Single‐PhaseTransformerModel
Chapter3:Transformers
• Two‐winding, single‐phase transformers shown below
Equivalent circuit of power transformer
Tests and performance
Three‐phase connections
Per‐unit system
3 Transformers
Notes on Power System Analysis
1
3 Transformers
V1
V2
Notes on Power System Analysis
V1
3 Transformers
jXm
Im
3
3 Transformers
Notes on Power System Analysis
4
TransformerEquivalentCircuit
• The approximate equivalent circuit:
I1 R
I2 R2 jX2
II'2
Ie
Rc
V2
Ic
N1:N2
Ideal
Notes on Power System Analysis
V2
Ideal transformer model:
V1/V2 = N1/N2 and N1 I1 = N2 I2
• Tee equivalent circuit is not derived here
V1
I2
N1 : N2
TransformerEquivalentCircuit
I1 R1 jX1
m
I1
N1 : N2
3 Transformers
2
Transformers
• Primary is input side and secondary is output side (regardless of voltage level)
I2
N1 < N2
Step-up
Notes on Power System Analysis
Transformers
I1
N1 > N2
Step-down
N1 = N2
1:1 turns ratio
V1
5
3 Transformers
jX
jXm
Im
I2
II’2
Ie
V2
Rc V’
Ic 2
N1:N2
Ideal
Notes on Power System Analysis
6
1
PhasorDiagramofTransformer
V1
Im
I’2 V’2
Neglect losses and magnetizing current
V1
(R+jX) I1
V’2
I1
jX I1
I1 =I’2
This neglect core loss.
How would this change if Im is negligible? How would it change if R is negligible?
3 Transformers
Notes on Power System Analysis
7
Three‐PhaseTransformer
Connections
9
3 Transformers
One‐linediagram
Notes on Power System Analysis
20 kV 230 kV
230 kV 13.2 kV
G
Gen
Gen
3 Transformers
10
One‐linediagram
• Generator step‐up transformer (delta‐
wye), transmission line, and substation step‐down transformer (delta‐wye).
20 kV 230 kV
8
• Convention: primary‐secondary, so delta‐wye means the delta is on the primary (input) side
To avoid confusion, we say a step‐up
up • To avoid confusion, we say a step
or step‐down transformer bank, or draw one‐line diagrams that show the transformer connections
– Wye‐wye connection
y
y
– Delta‐delta connection
– Wye‐delta connection
– Delta‐wye connection
Notes on Power System Analysis
Notes on Power System Analysis
Three‐phasetransformerbanks
• Primary and secondary can be connected in wye or delta:
3 Transformers
3 Transformers
L d
Load
230 kV 13.2 kV
Load
Notes on Power System Analysis
11
3 Transformers
Notes on Power System Analysis
12
2
Three‐phasetransformers
Three‐phasetransformers
• One option: connect three single‐
phase transformers in a three‐phase bank
• Another option: use a three‐phase Another option: use a three phase
transformer (all in one tank)
3 Transformers
Notes on Power System Analysis
• Three‐phase transformer – Core‐form construction is common in three‐phase transformers of low kVA and kV ratings
kV ratings
– Shell‐form construction is common in high kVA and kV ratings, and in single‐phase transformers
13
3 Transformers
Notes on Power System Analysis
Core‐formandshell‐form
constructionoftransformers
Construction
Not to scale
• Note that two different winding designs are shown.
Single Phase
C
Core‐Form
F
Sh ll F
Shell‐Form
• Either winding design may be used with either shell‐form or core‐form construction.
