TRANSFORMER AC WINDING RESISTANCE AND
DERATING WHEN SUPPLYING HARMONIC-RICH
CURRENT
By
Jian Zheng
A Thesis
Submitted in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
Michigan Technological University
2000
ABSTRACT
Transformer loading with harmonic-rich current and subsequent overheating is an
ongoing concern of electric utilities and consumers. UL Standards 1561 and 1562 suggest
using a K-factor for determination of transformer capacity with nonlinear loads.
This work focuses at investigating the concept of K-factor and the relationship between K-factor, transformer derating, and the transformer winding eddy-current loss.
The relationship between K-factor and AC winding resistance is investigated. Laboratory
test procedures for measuring the AC winding resistance of two type of distribution transformers are developed and explained. Test procedures for checking the linearity and superposition assumptions are also developed.
From the test results, it is found that linearity and superposition holds very well
for the test transformers while the K-factor overestimates the losses in transformer windings. The difference between K-factor results and lab test results is explained. Another
approach for estimating the total stray loss in transformer winding, the Harmonic Loss
Factor, is discussed and found to be a better solution.
i
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my advisor, Dr. Leonard Bohmann, for his insights and direction through this research. He has
been an excellent advisor, patient, and helpful all through the time.
I take this opportunity to thank Dr. Bruce Mork, for his help and suggestions during the course of the research.
Special thanks to my committee members: Dr. Noel Schulz and Dr.
Konrad Heuvers for their time spent on reviewing this work. Their insights
and suggestions are greatly appreciated.
Besides the professors I have listed, I would also like to thank all of
the faculty and staff of the Electrical Engineering Department, especially
Scott Ackerman, John Miller, and Chuck Sannes, for being so helpful.
Finally, I wish to thank my family and friends for all the support they
have provided. Your support made my stay at Michigan Tech one that I will
never forget and always cherish.
ii
TABLE OF CONTENTS
ABSTRACT………………………………………………………………………………i
ACKNOWLEDGEMENTS………………………………………………………………ii
TABLE OF CONTENTS...………………………………………………………………iii
LIST OF FIGURES AND TABLES …..…………………………………………………v
CHAPTER 1 INTRODUCTION................................................................................................................. 1
CHAPTER 2 INTRODUCTION TO TEST TRANSFORMER AND K-FACTOR ............................... 4
2.1 SINGLE PHASE TRANSFORMER MODEL................................................................................................. 4
2.2 THE TEST TRANSFORMERS.................................................................................................................... 5
2.3 TRANSFORMER LOSSES AND THE AC WINDING RESISTANCE ............................................................... 7
2.4 K-FACTOR ............................................................................................................................................ 9
2.5 HARMONIC LOSS FACTOR ................................................................................................................... 12
CHAPTER 3 LABORATORY TESTS..................................................................................................... 14
3.1 MEASUREMENT CONSIDERATION ....................................................................................................... 14
3.2 TEST DEVICES ..................................................................................................................................... 15
3.2.1 Power source ............................................................................................................................. 15
3.2.2 UPC-32 ...................................................................................................................................... 16
3.2.3 Oscilloscope............................................................................................................................... 17
3.3 SHORT-CIRCUIT TESTS......................................................................................................................... 17
3.3.1 2KVA distribution Transformer ................................................................................................ 17
3.3.2 10 KVA distribution Transformer .............................................................................................. 18
3.3.3 Data Recording.......................................................................................................................... 18
3.4 HARMONIC TEST ................................................................................................................................. 19
3.5 DATA SAMPLING AND DFT ................................................................................................................ 19
3.6. SPECIAL CONSIDERATION IN THE TESTS ............................................................................................. 22
CHAPTER 4 TEST RESULTS AND ANALYSIS................................................................................... 23
4.1 LINEARITY .......................................................................................................................................... 23
4.2 SUPERPOSITION ................................................................................................................................... 24
4.3 TEMPERATURE VARIATIONS ............................................................................................................... 25
4.4 SHORT CIRCUIT TEST RESULTS .......................................................................................................... 26
4.4.1 10 KVA distribution transformer ............................................................................................... 26
4.4.2 2 KVA distribution transformer ................................................................................................. 32
4.5 TEST RESULTS ANALYSIS .................................................................................................................... 38
CHAPTER 5 CONCLUSIONS AND RECOMMENDATION ............................................................. 41
5.1 CONCLUSIONS ................................................................................................................................ 41
5.2 RECOMMENDATIONS FOR FUTURE WORK ............................................................................................ 42
REFERENCE: ............................................................................................................................................ 43
APPENDIX A 10 KVA DISTRIBUTION XFMR SHORT CIRCUIT TEST RESULTS .................... 46
APPENDIX B 2 KVA DISTRIBUTION TRANSFORMER SHORT CIRCUIT TEST RESULTS ... 50
APPENDIX C HARMONIC GROUP TEST RESULTS ........................................................................ 54
iii
APPENDIX D MATLAB PROGRAM FOR ANALYSIS OF 2 KVA TRANSFORMER SHORT
CIRCUIT TEST RESULTS ...................................................................................................................... 55
APPENDIX E MATLAB PROGRAM FOR ANALYSIS OF 10 KVA TRANSFORMER SHORT
CIRCUIT TEST RESULTS ...................................................................................................................... 60
APPENDIX F INSTRUCTIONS FOR DOING SHORT CIRCUIT TEST MANUALLY .................. 66
APPENDIX G LABORATORY EQUIPMENT AND COMPUTER RESOURCES ........................... 67
iv
LIST OF FIGURES AND TABLES
FIGURE 2-1 CORE AND SHELL FORMS WITH WINDINGS ................................................................................... 4
FIGURE 2-2 SIMPLIFIED SINGLE-PHASE TRANSFORMER MODEL ..................................................................... 5
FIGURE 2-3 FOUR WINDING CORE-SECTION WITH MAIN LEAKAGE PATHS SHOWN ........................................ 5
FIGURE 2-4 10KVA, AMORPHOUS STEEL CORE SINGLE-PHASE DISTRIBUTION TRANSFORMER ........................ 6
FIGURE 2-5 WINDING EDDY-CURRENT INDUCED BY MAGNETIC FLUX IN THE WINDING CONDUCTORS ............. 8
FIGURE 3-1. LABORATORY SETUP FOR SHORT-CIRCUIT TESTS ON 2 KVA DISTRIBUTION XFMR................ 17
FIGURE 3-2. LAB SETUP FOR SHORT-CIRCUIT TESTS OF THE 10 KVA DISTRIBUTION XFMR....................... 18
FIGURE 3-3. LINE SPECTRUM......................................................................................................................... 21
TABLE 3-1TIME STEP VALUES AND CORRESPONDING DFT FREQUENCY SPACINGS FOR DIFFERENT NUMBERS
OF POINTS TRANSFORMED. ................................................................................................................... 20
FIGURE 4-1 TEMPERATURE EFFECT ON THE WINDING RESISTANCE ............................................................... 25
FIGURE 4-2 SHORT-CIRCUIT TEST RESULTS: R.X VS. FREQUENCY (10KVA TRANSFORMER) ...................... 27
FIGURE 4-3 2ND FIT FOR RAC (FH/F1)2 FROM 60 HZ TO 2940 HZ (25 POINTS)............................................... 28
FIGURE 4-4 OPTIMAL FIT FOR 10 KVA RAC FROM 60 - 2940 HZ ( ALL THE 25 POINTS) ............................... 29
FIGURE 4-5 TOTAL FIT ERROR WHILE TRANSITION POINT MOVES. (SQUARE/NON-SQUARE) .......................... 30
FIGURE 4-6 TOTAL FIT ERROR WHILE TRANSITION POINT MOVES (BOTH SECTIONS ARE OPTIMAL FIT) .......... 31
FIGURE 4-7 2KVA XFMR AC WINDING RESISTANCE (AUTOMATIC TEST RESULTS).................................... 33
FIGURE 4-8 ONE SECTION FIT FOR 2KVA XFMR AC WINDING RESISTANCE DATA .................................... 34
FIGURE 4-9 ONE SECTION OPTIMAL FIT FOR 2 KVA XFMR RAC (60-1680 HZ)........................................... 35
FIGURE 4-10 THE TOTAL FITTING ERROR WHILE THE TRANSITION POINTS BETWEEN 2ND ORDER FIT AND
OPTIMAL FIT MOVES ............................................................................................................................. 36
FIGURE 4-11 THE TOTAL FITTING ERROR WHILE THE TRANSITION POINTS BETWEEN TWO OPTIMAL FIT
REGIMES MOVES .................................................................................................................................. 37
TABLE 4-1 LINEARITY CHECK ON 10 KVA TRANSFORMER ........................................................................... 23
TABLE 4-2 SUPERPOSITION CHECK RESULTS ................................................................................................ 24
TABLE 4-3 MEASURED AC WINDING RESISTANCE AND REACTANCE AT DIFFERENT FREQUENCIES. .............. 26
TABLE 4-4 FITTING METHODS COMPARISON FOR 10 KVA TRANSFORMER DATA .......................................... 32
TABLE 4-5 MEASURED 2 KVA TRANSFORMER AC WINDING RESISTANCE ................................................... 32
TABLE 4-6 FITTING METHODS COMPARISON FOR 2 KVA TRANSFORMER DATA............................................ 38
TABLE A-1 10 KVA DISTRIBUTION TRANSFORMER TEST NO.1 ................................................................... 46
TABLE A-2 10 KVA DISTRIBUTION TRANSFORMER TEST NO.2.................................................................... 47
TABLE A-3 10 KVA DISTRIBUTION TRANSFORMER TEST NO.3.................................................................... 48
TABLE A-4 10 KVA DISTRIBUTION TRANSFORMER RDC TEST RESULTS ...................................................... 49
TABLE B-1 2 KVA MANUAL SHORT CIRCUIT TEST RESULTS....................................................................... 50
TABLE B-2 2 KVA AUTOMATIC SHORT CIRCUIT TEST RESULTS [18] .......................................................... 51
TABLE B-3 2 KVA DC VALUE TEST RESULTS ............................................................................................. 53
TABLE C-1 2 KVA DISTRIBUTION TRANSFORMER HARMONIC GROUP TEST RESULTS 1.............................. 54
TABLE C-2 KVA DISTRIBUTION TRANSFORMER HARMONIC GROUP TEST RESULTS 2................................. 54
TABLE C-3 DFT ACCURACY CHECK (10 KVA TRANSFORMER)................................................................... 54
v
________________________________________________
CHAPTER 1
INTRODUCTION
________________________________________________
With the ever-increasing use of solid state electronics in electrical load devices,
such as switching power supplies, variable-speed drives and many types of office
equipment [6], the power system network is being subjected to higher levels of harmonic
currents. One result of this trend is excessive internal heating in power distribution
transformers that are loaded with harmonic-rich current.
The transformer manufacturers have improved their design in response to these
heating problems. Design changes include enlarging the primary winding to withstand the
inherent triplen harmonic circulating currents, doubling the secondary neutral conductor
to carry the triplen1 harmonic currents, designing the magnetic core with a lower normal
flux density by using higher grades of iron, and using smaller, insulated secondary
conductors wired in parallel and transposed to reduce the heating from the skin effect and
associated AC resistance.
Several methods of estimating the harmonic load content are available. CrestFactor and Percent Total Harmonic Distortion (%THD) are the two common methods.
1
Triplen harmonics are created by non-linear loads. They flow in the neutral conductor and windings of the
power transformer. They are odd harmonics devisable by three, including the 3rd, 9th, 15th, and 21st.
1
The third method “K-Factor” can be used to estimate the additional heat created by nonsinusoidal loads
The crest factor is a measure of the peak value of the waveform compared to the
true RMS value
Crest − Factor =
Peak Magnitude of the Current Waveform
True RMS of the current
(1.1)
The %THD is a ratio of the root-mean-square (RMS) value of the harmonic
current to the RMS value of the fundamental.
∞
%THD =
∑ (I
h=2
h
)2
I1
(1.2)
It is a measure of the additional harmonic current contribution to the total RMS
current.
Both of the above methods are limited because frequency characteristics of the
transformer are not considered.
