Solutions to the Equations Describing the Behavior of a Phase-Shifting
Transformer
Kathleen O’Brien
December 9, 1999
NUMERICAL ANALYSIS FOR ENGINEERING
MEAE-4960H02
Table of Contents
List of Symbols Used
3
Introduction
5
Problem Description
5
Formulation
7
Numerical Approaches
Flux and Voltage Equations
Saturation (B-H) Curve
10
11
Results
11
Error Analysis
15
Discussion and Conclusions
16
References
17
Appendices
18
2
List of Symbols
u o = the permeability of free space
ur = the permeability of either the magnetic path or leakage path material
url = permeability of leakage material
l  mean length of either the magnetic path or leakage material path
A = the cross-sectional area of either the magnetic core or the leakage path
material.
Np = primary turns per phase (same for all primary windings)
Ns = secondary turns per phase (same for all secondary windings)
Nshift = shift turns per phase (same for all shift windings)
Rm = magnetizing reluctance
vac = voltage from input phase a to input phase c
vba = voltage from input phase b to input phase a
vcb = voltage from input phase c to input phase b
rac = resistance from input phase a to input phase c
rba = resistance from input phase b to input phase a
rcb = resistance from input phase c to input phase b
iac = current between input phase a and input phase c
iba = current between input phase b and input phase a
icb = current between input phase c and input phase b
ac = flux between input phase a and input phase c
ba = flux between input phase b and input phase a
cb = flux between input phase c and input phase b
vas = voltage from secondary phase a to neutral
vbs = voltage from secondary phase b to neutral
vcs = voltage from secondary phase c to neutral
ras = resistance from secondary phase a to neutral
rbs = resistance from secondary phase b to neutral
rcs = resistance from secondary phase c to neutral
ias = current between secondary phase a and neutral
ibs = current between secondary phase b and neutral
ics = current between secondary phase c and neutral
as = flux between secondary phase a and neutral
bs = flux between secondary phase b and neutral
cs = flux between secondary phase c and neutral
3
vashift = voltage across shift winding a
vbshift = voltage across shift winding b
vcshift = voltage across shift winding c
rashift = resistance of shift winding a
rbshift = resistance of shift winding b
rcshift = resistance of shift winding c
iashift = current through shift winding a
ibshift = current through shift winding b
icshift = current through shift winding c
ashift = flux through shift winding a
bshift = flux through shift winding b
cshift = flux through shift winding c
4
Introduction
Today there are many pieces of widely used electronic equipment that produce
harmonics. These harmonics are known to cause serious operational problems.
By integrating phase shifting into a transformer with single or multiple outputs,
substantial reduction of triplen, 5th and 7th harmonics can be achieved. The net
result is that the electrical distribution system becomes electromagnetically
compatible with the electronic loads (e.g. personal computers) it has to supply.
Traditional electrical system design had very little need to deal with harmonics
because the loads typically designed for were linear in nature. Over the years, as
more and more research and practical experience was gathered with linear
loads, the design process became more and more predictable. With the
proliferation of variable speed drives, electronic ballasts, personal computers and
other electronic equipment, electrical system design strategies need to be
adjusted. Because in many cases a major portion of the loads today are
nonlinear in nature, the loading due to harmonics created by these loads must
also be taken into consideration.
Phase-shifting transformers of different configurations have been used for
decades in industrial and computer facilities to treat harmonics produced by
loads that are balanced and connected phase to phase e.g. 5 th, 7th, 11th, 13th
harmonics.
This paper describes a numerical solution to equations describing the behavior of
a delta/wye phase-shifting transformer. A dynamic model of a 3-phase, multiple
secondary phase-shifted, delta/wye transformer was developed. In order to
incorporate the effects of saturation in the core of the transformer, curve fitting
was done to obtain an equation used to describe the relationship between the B
(magnetic field) and the H (magnetic field intensity) fields.
The goal of this project was to numerically solve the equations describing the
dynamic behavior of the transformer and obtain graphs of voltage versus time
which closely match the expected results.
This paper will describe the procedure used to solve these equations. It will
include a discussion of the errors involved as well as analyses of both the results
and the methods used.
Problem Description
The three-phase secondary of this transformer is to be phase-shifted relative to
the input by 20 degrees. In order to achieve this effect, the transformer windings
are applied to the core in an unusual way in that a shift winding is added to the
transformer. This winding can be applied to either the primary or the secondary.
In this case, I chose to apply it to the secondary. This was an arbitrary choice.
5
A shift winding is applied to each of the three phases. Each shift winding is
connected to the secondary winding of another phase. This results in a flux in
each phase of the secondary which is phase shifted from the input. This can be
seen more clearly in Figure 1 and Figure 2 which show the electrical and
magnetic connections of the transformer.
out a
out b
in a
Vas
Vac
Vbs
Vashift
Vba
Vbshift
n
in c
in b
Vcb
Vcshift
Vcs
out c
PRIMARY
ONE SECONDARY
Figure 1: Electrical Connection Diagram
in a
in b
Vac
in c
Vba
Vas
Vcb
Vbs
Vcs
out b
out a
out c
Vashift
Vbshift
Vcshift
n
Figure 2 - Magnetic diagram (note that the arrows indicate the direction of the
induced flux)
6
In order to solve the equations describing the dynamic behavior of the
transformer and obtain graphs of voltage versus time it was necessary to use
transformer voltage and magnetic equations, including Faraday’s law and
Ampere’s law, to describe the behavior of the transformer.
The model uses standard transformer parameters. Since I needed data
describing these parameters in order to obtain a realistic solution and in order to
have something to compare my results to, I obtained some data from a
transformer vendor. I found data describing the number of primary, secondary,
and shift winding turns and resistance values, magnetic and leakage path
lengths, cross sectional areas of the core and leakage paths, permeability of
magnetic core and leakage materials, and a magnetization curve of the core
material.
Formulation
The voltage equations for this model can be written as follows:
Primary winding set:
d
 N  
dt ac ac
d
vba  rba  iba   N ba ba 
dt
d
vcb  rcb  icb   N cb  cb 
dt
Secondary winding set:
d
vas  ras  ias   N as  as 
dt
d
vbs  rbs  ibs   N bs  bs 
dt
d
vcs  rcs  ics   N cs  cs 
dt
vac  rac iac 
7
Shift winding set:



d
N  ashift
dt ashift
d
 rbshift  ibshift 
N  bshift
dt bshift
d
 rcshift  icshift 
N  cshift
dt cshift
vashift  rashift  iashift 
vbshift
vcshift



The flux equations for this model can be written as follows:
Core primary set:
ac 
ba 
cb 
N
ac
N
ba












N ashift iashift
iac   N as ias 
 Nac iac 



Rm
Rm
Rm
Rlac
Nbshift ibshift
iba   Nbs ibs 
 Nba iba 



Rm
Rm
Rm
Rlba
N
N cshift icshift
icb   N cs ics 
 Ncb icb 



Rm
Rm
Rm
Rlcb
cb
Core Secondary set:
as 
bs 
cs 
N
as
N ashift iashift
ias   N ac iac 
 Nas ias 



Rm
Rm
Rm
Rlas
N
bs
N
Nbshift ibshift
ibs   Nba iba 
 Nbs ibs 



Rm
Rm
Rm
Rlbs
N cshift icshift
ics   N cb icb 
 Ncs ics 



Rm
Rm
Rm
Rlcs
cs
8
Core shift set:
ashift 
bshift 
cshift 
N
N
ashift
iashift
Rm
bshift
N
ibshift
Rm
cshift
icshift
Rm
  N
  N






N ashift iashift
ias   N ac iac 


Rm
Rm
Rlashift
as
Nbshift ibshift
ibs   Nba iba 


Rm
Rm
Rlbshift
bs
  N
N cshift icshift
ics   N cb icb 


Rm
Rm
Rlcshift
cs
The reluctances are calculated using the general equation below.
R
l
u0  ur  A
There are nine voltage equations and nine flux equations. These equations
must be solved simultaneously. The voltage equations are first-order ordinary
differential equations. The flux equations are algebraic equations. The problem
is complicated by the fact that the flux, current, and voltage vary with time.
In solving this set of equations, I assumed that I knew both the input voltage and
the output current (an ideal current sink). This leaves 18 equations and 18
unknowns.
The equations were originally written assuming linear operation, neglecting the
effects of saturation in the core. Saturation has an effect on the permeability of
the core and therefore on the magnetizing reluctance. Magnetizing reluctance is
used in the flux equations and therefore has a direct effect on the results. I
decided to add this effect both because it is important and because it is
interesting from a numeric point-of-view. This was done by picking points off of
the magnetization curve of the core material and curve-fitting them to obtain an
equation that would accurately describe the B-H curve.
To incorporate saturation into the model, the following magnetic equations are
used in conjunction with the curve-fitting approach:
B  u0 ur H