a
Three Phase
a
14
b c
b
c
3 Transformers
Notes on Power System Analysis
15
Schematicviewofthree‐phase
transformerconnections
a
c
b
3 Transformers
n’
a
c
b
b’
c’
Notes on Power System Analysis
a’
b’
cc’
Delta‐delta connection
Wye‐wye connection
3 Transformers
16
Schematicviewofthree‐phase
transformerconnections
a’
n
Notes on Power System Analysis
17
3 Transformers
Notes on Power System Analysis
18
3
Delta‐wyeconnections
a
n’
c
b
Delta‐wyeconnections
a’
b’
c’
Delta‐wye connection (delta leading by 30°). Typical delta‐wye stepdown bank
3 Transformers
Notes on Power System Analysis
a’
b’
n’
c’’
a
c
b
19
StandardDelta‐Wye Connection
Delta‐wye connection (wye leading by 30°). Typical delta‐wye step‐up bank
3 Transformers
Notes on Power System Analysis
20
Three‐phasetransformerbanks
• The two connections shown on the previous pages are the standard connections in the US
– The high‐voltage side leads the low‐
voltage side by 30°, regardless which side is primary or which side is delta
• Many transformers in the real world do not follow the ANSI standard
• We assume positive (ABC) phase sequence, or else sign of phase shift is reversed
• Delta‐wye transformer banks are common
– Local standards may apply
3 Transformers
Notes on Power System Analysis
21
3 Transformers
Delta‐wyebanks
– Provides a neutral on the wye side (usually the secondary but not always)
(usually the secondary, but not always). – Isolates the grounds on the secondary from the grounds on the primary.
Notes on Power System Analysis
22
Typicalsystem
• Delta‐wye transformer banks are common
3 Transformers
Notes on Power System Analysis
23
20 kV 230 kV
Generator
3 Transformers
230 kV 13.2 kV
Transmission
Lines
Notes on Power System Analysis
Distribution
Feeders
24
4
Typicalsystem
Turnsratioofdelta‐wyebank
• Generator grounded through resistor to limit ground fault current
• Medium‐voltage distribution circuits solidly grounded at stepdown
solidly grounded at stepdown
transformer
• Relays and circuit breakers at stations and fuses on distribution lines clear heavy faults quickly
3 Transformers
Notes on Power System Analysis
25
– Let each single‐phase transformer in a delta‐wye bank have a turns ratio of N1/N2
– The bank “turns ratio” is the line‐line voltage ratio of N1/(3 N2) with a phase shift of 30° with the high‐voltage side leading (phase sequence abc)
3 Transformers
Per‐PhaseAnalysis
– Represent one phase of equivalent Y, with neutral as return path
– Divide all delta impedances by 3 to get equivalent Y impedances
– Divide all line‐line voltages by 3 to get equivalent Y line‐neutral voltages
– For 3‐phase transformer banks, use ideal transformer with turns ratio equal to nominal line‐line voltage ratio
– We may include phase shift due to delta‐wye
banks or not, depending on the situation
3 Transformers
Notes on Power System Analysis
27
line
model
j XD/3
/
j /6
1:ej/6
20:230
3 Transformers
load
model
j XY ej/6:1
phase shift
Notes on Power System Analysis
26
Typicalexample
20 kV 230 kV
230 kV 13.2 kV
Gen
Load
3 Transformers
Notes on Power System Analysis
28
Normalsystem
Typicalexample
generator
model
Notes on Power System Analysis
230:13.2
29
• Normal System is one in which the products of the complex voltage ratios of transformer around every loop is 1
– Nominal voltage ratios match up around each loop
– Phase shifts are zero around each loop
– Many systems are only approximately “Normal Systems”
– Deviations are usually small
3 Transformers
Notes on Power System Analysis
30
5
Normalsystem
Examplesystem
20 kV:230 kV
230kV:44 kV
230kV:46 kV
• In a normal system, leave out the phase shifts due to delta‐wye banks (balanced conditions to be analyzed gp p