The third method, K-factor, is defined as the sum of the squares of the per unit
harmonic current times the harmonic number squared:
∞
K = ∑ ( I h ( pu ) ) 2 h 2
h =1
(1.3)
where Ih(pu) is the harmonic current expressed in per unit based upon the
magnitude of the fundamental current and h is the harmonic number.
2
K-factor was introduced in UL standards 1561 [14] and 1562 [15] for rating
transformers based on their capability to handle load currents with significant harmonic
content.
Field application of K-factor requires knowledge of the fundamental and
harmonic load current magnitudes expected. Several manufacturers have utilized this
standard to market transformers that are specifically designed to carry the additional
harmonic currents.
This thesis is aimed at investigating the concept of K-factor and the relationship
between K-factor, derating, and the winding eddy-current loss of harmonic currents.
Chapter 2 presents the structure of the transformer under study, the K-factor theory and
existing work. Chapter 3 documents the test procedure and data processing methods
developed for determining the winding eddy-current loss, AC winding resistance and Kfactor. Chapter 4 compares the results of the measurements with the ideal results of the
K-factor theory and explains the difference. Chapter 5 provides conclusions and
recommendations for further research.
3
________________________________________________
CHAPTER 2
INTRODUCTION TO TEST TRANSFORMER AND K-FACTOR
________________________________________________
This chapter includes a general discussion of the single-phase transformer, a
description of the test transformers and the definition of the K-factor.
2.1 SINGLE PHASE TRANSFORMER MODEL.
There are two basic core designs for single-phase transformer: core form and shell
form.
φ
a) Core
φ/2
b) Shell
Figure 2-1 Core and Shell forms with Windings
Due to insulation requirements, the low voltage (LV) winding normally appears
closest to the core, while the high voltage (HV) winding appears outside. The windings
are usually referred to as primary and secondary winding(s) as denoted by the P and S. In
the shell form, the flux generated in the core by the windings splits equally in both "legs"
of the core. Winding configurations may vary with core design and include concentric
4
windings, pancake windings and assemblies on separate legs. A commonly used
equivalent circuit for a single-phase model is shown below:
P2
Lp
Ls
Rp
Rs
S2
Lc
Rc
S1
P1
Figure 2- 2 Simplified Single-Phase Transformer Model
This model is sufficient to model the short circuit behavior of a single-phase
transformer. It includes the winding resistance and leakage as well as the core losses so it
is widely used for all core and winding configurations of the single-phase two-winding
variety.
2.2 THE TEST TRANSFORMERS
There are two transformers selected for this project.
φ/2
φ/2
φ
Figure 2- 3 Four-Winding Core-Section with Main Leakage Paths Shown
The first one is a four-winding shell-type single-phase variety. A cross-section
view of this transformer is shown in Figure 2-3 [20].
This dry-type 2-kVA transformer can be connected 480-240V or 240-120V
depending on a series or parallel connection of the windings. Amperage rating for 120V
5
winding connection is 8.33A while it is 4.17A for the 240 V winding connection. In this
particular design, the high voltage windings are nearest to the core while the low voltage
winding are next to the high voltage windings.
The second one is a 10kVA, amorphous steel core-type single-phase polemounted distribution transformer, shown in Figure 2.4 [19].
This transformer is rated 7200-120/240-V, 10 kVA, and has an amorphous steel
core. This transformer consists of two low-voltage and two high-voltage windings
Tank
High Voltage
Low Voltage winding
Low Voltage winding
High Voltage
High Voltage
Low Voltage winding
Amorphous
Low Voltage winding
High Voltage
Oil
Core type
Figure 2- 4 10kVA, amorphous steel core single-phase distribution transformer
which are concentrically wound about the magnetic core. The low-voltage (secondary)
winding is placed closest to the core, with the high-voltage (primary) winding is outside.
The two high-voltage windings of the transformer are permanently connected in series. A
center tap in secondary winding can be used to provide different output voltages.
The core of the test transformer is made of wound amorphous steel ribbons and
has a core-type structure. Amorphous steel is made by rapidly cooling the metal at a rate
of 106 K/s. Thinner gauge steel, lower electrical conductivity, and a disorderly crystalline
6
structure are characteristics that separate amorphous from silicon steel. Compared to a
typical silicon steel core, an amorphous core offers an impressive reduction in average
core losses of up to 60-70% [19]. The reduction in average power losses due to hysteresis
can be attributed to the disorderly crystalline structure. The reduction in eddy current
losses is due to the thinner laminations and lower electrical conductivity.
2.3 TRANSFORMER LOSSES AND THE AC WINDING RESISTANCE
In ANSI/IEEE C57.110-1986 [1], transformer losses are categorized as: no-load
loss (excitation loss); load loss (impedance loss); and total loss (the sum of no-load loss
and load loss). Load loss is subdivided into I2R loss and “stray loss.” [1].
Ptotal = Pno-load + Pload
= Pno-load + (I2R + Pstray)
(2.1)
where Ptotal is the total loss, Pno-load is the no-load loss, Pload is the load loss and the
Pstray is the stray loss
“Stray Loss” is the loss caused by stray electromagnetic flux in the windings,
core, core clamps, magnetic shields, enclosure or tank walls, etc. Thus, the stray loss can
be subdivided into winding stray loss and stray loss in components other than the
windings (POSL).
The winding conductor strand eddy-current loss is caused by the time variation of
the leakage flux through the winding conductors [21], as shown in Figure 2-5. The other
7
stray loss is caused by the same mechanism within the tank wall, core clamps, etc.
Magnetic flux in
the winding
Magnetic flux
in the core
Core
Winding
Winding eddycurrent
Figure 2-5 Winding eddy-current induced by magnetic flux in the winding conductors
The total load loss can be stated as:
PLoad = I2RDC + PEC + POSL
(2.2)
where PEC is the winding eddy-current loss and POSL is the other stray loss.
The AC winding resistance RAC is defined as
RAC = Pload/I2
(2.3)
According to [1], all of the stray loss is assumed to be winding eddy current loss
and winding eddy-current loss for sinusoidal currents is approximately proportional to the
square of the frequency. The total load loss (copper loss) can be stated as
Pload = I2RDC + PEC = I2RDC + I2REC-R(fh/f1)2
(2.4)
where REC-R is the equivalent resistance corresponding to the eddy-current loss.
So the AC winding resistance RAC can be defined as
RAC = Pload/I2 = RDC + REC(fh/f1)2
(2.5)
By measuring the copper loss and the rms current, RAC can be measured.
8
2.4 K-FACTOR
UL standards 1561 [14] and 1562 [15] introduced a term called the K-factor for
rating transformers based on their capability to handle load currents with significant
harmonic content. This method is based on the ANSI/IEEE C57.110-1986 standard,
Recommended Practice for Establishing Transformer Capability When Supplying
Nonsinusoidal Load Currents [1].
The K-factor is an estimate of the ratio of: (a) the heating in a transformer due to
winding eddy currents when it is loaded with a given nonsinusoidal current to (b) the
winding eddy-current heating caused by a sinusoidal current at the rated line frequency
which has the same RMS value as the nonsinusoidal current. For example, if the current
in a transformer winding is 100 A, and this current has a K-factor of 10, then the eddy
current losses in that winding will be approximately 10 times what they would be for a
100 A sinusoidal current at the rated line frequency.
Although the K-factor formula was defined for transformer currents, K-factors of
individual load currents are sometimes computed. This practice can be misleading
because, in general, K-factors measured at transformers are significantly lower than the
relatively high K-factors commonly measured at the input of individual electronic
devices. The reduction is primarily due to other sinusoidal load currents, power system
impedance and the essentially random phase angles of the harmonic currents produced by
various loads.
The AC loss in a transformer winding is mainly due to the sum of the I2R losses
produced by the fundamental and harmonic components of the current, recognizing that
for each component, R depends on the frequency of that component. For lower-order
9
harmonics, the frequency dependence of the winding resistance is primarily due to the
proximity effect, a phenomenon that occurs in coils because the magnetic field
surrounding each conductor in a coil depends on the fields produced by other conductors.
The proximity effect produces greater losses than those predicted by the skin effect,
which is dominant at higher frequencies [2].
The K-factor formula does not account for the core eddy current losses and other
losses that occur in transformer cores. Core losses due to harmonics depend primarily on
the voltage distortion across the transformer windings. The voltage distortion appearing
across the windings of a transformer carrying harmonic currents depends on the
impedance of the transformer, the impedance of the system feeding the transformer, and
the voltage distortion of that system. Although K-rated transformers are usually
constructed to withstand more voltage distortion than other transformers, this capability
cannot be directly determined from K ratings [2].
The K-factor formula is based on the assumption that the winding eddy current
loss produced by each harmonic component of a nonsinusoidal current is proportional to
the square of the harmonic order as well as being proportional to the square of the
magnitude of the harmonic component. UL defines K-factor as follows: [1]
(1) “K-FACTOR – A rating optionally applied to a transformer indicating its
suitability for use with loads that draw nonsinusoidal currents.”
(2) “The K-factor equals
∞
K = ∑ ( I h ( pu ) ) 2 h 2
h =1
(2.6)
where Ih(pu) is the rms current at harmonic “h” (per unit of rated rms load
current) and h is the harmonic order.”
10
(3) “K-factor rated transformers have not been evaluated for use with harmonic
loads where the rms current of any singular harmonic greater than the tenth
harmonic is greater than 1/h of the fundamental rms current.”
K-factor definition is based on the following two assumptions:
(a) Winding eddy-current loss (PEC) is proportional to the square of the load
current and the square of the frequency.
(b) Superposition of eddy current losses will apply, which will permit the direct
addition of eddy losses due to the various harmonics.
According to [1], suppose the eddy current loss under rated conditions is
PEC − R = REC − R ⋅ I R2
(2.7)
where PEC-R is the eddy current loss under rated conditions and
REC-R = RAC-R - RDC
(2.8)
where RAC-R is the AC Winding resistance at rated frequency (60 Hz).
From the first assumption, the eddy-current loss due to harmonic component is
PEC ( h ) = REC − R ⋅ (
fh 2 2
I
) I h = REC − R ⋅ h 2 I h2 = PEC − R ( h ) 2 h 2
f1
IR
(2.9)
where PEC(h) is the eddy current loss due to harmonic current of order h, IR is the
rated load current, fh is the harmonic frequency at order h and f1 is the fundamental
frequency.
According to the second assumption, the eddy-current loss due to the total
nonsinusoidal load current is
PEC =
h = hmax
h = hmax
h =1
h =1
∑ PEC ( h) = PEC − R
Ih
∑ (I
) 2 h 2 = PEC − R ⋅ K
(2.10)
R
From (2.10), it is clear where the definition of K-factor comes from.
11
In subsequent chapters, it is found that the K-factor assumption is too restrictive.
So I suggest that the assumption that the winding eddy-current loss (PEC) is proportional
to the square of the frequency should be relaxed if it is made proportional to an arbitrarily
power ε, then the formula becomes
PEC ( h ) = REC − R ⋅ h ε I h2 = REC ( h ) ⋅ I h2 = PEC − R (
Ih 2 ε
) h
IR
(2.11)
where
REC ( h ) = REC − R ⋅ h ε = R AC ( h ) − RDC
(2.12)
RAC(h) is the AC winding resistance at harmonic order h and ε is an exponent other
than 2
Then an alternative K factor definition Kε could be defined as
∞
K ε = ∑ ( I h ( pu ) ) 2 h ε
(2.13)
h =1
2.5 HARMONIC LOSS FACTOR
The Harmonic Loss factor, as defined by IEEE Std C57.110-1998 [17], is given below
h = hmax
FHL =
∑
Ih h
h =1
h = hmax
∑
h =1
2
Ih
2
h = hmax
2
=
Ih
∑ [I
]2 h 2
h =1
1
h = hmax
∑
h =1
I
[ h ]2
I1
(2.14)
where I1 is the fundamental harmonic current.
From (2.7) – (2.9), the K-factor was derived based on the assumption that the
measured application currents are taken at rated currents of the transformer. This is
seldom encountered in the field. This is where the FHL comes in handy because it can be
12
calculated in terms of the actual rms values of the harmonic currents and the quantity Ih/I1
may be directly read on a meter.