B
A
Ni
H
l
9
Using these equations allows a few different methods for checking the
magnetization curve.
Numerical Approaches
Voltage and Flux Equations
Solving the voltage and flux equations proved to be an extremely difficult task.
The problems I encountered were due to a few things:
1. The number of equations to be solved was high.
2. The equations were of different types (algebraic and differential).
3. The solutions and inputs vary with time, creating the need for computationally
intensive code.
I started with Matlab and attempted to solve the flux equations symbolically and
then apply those results to the voltage equations by simple substitution. This
would give me nine first-order differential equations. I wanted to use the RungeKutta Method for Systems of Differential Equations from chapter 5 of the text to
solve these equations. I would need to set up a time-loop to obtain solutions at
enough points to get accurate plots of voltage vs. time.
I was able to find symbolic solutions to the flux equations and to substitute those
solutions into the voltage equations to obtain nine first-order differential
equations. I then attempted to use the Runge-Kutta Method for Systems of
Differential Equations to solve these nine equations. I used initial conditions of
zero, as would be the case in initially applying voltage to the transformer. I was
not able to obtain a useful result. The program either froze or gave me an error
message for an internal function in one of the Matlab toolboxes.
I then tried to use the “dsolve” function that is described on pages 323 and 324 of
the text. This seemed to work, but it took too long to solve to be practical. One
time point took over a half an hour to solve, and I needed at least a few hundred
for a good solution.
I then looked at some different software. I tried a software package called Saber,
which allows the user to write code (in a language called MAST) for the solution
of both algebraic and differential equations and then solve them simultaneously
over time. Since this was exactly what was needed, I entered all of the equations
and tried again. After some debugging and some research into exactly how the
equations where solved, I was able to get what I consider to be some good
results. Although the code used to solve the differential equations is encrypted
and I was not able to alter it, the documentation that accompanies the software
states that the solutions are obtained by calling a C-routine using the RungeKutta method. I do not know why this routine was able to solve the equations at
a much faster rate.
10
Saturation (B-H) Curve
I used Matlab to find an appropriate polynomial to describe the saturation effects
in the transformer. This was done by picking points off of the magnetization (BH) curve of the core material and curve-fitting them to obtain an equation that
would accurately describe the B-H curve. I used the ‘polyfit’ and ‘polyval’
functions in Matlab to do this. I also tried to use Newton’s Interpolatory DividedDifference Formula from chapter 3 of the text. However, this gave results which
did not come close to matching the actual magnetizations curve for the core.
Although I originally thought of the saturation issue as secondary to the main
problem, it became an interesting issue. The magnetization (B-H) curves for
transformer cores are usually very steep. This was the case here. Because of
the inherent steepness of the curve, it was not easy to find an equation that could
