Electric Machines and Power Systems and Equivalent Circuit Report Your review should include the following details  Thorough technical review  Cleary men

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Upload an e-copy in pdf format on canvas. Chapter 5 Power Transformers
5.1 Single phase transformers
5.1.1 Equivalent circuit and equations
The task of a power transformer is to transfer the electrical energy from a circuit,
operating at one voltage level, to another circuit, operating at another voltage level.
The main electrical and magnetical components of a single phase transformer are
shown in the following figure:
Depending on the solution task, different versions of the equivalent circuit can be
used, and these equivalent circuit version are valid for any sequence. The equivalent
circuits shown are valid for a single-phase transformer, or for one phase of a threephase transformer, for a three-phase symmetrical steady-state operation.
The related data of a transformer are:
-The rated apparent power, S rated
-the rated voltages v1rated /v2rated (for two windings transformers)
1
-Rated frequency
Also it may be specified
-The short-circuit voltage, Vsc %
-the no-load current, Inl %
-Complete equivalent circuit with the ideal transformer included:
The circuit equations are:
V1 = (R1 + jX1e )I1 + E1 (1)
V2 = E2 − (R 2 + jX2 )I2 (2) (Characteristics for this equivalent circuit is that all
quantities have the actual values.)
E1
E2
I1
I2
N
= N1
N
2
= N2
1
Ie = Ic + Im
I1 = I2 + Ie
(3)
(4)
(5)
(6)
2
-Complete equivalent circuit with all quantities referred to one side (without ideal
transformer).
* When quantities from side 2 are referred to side 1
Where,
N
V2′ = V2 (N1 )
I2 = I2 (
2
N1
)
N2
R′2 = R 2 (
N1 2
)
N2
′
X2e
= X2e (
N
N1 2
)
N2
(N1) the actual turn ratio!!!
2
Here the quantities at side 2 do not have the actual value. They are “referred” to side
1.
The circuit equations are:
′′ )I
V1 ′′ = V2 + (R′′1 + jX1e
1 ′′ + (R 2 + jX 2e )I2
Ie = Ic ′′ + Im ′′
I1 ′′ = I2 + Ie ′′
Approximate or gamma equivalent circuit with all quantities referred to the same
side.
3
In this case the shunt branches are connected to one of the transformer terminals.
The side selected for the shunt branches location depends on the solution
convenience.
For example, the equivalent circuit with all quantities referred to side 1, and with
the shunt branches connected to side 1:
Comments:
–
Since >> , many times is disregarded
More than that, since ≫ 1 , 1 , 2 , 2 , frequently both shunt branches are
neglected.
As the transformer rated power is higher, 1 2 ′ become much lower
than 1 2 ′, and resistance (series resistances) are neglected.
If it is desired or needed then to the equivalent circuit with all quantities referred
to one side it can be attached the corresponding ideal transformer, connected to
that terminal from where parameters have been referred.
On these equivalent circuits, the potential phase-angle difference between the two
sides of the transformer is not included.
5.1.2 Tests and parameters
For defining the equivalent circuit parameters the transformer is tested at opencircuit and short-circuit, and also the windings resistance is measured in dc.
Based on the tests measurement, parameters are calculated, and the result as
referred to the side which was energized and where the measurements have been
made. With the measurements data, parameters are defined, accepting the fact
that Rc, Xm >> R1, R2, X2e, X1e.
4
Tests and the parameters calculation will be considered here for the equivalent
circuit shown on page 2 (actually does not matter which circuit is considered, the
only requirement is to be one with parameters referred to one side.)
For the given equivalent circuit, let’s assume that side 1 is the higher voltage side,
and side 2 is the lower voltage side.
At the short-circuit test; to one side terminals a voltage source is applied (this is
‘energized’ side), and to the other side. The terminals are short-circuited.
Usually the voltage source is applied to the higher voltage side (energized side),
and the voltage of the source is adjusted until the current is absorbed by
transformer is the rated current at the energized side!!
At the energized side are measured the terminal voltage, the terminal current,
and the active power absorbed by the transformer. For example, if for the given
transformer side 1 is energized by the transformer.
Comments:
V1 = V1sc ≪ V1rated
I1 = I1sc = I1.rated
P1 = P1sc
Then the equivalent sense parameters referred to side 1 are calculated as:
S1sc = V1sc I1sc → p1sc = �s21sc − p1sc 2 , and calculated with the measured data.
Measured:
P
(R1 + R 2 ′) = I 1sc2
1sc
(X1e + X2e ′) =
Q1sc.calc.