The relationship between K-factor and FHL is
 h = hmax 2 
 ∑ Ih 
K − factor =  h =12  FHL
 IR 




(2.15)
An important improvement the Harmonic Loss factor made is separating other
stray loss (POSL) from winding stray loss (PEC)
Pload = I2RDC + PEC + POSL
(2.2)
According to [17], a Harmonic Loss Factor for other stray losses is defined as
h = hmax
FHL− STR =
∑
h =1
h = hmax
∑
h =1
h = hmax
2
I h h 0.8
Ih
2
=
Ih
∑ [I
]2 h 0.8
h =1
1
h = hmax
∑
h =1
I
[ h ]2
I1
(2.16)
based on the assumption that the other stray losses are proportional to the square of the
load current and the harmonic frequency to the 0.8 power.
Because the other stray losses can not be ignored, in [17], an assumption is made
to estimate the portion of the other stray losses.
a) 67% of the total stray loss at rated frequency is assumed to be winding eddy
losses for dry-type transformers and 33% of the total stray loss at rated
frequency is assumed to be the other stray loss.
b) 33% of the total stay loss at rated frequency is assumed to be winding eddy
losses for oil-filled transformers and 67% of the total stray loss at rated
frequency is assumed to be the other stray loss
13
________________________________________________
CHAPTER 3
LABORATORY TESTS
________________________________________________
This chapter presents the test procedure and data processing methods developed
for determining the winding eddy-current loss, AC winding resistance and K-factor.
Detailed test procedure can be found in Appendix F and [18].
3.1 MEASUREMENT CONSIDERATION
The determination of parameters for transformer equivalent circuit models has
typically been based on meter measurements. Voltages and currents are measured with
RMS meters, and power is measured with an average reading wattmeter. Significant
measurement errors are possible for harmonic study. Only a “true RMS” meter can take
measurements which correctly include the effect of all harmonics within the meter’s
bandwidth. However the information about the harmonic content is lost.
In order to improve the accuracy of the measurement results, a digital storage
oscilloscope was used to record the waveforms of the voltage and current. A voltage
probe of ratio 1:100 was used. Hall effect current probe with a 1:1 ratio was used to
obtain current waveforms. The digital scope could save the sampled data on floppy
diskette. This allowed waveform data to be transported to a PC for analysis using the
VuPoint software, which was capable of many signal- processing operations.
14
3.2 TEST DEVICES
3.2.1 Power source
The power source used, AMX-3120, is a product of Pacific Power Source
Corporation (PPSC). It is a high-performance AC power conversion equipment. For our
test purposes, 3-phase voltage with programmable harmonic contents can be generated
from this device. It is configured with an interchangeable digital controller called the
Universal Programmable Controller (UPC). This programmable controller not only
allows control of voltage and frequency, but also allows the user to simulate virtually any
transient (including sub-cycle waveform disturbance). Main features of the power source
are: [11]
•
Capable of 1, 2 or 3 phase operation
•
Master Slave arrangement to obtain precise control
•
Standard output range is 0-135 VAC(1-n )
•
Phase separation fixed @ 180° for 2-phase operation
•
Phase separation is programmable for 3-phase operation. Default is 120°
•
Output power rating is 12 kVA
•
Output can be direct coupled or transformer coupled. Voltage ratios of up to
2.5:1 are available
•
Output Bandwidth is 20 – 5000 Hz
•
Sophisticated programmable controller (UPC32)
•
GPIB or Serial I/O communication capability
•
External Sense input – This is required for precise control of the output
voltage of the power source. The line drops are taken care of by using external
sense inputs.
15
3.2.2 UPC-32
UPC –32 is a programmable controller designed to directly plug into Pacific
Power Source Corporation’s AMX/ASX Series Power source. It is a highly versatile
single, two or three phase signal generator and can be remotely controlled from a PC
either through a GPIB interface or through a serial interface. Main features of the UPC32 are: [12]
•
Operations in 4 modes:
1. Manual Operate: Control by user manually
2. Program Operate: Control by the program stored by user
3. Program Edit: Storing of the program by the user
4. Setup: To setup all the auxiliary functions of the source
•
Magnitude range: 0% - 99% of the fundamental voltage ( with a maximum output of
135 V) and a resolution of 0.1%
•
Phase Angle: 0° - 359.9°, resolution 0.1°
•
Calculation time: 45 sec + 10 sec for each non-zero magnitude of the harmonic
•
99 user programs that contain steady state and transient parameters can be stored
•
Harmonic content of voltage signal is programmable. Harmonic range is 2 through 51
•
Continuous Self Calibration (CSC) is used to maintain a constant output voltage at
the metering point based on the metered voltage at that point. Therefore, accurate
calibration of the metering functions is essential for CSC to operate accurately.
•
Control – Local/Remote. In remote control mode, the source can be either controlled
through GPIB or through serial communication.
16
3.2.3 Oscilloscope
The oscilloscope used is a Nicolet Pro20, a digital oscilloscope from Nicolet
Technologies Inc. It is an oscilloscope with 4 channels, each having
•
1MegaSamples/s of maximum sample rate
•
12 bit vertical resolution and
•
Differential type amplifier
The Nicolet Pro20 can be configured with a wide variety of input channels and
can simultaneously collect from low and high-speed channels.
3.3 SHORT-CIRCUIT TESTS
3.3.1
2 kVA Distribution Transformer
The laboratory setup used to perform the short-circuit test of the 2 kVA dry-type
Transformer is shown below:
Master
AMX 3120 AC
Power Source
transformer
H4
X1
Slave
H3
UPC-32
X2
H2
X3
H1
X4
Current Amplifier
Nicolet Pro 20
Oscilloscope
Figure 3-1. Laboratory Setup for Short-circuit Tests on 2 kVA Distribution XFMR
17
This transformer is a 2 kVA single phase, dry type, 4winding 120/240 Volt
general purpose transformer. It is excited at the high-voltage winding (H4-H3) with lowvoltage winding (X1-X2) short circuited.
3.3.2 10 kVA Distribution Transformer
The laboratory setup used to perform the short-circuit test of the 10 kVA
Distribution Transformer is shown below:
AMX 3120 AC
Power Source
transformer
X3
H1
X2
H2
X1
Current Amplifier
Nicolet Pro 20
Oscilloscope
Figure 3-2. Lab Setup for Short-circuit Tests of the 10 kVA Distribution XFMR
This test transformer is a single-phase pole-mounted distribution transformer. It is
rated 7200-120/240-V, 10-kVA with an amorphous steel core.
3.3.3 Data Recording
The sampled waveform data of voltage and current are saved to floppy diskette.
This allowed waveform data to be transported to a PC for analysis using the VuPoint
software, which was capable of many signal-processing operations.
18
The average power is calculated from v(t) and i(t):
T
1
P = ∫ v(t )i (t )dt
T 0
which can be acquired using the statistic function Mean in the Vupoint program.
The apparent power is
S = VRMS*IRMS
the reactive power is
Q = S 2 − P2
so the equivalent winding resistance and reactance are
Rsc =
P
I
2
RMS
X sc =
Q
2
I RMS
3.4 HARMONIC TEST
The laboratory setup used to perform the harmonic test is the same as short-circuit
tests above. The only difference is that the voltage applied to the transformer in this test
consists of a group of harmonics at different frequencies. It can be implemented by
programming the UPC-32 in the power source.
3.5 DATA SAMPLING AND DFT
The harmonic test requires FFTs (Fast Fourier Transform) of the current
waveform data to obtain frequency spectra of DFTS (Discrete Fourier Transforms).
Software called VuPoint was used to perform FFTs on laboratory measurements. To
19
obtain a discrete spectrum or “line spectrum” for periodic waveforms, the waveform data
must meet the following requirements before transform:
•
The waveform data must cover the range of an integral number of cycles.
•
No windowing can be used.
•
The number of data points (NPTS) must equal to 2N ( N is a positive integer)
Some possible combinations are listed in the following table:
∆t (µs)
91.533
100.00
122.07
244.14
NPTS
8192
2048
4096
8192
2048
4096
8192
∆f (Hz)
1.333
4.883
2.441
1.220
4.0
2.0
1.0
No. of 60 Hz Cycles
45.00
12.29
24.57
49.15
15.00
30.00
60.00
Total Time(sec)
0.75
0.2048
0.4096
0.8192
0.25
0.5
1.0
1024
2048
4096
4.0
2.0
1.0
15.00
30.00
60.00
0.25
0.5
1.0
Table 3-1Time step values and corresponding DFT frequency spacings for different numbers of
points transformed. [13]
The relationship between the time step, numbers of points and DFT frequency
spacing is:
∆f =
1
∆t ⋅ NPTS
where
∆f is the DFT frequency spacing
∆t is the time step
NPTS is the numbers of points
These conditions require a sampling interval that is unavailable to the Nicolet
Pro 20 oscilloscope used. The sampling rate of Nicolet Pro 20 oscilloscope is 1 µS
and the available time setting is
20
∆T = 1, 2, 5 µS;
10, 20, 50 µS;
100, 200, 500 µS
…
Figure 3-3. Line spectrum
For the test purpose, a sweep length of 8192 points and time setting of 200 µS
were chosen. The total length of the waveform is 1.6384 second. Then the data was cut
off and re-sampled in VuPoint to meet the FFT requirements.
VuPoint provides several different windowing possibilities: none (rectangular),
cosine-tapered rectangular, Bartlett, Hanning and Parzen. If the data being transformed
21
was an integer number of waveform cycles, a rectangular window with no tapering was
sufficient.
For the processing of harmonic test data, the last row in Table 3.1 was used. In
VuPoint, the data was first interpolated to a sampling time of 244.14 µS, then cut off to
only 1 second long. The FFT result is a perfect discrete spectrum (line spectrum). (Figure
3.3)
3.6. SPECIAL CONSIDERATION IN THE TESTS
The AM 503 current probe used requires a degauss function before
measurements. It removes any residual magnetism from the attached current probe and it
initiates an operation to remove any undesired DC offsets from probe circuitry. This
operation is recommend each time a new measurement is started or any setting on the
probe is changed.
The short-circuit tests for the 2 kVA transformer were done both manually and
automatically. The manual test was done continuously. At each frequency, the test was
repeated three times and the average value was recorded. The automatic test procedure is
discussed thoroughly in [18].
The short-circuit tests for the 10 kVA transformer was done only manually
because of the limitation of the voltage output of the Power source. At each frequency
point, only one measurement was made. But the whole test sequence was repeated three
times in different order. The first and the third test were done from high frequency to low
frequency while the second was done from low frequency to high frequency. The average
values were used for analysis.
22
________________________________________________
CHAPTER 4
TEST RESULTS AND ANALYSIS
________________________________________________
This chapter presents the test results obtained from the test setup developed in
chapter 3. Detailed analysis of the test results is provided. The test results can be found in
Appendix A through Appendix C.
4.1 LINEARITY
Because the voltage limitation of the power source, when doing the short-circuit
test, the rated current of the test transformer may not be reached at high frequencies.
Because only resistance is our concern, an alternative way is to do the short-circuit test at
lower voltage level if the linearity of the resistance holds. A check on the linearity of
resistance is needed. The test results of the 10 kVA transformer at 60 Hz is shown in
Table 4.1
I rms (A) V rms (V)
0.04175
4.075
0.06375
6.064
0.0847
8.049
0.10565
10.041
0.1264
12.09
0.053275
5.06
0.074525
7.05
0.094525
9.05
P(W)
0.125145
0.289843
0.49408
0.767283
1.120973
0.196378
0.386867
0.625613
R(ohm)
71.79605
71.31854
68.87006
68.74113
70.16183
69.19024
69.65587
70.01841
X(ohm)
66.12127
62.94266
65.47919
65.63003
65.00767
65.06689
64.00824
65.29875
Table 4-1 Linearity check on 10 kVA transformer
Because the error of R is between ± 2.6%, the linearity of the resistance holds
very well.
23
4.2 SUPERPOSITION
As described in Chapter 2, the UL definition of the K-factor is based on several
assumptions. One of them is that superposition of eddy current losses will apply, which
will permit the direct addition of eddy losses due to the various harmonic. This
assumption could be checked by a test described below:
First, a group of harmonics is applied to the transformer together. The voltage and
current waveforms are recorded. The load loss is measured as Pgroup. An FFT is then used
on the voltage waveform to get the amplitude and the frequency of the individual
harmonics in the group. Then individual harmonic in the group is applied to the
transformer one by one at the same amplitude and frequency, the load losses are recorded
as Pindividual. If the sum of the Pindividual is equal to Pgroup, the superposition assumption is
correct.