accurately describe it. It quickly became clear that the equation would be of a
high order. This is acceptable to a point, but leads to problems when the
equation must be evaluated during each time-step as it is used to calculate the
magnetizing reluctance.
Because of these issues, I was forced to compromise between an order which
would give a lot of accuracy and one which could be evaluated in a timely
manner. After several trial runs, I chose an 8th order equation to serve this
purpose.
Results
The transformer was designed to have a ratio of .129. It was supposed to have a
primary voltage of 3396 V line-to-neutral and a secondary voltage of 437 V lineto-neutral. It was designed to shift by –20 degrees. The results are in the form
of plots based on time.
Figure 3 shows the primary voltage with the secondary voltage laid on top of it.
In this figure you can see that the voltages are approximately correct and that
there is a noticeable phase-shift between the input and output.
11
Figure 3: Primary and Secondary Voltages
Figure 4 shows the primary and secondary voltages for phase A only. This is
done to get an accurate calculation of the phase shift.
12
Figure 4: Primary and Secondary Voltages for Phase A
Figure 5 shows the magnetization curve represented by the curve-fit that was
done.
13
B vs. H
2
1.5
1
B (Tesla)
0.5
0
-0.5
-1
-1.5
-2
-300
-200
-100
0
H (A/m)
100
200
300
Figure 5: Curve-fit of magnetization data
In order to put this into the transformer model, I had to solve for ur (permeability
of the core). Figure 6 shows the results of this.
14
4
3
Permeability vs. B
x 10
2.5
2
1.5
ur
1
0.5
0
-0.5
-1
-1.5
-2
-1.5
-1
-0.5
0
B
0.5
1
1.5
2
Figure 6: ur vs. B
Error Analysis
The voltage plots gave me results that were about 5% different from what I
expected. I attribute that mostly to the lack of accuracy in the magnetization
curve. Although there are several other possible sources of error such as the
fact that I ignored leakage effects and hysteresis effects, and to the fact that real
transformers have variations in the ratings of the components. I think that 5%
error is fairly acceptable in this case.
The magnetization curve matched the data for the core fairly closely. It would
have been more accurate, of course, had I used a higher order equation. This
added to the complexity and therefore to the length of time it took to get a
solution.
When the equation obtained by the curve-fit is evaluated, Matlab allows the user
to see the error bounds at each step. For this case, the minimum error bound
was +/- .57 and the maximum was +/- .78. This is a large error, but I could not
find a more accurate curve-fit that would allow the voltage and flux equations to
be evaluated with reasonable speed.
15
Discussion and Conclusions
The results for this project came out fairly well. The solution seemed to match
the expected results. However, I believe that the project could have been made
much easier and more rewarding if I had planned better. I should have chosen a
simpler set of equations to solve.
I would have preferred a simpler set of equations with more time to investigate
different numerical methods for solving them. Because of the complexity of
these equations I was left with only one available method. I would have liked to
have chosen several methods and to have taken the time to compare those
methods.
16
References
Burden, R., and Faires, J.D., Numerical Analysis, Sixth Ed., Brooks/Cole
Publishing Company, Pacific Grove, CA, 1997.