I1sc 2
5
At the open circuit test one side is energized usually the lower voltage side and
the other side is open.
The voltage applied at the energized side is set to the rated voltage at that side.
Then the terminal voltage, and the absorbed current and active power at the
energized side are measured. For example, if for the given transformer the
energized side is slide 2:
Comments:
V2 = V2ec = V2.rated
I2 = I2oc ≪ I2.rated
Then the shunt parameters of the equivalent circuits as referred to side 2, are
calculated as:
R′′c
V 2 OC
=
P2oc
V2 2oc
′′
Xm
=Q
20.calc.
S2oc = V2oc I2oc = Q2oc = �S 2 2oc − P2oc 2  Calculated with the measured data.
If the intention is to build the complete equivalent circuit (with all quantities
referred to one side) the windings resistance dc measurements must be applied.
On the dc, it is measured (R1 dc ) and (R 2 dc )
P
R
Then with (R1 + R 2 ′) = I21sc and with R1 =
1sc
2
For leakage reactance it is assumed that
′
1 = 2
=
( 1 + 2 ′)
2
R1dc
N
R2dc ( 1 )2
it results (R1) and (R2’)
N2
On the next table it is shown the impedance value for different transformer sizes
and rated voltages.
6
5.1.3 Terminal marks and polarity:
According to the USA standards, the higher voltage side is labelled as “HV” or “H”,
and the lower voltage side is labelled as “LV” or “X”.
Winding terminals are labelled as “LV” or “X”.
7
Winding terminals are labelled as H1, H2, X1, X2,
Also the standard requires that the winding’s dot to be at the same side of the
label number (for example the dots are both located X1 and H1).
The polarity can be defined by the following test:
8
Implicitly, only by the winding terminal labels, H1, …. , X0, it is known that the
dot position is as shown above.
It is reminded the meaning of the dot mark: if on both sides of the transformer
windings (pair of windings) the terminal with dot has the same polarity (+ or -),
then the terminal voltages defined at each side are in phase; if at one terminal the
positive reference of the current is defined as entering (or leaving) the dot
terminal, and at the other terminal the positive reference of the current is defined
as leaving (or entering) the dot terminal, the terminal currents are in phase
(neglecting the existing current).
5.1.4 (Autotransformer) (AT)
Initially it is considered a single-phase transformer with the rated data specified
on the following figure:
9
The transformer turn ratio is defined as:
t tr =
Vtr.1.rated Itr.2.rated N1
=
=
Vtr.2.rated Itr.1.rated N2
And the transformer rated apparent power is defined as
Str.rated = Vtr.1.rated ∗ Itr.1.rated = Vtr.2.rated ∗ Itr.2.rated .
This single-phase transformer can be converted to an autotransformer. Typical for an
autotransformer is that along with the magnetical coupling between windings, there
is an electrical connection between them. Here a single-phase case will be shown and
discussed. The same rules apply for a single-phase of a three-phase transformer
/autotransformer.
Depending how the windings are electrically connected, there are different versions
to convert the transformer to autotransformer, and on what follows are shown
versions when the windings’ voltage are additive.
It will be analyzed in detail on of these versions, specifically version A, the purpose
is define the corresponding relations between the transformer and the corresponding
autotransformer data.
10
Equations that will be applied are valid by neglecting the existing current. Also, when
the given transformer is converted to an autotransformer it must be taken care that
the rated voltage and current for each winding to be the same as it was for the
corresponding transformer.
It is also added that actually the autotransformer is not obtained by using an existing
transformer. It is built from the beginning as an autotransformer. Here the intention
is to compare the advantages of an autotransformer with respect to a similar
transformer.
For version A the following equation are valid:
The auto transformer rate apparent power is defined by the equations
SAT.1..rated = SAt.2.rated, that is
VAT.1.rated ∗ IAT.1.rated = VAT.2.rated ∗ IAT.2.rated , from where
IAT.1.rated =
VAT.2.rated ∗ IAT.2.rated (Vtr.1.rated + Vtr.2.rated )
=
∗ Itr.2.rated
VAT.1.rated
Vtr.1.rated
= (Itr.1.rated + Itr.2.rated )
If it is defined the autotransformer turn ration as from left to right side, as it was
defined for transformer, it results:
V
t AT = VAT.1.rated = N
AT.2.rated
N1
1 +N2
t
t
= 1+ttr and t tr = 1+tAT
tr
11
AT
The ratio of the autotransformer to the transformer rated power results as
SAT.rated (Vtr.1.rated + Vtr.2.rated ) ∗ Itr.2.rated
1
=
= (1 + t tr ) =
Str.rated
Vtr.2.rated
1 − t AT
As it can be seen, as compared with a transformer, the autotransformer has the
advantage that has a higher rated power for the same core and conductor’s
assumption. It means that for the same rated apparent power an autotransformer
is physically smaller, has lower losses, smaller existing current, and smaller series
impedance (that can be a disadvantage).