The test results of 2 kVA transformer is presented in Table 4.2
Harmonic Groups (Voltage: 75% 3rd; 50% 5th; 25% 7th)
IRMS=8.372 A
Vrms = 5.9295 V
P = 46.02 W
FFT Analysis results
Harmonic Order (h)
1
3
5
7
Frequency (Hz)
60.0
180.0
300.0
420.0
I raw (mv)
182.9
124.6
70.73
29.7
∑
Individual Harmonic Test Results
Harmonic Order(h) Frequency (Hz)
1
60.0
3
180.0
5
300.0
7
420.0
I raw (mv)
184.0
124.16
72.48
32.13
∑
Ih (A)
6.466
4.405
2.501
1.05
2
=8.281
A
Error = 1.1%
Ih
77.2%
67 %
38.8%
17.2%
Ih(A)
6.505
4.386
2.566
1.137
2
Error = 0.5%
I h =8.332 A
P (w)
27.466
12.77
4.42
0.912
Total =45.568
Table 4-2 Superposition Check Results
24
The error is (45.568 – 46.02)/46.02*100% = 0.98% which is small enough to
verify the superposition assumption is correct.
4.3 TEMPERATURE VARIATIONS
Temperature effect on the Winding Resistance
450
High to low frequency
Low to High frequency
400
350
300
c
a
R
250
200
150
100
500
1000
1500
f (Hz)
2000
2500
Figure 4-1 Temperature effect on the winding Resistance
Because when short-circuit tests are made continuously, heat may accumulate in the
transformer and the temperature in the transformer winding conductor may rise which
will cause the increase of the resistance. To test how much the effect will be, two set of
short-circuit tests were performed on the same 10 kVA transformer. The first set did the
test from high frequency (2940 Hz) to low frequency (60 Hz) while the second test set
was done from low frequency (60 Hz) to high frequency (2940 Hz). The results are
25
plotted in Figure 4.1. There is a small difference between the two sets of test results. The
difference is small enough to be ignored.
4.4 SHORT CIRCUIT TEST RESULTS
4.4.1 10 kVA Distribution transformer
Harmonic
Order
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
frequency
R (Ohm)
X (Ohm)
60
180
300
420
540
660
780
900
1020
1140
1260
1380
1500
1620
1740
1860
1980
2100
2220
2340
2460
2580
2700
2820
2940
69.14738
77.88052
86.78983
102.1515
118.3543
137.0394
153.8812
175.5246
192.0341
212.4019
228.5787
250.3567
262.9465
284.549
295.4238
310.6725
329.7242
344.7513
361.3698
377.7148
395.8492
413.6822
432.4653
450.7207
471.7872
64.78765
207.9448
343.4697
476.3446
608.7855
732.4207
858.2045
979.6188
1098.353
1217.341
1336.227
1450.54
1569.469
1687.727
1805.996
1927.585
2046.201
2166.064
2287.776
2414.01
2535.294
2661.516
2789.146
2912.255
3050.468
Table 4-3 Measured AC winding resistance and reactance at different frequencies.
The test was repeated for three times. Please see Appendix A for original data. In
Table 4.3, the quantities are the average values of these measurements.
26
The resistance and the reactance are plotted in Figure 4-2.
The first and the third test were done from high frequency to low frequency while
the second test was from low frequency to high frequency. Although temperature effect
has been considered to be small enough, this kind of test scheme can reduce possible
error resulted from test sequence.
Short-Circuit Test:R,X versus Frequency (10 KVA Distribution XFMR)
3000
R
X
2500
2000
s
m
h
O
1500
1000
500
0
0
500
1000
1500
Frequency(Hz)
2000
2500
3000
Figure 4-2 Short-circuit Test Results: R.X vs. Frequency (10kVA Transformer)
4.4.1.1 2nd order fit for the AC Winding Resistance RAC (10 kVA, all points)
According to (2.7), a least square fit of a second order polynomial is used for all
the data points in Table 4.2. The original RAC curve and the fit curve are shown in Figure
4.3.
27
From Figure 4.3, it is very clear that the second order polynomial is not a good
choice for fitting the test data. Using this fit, the RAC would be:
RAC = 127.7 + 0.1605(fh/f1)2
(4.1)
The total fitting error is 150.5.
2nd order fit for Rac (fh/f1)2 from 60 Hz to 2940 Hz ( 25 points)
500
Test data
Fit data
450
400
350
300
c
a
R
250
200
150
100
50
0
0
500
1000
1500
f (Hz)
2000
2500
3000
Figure 4-3 2nd fit for Rac (fh/f1)2 from 60 Hz to 2940 Hz (25 points)
4.4.1.2 One Section Optimal fit for the AC Winding Resistance RAC ( 10 kVA)
From Figure 4.3, it is clear that the second order polynomial for the whole data set
is not a good match. Next step would be to use an optimal fit of a constant plus a 2nd term
with an unknown exponent. It would have the form of
RAC = RDC + REC(fh/f1)x
(4.2)
It turned out that the optimal exponent found is 1.03 and the total error is 20.32.
The original RAC curve and the fit curve are shown in Figure 4-4
28
RAC = 52.69 + 7.464(fh/f1)1.034
(4.3)
Optimal fit for 10 kVA Rac (fh/f1) expo from 60 Hz to 2940 Hz ( 25 points)
450
Test data
Fit data
400
350
300
250
c
a
R
200
150
100
50
0
0
500
1000
1500
f (Hz)
2000
2500
3000
Figure 4-4 Optimal fit for 10 kVA Rac from 60 - 2940 Hz ( all the 25 points)
4.4.1.3 Two-section fit for the AC Winding Resistance RAC ( 10 kVA)
According to [2], the K-factor formulas overestimate the high-frequency losses in
transformer winding because of the assumption that the eddy current losses are
proportional to the square of the frequency for all frequencies. In fact, for high enough
frequencies, winding eddy current losses in transformers are asymptotically proportional
to the square root of the frequency instead of the square of the frequency.
One improvement suggested from this explanation is to use a two-section fit for
the RAC test data. It is necessary to find where the transition between the 2nd order
polynomial and non-2nd order regimes occurs.
29
It would have the form:
RAC = RDC + Rco1(fh/f1)2
(fh ≤ transition frequency )
RAC = RDC + Rco2(fh/f1)x
(fh > transition frequency )
(4.4)
So the transition point is moved from the 3rd point to the 22nd point in the data to
find a best position (minimum fit error). The first part is the 2nd order polynomial fit
while the second part is a non-2nd order polynomial optimal fit.
Total Error (transition point moves from the 3rd - 22nd(300 Hz- 2580 Hz) 10 kVA
120
100
80
r
or
r
E
l
at
o
T
60
40
20
0
0
500
1000
1500
f (Hz)
2000
2500
Figure 4-5 Total fit error while transition point moves. (Square/non-square)
From Figure 4.5, it can be seen there is no optimal transition point found when the
2nd order polynomial/non-2nd order two-section pattern is used.
Naturally, further improvement is to use optimal fit for both sections. It would
have the form:
30
RAC = RDC + Rco1(fh/f1)x1
(fh ≤ transition frequency )
RAC = RDC + Rco2(fh/f1)x2
(fh > transition frequency )
(4.5)
In Figure 4.6, an optimal transition point is found on 1260 Hz (the 11th point in
the data serial).
The optimal fit for the first part is:
RAC = 63.95 + 3.168(fh/f1)1.304 (60 Hz ≤ fh < 1380 Hz)
(4.6)
The optimal fit for the second part is:
RAC = 245.9 + 5.621(fh/f1)1.118 (1380 Hz ≤ fh ≤ 2940 Hz)
(4.7)
Total Error (transition point moves from the 3rd - 22nd(300 Hz- 2580 Hz) 10 kVA
20
18
16
14
r
or
r
E
l
at
o
T
12
10
8
6
4
2
0
0
500
1000
1500
f (Hz)
2000
2500
Figure 4-6 Total fit error while transition point moves (both sections are optimal fit)
4.4.1.4 Summary of the fitting tests for the RAC (10 kVA)
The results from these different fitting methods for the 10 kVA transformer AC
winding resistance are summarized in Table below
31
Exponent
Fitting Method
Section 1
One section (total 25 points)
One Section (total 25points)
Two sections(first fixed at 2)
Two sections (both optimal)
Error
Section 2
2
Optimal found = 1.034
Best transition points not found
1.304
1.118
150.5
20.32
N/A
13.0
Table 4-4 Fitting methods comparison for 10 kVA Transformer data
4.4.2
2 kVA distribution transformer
The short-circuit tests for the 2 kVA distribution transformer were done both
automatically and manually. The AC winding resistance derived from the automated
tests is plotted in Figure 4.7.
Because the frequency step used in the automated tests is 30 Hz, only part of the
data set will be used for the following fitting experiment. (Table 4.5)
Harmonic orders
1
2
3
4
5
6
7
8
9
10
11
12
13
14
16
17
18
19
20
21
22
23
24
25
26
27
28
Frequency
60
120
180
240
300
360
420
480
540
600
660
720
780
840
960
1020
1080
1140
1200
1260
1320
1380
1440
1500
1560
1620
1680
Resistance
0.72
0.73
0.73
0.74
0.76
0.76
0.77
0.77
0.78
0.8
0.81
0.83
0.85
0.86
0.9
0.93
0.95
0.97
0.99
1.01
1.04
1.06
1.08
1.11
1.13
1.17
1.19
Table 4-5 Measured 2 kVA Transformer AC winding resistance
32
2 KVA XFMR AC Winding Resistance (automatic test results)
1.8
1.6
1.4
1.2
1
s
m
h
O
0.8
0.6
0.4
0.2
0
R
0
500
1000
1500
Frequency(Hz)
2000
2500
3000
Figure 4-7 2kVA XFMR AC Winding Resistance (automatic test results)
4.4.2.1 2nd Order Polynomial Fit for the AC Winding Resistance RAC (2 kVA
distribution XFMR)
First, a 2nd order polynomial fit is used for all the data points in Table 4.5.
The original RAC curve and the fit curve are shown in Figure 4.8.
33
One section 2nd order fit for 2 kVA XFMR (60 - 1680 Hz) ( 27 points)
1.2
Test data
Fit data
1
0.8
c
a
R
0.6
0.4
0.2
0
0
200
400
600
800
1000
f (Hz)
1200
1400
1600
Figure 4-8 One Section fit for 2kVA XFMR AC Winding Resistance data
The fit formula for RAC is:
RAC = 0.7387 + 0.0006(fh/f1)2
(4.8)
The total error of this fit is 0.0536
4.4.2.2 Optimal fit for the AC Winding Resistance RAC (2 kVA distribution XFMR)
From Figure 4.8, it can be seen that the 2nd order polynomial fit for the whole data
set is not a good match. Next step would be to use optimal fit for the whole length of
data.
It turned out that the optimal exponent found is 1.7087 and the total error is
0.0278. The original RAC curve and the fit curve are shown in Figure 4.9.
RAC = 0.7218 + 0.0016(fh/f1)1.709
(4.9)
34
One section optimal fit for 2 kVA XFMR Rac (60 - 1680) Hz ( 27 points)
1.2
Test data
Fit data
1
0.8
c
a
R
0.6
0.4
0.2
0
0
200
400
600
800
1000
f (Hz)
1200
1400
1600
Figure 4-9 One Section Optimal fit for 2 kVA XFMR RAC (60-1680 Hz)
4.4.2.3 Two-section fit for the AC Winding Resistance RAC ( 2 kVA)
According to [2], a two-section fit is used for the whole data set of RAC. The first
section will use a 2nd order polynomial fit while the second part uses a non-2nd order
optimal fit. The transition point is moved from the 3rd point to the 27th point to find the
best fit.