Guru and Hiziroglu, Electric Machinery and Transformers, Second Ed., Oxford
University Press, New York, NY, 1988.
Kreyszig, Erwin, Advanced Engineering Mathematics, Third Ed., John Wiley and
Sons, New York, NY, 1972.
Olsun Transformers, Phase-Shifting Transformer Parameters, Fax
17
Appendices
A. Code for B-H Curve Fit (with results as comments)
% B-H curve
%Define B and H vectors
B=[-1.7 -1.7 -1.7 -1.65 -1.6 -1.55 -1.5
.4 -.2 0 .2 .4 .6 .8 1 1.2 1.4 1.45 1.5
H=[-300 -240 -180 -150 -127 -90 -75 -50
2 2.5 3 5 7 25 50 75 90 127 150 180 240
order=8
-1.45 -1.4 -1.2 -1.0 -.8 -.6 1.55 1.6 1.65 1.7 1.7 1.7];
-25 -7 -5 -3 -2.5 -2 -1.5 0 1.5
300];
%order of equation desired
%Fit into an equation (H as a function
of B)
[p,s]=polyfit(H,B,order);
[Bgraph,delta]=polyval(p,H,s);
Hgraph=H;
%Find the Derivative (dB/dH)
Mult=[order:-1:0];
Preder=Mult.*p;
Der=Preder(:,1:order);
%Evaluate for values of H
preur=polyval(Der,H);
%Divide by uo to get ur only
ur=preur/(4e-7*pi);
%Fit into an equation (ur as a function
of B)
eqn=polyfit(B,ur,order);
%Evaluate for values of B
f=polyval(eqn,B);
%This plots permeability vs. B
plot(B,f)
title('Permeability vs. B')
ylabel('ur')
xlabel('B')
%This will plot B vs. H
figure
plot(H,B)
title('B vs. H')
ylabel('B (Tesla)')
xlabel('H (A/m)')
%This will plot permeability vs. B
%plot(B,ur)
%ylabel('ur')
%xlabel('B')
18
%test to see if this method of finding a derivative works
%B=[-1 0 1]
%H=[-100 0 100]
%p=polyfit(H,B,1);
%Mult=[1 0]
%PreD=Mult.*p;
%Der=PreD(:,1)
%Coefficients for ur as a function of B (8th order polynomial,
descending)
%
3.666417656132491e+03
2.434872013692225e+04
-1.716999686421038e-10
-
%
9.846624911062185e-10
1.634443854047623e-09
4.274136410855257e+04
-
%
-2.159666828973363e+04
2.523839057413041e+04
6.842255905983909e-10
%Coefficients for B as a function of H (8th order polynomial,
descending)
%
-1.938028737605247e-34
6.605923063965146e-29
-1.369495361495544e-16
%
2.387241071265011e-11
1.307688202372934e-06
-5.813297199300107e-24
-
%
1.041641795984777e-19
1.952394285266943e-16
2.978512509933560e-02
-
19
B. Transformer Code
# Single 3-phase Delta/ Wye, Secondary Phase-shifted -20 degree
Transformer
# Kathleen O'Brien -- NAE
# This program models a single, 3-phase Delta primary/ phase-shifted
wye secondary
# transformer. It transforms a 3-phase generated source voltage to a
three-phase
# output voltage shifted in phase by -20 electrical degrees.
#
# Saturation effects included.
#
# i(x)p = current into phase (a,b,c) primary winding
# i(x)s = current out of phase (a,b,c) of secondary winding
# Np = primary turns per phase (same for all primary windings)
# Ns = secondary turns per phase (same for all secondary windings)
# Nshift = shift turns per phase (same for all shift windings)
# r = windings resistance per phase
# Rm = magnetizing reluctance
# Rl = leakage reluctance
# l = mean length of magnetic path
# A = cross-section area
# url = permeability of leakage material
# ur = permeability of core
# uo = permeability of free space
# phshft = change in phase angle between primary and secondary voltages
resulting
#
from shift winding voltage (in degrees)
element template minus_delta_y_sec_phshft_xfmr ina inb inc outa outb
outc n
electrical ina, inb, inc, outa, outb, outc, n
unit {"Tesla-m2","Webers","flux"} phi
{
electrical
vshifta, vshiftb, vshiftc
branch
vac
= v(ina,inc), iac = i(ina->inc)
# primary input V,I
branch
vba
= v(inb,ina), iba = i(inb->ina)
branch
vcb
= v(inc,inb), icb = i(inc->inb)
branch