The autotransformer has the advantage that is lost the electrical isolation
between the windings and the corresponding network to which the
autotransformer is connected. The autotransformer’s tests and equivalent circuit
are similar as for a transformer.
5.2 Three-phase transformers
5.2.1 Core structure and terminal marks
The core of a three-phase transformer can be composed of three single-phase
transformer cores, or one single core with three or more columns:
12
At the higher voltage side, the terminals are marked as H1, H2, H3, and at the
lower voltage side the terminals are marked as X1, X2, X3.
It is also customary that phases are defined to the higher voltage side with capital
letter (A, B, C) and to the lower voltage side with lower case.
The USA standards requires that for Y-Y and ∆ − ∆ windings connection, the
voltages to neutral from terminals H1, H2, and H3 are in phase with voltages to
neutral from terminals X1, X2, and X3, respectively.
For Y/∆ or ∆/ connections the terminals are such labelled that for the positive
sequence, the voltages from terminals H1, H2, H3 to neutral lead the voltage from
terminals X1, X2, X3 to neutral respectively by 30 degree.
13
5.2.2 Equivalent circuit, tests and parameters for the positive (or
negative) sequence of two-windings transformer.
Since the equivalent circuit is built for one phase of a corresponding actual or
equivalent Y connection, it can have one of the forms shown in section 5.1.1, pages
2 – 4.
For example, the complete equivalent circuit with all quantities referred to side 1
(see the figure on page 2)
For one phase of a three-phase transformer is
Disregarding the windings connection, to refer quantities from one side to the other
side of the transformer it is used the turn ratio with the number of turns defined on
both sides as for an actual or equivalent Y connection.
It is the same as using the corresponding line-to-line voltage ratio.
For example, if the transformer is operating at rated turn ratio, and the line-to-line
rated voltages are V1rated and V2rated, the referred quantities shown on the above
equivalent circuit.
V
V
V
V
′
′
R′2 = R 2 (V1.rated )2 , X2e
= X2e (V1.rated )2 , V2.ph
= V2.ph (V1.rated )1 , I2′ = I2 �V2.rated �. (Take care,
2.rated
2.rated
2.rated
1.rated
being a three-phase transformer, the rated voltages are line-to-line voltages.)
The equation that describe the steady-state operation of a given equivalent circuit
are the same as those shown on pages 2 and 3. This time the equations are referring
to one phase of a three phase transformers with the windings Y connected.
For defining the equivalent circuit parameters there are the same tests as for singlephase transformers (see pages 4-6):
Some comments are added:
14
This time the tests are three-phase symmetrical tests.
The measured quantities could be per phase (possible for a Y-connection with neutral
accessible) or per total three-phase transformer (that is the total three-phase power
(absorbed active) the line-to-line voltage, and the line current at the energized side)
-the parameters can be defined per winding (and in this case, if windings are
∆ connected, to obtain parameters/phase of the equivalent Y connection, it must be
applied to the ∆-to-Y connection) or directly per phase of the equivalent or actual Y
connection it is specified that if for defining parameters are used the total three-phase
absorbed active power, the line-to-line voltages, and the line-currents, then it will
result directly, parameters per phase of the equivalent Y connections !!
The last version is applied here:
–
–
The same test rules as for the single-phase transformer apply here (Isc =
Irated; Voc = Vrated; usually the short circuit test is with the higher voltage
side energized, and the open circuit test is with the lower voltage side
energized).
Regarding the dc windings resistance measurement there are two possible
cases.
If windings are Y-connected with the neutral accessible, the dc measurement
can be made between terminal and neutral. Then the winding’s conductor’s dc
resistance is R dc = R dc.measured
If the dc measurement is made between two terminals (it could be for a Yconnection with the neutral accessible, and must be for a Y-connection with
the neutral accessible, or for a ∆ − connection)
15
Then the dc resistance for one phase of the equivalent Y-connection is:
R dc =
R dc.measured
2
As for the single-phase transformer, parameters calculated by using the test data are
referred to the side that was energized for that test.
The parameters value per phase are defined as for the single-phase transformers (see
pages 6 and 7) now it must be counted that the power is three-phase, the voltage is
line-to-line, and the current is line current.