35
Total Error when the transition point moves from 3 - 27
0.06
0.05
0.04
r
or
r
E
l
at
o
T
0.03
0.02
0.01
0
5
10
15
Harmonic order
20
25
Figure 4-10 The total fitting error while the transition points between 2nd order fit and optimal fit
moves
The total error while the transition points moves is plotted in Figure 4.10. It can
be observed that the minimum error is found when the transition point is at 1080 Hz and
the minimum error is 0.0342.
The two-section fit curve is:
RAC = 0.7296 + 0.007(fh/f1)2
(60Hz ≤ fh < 1140 Hz)
(4.10)
RAC = 0.9554 + 0.0152(fh/f1)1.189 (1140Hz ≤ fh ≤ 1680 Hz)
The same process was repeated for a two-section fit which both sections use an
optimal fit.
36
The total error while the transition points moves is plotted in Figure 4.11. It can
be observed that the minimum error is found when the transition point is at 1560 Hz and
the minimum error is 0.0271.
Total Error when the transition point moves from 3 - 27
0.06
0.05
0.04
r
or
r
E
l
at
o
T
0.03
0.02
0.01
0
5
10
15
Harmonic order
20
25
Figure 4-11 The total fitting error while the transition points between two optimal fit regimes moves
The two-section fit curve is:
RAC = 0.7218 + 0.0016(fh/f1)1.706 (60Hz ≤ fh < 1560 Hz)
(4.11)
RAC = 1.1594 + 0.0106(fh/f1)1.531 (1560Hz ≤ fh ≤ 1680 Hz)
4.4.2.4 Summary of the fitting tests for the RAC ( 2 kVA)
The results from these different fitting methods for the 2 kVA transformer AC
winding resistance are summarized in Table 4.6.
37
Fitting Method
Section 1
Exponent
Section 2
Error
One section (total 27 points)
2
0.0536
One Section (total 27points)
Two sections(first fixed at 2)
Optimal exponent found = 1.709
2
1.189
0.0278
0.0342
Two sections (both optimal)
1.706
1.531
0.0271
Table 4-6 Fitting methods comparison for 2 kVA Transformer data
4.5 TEST RESULTS ANALYSIS
The K-factor is an estimate of the ratio of the heating in a transformer due to
winding eddy currents when it is loaded with a given nonsinusoidal current to the
winding eddy-current heating caused by a sinusoidal current at the rated line frequency
which has the same RMS value as the nonsinusoidal current. [2]
Transformers with K-factor ratings are constructed so that their winding eddy
current losses are very low for sinusoidal currents at the rated line frequency. This allows
them to have acceptable losses when they are fully loaded with non-sinusoidal currents
that have a K-factor less than or equal to the K-rating of the transformer.
As stated in chapter 2, the K-factor formula is based on the assumption that the
winding eddy current loss produced by each harmonic component of a nonsinusoidal
current is proportional to the square of the harmonic order as well as being proportional
to the square of the magnitude of the harmonic component. However, this assumption is
not always true which can be seen from the test results presented in this chapter.
For example, the best fit curve for the 10 kVA distribution transformer tested is
RAC = 63.9518 + 3.1681(fh/f1)1.304 (60 Hz ≤ fh < 1380 Hz)
(4.6)
RAC = 245.9214 + 5.621(fh/f1)1.118 (1380 Hz ≤ fh ≤ 2940 Hz)
(4.7)
38
So at lower frequencies the exponent of (fh/f)ε is about ε = 1.3037 and at higher
frequencies this exponent is even smaller.
A better approach can be obtained by relaxing the limitation in the definition of
the K-factor. The power of the harmonic order should not be limited to 2. The Kε
definition is more appropriate.
∞
K ε = ∑ ( I h ( pu ) ) 2 h ε
(2.13)
h =1
From (2.7)-(2.10), the K-factor calculated for this transformer is apparently
conservative in the sense of derating.
Part of the reason that the exponent ε is less than 2 is that in (2.2)
Pload = I2RDC + PEC + POSL
(4.2)
The other stray loss (POSL) was ignored when defining K-factor. So the actual
winding eddy-current loss is
PEC-A=(PEC + POSL)
(4.12)
Because POSL are proportional to the square of the load current while not
proportional to the square of the harmonic frequency, the total PEC-A is not proportional to
the square of the harmonic frequency.
Another weak point of the K-factor formula is that it overestimates the highfrequency losses in transformer windings. According to formulas in [16], for high enough
frequencies, winding eddy current losses in transformers are not proportional to the
square of the frequency. The geometry of the windings in a given transformer determines
when the transition between the square and the non-square regimes occurs.
As stated in Chapter 2, an important improvement the Harmonic Loss Factor
made is separating other stray loss (POSL) from winding stray loss (PEC)
39
Because the other stray losses can not be ignored, in [17], an assumption is made
to estimate the portion of the other stray losses.
a) 67% of the total stray loss is assumed to be winding eddy losses for dry-type
transformers and 33% of the total stray loss is assumed to be the other stray
loss.
b) 33% of the total stay loss is assumed to be winding eddy losses for oil-filled
transformers and 67% of the total stray loss is assumed to be the other stray
loss.
This assumption can be checked using an optimal search. Using the assumption
that the winding eddy-current loss vary with the square of the frequency and the other
stray loss vary with the frequency raised to the 0.8 power, the fit formula is:
RAC = RDC + β1h2 + β2h0.8
It is found these assumptions are not accurate for the tested transformer but it can
help explain the difference of the exponent of (fh/f)ε between 2 kVA dry-type transformer
and 10 kVA oil-filled transformer.
For the 2 kVA dry-type transformer, at low frequencies
RAC = 0.7218 + 0.0016(fh/f1)1.706 (60Hz ≤ fh < 1560 Hz)
(4.11)
For the 10 kVA oil-filled transformer, at low frequencies
RAC = 63.95 + 3.168(fh/f1)1.304 (60 Hz ≤ fh < 1380 Hz)
(4.6)
In the dry-type transformer, the winding eddy losses, which is assumed
proportional to the square of the frequency, takes a larger part in the total stray loss than
in the oil-filled transformer, the exponent of (fh/f)ε found is larger.
40
________________________________________________
CHAPTER 5
CONCLUSIONS AND RECOMMENDATION
_______________________________________________________
This chapter presents the conclusions drawn from this work. Topics including
closing comments regarding the lab tests, the K-factor concept and the Harmonic Loss
Factor (FHL). Recommendations for future work are also provided.
5.1 CONCLUSIONS
•
The K-factor does not apply to the two tested transformer and overestimates the
losses in transformer windings because the winding eddy current losses in
transformers tested are not proportional to the square of the frequency, instead, they
are proportional to a power of the frequency which is less than 2.
•
For the two transformers tested, the eddy-current loss is a function of frequency with
power less than 2 so an alternative definition of the K factor, Kε, in which the
exponent ε is less than 2 is better.
•
The Harmonic Loss Factor is a better approach for estimating transformer load loss.
Compared with the K-factor, the Harmonic Loss Factor is a function of the harmonic
current distribution and is independent of the relative magnitude while the K-factor is
dependent on both the magnitude and distribution of the harmonics. Harmonic Loss
Factor also has a separate definition for the other stray losses assuming that they are
41
proportional to the square of the load current magnitude and the harmonic frequency
to the 0.8 power.
5.2 RECOMMENDATIONS FOR FUTURE WORK
More laboratory tests on different transformers are needed. A detailed study of
transformer structure such as the geometry of the windings is necessary for further study.
A laboratory test method for separating winding eddy current losses from stray
losses in components other than windings are important in the future work.
42
Reference:
[1] "An American National Standard: IEEE Recommended Practice for Establishing
Transformer Capability When Supplying Nonsinusoidal Load Currents."
ANSI/IEEE C57.110-1986
[2] Bryce Hesterman, "Time-Domain K-Factor Computation Methods", 29th
International Power Conversion Conference, September 1994, pp.406-417
[3] Tom Shaughnessy, "Use Derating and K-Factor Calculation Carefully", Power
Quality Assurance, March/April 1994, pp.36-41.
[4] E.F.Fuchs, D.Yildirim, and W.M.Grady, "Measurement of Eddy-Current Loss
Coefficient PEC-R, Derating of Single-Phase Transformers, and Comparison with KFactor Approach", IEEE Trans on Power Delivery, Paper # 99WM104, accepted
for publication.
[5] D.Yildirim and E.F.Fuchs, “Measured Transformer Derating and Comparison with
Harmonic Loss Factor (FHL) Approach”, PE-084-PWRD-0-03-1999.
[6] Jerome M. Frank, “Origin, Development, and Design of K-Factor Transformers”,
IEEE Industry Applications Magazine, September/October, 1997, pp67-69
[7] A.W.Galli and M.D.Cox, “Temperature Rise of Small Oil-filled Distribution
Transformers Supplying Nonsinusoidal Load Currents”, IEEE Transaction on
Power Delivery, January 1996, Vol.11, No.1, pp. 283-291
[8] M.T.Bishop, J.F.Baranowshki, D.Heath and S.J.Benna, “Evaluating HarmonicInduced Transformer Heating,” IEEE Transaction on Power Delivery”, January
1996, Vol.11, No.1, pp. 305-311.
43
[9] Keith H. Sueker, “Comments on ‘Harmonics: The Effects on Power Quality and
Transformers’”, IEEE Transaction on Industry Applications, March/April 1995,
Vol.31, No.2, pp. 405-406.
[10] Gregory W. Massey, “Estimation Methods for Power System Harmonic Effects on
Power Distribution Transformers”, IEEE Transaction on Industry Applications,
March/April 1994, Vol. 30, No.2, pp. 485-489.
[11] “AMX Series AC Power Source Operation Manual”, Pacific Power Source, Oct,
1996.
[12] “UPC-32/UPC-12 Operation Manual”, Pacific Power Source, Jan, 1995.
[13] Bruce Andrew Mork, “Ferroresonance and Chaos: Observation and Simulation of
Ferroresonance in a Five-Legged core distribution transformer”, Ph.D. Thesis, May
1992, Fargo, North Dakota, pp240.
[14] Standard UL1561, “Dry-Type General Purpose and Power Transformers”, April 22,
1994.
[15] Standard UL1562, “Transformers, Distribution, Dry-Type-Over 600 Volts”, 1994
[16] P.L.Dowell, “Effects of Eddy Currents in Transformer Windings” Proceedings of
the IEE, Vol 112, No.8 Aug. 1966, pp. 1387-1394.
[17] "ANSI/IEEE Recommended Practice for Establishing Transformer Capability When
Supplying Nonsinusoidal Load Currents." ANSI/IEEE C57.110/D7-February 1998,
Institute of Electrical and Electronics Engineers, Inc., New York, NY, 1998.
[18] Manjunatha Rao, “Development of a Laboratory Test Setup Using LabView for a
Power Quality Study”, MS Report, Michigan Tech University, 1999.
44
[19] Michael J. Gaffney, “Amorphous Core Transformer Model for Transient
Simulation”, MS Thesis, Michigan Tech University, 1996.
[20] Michael A. Bjorge, “Investigation of Short-Circuit Models for A Four-Winding
Transformer”, MS Thesis, Michigan Tech University, 1996.
[21] Richard L. Bean, “Transformers for the Electric Power Industry”, McGraw-Hill
Book Company, Inc., 1959.