y
using per‐phase analysis)
Gen
230kV:46 kV
Normal system shown. If one of the 230:46 kV transformer ratings is changed to 230:44 kV, then the system would not be normal
3 Transformers
Notes on Power System Analysis
31
– The phase of the voltage is shifted the same as the phase of the current, so the power factor is preserved
– Warning: remember the phase shift is really there
3 Transformers
Non‐NormalSystem
– phase shifting transformers used to shift average power loading
average power loading
– off‐nominal voltage ratios used to shift reactive power loading or to induce changes in voltage magnitude
Notes on Power System Analysis
33
Quantity in per unit = actual quantity/base value
3 Transformers
Notes on Power System Analysis
34
V1 = Z1 I1 + (N1/N2) V2 and V1/Vb1 = Z1(Ib1/Vb1) (I1/Ib1) + (N1/N2) V2 /Vb1
Let Zb1=Vb
Let Zb
Vb1/Ib1 and Vb
and Vb2=(N
(N2/N1)Vb1
Then V1/Vb1 = Z1/Zb1 (I1/Ib1) + V2 /Vb2
V1pu = Z1pu I1pu + V2pu
V = Z I where I is in [A], Z in [], and V in [V]
Let the base quantities be Ib, Zb, and Vb, also in SI units: V
l i SI it Vb = ZZbIb
V/Vb = (Z/Zb) (I/Ib) or Vpu = Zpu Ipu
Notes on Power System Analysis
• It is convenient to use a per‐unit system by normalizing with respect to a base quantity
a base quantity
Per‐unitequationsoftransformer
Per‐UnitSystem
3 Transformers
32
Per‐UnitSystem
• In some cases, the system is not be “normal”:
3 Transformers
Notes on Power System Analysis
35
3 Transformers
Notes on Power System Analysis
36
6
Per‐unitequationsoftransformer
Per‐unittransformermodel
I1 R
I1 = (N2/N1) I2
Let Ib2 = (N1/N2) Ib1
then
I1/Ib1 = (N2/N1) I2/Ib1 = I2/Ib2
I1pu = I2pu
jX I2
V2
V1
I1 R
V1
jX
jXm
Neglects magnetizing current
I2
Includes V2 magnetizing current
All quantities in per unit for both circuits
3 Transformers
Notes on Power System Analysis
37
3 Transformers
Basequantities
– Base current Ib = Sb/Vb
– Base impedance Zb = Vb/Ib = Vb2/Sb
Notes on Power System Analysis
39
– Transformer impedances are usually given in % on a rating base. For example 15% = 0.15 pu means 0.15 x Zb in ohms
– Same per
Same per‐unit
unit value on either side of value on either side of
transformer gives different value in ohms since Zb varies as Vb2
3 Transformers
• Regardless of whether delta or wye:
– Choose base voltages in same ratio as the nominal line‐line voltages
– Choose base VA same on both sides
h
b
b h d
– Result is the per‐phase equivalent circuit in per‐unit system has no ideal transformer in equivalent circuit
– See examples in text
Notes on Power System Analysis
Notes on Power System Analysis
40
ChangeofBase
PerUnitforThree‐Phase
3 Transformers
38
Basequantities
• Single‐phase transformer: Base voltage in [V] or [kV] and base apparent power in [kVA] or [MVA] are given: 3 Transformers
Notes on Power System Analysis
41
• Changing the base will change the per‐
unit value, but not the actual value in SI units (volts, ohms, amps). Let Z be i
in pu:
Znew=ZoldZbold/Zbnew
Znew=Zold(Vbold2/Sbold)(Sbnew/Vbnew2)
Znew=Zold(Sbnew/Sbold)(Vbold/Vbnew)2
3 Transformers
Notes on Power System Analysis
42
7
Per‐UnitAnalysisofaNormalSystem
– All other base voltages are set by the transformer ratios (rated line‐line voltage ratios)
– Express all impedances per unit of the appropriate base, and draw the per‐unit diagram
– Solve the problem in per unit, and convert back to SI units
• Note that phase shifts due to transformers are often ignored. The procedure follows:
– Select a common volt‐ampere base for Select a common volt ampere base for
the system
– Select one base voltage (usually a nominal or rated voltage in one part of the system)
3 Transformers