With the same assumptions (Rc, Xm >> R1, R’2, X1e, X’2e), and for a similar case as
shown on pages 6 and 7, parameters result
16
Comments shown on page 4 for the single-phase transformer apply for the threephase transformer.
Just if it’s a phase angle difference between voltage to neutral on one side as with
respect to the other side, here the difference was disregarded.
If the phase angle difference is to be considered, a phase shifter must be included in
the equivalent circuit.
As if for any the phase power component, the single-phase impedance diagram
contains parameters for a branch of an actual or equivalent Y connection.
5.2.3 Three-windings transformer:
A power transformer can be provided with a third winding per phase, which has a
rated voltage different from the other two windings and a rated power which
generally is different of rated powers of the other two windings.
Regarding the windings rated power, it could be the same for all three windings, it
could be the same for two windings (for the higher and the intermediate voltage
levels), or it can be different for each voltage level.
The third winding is used to supply a local distribution network, or to supply the
shunt capacitors or reactors for correcting the power factor, or to provide a closed
path for the zero sequence currents if windings are ∆ −connected.
The transformer terminals are marked as H (H1, H2, H3) for the higher voltage side,
as X(X1, X2, X3), for the intermediate voltage level, and as Y (Y1, Y2, Y3) for the
lower voltage level.
When the direction of the power flow is known, the windings or the voltage levels can
be called “primary”, “secondary”, and” tertiary”
The series parameters of the transformer are defined with the short-circuit tests. At
the short-circuit test one voltage level is energized, the other one is short-circuited,
and the third-one is open circuited.
Repeating the test for each pair of windings, three equations with three unknowns
obtained. As for any other transformer, parameters that result based on a test are
referred to that side that was energized for that test.
For the series impedance per phase of the voltage level “α”, referred to the voltage
level “ϒ”, it will be used the notation:
17
For the equivalent impedance of two windings (Voltage levels) “α” and “β”, defined as
referred to the windings (voltage level) “ϒ”, it will be used the notation ( )
Then let’s assume that it is desired to build the equivalent circuit with all parameters
referred to the higher voltage side. Then the following tests apply, and the following
parameters are calculated.
*) higher voltage side energized, Intermediate voltage side short-circuit, and lower
voltage side open-circuited.
Based on the measurement data: Zhx (H) is calculated, the equals:
ZHX (H) = ZH (H) + ZX (H)
(1)
ZHY (H) = ZH (H) + ZY (H)
(2)
*) higher voltage side energized, lower voltage side short-circuited, intermediate
voltage side open. It results;
*) Intermediate voltage side energized, lower voltage side short-circuited, higher
voltage side open. It results;
ZXY (X) = ZX (X) + ZY (X)
(3’)
If (3’) is multiplied with (NH/NX)^2, where NH and NX are the number of turns on
the higher and the intermediate voltage level, respectively (number of turns are for
the winding per phase of the actual or equivalent Y-connection; instead of the number
of the transition be used line-to-line voltages), the equation is converted to that with
parameters referred to the higher voltage:
ZXY (H) = ZX (H) + ZY (H)
(3)
With (1), (2) and (3) the parameters per phase and voltage level, in this case all as
referred to the higher voltage side, H, result as:
ZH (H) =
1
�Z (H) + ZHY (H) − ZXY (H)�
2 HX
18
ZX (H) =
ZY (H) =
1
(Z (H) + ZXY (H) − ZHY (H))
2 HX
1
(Z (H) + ZXY (H) − ZHX (H))
2 HY
The equivalent circuit with the existing current neglected looks like:
Comments:
Usually the H and X windings are Y, and the Y winding is ∆-connected.
The junction point, N, of three branches ZH(H), ZX(H) and ZY(H) do not represent
and has relation to the actual system’s neutral.
Some of the parameters per phase and winding can result as of negative or zero
value!!
When parameters are connected in per unit. It must be used a common base voltage
and apparent power for all voltage levels.
Frequently, the impedances ( ) are given in the present (that is similar to be in
related per unit) then it must be understood that ( ) and ( ) in percent are
defined with the related voltage and power at higher voltage side, and ( ) is
defined with the related voltage and power at the intermediate voltage side.
The shunt branches are defined as for a single or a three-phase two windings
transformer by an open-circuit test with that side energized to which the shunt (and
series branches) are referred.
5.2.4 phase shift for y/∆ transformers.
According to the USA standards, for the Y/Y and the ∆/∆ windings connection the
voltage to neutral from the terminal H1…
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