45
APPENDIX A: 10 KVA DISTRIBUTION XFMR SHORT CIRCUIT TEST RESULTS
Table A-1 10 KVA Distribution Transformer Test No.1
Order
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
Freq Ipp(mv)
Iraw
(Hz)
rms(mv)
60
2.6
0.867
180
2.752
0.914
300
2.856
0.97
420
2.84
0.952
540
2.928
0.974
660
2.776 0.9245
780
2.736
0.926
900
2.96
0.912
1020
2.744
0.908
1140
2.656 0.9185
1260
2.888 0.9725
1380 2.8488 0.9605
1500
2.856
0.956
1620
2.864 0.9735
1740
2.92
0.998
1860
2.904
0.987
1980
2.952
1
2100
2.76 0.9455
2220
2.84 0.9645
2340
2.928
0.997
2460
3.064
1.029
2580
2.992 1.0185
2700
2.936 1.0055
2820
2.824
0.962
2940
2.904
0.977
Irms(A)
0.04335
0.0457
0.0485
0.0476
0.0487
0.046225
0.0463
0.0456
0.0454
0.045925
0.048625
0.048025
0.0478
0.048675
0.0499
0.04935
0.05
0.047275
0.048225
0.04985
0.05145
0.050925
0.050275
0.0481
0.04885
Vpp(mv)
0.11648
0.288
0.484
0.656
0.856
0.968
1.1392
1.2848
1.424
1.5968
1.8544
1.9984
2.1424
2.3424
2.5728
2.7136
2.912
2.912
3.1424
3.4272
3.7088
3.8496
3.9904
3.9872
4.2656
Vraw
V rms (v)
Mean
P(W)
S(VA)
Q (VAR) R(Ohm) X(Ohm)
rms (v)
(Vpp*Ipp)
0.04065
4.065 25.79968 0.128998 0.176218 0.12005 68.6446 63.88304
0.10125
10.125
32.256 0.16128 0.462713 0.433695 77.2233 207.6597
0.1706
17.06 39.6928 0.198464 0.82741 0.803255 84.372 341.4839
0.231
23.1
46.848 0.23424 1.09956 1.07432 103.383 474.1544
0.3014
30.14
56.217 0.281085 1.467818 1.440653 118.517 607.4373
0.3417
34.17 58.6368 0.293184 1.579508 1.55206 137.21 726.3645
0.4021
40.21 66.0992 0.330496 1.861723 1.832153 154.172 854.6726
0.4531
45.31 71.9104 0.359552 2.066136 2.034611 172.915 978.4792
0.5034
50.34 78.8224 0.394112 2.285436 2.251198 191.209
1092.2
0.5638
56.38
90.24
0.4512 2.589252 2.549636 213.93 1208.871
0.6552
65.52 106.6752 0.533376 3.18591 3.140945 225.587 1328.437
0.7062
70.62 113.4592 0.567296 3.391526 3.343743 245.966 1449.767
0.7574
75.74 119.6032 0.598016 3.620372 3.57064 261.732 1562.753
0.8284
82.84 130.9184 0.654592 4.032237 3.978749 276.286 1679.325
0.9088
90.88 149.4016 0.747008 4.534912 4.472964 300.002 1796.364
0.9592
95.92 148.1728 0.740864 4.733652 4.675316 304.203 1919.715
1.0294
102.94 163.2256 0.816128
5.147 5.081884 326.451 2032.754
1.0298
102.98 153.0368 0.765184 4.86838 4.80787 342.376 2151.244
1.1102
111.02 166.7072 0.833536 5.35394 5.288656 358.41 2274.055
1.2114
121.14 186.5728 0.932864 6.038829 5.966341 375.395 2400.92
1.3118
131.18 207.7696 1.038848 6.749211 6.668781 392.447 2519.276
1.3618
136.18 212.8896 1.064448 6.934967 6.852789 410.452 2642.441
1.4104
141.04 216.6784 1.083392 7.090786 7.007532 428.629 2772.432
1.4096
140.96
207.36
1.0368 6.780176 6.700435 448.131 2896.095
1.4998
149.98 224.0512 1.120256 7.326523 7.240371 469.449 3034.112
46
Table A-2 10 KVA distribution Transformer Test No.2
Order
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
Freq
Iraw
(Hz) rms(mv)
60 0.9695
180 0.5445
300
0.901
420 1.0255
540 1.0335
660
0.939
780 0.9215
900 0.9075
1020 0.9005
1140
0.89
1260
0.933
1380
0.955
1500 0.9845
1620 0.9985
1740 1.0095
1860 1.0315
1980
1.067
2100 1.0795
2220 1.0915
2340 1.0695
2460
1.054
2580 1.0415
2700
1.016
2820
0.955
2940 0.9665
Irms(A)
0.048475
0.027225
0.04505
0.051275
0.051675
0.04695
0.046075
0.045375
0.045025
0.0445
0.04665
0.04775
0.049225
0.049925
0.050475
0.051575
0.05335
0.053975
0.054575
0.053475
0.0527
0.052075
0.0508
0.04775
0.048325
Vraw
V rms (v)
Mean
P(W)
S(VA)
Q (VAR)
rms (v)
(Vpp*Ipp)
0.04574
4.574 31.9948 0.159974 0.22172465 0.153526
0.06074
6.074 11.89376 0.059469 0.16536465 0.154301
0.1608
16.08 35.8912 0.179456
0.724404 0.701824
0.25075
25.075 52.6592 0.263296 1.28572063 1.258472
0.3213
32.13 64.0256 0.320128 1.66031775 1.629163
0.35195
35.195 60.2624 0.301312 1.65240525 1.624701
0.402
40.2 65.0752 0.325376
1.852215 1.823412
0.4529
45.29
72.832 0.36416 2.05503375 2.022511
0.50315
50.315 78.7456 0.393728 2.26543288 2.230956
0.55415
55.415 85.4016 0.427008 2.4659675 2.428716
0.6351
63.51 100.5568 0.502784 2.9627415 2.919768
0.7058
70.58 113.9712 0.569856
3.370195 3.321668
0.7869
78.69 128.6656 0.643328 3.87351525 3.819719
0.8575
85.75 141.2096 0.706048 4.28106875 4.222445
0.9287
92.87 151.9104 0.759552 4.68761325 4.625667
1.0092
100.92 167.1168 0.835584
5.204949 5.13744
1.1101
111.01 188.6208 0.943104 5.9223835 5.846809
1.1908
119.08 203.6736 1.018368
6.427343 6.346154
1.2706
127.06 217.4976 1.087488 6.9342995 6.848495
1.3118
131.18 217.3952 1.086976 7.0148505 6.930123
1.3616
136.16 222.6176 1.113088
7.175632 7.088775
1.4112
141.12 225.792 1.12896
7.348824 7.261588
1.441
144.1 223.9488 1.119744
7.32028 7.234132
1.4096
140.96 206.4384 1.032192
6.73084 6.651224
1.5
150 221.4912 1.107456
7.24875 7.163653
R(Ohm)
68.07909
80.23307
88.42364
100.1458
119.8844
136.6927
153.2692
176.872
194.2177
215.6334
231.0352
249.93
265.4978
283.2684
298.1294
314.1316
331.3529
349.5582
365.1211
380.1179
400.7821
416.313
433.9017
452.7034
474.2232
X(Ohm)
65.33493
208.1777
345.8108
478.6657
610.1037
737.0591
858.9227
982.3305
1100.484
1226.469
1341.668
1456.832
1576.376
1694.057
1815.607
1931.383
2054.235
2178.338
2299.364
2423.48
2552.407
2677.769
2803.232
2917.124
3067.544
47
Table A-3 10 KVA distribution Transformer Test No.3
Order
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
Freq
Iraw
(Hz) rms(mv)
60 3.1305
180
2.264
300 1.6955
420 1.4425
540
1.297
660 1.2125
780
1.151
900
1.114
1020
1.083
1140
1.062
1260 1.0415
1380 1.0285
1500
1.015
1620
1
1740 0.9945
1860 0.9805
1980
1.069
2100
1.058
2220 1.0455
2340
1.031
2460 1.0225
2580
1.015
2700 0.9985
2820 0.9875
2940
0.972
Irms(A)
0.156525
0.1132
0.084775
0.072125
0.06485
0.060625
0.05755
0.0557
0.05415
0.0531
0.052075
0.051425
0.05075
0.05
0.049725
0.049025
0.05345
0.0529
0.052275
0.05155
0.051125
0.05075
0.049925
0.049375
0.0486
Vraw
V rms (v)
Mean
P(W)
rms (v)
(Vpp*Ipp)
0.1505
15.05 346.5216 1.732608
0.25075
25.075 195.2512 0.976256
0.3002
30.02 125.8752 0.629376
0.3514
35.14 107.0848 0.535424
0.402
40.2 98.1248 0.490624
0.4526
45.26 100.864 0.50432
0.5034
50.34 102.144 0.51072
0.5536
55.36 109.696 0.54848
0.6058
60.58 111.8208 0.559104
0.6554
65.54 117.0944 0.585472
0.7072
70.72 124.2624 0.621312
0.7546
75.46 134.9632 0.674816
0.8074
80.74 134.7584 0.673792
0.8576
85.76 147.0464 0.735232
0.9094
90.94 142.4896 0.712448
0.9594
95.94 150.784 0.75392
1.1108
111.08 189.3376 0.946688
1.1614
116.14 191.5904 0.957952
1.2118
121.18 197.0688 0.985344
1.2614
126.14 200.704 1.00352
1.3112
131.12 206.1312 1.030656
1.3684
136.84 213.4016 1.067008
1.4106
141.06 216.7808 1.083904
1.4606
146.06 220.0576 1.100288
1.4998
149.98 222.8224 1.114112
S(VA)
Q (VAR)
2.355701
2.83849
2.544946
2.534473
2.60697
2.743888
2.897067
3.083552
3.280407
3.480174
3.682744
3.880531
4.097555
4.288
4.521992
4.703459
5.937226
6.143806
6.334685
6.502517
6.70351
6.94463
7.042421
7.211713
7.289028
1.596057
2.665324
2.465894
2.477271
2.560387
2.697143
2.851695
3.03438
3.23241
3.430573
3.629955
3.821406
4.041777
4.224497
4.465515
4.642642
5.861266
6.068664
6.257581
6.424615
6.623805
6.86217
6.958508
7.127283
7.20338
R(Ohm)
70.71848
76.18524
87.57388
102.9263
116.6618
137.2153
154.2028
176.787
190.6758
207.6429
229.1138
255.1742
261.6096
294.0928
288.14
313.6823
331.3687
342.3201
360.5785
377.6319
394.3184
414.2815
434.8652
451.3279
471.6896
X(Ohm)
65.14498
207.997
343.1143
476.2137
608.8154
733.8384
861.0183
978.0467
1102.375
1216.684
1338.575
1445.022
1569.279
1689.799
1806.018
1931.657
2051.616
2168.611
2289.91
2417.63
2534.2
2664.338
2791.772
2923.545
3049.747
48
Table A-4 10 KVA distribution Transformer Rdc Test Results
RHV (Ω)
35
RLV (Ω)
0.3
Turns Ratio
30:1
RDC (Ω)
44
49
APPENDIX B
2 KVA Distribution Transformer Short Circuit Test Results
Table B-1 Manual Short Circuit Test Results
Rec
No.
Frequency
(Hz)
Peak-Peak
Voltage
Voltage
Probe Sclae
Peak_Peak
Current
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
30
60
120
180
240
300
360
420
480
540
600
660
720
780
840
900
960
1020
1080
1140
1200
1260
1320
1380
1440
1500
1560
1620
1680
1740
1800
1860
1920
1980
2040
2100
2160
2220
2280
2340
2400
2460
2520
2580
214.72
183.04
184.32
200.8
217.28
225.12
246.4
267.52
289.92
307.52
327.04
349.76
379.52
403.84
428.8
455.04
482.88
509.76
538.56
562.88
587.84
624
651.2
678.4
704.8
731.2
757.6
785.6
810.4
838.4
866.4
877.6
903.2
929.6
958.4
984.8
1013.6
1041.6
1049.6
1078.4
1105.6
1132.8
1164
1191.2
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
471.68
470.08
458.56
468.48
472.64
456.32
461.44
463.36
465.28
459.2
455.04
454.72
462.4
463.04
462.72
464.64
467.2
468.8
470.4
469.44
474.88
474.56
474.56
474.88
473.28
473.6
472.64
472.64
472
472.32
473.28
468.16
467.84
468.48
469.44
470.72
471.04
464.96
460.8
462.72
463.68
464.64
466.88
467.52
Current
Probe
Scale
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
Curren
t xfmr
ratio
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Power
12.5526
10.6717
10.140057
10.7512
11.4762
10.4104
11.0166
11.5032
11.8079
12.0439
11.0543
12.0209
12.1356
12.2978
13.0515
12.4961
13.7347
13.5758
13.8396
14.685
13.4545
14.508
14.5961
14.7538
15.6918
15.2228
16.3164
16.4086
16.4864
17.5677
17.1418
17.3752
17.5923
18.1084
18.5713
18.9768
19.5543
19.6813
19.7427
20.2752
20.8323
21.2664
21.9628
22.487
50
44
45
46
47
48
49
50
2640
2700
2760
2820
2880
2940
3000
1217.6
1244
1257.6
1289.6
1321.6
1348
1377.6
100
100
100
100
100
100
100
468.16
468.8
464
466.88
469.12
469.12
470.72
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1
1
1
22.8884
23.4004
23.38
24.0353
24.7685
25.3379
25.8908
Table B-2 Automatic Short Circuit Test Results [18]
frequency(Hz)
30
60
90
120
150
180
210
240
270
300
330
360
390
420
450
480
510
540
570
600
630
660
690
720
750
780
810
840
930
960
990
1020
1050
1080
1110
1140
1170
1200
1230
Power(W)
48.8
49.6
49.5
50.3
50.5
50.3
51
51.3
51.3
51.3
52.2
52.9
52.2
52.6
53
52.8
53
53.1
54.3
54.4
55.1
56.2
56.2
57.1
57.8
58
58.6
59.2
60.9
61.8
62.8
63.4
64.2
65.3
65.2
66
67
67.8
68.8
Irms(A)
8.25
8.32
8.29
8.32
8.3
8.27
8.32
8.32
8.28
8.24
8.29
8.32
8.24
8.25
8.24
8.3
8.28
8.25
8.28
8.25
8.27
8.3
8.26
8.28
8.29
8.26
8.28
8.29
8.27
8.27
8.27
8.28
8.28
8.29
8.26
8.26
8.28
8.27
8.27
Vrms(V)
5.88
5.97
6.06
6.29
6.49
6.74
7.01
7.35
7.63
7.96
8.35
8.78
9.05
9.46
9.87
10.3
10.66
11.06
11.54
11.97
12.47
12.97
13.35
13.87
14.35
14.75
15.25
15.73
17.11
17.6
18.12
18.6
19.1
19.6
20
20.48
20.95
21.47
21.97
R (Ω)
0.72
0.72
0.72
0.73
0.73
0.73
0.74
0.74
0.75
0.76
0.76
0.76
0.77
0.77
0.78
0.77
0.77
0.78
0.79
0.8
0.81
0.81
0.82
0.83
0.84
0.85
0.85
0.86
0.89
0.9
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1.01
51
1260
1290
1320
1350
1380
1410
1440
1470
1500
1530
1560
1590
1620
1650
1680
1710
1730
1760
1790
1820
1850
1880
1910
1940
1970
2000
2030
2060
2090
2120
2150
2180
2210
2240
2270
2310
2340
2370
2400
2430
2460
2490
2520
2550
2580
2610
2640
2670
2700
2730
2760
2790
69.4
70.5
71.3
72
73.1
73.9
74.7
75.2
76.1
76.8
78.5
79.6
80.6
80.8
81.6
83
83.5
84.4
85.1
86.7
86.9
88.2
88.6
89.5
90.9
91.2
92.7
94
94.3
95.5
96.5
97.3
98.4
99.6
100.4
101.6
102.9
104.3
105.3
106.3
107.5
108.2
108.5
109.9
111.1
112.5
113.2
113.7
115
116.3
117.3
118.5
8.28
8.28
8.28
8.29
8.3
8.3
8.3
8.25
8.28
8.28
8.32
8.32
8.3
8.28
8.28
8.29
8.3
8.3
8.28
8.32
8.29
8.29
8.3
8.3
8.32
8.28
8.3
8.3
8.28
8.29
8.3
8.3
8.3
8.3
8.3
8.29
8.29
8.32
8.32
8.29
8.29
8.32
8.3
8.29
8.3
8.3
8.29
8.29
8.28
8.3
8.29
8.28
22.45
22.94
23.44
23.92
24.44
24.94
25.43
25.82
26.32
26.81
27.38
27.88
28.37
28.76
29.26
29.73
30.14
30.61
31.02
31.61
32
32.47
32.99
33.44
33.94
34.35
34.94
35.41
35.8
36.29
36.79
37.27
37.72
38.22
38.74
39.26
39.78
40.39
40.87
41.23
41.72
42.11
42.58
43.06
43.54
44.06
44.53
44.44
45.03
45.53
46.02
46.39
1.01
1.03
1.04
1.05
1.06
1.07
1.08
1.11
1.11
1.12
1.13
1.15
1.17
1.18
1.19
1.21
1.21
1.22
1.24
1.25
1.26
1.28
1.28
1.3
1.32
1.33
1.34
1.36
1.37
1.39
1.4
1.41
1.43
1.44
1.46
1.48
1.5
1.51
1.52
1.55
1.56
1.56
1.57
1.6
1.61
1.63
1.65
1.65
1.68
1.69
1.71
1.73
52
2820
2850
2880
2910
2940
2970
3000
119.4
120
120.7
121.8
122.2
122.8
123.7
8.29
8.29
8.29
8.29
8.26
8.28
8.3
46.39
46.39
46.39
46.39
46.39
46.39
46.39
1.74
1.75
1.76
1.77
1.79
1.79
1.79
Table B-3 DC Value Test Results
RHV (Ω)
0.6
RLV (Ω)
0.4
Turns Ratio
2:1
RDC (Ω)
1.4
53
APPENDIX C
Harmonic Group Test Results
Table C-1 2 KVA Distribution Transformer Harmonic Group Test Results 1
Harmonic Groups (75% 3nd; 50% 5th; 25% 7th)
IRMS=8.372 A
Vrms = 5.9295 V
FFT Analysis results
Harmonic Order
Frequency (Hz)
I raw (mv)
1
60.0
187.03
3
180.0
125.46
5
300.0
72.58
7
420.0
32.13
Individual Harmonic Test Results
Frequency (Hz)
I raw (mv)
1
60.0
186.60
3
180.0
127.04
5
300.0
72.48
7
420.0
32.13
P = 46.02 W
I (A)
6.613
4.441
2.569
1.137
67 %
38.8%
17.2%
I(A)
6.606
4.497
2.566
1.137
Total
P (w)
28.38
13.36
4.42
0.912
47.07
Table C-2 KVA Distribution Transformer Harmonic Group Test Results 2
Harmonic Groups (75% 3nd; 50% 5th)
IRMS=8.23 A
Vrms = 5.328 V
FFT Analysis results
Harmonic Order
Frequency (Hz)
I raw (mv)
1
60.0
184.11
3
180.0
125.40
5
300.0
71.2
Individual Harmonic Test Results
Frequency (Hz)
I raw (mv)
1
60
184.0
3
180
126.4
5
300
72.0
P = 45.20 W
I (A)
6.613
4.441
2.569
67 %
38.8%
I(A)
6.678
4.475
2.549
Total
P (w)
27.5
13.3
4.38
45.18
Table C- 3 DFT Accuracy Check (10 KVA Transformer)
f1(Hz)
60
60
60
60
60
60
f2(Hz)
180
300
420
540
660
780
I 1_rms(A)
0.0927243
0.1001914
0.1003674
0.100342
0.1000747
0.1004785
I 2 rms (A)
0.040763292
0.029110879
0.021298268
0.016760269
0.01371172
0.011482283
P (W)
1.508096
1.612288
1.551105
1.534208
1.515008
1.509376
I rms (A)
FFT error(%)
0.1014
-0.1095
0.104525
-0.1819
0.102775
-0.1680
0.1019
-0.1648
0.101175
-0.1633
0.1013
-0.1654
54
APPENDIX D
Matlab Program for Analysis of 2 KVA Transformer Short Circuit
Test Results
List D.1 Program for finding the best one-section fit curve
%=====================================================================
% find the best fit curve for 2KVA XFMR automatic short-circuit data %
% (R)
%
%
% Newobj.m
is used to find the best exponent fit for R array without
%
DC point
%
Matlab fmin function used as object function
%
take the exponent as input parameter, then do
%
Linear Regression with (fh/f1)^expo up to the points
%
specified by N1;
%
the error was the return value so fmin
%
can find the optimal exponent value.
% autofit22.m call the fmin (will use Newobj.m )
%
% Usage: change the N0 and N1 to decide how many point you want to be
%
used in the fitting.
%======================================================================
clear all;
close all;
global r f N0 N1;
N0 = 10;
% global, the number of points used for fitting
% in Newobj.m, full length is 27 points
% ----------------load resistance array
load TwokRaut2;
dat1 = twokRaut2;
temp = size(dat1);
N = temp(1);
%how many test records
f = dat1(1:N, 1)';
r = dat1(1:N, 2)';
%--------------------------------------------------------%
Without DC value: R start from 60 Hz
%--------------------------------------------------------%-----------------------------% Case 1: expo = 2
%-----------------------------disp('------Without DC value: R start from 60 Hz, expo = 2')
expo = 2;
for i=1:N0
h(i) =(f(i)/60)^expo;
end;
55
f11 = f(1:N0);
r11 = r(1:N0);
[p,s]= polyfit(h,r11,1)
err11 = getfield(s,'normr')
y0 = polyval(p,h);
plot(f11,r11,'r.:',f11,y0),grid;
% r: real data; y0: fit data
xlabel('f (Hz)');
ylabel('Rac');
legend('Test data','Fit data',2)
title('One section Square fit for 2 kVA XFMR (60 - 1680 Hz) ( 27
points) ');
axis tight;
%---------------------------------------------% Case 2: expo = optimal output of the Newobj
%---------------------------------------------output2 = fmin('Newobj',0.5,2)
% Newobj: without DC point
%
all points N = 25 points
% N0
%how many points are included from 60Hz
disp('------Without DC value: R start from 60 Hz, expo = best')
expo = output2
for i=1:N0
h2(i) =(f(i)/60)^expo;
end;
f12 = f(1:N0);
r12 = r(1:N0);
[p2,s2]= polyfit(h2,r12,1)
err12 = getfield(s2,'normr')
y2 = polyval(p2,h2);
figure;
plot(f12,r11,'r.:',f12,y2),grid;
% r: real data; y2: fit data
xlabel('f (Hz)');
ylabel('Rac');
legend('Test data','Fit data',2)
title('One section optimal fit for 2 kVA XFMR Rac (60 - 1680) Hz ( 27
points) ');
axis tight;
56
List D.2 Object function used in finding best one-section fit curve
%================================================================
% Object Function for finding the best one-section fit curve
%
%================================================================
function [err] = myObj(expo)
global r f N0 N1;
% Linear Regression with (fh/f1)^expo up to only N0 points
% Started from 60 Hz, without the DC point
for i=1:N0
h4(i) = (f(i)/60)^expo;
end;
r4 = r(1:N0);
[p4,s4] = polyfit(h4,r4,1);
err = getfield(s4,'normr');
List D.3 Program for finding the transition point of the two-section fit and the best
Curves.
%================================================================
% Find the best transition points and fit curves for 2KVA XFMR Short
% Circuit test data ( AC winding resistance R )
%
% R array has 25 point in total, without the
DC point
%
% change the variable "expo1" can set if the first section fit
%
is using 2nd order or a optimal value.
% the total fit error is in array err
%================================================================
clear all;
close all;
global r f N0 N;
% ----------------load resistance array
load TwokRaut2;
dat1 = TwokRaut2;
temp = size(dat1);
N = temp(1);
%how many test records
f = dat1(1:N, 1)';
r = dat1(1:N, 2)';
k = 1;
for N0 = 3: N-1;
for fitting
% N0 is global, the number of points used
% the first part, assume square.
expo1(k) = fmin('firstpart', 0.5, 2);
57
%expo1(k) = 2;
for i=1:N0
h(i) =(f(i)/60)^expo1(k);
end;
f1 = f(1:N0);
r1 = r(1:N0);
[p,s]= polyfit(h,r1,1);
err1(k) = getfield(s,'normr'); % the error of firs part square fit
pp_a(k) = p(1);
%keep the results
pp_b(k) = p(2);
% the second part
expo2(k) = fmin('secpart',0.5,2);
for i=N0+1:N
h2(i-N0) = (f(i-N0)/60)^expo2(k);
end;
h22 = h2(1:N-N0);
r2 = r(N0+1:N);
[p1,s1] = polyfit(h22,r2,1);
err2(k) = getfield(s1,'normr');
pp1_a(k) = p1(1);
%keep the results;
pp1_b(k) = p1(2);
err(k) = err1(k) + err2(k);
k = k + 1;
end;
x = f(3:N-1);
hx = x/60;
plot(hx,err,'r.:'),grid;
xlabel('Harmonic order');
ylabel('Total Error');
axis([3 27 0 0.06]);
title('Total Error when the transition point moves from 3 - 27 ');
figure;
plot(x,expo1,x,expo2),grid;
% r: real data; y0: fit data
xlabel('f (Hz)');
ylabel('exponent');
legend('firt part','second part',2)
title('exponent found ');
axis tight;
58
List D.4 firstpart.m
%====================================================================
% Object function used for find the best fitting curve for the points
% group from 1 -> N0
%
%====================================================================
function [err] = firstpart(expo)
global r f N0 N;
% Linear Regression with (fh/f1)^expo up to only N0 points
% Started from 60 Hz, without the DC point
for i=1:N0
h(i) = (f(i)/60)^expo;
end;
r1 = r(1:N0);
[p1,s1] = polyfit(h,r1,1);
err = getfield(s1,'normr');
List D.5 secpart.m
%======================================================================
% Object function used for find the best fitting curve for the points
% from N0 +1 -> N
%
%======================================================================
function [err] = secpart(expo)
global r f N0 N;
% Linear Regression with (fh/f1)^expo up to N0 points
% Started from 60 Hz, without the DC point
for i=N0+1:N
h(i-N0) = (f(i-N0)/60)^expo;
end;
r1 = r(N0+1:N);
[p1,s1] = polyfit(h,r1,1);
err = getfield(s1,'normr');
59
APPENDIX E
Matlab Program for Analysis of 10 KVA Transformer Short
Circuit Test Results
List E.1 Program for finding the best one-section fit curve
%================================================================
% find the best fit for 10 kVA s-c data (R)
% the data starts from 60 Hz, 25 points in total
%
% Newobj.m
is used to find the best exponent fit for R array
%
without DC point
% Newobj2.m based on Newobj.m, insert the DC value to the R array
%
%
both used Matlab fmin function as object function
%
take the exponent as input parameter, then do
%
Linear Regression with (fh/f1)^expo up to the points
%
specified by N1;
%
the error was the return value so fmin
%
can find the optimal exponent value.
% autofit10.m call the fmin (will use Newobj.m and Newobj2.m )
%
%
case 1: Without DC value ( Newobj.m is used )
%
first fit it to square,(so expo is set to =2)
%
then use the best expo results (the output2) for
%
fitting
%
case 2: repeat above 2 tests with DC value inserted at
%
the head of the number in the R array.
%
(Newobj2.m is used)
% Usage: change the N0 and N1 to decide how many point you want to be
%
used in the fitting.
%
%
The two files above process the average R value from
%
dat12_1, dat12_3, dat12_4 and
%================================================================
clear all;
close all;
global r f N0 N1;
N0 = 5;
N1 = 26;
% global, the number of points used for fitting
% in Newobj.m ( without the DC point) (max = 25 )
% global, the number of points used for fitting
% in Newobj2.m ( with the DC point)(max = 26 )
% ----------------find the average value of r
load dat12_1;
load dat12_3;
load dat12_4;
%--------- Data 12_1
dat1 = dat12_1;
temp = size(dat1);
N = temp(1);
%how many test records
Hord = dat1(1:N,1)';
%harmonic order
f = dat1(1:N,2)';
%frequecy (Hz)
60
P = dat1(1:N,10)';
% Power (W)
r1 = dat1(1:N,13)';
%--------Data 12_3 -----------------------------dat3 = dat12_3;
r3 = dat3(1:N,11)';
% different column with data12_1
%--------Data 12_4 -----------------------------dat4 = dat12_4;
r4 = dat4(1:N,11)';
% different column with data12_1
%--------- Average value of these 3 data set ------------r = (r1 + r3 + r4)/3;
%--------------------------------------------------------%
Without DC value: R start from 60 Hz
%--------------------------------------------------------%-----------------------------% Case 1: expo = 2
%-----------------------------%
all points N = 25 points
%how many points are included from 60Hz upward
disp('------Without DC value: R start from 60 Hz, expo = 2')
expo = 2;
for i=1:N0
h(i) =(f(i)/60)^expo;
end;
f11 = f(1:N0);
r11 = r(1:N0);
[p,s]= polyfit(h,r11,1)
err11 = getfield(s,'normr')
y0 = polyval(p,h);
plot(f11,r11,'r.:',f11,y0),grid;
% r: real data; y0: fit data
xlabel('f (Hz)');
ylabel('Rac');
legend('Test data','Fit data',2)
title('Square fit for Rac (fh/f1)^2 from 60 Hz to 2940 Hz ( 25 points)
');
axis tight;
%----------------------------% Case 2: expo = output2
%----------------------------output2 = fmin('Newobj',0.5,2)
% myobj: without DC point
%
all points N = 25 points
% N0
%how many points are included from 60Hz
disp('------Without DC value: R start from 60 Hz, expo = best')
expo = output2
for i=1:N0
h2(i) =(f(i)/60)^expo;
end;
f12 = f(1:N0);
r12 = r(1:N0);
[p2,s2]= polyfit(h2,r12,1)
err12 = getfield(s2,'normr')
61
y2 = polyval(p2,h2);
figure;
plot(f12,r11,'r.:',f12,y2),grid;
% r: real data; y2: fit data
xlabel('f (Hz)');
ylabel('Rac');
legend('Test data','Fit data',2)
title('Optimal fit for 10 kVA Rac (fh/f1) ^ expo from 60 Hz to 2940 Hz
( 25 points) ');
axis tight;
%-------------------------------------------------------------%
With DC value: Rdc = 30 ohm, R start from 60 Hz
%
(total 26 points)
%
expo = output2 ( the best value fmin found)
%-------------------------------------------------------------% insert the DC point to f, R array
rNew(1) = 30;
%R dc = 30 Ohm
rNew(2:26) = r(1:25);
fNew(1) = 0;
fNew(2:26) = f(1:25);
%---------------------% Case 1:
expo = 2
%---------------------% Linear Regression with (fh/f1)^2 up to N1 points)
%N1 ;
f3 = fNew(1:N1);
disp('------With DC value: R start from 0 Hz, expo = 2')
expo = 2;
for i=1:N1
h3(i) = (f3(i)/60)^expo;
end;
%use the fit results
r3 = rNew(1:N1);
[p3,s3] = polyfit(h3,r3,1)
err21 = getfield(s3,'normr')
y3 = polyval(p3,h3);
figure;
plot(f3,r3,'r.:',f3,y3),grid;
xlabel('f (Hz)');
ylabel('Rac');
legend('Test data','Fit data',2)
title('Linear regression for Rac (fh/f1)^2 including DC value ');
axis tight;
%---------------------% Case 2:
expo = output2
%---------------------output2 = fmin('Newobj2',0.5,2)
% myobj2: including DC point
% Linear Regression with (fh/f1)^expo up to N1 points
% N1
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f4 = fNew(1:N1);
disp('------With DC value: R start from 0 Hz, expo = best')
expo = output2
for i=1:N1
h4(i) = (f4(i)/60)^expo;
end;
%expo = output2
%use the fit results
r4 = rNew(1:N1);
[p4,s4] = polyfit(h4,r4,1)
err22 = getfield(s4,'normr')
y4 = polyval(p4,h4);
figure;
plot(f4,r4,'r.:',f4,y4),grid;
xlabel('f (Hz)');
ylabel('Rac');
legend('Test data','Fit data',2)
title('Linear regression for Rac (fh/f1)^expo including DC value ');
axis tight;
List E.2 Program for finding the transition point of the two-section fit and the best
Curves.
%================================================================
% Find the best transition points and fit curves for 2KVA XFMR Short
% Circuit test data ( AC winding resistance R )
%
% change the variable "expo1" can set if the first section fit
%
is using 2nd order or a optimal value.
% the total fit error is in array err
%================================================================
clear all;
close all;
global r f N0 N;
% ----------------find the average value of r
load dat12_1;
load dat12_3;
load dat12_4;
%--------Data 12_1
dat1 = dat12_1;
temp = size(dat1);
N = temp(1);
%how many test records
Hord = dat1(1:N,1)';
%harmonic order
f = dat1(1:N,2)';
%frequecy (Hz)
P = dat1(1:N,10)';
% Power (W)
r1 = dat1(1:N,13)';
%--------Data 12_3 -----------------------------dat3 = dat12_3;
r3 = dat3(1:N,11)';
% different column with data12_1
%--------Data 12_4 -----------------------------dat4 = dat12_4;
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r4 = dat4(1:N,11)';
% different column with data12_1
%--------- Average value of these 3 data set ------------r = (r1 + r3 + r4)/3;
k = 1;
for N0 = 3:22;
%N0 is global, the number of points used for fitting
% the first part, assume square.
%expo1(k) = fmin('firstpart', 0.5, 2);
expo1(k) = 2;
for i=1:N0
h(i) =(f(i)/60)^expo1(k);
end;
f1 = f(1:N0);
r1 = r(1:N0);
[p,s]= polyfit(h,r1,1);
err1(k) = getfield(s,'normr'); % the error of firs part square fit
p_a(k) = p(1);
%keep the data;
p_b(k) = p(2);
% the second part
expo2(k) = fmin('secpart',0.5,2);
for i=N0+1:N
h2(i-N0) = (f(i-N0)/60)^expo2(k);
end;
h22 = h2(1:N-N0);
r2 = r(N0+1:N);
[p1,s1] = polyfit(h22,r2,1);
err2(k) = getfield(s1,'normr');
p1_a(k) = p1(1);
%keep the data;
p1_b(k) = p1(2);
err(k) = err1(k) + err2(k);
k = k + 1;
end;
err
err1
err2
expo1
expo2
x = f(3:22);
plot(x,err,'ro:'),grid;
xlabel('f (Hz)');
ylabel('Total Error');
axis manual;
axis([0 2700 0 140]);
title('Total Error (transition point moves from the 3rd - 22nd(300 Hz2580 Hz) 10 kVA');
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figure;
plot(x,expo1,x,expo2),grid;
% r: real data; y0: fit data
xlabel('f (Hz)');
ylabel('exponent');
legend('firt part','second part',2)
title('exponent found ');
axis tight;
% find the p, s for the second part best fit
N0 = 11; % found
for i=N0+1:N
h3(i-N0) = (f(i-N0)/60)^expo2(9);
end;
r1 = r(N0+1:N);
[p1,s1] = polyfit(h3,r1,1)
errSec = getfield(s1,'normr');
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APPENDIX F
Instructions for Doing Short Circuit Test Manually
Connect Test Device correctly. Please refer to Chapter 3 for different test configuration and
connection diagram.
Start up the AMX 3120 Power Source
•
•
•
•
Make sure the Output Power switch of the Master Power source is turned off
Make sure the Output Power switch of the Slave Power source is turned off
Turn on the Input Power switch of the Master Power source
Turn on the Input Power switch of the Slave Power source
Start up the Nicolet Pro20 Oscilloscope
•
•
Turn on the power switch of Oscilloscope
Set the sweep length to 8192 points under Menu\Acquisition\Sweep length. Refer to
Chapter 3 for detailed explanation of the setting.
Start up the AM 503A Current Probe
•
•
•
Turn on the power of the current probe
Degauss the Probe
Set the Currnt/Division setting to 0.5 A/DIV
Apply the Voltage to test transformer
•
•
•
•
Close the switch on the test bench.
On the UPC32 Panel of the AMX 3120 Power Source, choose the correct Voltage and
Frequency or harmonic groups
Turn on the Output Power Enable switch on the UPC32 Panel
Turn on the Output Power switch of the Master Power Source
Record the test data
•
•
•
Press Autosetup of the oscilloscope.
Set the Time setting to 200 µs.
Store the data to floppy disk
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Appendix G
Laboratory Equipment and Computer Resources
Test Transformer:
(1) 2 KVA single phase, dry type,4 winding 120/240 Volt transformer Square D cat. No.
2S1F
(2) 10 KVA amorphous steel core single-phase oil filled distribution transformer
7200-120/240-V
Lab Equipment
Pacific Source AMX 3120 AC Power Source
Nicolet Pro 20S digital oscilloscope
Tektronix model AM503S current probe and amplifier
Computer Hardware
Gateway 2000 486 DX2 66MHz Computer
Computer Software
Vu-Point II (version 3.14)
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