# secondary
branch
branch
vac_out
output V
vba_out
vcb_out
= v(outa,outc)
= v(outb,outa)
= v(outc,outb)
branch
vas
= v(outa,vshiftc),ias = i(outa->vshiftc)
secondary coil V,I
branch
vbs
= v(outb,vshifta),ibs = i(outb->vshifta)
branch
vcs
= v(outc,vshiftb),ics = i(outc->vshiftb)
#
20
branch
V,I
branch
branch
vashift = v(vshifta,n),iashift = i(vshifta->n)
# shift coil
vbshift = v(vshiftb,n),ibshift = i(vshiftb->n)
vcshift = v(vshiftc,n),icshift = i(vshiftc->n)
var phi
phi_ashift,
phi_ac, phi_ba, phi_cb, phi_as, phi_bs, phi_cs,
phi_bshift, phi_cshift
var nu
B, H
val nu
Rlp, Rls, Rm, Rlshift, rac, rba, rcb, ras, rbs, rcs,
rashift, rbshift, rcshift, Nshift, Np, Ns, rshift,
lp, ls, lshift, lm, url, ur, A, Al
number
pi, uo
parameters
{
pi
uo
= 3.1415962
= 4*pi*1e-7
}
values
{
Nshift
Np
=
Ns
=
A
=
Al
=
lp
=
ls
=
lshift
lm
=
url
=
= 16
514
30
.021
.01
1
1
= 1
5.22
1
##### magnetizing curve saturation effect #####
ur
=3.666417656132491e+03*B**8 -1.716999686421038e-10*B**7 2.434872013692225e+04*B**6 +9.846624911062185e-10*B**5
+4.274136410855257e+04*B**4 -1.634443854047623e-09*B**32.159666828973363e+04*B**2+ 6.842255905983909e-10*B
+2.523839057413041e+04
###############################################
Rlp
= lp/(uo*url*Al)
Rls
= ls/(uo*url*Al)
Rlshift
= lshift/(uo*url*Al)
Rm
= lm/(uo*ur*A)
rac
rba
rcb
ras
=
=
=
=
1.452
1.452
1.452
.0142
21
rbs
= .0142
rcs
= .0142
rashift
= .0076
rbshift
= .0076
rcshift
= .0076
}
equations {
iac: vac
iba: vba
icb: vcb
= rac*iac + d_by_dt(Np*phi_ac)
= rba*iba + d_by_dt(Np*phi_ba)
= rcb*icb + d_by_dt(Np*phi_cb)
ias: vas
ibs: vbs
ics: vcs
= ras*ias + d_by_dt(Ns*phi_as)
= rbs*ibs + d_by_dt(Ns*phi_bs)
= rcs*ics + d_by_dt(Ns*phi_cs)
# voltage
# expressions
iashift: vashift
= rashift*iashift +
d_by_dt(Nshift*phi_ashift)
ibshift: vbshift = rbshift*ibshift + d_by_dt(Nshift*phi_bshift)
icshift: vcshift = rcshift*icshift + d_by_dt(Nshift*phi_cshift)
phi_ac: phi_ac = (Np*iac)/Rm + (Ns*ias)/Rm - (Nshift*iashift)/Rm
+ (Np*iac)/Rlp
# flux
phi_ba: phi_ba = (Np*iba)/Rm + (Ns*ibs)/Rm - (Nshift*ibshift)/Rm
+ (Np*iba)/Rlp
# expressions
phi_cb: phi_cb = (Np*icb)/Rm + (Ns*ics)/Rm - (Nshift*icshift)/Rm
+ (Np*icb)/Rlp
phi_as: phi_as
+ (Ns*ias)/Rls
phi_bs: phi_bs
+ (Ns*ibs)/Rls
phi_cs: phi_cs
+ (Ns*ics)/Rls
= (Ns*ias)/Rm + (Np*iac)/Rm - (Nshift*iashift)/Rm
= (Ns*ibs)/Rm + (Np*iba)/Rm - (Nshift*ibshift)/Rm
= (Ns*ics)/Rm + (Np*icb)/Rm - (Nshift*icshift)/Rm
phi_ashift: phi_ashift = (Nshift*iashift)/Rm - (Ns*ias)/Rm (Np*iac)/Rm + (Nshift*iashift)/Rlshift
phi_bshift: phi_bshift = (Nshift*ibshift)/Rm - (Ns*ibs)/Rm (Np*iba)/Rm + (Nshift*ibshift)/Rlshift
phi_cshift: phi_cshift = (Nshift*icshift)/Rm - (Ns*ics)/Rm (Np*icb)/Rm + (Nshift*icshift)/Rlshift
###########################################
#
H: B
= (uo*ur)*H
B: B
= phi_ac/A
H: H
= Np*iac/lm
###########################################
}
}
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C. Symbolic Expressions Obtained for the Current Equations.
%iac = (phi_ac*Rm-Ns*ias+Nshift*iashift)/Np/(Rlp+Rm)*Rlp
%iba =(phi_ba*Rm-Ns*ibs+Nshift*ibshift)/Np/(Rlp+Rm)*Rlp
%icb =(phi_cb*Rm-Ns*ics+Nshift*icshift)/Np/(Rlp+Rm)*Rlp
%ias =-(-phi_as*Rm+Np*iac-Nshift*iashift)/Ns/(Rls+Rm)*Rls
%ibs =-(-phi_bs*Rm+Np*iba-Nshift*ibshift)/Ns/(Rls+Rm)*Rls
%ics =-(-phi_cs*Rm+Np*icb-Nshift*icshift)/Ns/(Rls+Rm)*Rls
%iashift =(phi_ashift*Rm+Ns*ias+Np*iac)/Nshift/(Rlshift+Rm)*Rlshift
%ibshift =(phi_bshift*Rm+Ns*ibs+Np*iba)/Nshift/(Rlshift+Rm)*Rlshift
%icshift =(phi_cshift*Rm+Ns*ics+Np*icb)/Nshift/(Rlshift+Rm)*Rlshift
23