Notes on Power System Analysis
Per‐UnitAnalysisofaNormalSystem
43
3 Transformers
Notes on Power System Analysis
Per‐unitnetworksforthree‐phase
two‐windingtransformers
Wye‐wye
ZN
• Here we discuss only the per‐phase (positive sequence) network for the transformer
• Shorthand notation for transformer equivalent circuit is explained below
3 Transformers
H
Notes on Power System Analysis
X
VH
Y
45
3 Transformers
H
VH
ZX
Y
ZH
VH1
VX
ZX
Y
VX1
Notes on Power System Analysis
ZH
X
VX
VH
ZN
46
j/6
ZX e :1
Y
VX
Zn
Wye‐delta (typical for delta‐wye stepup with H side leading X side by 30) 3 Transformers
ZH
Delta‐delta
ZX ej/6:1
ZH
Zn
44
Notes on Power System Analysis
Delta‐wye (typical for delta‐wye stepdown
with H side leading X side by 30) 47
3 Transformers
Notes on Power System Analysis
48
8
Three‐WindingTransformers
Three‐WindingTransformers
• Consider the single‐phase three‐
winding transformer shown below
I1
• Per unit equivalent circuit: Y represents magnetizing branch, often omitted
I2
V2
V1
I1 Z1
V3
3 Transformers
Notes on Power System Analysis
V1
49
• Measure short‐circuit characteristics of each pair of coils with third coil open
Notes on Power System Analysis
51
Three‐phasethree‐winding
transformers
• Consider a YY connection with both Y's grounded
– Per‐phase (positive sequence) networks Per phase (positive sequence) networks
are same as that of single‐phase case, with the phase shift to the delta
3 Transformers
Notes on Power System Analysis
3 Transformers
53
I3
Z 3 V V2
3
Y
Notes on Power System Analysis
Z12 = Z1+Z2
Z12 = Z measured at 1 with 2 shorted and 3 open
Z13 = Z measured at 1 with 3 shorted and 2 open
Z23 = Z measured at 2 with 3 shorted and 1 open
3 Transformers
I2
Z2
50
Z13 = Z1+Z3 Z23 = Z2+Z3 Solve for Z1, Z2, Z3
Z1 = (Z12+Z13‐Z23)/2
Z2 = (Z12+Z23‐Z13)/2
Z3 = (Z13+Z23‐Z12)/2
3 Transformers
Notes on Power System Analysis
52
Per‐phase (positive sequence) diagram for YY stepdown
H = primary (high‐voltage side) Y grounded
X = secondary (low‐voltage side) Y grounded
T = tertiary (another low‐voltage side) 
ZX IX
IH ZH
VH
3 Transformers
Ye
IT
Z T V VX
T
Notes on Power System Analysis
54
9
• Remarks:
– A transformer with a path to ground, besides the high‐impedance magnetizing branch, is a grounding transformer
– The YY and Y transformers are grounding transformers, if the Y's are grounded
– The YY transformer is not a grounding The YY transformer is not a grounding
transformer – The YY and YY transformers pass other system grounds from one side to the other, if the Y's are grounded
– The Y and Y isolate grounds
3 Transformers
Notes on Power System Analysis
55
Autotransformers
• Autotransformers are efficient and economical for voltage ratios above 1:1 and below about 3:1
– Commonly used in Y connections l
d
– Often have a delta tertiary winding
3 Transformers
Notes on Power System Analysis
56
Three‐phaseautotransformer
Autotransformers
• Widely used to connect systems of different voltage (e.g. 345 kV to 230 kV or 230 kV to 115 kV)
• Often has a Y grounded connection Often has a Y grounded connection
with a delta tertiary
Tertiary
3 Transformers
Notes on Power System Analysis
– Equivalent circuit is same as YY
transformer
57
3 Transformers
Notes on Power System Analysis
58
Regulatingtransformersandoff‐
nominaltaps
• Transformers that automatically adjust voltage magnitude or phase shift will often have off‐nominal taps
– This is not a This is not a “normal”
normal system, so ideal system so ideal
transformers are not avoided
– voltage magnitude changes mainly affect reactive power
– phase angle changes mainly affect average power
3 Transformers
Notes on Power System Analysis
59
10