characteristic). It can be
a disadvantage in case of a short circuit fault in a power line supplied by an autotransformer
because the low leakage reactance will result with a large amount of fault current, requiring
a higher capacity (more expensive) circuit breaker for protection.
Refer to Example 11-2 in the Wildi's textbook to see how a two-winding transformer can be
reconfigured to work as an autotransformer and what effect it has on its power rating. Note
that the reconfiguration to step up the voltage may be practical only for low voltage
applications. High voltage transformers may not have sufficient insulation in their low
voltage winding to be connected "on top" of the high voltage winding, which would be
necessary to achieve the step-up.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 15
TC, 2009
5.2 Three-winding single-phase transformer
Figure 21: Equivalent circuit of a three-winding transformer.
MMF balance:
Input A-T = Output A-T → 332211 INININ →
'3
'23
1
32
1
21 III
N
NI
N
NI
Must then include leakage resistance and reactance for each winding. Three separate short
circuit tests allow the leakage impedances to be found:
Energise primary, S/C secondary → '21'
21 XXjRRZa
Energise primary, S/C tertiary → '31'
31 XXjRRZb
Energise secondary, S/C tertiary → '3'
2'
3'
2 XXjRRZc
From which:
cba ZZZjXR 2
111
bca ZZZjXR 2
1'2
'2
acb ZZZjXR 2
1'3
'3
5.3 Instrument transformers.
Instrument transformers are high precision transformers designed to allow measurements of
high voltage and high current. Voltage transformers (VTs) and current transformers (CTs) are
the two types of instrument transformers commonly used. They are designed for precise turns
ratio that changes very little with the load. At the same time, they are not to introduce a phase
shift between the primary and the secondary voltages, or currents.
The load is called the instrument transformer burden. The turns ratio is made to a certain
accuracy class, which describes the instrument maximal ratio error in percent. For example,
a class 0.5 instrument transformer has the accuracy of ±0.5%. Instrument transformers are
used for metering of electrical energy and in power system protection circuits. The more
precise instrument transformers are used for metering of electrical energy.
I1 I2
I3
V1
V3
V2
V1
V’2
V’3
I1 I2
I3 R1 jX1
R’2 jX’2
R’3 jX’3
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 16
TC, 2009
Voltage transformers.
They are also called potential transformers. The primary role of a VT is to allow
measurements of high voltage. At the same time, they isolate the measurement circuit from
the power circuit. There exist two main types of VTs:
Inductive voltage transformers
Capacitance voltage transformers (CVTs)
Excerpts from AS60044.2-2007:
5 Ratings
5.1 Standard values of rated voltages
5.1.1 Rated primary voltages
The standard values of rated primary voltage of three-phase transformers and of single-phase transformers for use in a single-phase system or between lines in a three-phase system shall be one of the values of rated system voltage designated as being usual values in IEC 60038. The standard values of rated primary voltage of a single-phase transformer connected between one line of a three-
phase system and earth or between a system neutral point and earth shall be 1/√3 times one of the
values of rated system voltage.
NOTE The performance of a voltage transformer as a measuring or protection transformer is based on the rated primary voltage, whereas the rated insulation level is based on one of the highest voltages for equipment of IEC 60038.
5.1.2 Rated secondary voltages
The rated secondary voltage shall be chosen according to the practice at the location where the transformer is to be used. The values given below are considered standard values for single-phase transformers in single-phase systems or connected line-to-line in three-phase systems and for three-phase transformers.
a) Based on the current practice of a group of European countries:
– 100 V and 110 V;
– 200 V for extended secondary circuits.
b) Based on the current practice in the United States and Canada:
– 120 V for distribution systems;
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 17
TC, 2009
– 115 V for transmission systems;
– 230 V for extended secondary circuits.
For single-phase transformers intended to be used phase-to-earth in three-phase systems where the
rated primary voltage is a number divided by √3, the rated secondary voltage shall be one of the fore-
mentioned values divided by √3, thus retaining the value of the rated transformation ratio.
NOTE 1 The rated secondary voltage for windings intended to produce a residual secondary voltage is given in 13.3.
NOTE 2 Whenever possible, the rated transformation ratio should be of a simple value. If one of the following values: 10 – 12 – 15 – 20 – 25 – 30 – 40 – 50 – 60 – 80 and their decimal multiples is used for the rated transformation ratio together with one of the rated secondary voltages of this subclause, the majority of the standard values of rated system voltage of IEC 60038 will be covered.
5.2 Standard values of rated output
The standard values of rated output at a power factor of 0,8 lagging, expressed in voltamperes, are:
The values underlined are preferred values. The rated output of a three-phase transformer shall be the rated output per phase.
NOTE For a given transformer, provided one of the values of rated output is standard and associated with a standard accuracy class, the declaration of other rated outputs, which may be non-standard values but associated with other standard accuracy classes, is not precluded.
11.2.2 Terminal identifiers
Markings shall be in accordance with Figures 6 to 15 as appropriate. Capital letters A, B, C and N denote the primary-winding terminals and the lower-case letters a, b, c and n denote the corresponding secondary-winding terminals.
The letters A, B and C denote fully insulated terminals and the letter N denotes a terminal intended to be earthed and the insulation of which is less than that of the other terminal(s). Letters da and dn denote the terminals of windings intended to supply a residual voltage.
11.2.3 Relative polarities
Terminals having corresponding capital and lower-case markings shall have the same polarity at the same instant.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 18
TC, 2009
Figure 22: 1-pole and 2-pole type of VT.
Figure 23: Potential transformer connected to a 69 kV line.
Secondary winding of a VT must be always grounded for safety, to avoid dangerous
potentials from capacitive coupling between primary and secondary windings.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 19
TC, 2009
Figure 24: Three-phase VT with two secondary windings: u-x for measurements and e-n for
protection.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 20
TC, 2009
Figure 25: Schematic of CVT.
Figure 26: Construction of CVT.
Current transformers
The primary role of a CT is to allow measurements of high currents for metering and
protection purposes. At the same time, CTs provide electrical insulation between the primary
and the measurement circuit.
As in any other transformer, also in the CT we have
1
2
2
1
N
N
I
I (43)
Equation (43) implies that there are fewer turns in the primary winding in comparison with
the secondary. For example, a 200A/5A CT will have 40 times more turns in the secondary
than in the primary. Normally, an ammeter is connected across the secondary terminals, and
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 21
TC, 2009
thus, the secondary voltage is very small. What happens if we disconnect the ammeter from
the circuit when the CT is still energised?
Figure 27: Secondary voltage induced when a CT is open-circuited.
Figure 27 illustrates why a CT can never be left open-circuited.
ECE3051, ELECTRICAL POWER ENGINEERING – LECTURE NOTES 22
TC, 2009
Ratings of current transformers
As per AS60044.1-2007:
The standard values of rated primary currents are:
