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Phase shifting transformers and passive harmonic filters

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Title:
Phase shifting transformers and passive harmonic filters interfacing for power electronic motor drive converters
Creator:
Phipps, James Keith
Publication Date:
Language:
English
Physical Description:
ix, 144 leaves : illustrations ; 29 cm

Subjects

Subjects / Keywords:
Electric transformers -- Design ( lcsh )
Electric filters, Passive -- Design ( lcsh )
Electric filters, Passive -- Design ( fast )
Electric transformers -- Design ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references.
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Electrical Engineering.
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by James Keith Phipps.

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Source Institution:
|University of Colorado Denver
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Auraria Library
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
28863479 ( OCLC )
ocm28863479
Classification:
LD1190.E54 1993m .P544 ( lcc )

Full Text
PHASE SHIFTING TRANSFORMERS
AND
PASSIVE HARMONIC FILTERS:
INTERFACING FOR POWER ELECTRONIC
MOTOR DRIVE CONVERTERS
by
James Keith Phipps
B.S.E.E., University of Colorado at Denver, 1988
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver in partial
fulfillment of the requirements for the degree of
Master of Science
Electrical Engineering
1993
, I
i


This thesis for the Master of Science degree by
James Keith Phipps
has been approved for the
Department of
Electrical Engineering
by
William R. Roemish
Date ^ ~


Phipps, James Keith (M.S., Electrical Engineering)
Phase Shifting Transformers and Passive Harmonic Filters:
Interfacing for Power Electronic Motor Drive Converters
Thesis directed by Professor Pankaj K. Sen
ABSTRACT
This thesis details the analysis and design of electric power systems with
power electronic industrial converter loads for the reduction and control of
harmonic distortion produced by their non-linear characteristics. A technique
for harmonic component cancellation resulting from transformer phase shifting
is discussed in theory and measurements are shown to support the theory.
Four different types of phase shifting transformer winding connections based
on three and four winding transformer core arrangements are presented. A
general mathematical approach for describing unbalanced harmonic conditions
is devised based on symmetrical component theory and the result is used to
predict symmetrical component phase shifts through transformers of arbitrary
winding displacement.
Four passive filter configurations are introduced and the associated
design equations are derived using a transfer function approach. A filter
design procedure is outlined based on the applicable harmonic limits
m


established by ANSI/IEEE Std 519-1993 and an application of the design
procedure is presented.
An example is presented where a phase shifting transformer is used in
conjunction with passive harmonic filters to reduce voltage and current
distortion from two independent ski lift motor drives.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
IV


CONTENTS
ACKNOWLEDGEMENTS ......................................... viii
l. INTRODUCTION.......................................... .. 1
H. PHASE SHIFTING TRANSFORMERS .......................... 13
2.1 Introduction ................................... 13
2.2 Harmonic Component Cancellation.................. 15
2.3 Typical Transformer Connections.................. 33
2.3.1 Forked-Wye Connection...................... 35
2.3.2 Extended-Delta Connection.................. 37
2.3.3 Closed-Polygon Connection.................. 38
2.3.4 Wye-Delta Two Winding Secondary............ 39
2.4 Symmetrical Component Harmonic Phase Shifting.... 40
m. PHASE SHIFTING APPLICATIONS ............................. 52
3.1 An Application for Distributed Converters........ 52
3.2 An Application for Ski-Lift Converters.......... 56
IV. FILTER TRANSFER FUNCTIONS and CONFIGURATIONS ... 63
4.1 Justification for Harmonic Control Measures..... 63
4.1.1 Power Transformers and Reactors............ 64
4.1.2 Power Capacitors .......................... 64
4.1.3 Power Cables .............................. 65


4.1.4 Protective Relaying and Metering................. 66
4.1.5 Generators and Motors............................ 66
4.2 Filter Placement .......................................... 69
4.3 Filter Transfer Functions.................................. 71
4.3.1 Filter Impedance Transfer Function .............. 72
4.3.2 Filter/System Impedance Transfer Function ....... 73
4.3.3 Filter/System Current Divider Transfer Function .... 74
'I
4.4 lst-Order High-Pass Filter.................................. 76
4.5 lst-Order Damped High-Pass Filter......................... 80
4.6 2nd-Order Series Resonant Filter ........................... 84
4.7 2nd-Order Damped Series Resonant Filter..................... 91
4.8 Higher-Order Filters........................................ 94
4.9 Line Reactors and Harmonic Traps............................ 97
V. DISTORTION LIMITS and FILTER DESIGN........................ 99
5.1 ANSI/IEEE-519 Harmonic Distortion Limits.................... 99
5.2 A Passive Harmonic Filter Design Procedure................. 103
5.2.1 Filter Design Procedure Steps.................... 104
5.2.2 Step #1: System Data............................ 105
5.2.3 Step #2: Define the PCC ........................... 106
5.2.4 Step #3: Calculate the Filter Design Template H^ 108
5.2.5 Step #4: Fit the Filter to the System, Hj and H^ .. 110
vi


5.2.6 Step #5: Check Component Loadings/Modeling ... 110
5.3 A Transfer Function Application Example............ 112
5.4 A Practical Filter Design.......................... 125
5.4.1 Filter System Configuration.................. 128
5.4.2 Filter Design Steps.......................... 129
5.4.3 Filter Performance Tests..................... 133
5.4.4 Final Measured Results....................... 134
VI. CONCLUSION.............................................. 138
REFERENCES.................................................... 142
. vii


ACKNOWLEDGEMENTS
I have finally made it to a sunny horizon on a long and continuing
journey which this thesis serves as a marker. Although the path to this point
seemed at times to be never ending and solitary, I owe thanks to many people
who helped me along the way.
My academic advisor, Dr. Pankaj K. Sen, was especially important
because of his guidance in helping me choose the road to power engineering
in my senior year as an undergraduate. His enthusiasm and sound practical
thinking in electric power engineering has given me a broad viewpoint from
which I have learned to practice systems engineering and consulting.
Much of the content of this thesis was grown from the seeds that all of
my professors helped sew. A significant amount of my thinking in this thesis
was influenced by Dr. Robert W. Erickson and his excellent presentation and
instruction of Power Electronics. I appreciate the many comments he made in
reviewing this thesis and I have tried to incorporate most of them.
I also wish to thank Dr. Ewald F. Fuchs for his interest and
encouragement in my pursuit of power system harmonics. His work in this
field is well known and I am honored to have him as a reader and an advisor.
VUl


I owe thanks to Dr. William R. Roemish for being a reader of my
thesis. I have the highest regards for Dr. Roemish and am inspired by his long
and distinguished career.
I am thankful for the opportunity to work in an organization where
independent thought is highly encouraged and the pursuit of higher education
is fully supported. This aspect is amplified further by the knowledge and
experience of Mr. John P. Nelson from whom I have learned a great deal and
have the highest respect for.
This thesis would have been nearly impossible without the many figures
and drawings which, as they say, are worth a thousand words. For this work, I
am greatful to Ms. Kristi Jones for her patience in dealing with my many
revisions and her expertise in CAD.
Finally, I am indebted to my wife for being understanding and forgiving
of the time sacrifice that I have made in getting to this horizon.
IX


CHAPTER I
INTRODUCTION
With the invention of the silicon controlled rectifier (SCR) by Bell
Laboratories in 1956 and the marketing of the power version by General
Electric in 1957, the era of solid state power electronics was bom. Since that
time, several other power electronic switching devices have been developed:
triacs, gate turn off thyristors (GTO), static induction thyristors (SITH), metal
oxide-controlled thyristors (MCT), power bipolar junction transistors (BJT),
field effect transistors (FET), metal oxide field effect transistors (MOSFET),
insulated gate bipolar junction transistors (IGBT), and static induction
transistors (SIT). Today, material science technology for these devices is
rapidly maturing into new groups such as silicon carbide, gallium arsenide, and
diamond. So far today, however, most power electronic devices are made of
silicon and used as switches in conjunction with traditional passive components
such as iron core inductors and power capacitors to store and process energy.
[1]
In signal processing applications, as shown in figure 1.1, signals are
applied to the input of a processing network. Control signals and power are
also input to feed the processing electronics. Resistors, capacitors and


Input V
Signals T'
VW0 Resistors
Capacitors
Linear Active
Semiconductors
Switched
Saturation/Cutoff
Semiconductors
Processed
Output
Signals
Control
Signals
Control
Power
Figure 1.1 Signal processing network
semiconductor electronics are used to process the signals and provide an
output. Both linear active region and switched saturation/cutoff region
semiconductor devices are used. Since the signal power level input is
relatively small, heating losses in the network are not significant and,
therefore, resistors and linear active region components are desired. The
performance of the signal processor is based mostly on speed and bandwidth;
therefore, inductive energy storage elements are usually avoided since they
tend to respond slowly to change and reduce the overall speed and bandwidth
of the system.
In power processing applications, as shown in figure 1.2, raw
unconditioned power is input to the power processing network along with
2


0o Inductors
Input jXj
Power
. v ri uucs:
(--- Capacitors j> Output
Power
Processed
O
Switched
Saturation/Cutoff
Semiconductors
f
f
Control
Signals
Control
Power
Figure 1.2 Power processing network
some control signals and control power. Semiconductor switches are used
along with energy storage devices to process the input power and provide an
output. Since the performance of the power processor is based mostly on
efficiency, only energy storage devices such as capacitors and inductors are
used along with switched saturation/cutoff region semiconductor devices so
that high energy conversion efficiency is obtained. Resistors and active region
semiconductor devices are avoided at all costs since they consume energy and
reduce efficiency. [2]
Power processing can be grouped into two main areas: 1) low power
and high frequency where size and cost are the main factors which dictate the
design; and 2) high power and low frequency where energy efficiency and
3


system reliability are the predominant concerns. There are four basic design
topologies of power processors or power converters: 1) rectifiers which convert
alternating current (ac) to direct current (dc), 2) inverters which convert dc to
ac, 3) choppers which convert dc to dc, and 4) cycloconverters which convert
ac to ac. In all converter designs, electrical energy conversion takes place
between a source and a load. Power processing currently covers a very broad
range of applications: automobile, information, home appliance,
transportation, industrial, and power utilities just to name a few. The energy
conversion range is also very broad; calculators and palm-top computers use
converters which operate in the milliwatt range while utility high voltage direct
current (HVDC) converter stations operate in the 100s of megawatt ranges.
When power processing semiconductor technology started around 1960, low
switching frequencies of around 102 kHz and power capacities of around 101
kW were typical. Since that time, and with the introduction of new
semiconductor devices, the switching frequencies and power capacities have
been increasing by approximately one order of magnitude per decade [3].
Today, these values are approaching 105 kHz and 104 kW. When SCRs were
first introduced in 1957 for example, continuous forward bias current ratings of
only 25 A and reverse bias voltage ratings of 300 V were typical. Today,
continuous current ratings of nearly 3,000 A and reverse bias voltage ratings of
around 4,000 V are available in single device SCRs [4].
4


The electrical energy consumption in the United States today is on the
order of 600 x 109 kWh/yr [5]. Of this consumption, approximately 70% to
80% consists of electric motors with the balance being lighting and in other
miscellaneous uses. With the advances in power electronic converters, more
and more of the proportion of motor loads are being applied using power
electronic motor drives which use phase controlled rectifiers to convert ac to
dc. By the year 1990, it was estimated that nearly 400,000 units were on-line
[5]; and by the year 2000, the potential energy savings resulting from the use
and operation of power electronic motor drives could reach as high as 170 x
109 kWh/yr [6]. The main benefit of using power electronic motor drives is in
the economics and energy reduction savings which is typically in the 20% to
50% range over that of conventional motor electro-mechanical drive
applications. Other typical benefits include smooth ramp-up and ramp-down
acceleration transitions, optimized mechanical load characteristics, equipment
reliability, reduced mechanical related maintenance, increased product
qualities, reduced audible noise levels, and reduced physical space
requirements. However, there are a few disadvantages associated with power
electronic motor drives, but perhaps the single greatest disadvantage comes for
the rectifier interface to the utility power system.
There are just a few basic types of electrical motors which are used
extensively in industry. These include, dc motors, ac induction motors and ac
5


synchronous motors. As shown in figures 1.3 through 1.7, several types of
power electronic motor drives are possible [7].
Id
Figure 1.3 DC motor drive using a 6-pulse, phase controlled, full-bridge
rectifier
Id
Figure 1.4 AC induction motor drive using a 6-pulse, phase controlled,
full-bridge rectifier and current source bridge inverter
6


Id
Figure 1.5 AC induction motor drive using a 6-pulse, phase controlled,
full-bridge rectifier and voltage source bridge inverter
Id
Figure 1.6 AC induction motor drive using a 6-pulse, phase controlled,
full-bridge rectifier and voltage PWM inverter
7



Figure 1.7 AC synchronous motor drive using a 6-pulse, phase
controlled, full-bridge rectifier and current source load
commutated inverter (LCI)
These motor drives share two things in common: 1) they drive electric
motors and save on energy costs by optimization of speed and torque, and 2)
they use 6-pulse phase controlled, full bridge rectifiers which produce distorted
voltage and current waveforms in the ac power system as shown in figure 1.8.
The utility system ideally consists of a constant voltage and frequency
(50 or 60 Hz) source of electrical energy. The power electronic motor drives
convert three-phase ac power to variable voltage dc power with 6-pulse type
controlled rectifiers. It is this conversion and rectification of electric power
which produces undesirable harmonic current and voltage distortion on the ac
power system. When the harmonic currents flow in the power system, they
cause problems such as severe voltage distortion, power factor correction
8


Figure 1.8 Typical three-phase ac voltage and current waveforms
produced by 6-pulse, full bridge, phase controlled rectifiers
capacitor parallel resonances, power transformer heating and overload, electric
meter errors, power cable failures, telephone and communication line
interference, and reduced motor efficiency by the additional motor stator and
rotor heating and pulsating mechanical torques resulting from the induced
9


negative sequence voltage harmonics. The electric utilities are finding that the
total number of large and small rectifier/converter systems which are
connected to their systems is significant, they are growing rapidly, and they are
making up a larger portion of their total load.
In recent years, with more and more large nonlinear power electronic
converters being utilized, power system waveform distortion has warranted the
development of stringent harmonic distortion control limits by several agencies
around the! world. In the United States, the development and application of
the IEEE-519 guidelines and standards has necessitated the need for sound
engineering and design of voltage and current waveform interfacing between
rectifier loads and the electric power source,
There are several reasons and incentives for reducing the harmonic
current and voltage distortion produced by controlled rectifiers in the power
system; perhaps the greatest incentives are 1) utility and engineering
specification imposed distortion limits (IEEE-519), 2) telephone interference
complaints, and 3) reducing power system equipment thermal and voltage
stresses produced by the distortion. Little effort has been made by
manufacturers, users, or utilities to integrate converter/rectifier loads into the
power system so that harmonic distortion is controlled and reduced. Motor
drive manufacturers are typically not concerned with power source interfacing
to reduce harmonic distortion produced by their rectifiers and have not yet
10


developed or marketed a low-harmonic/high power-factor drive system which
uses new technology such as resonant-link or controlled active-line-current-
shaping converters because the economic constraints have not yet fully
surfaced. These factors have challenged power system engineers to develop,
design and implement practical interfaces for these large power electronic
rectifier loads so that harmonic waveform distortion is reduced and the
efficient operation and energy quality of the electric power system is
maintained.
The theory and analysis describing the voltage and current waveforms
shown in figure 1.8 has been well established and documented in [8-16] and
will not be discussed to that level of detail in this thesis. The purpose of this
thesis is to detail the analysis and design techniques of applying phase shifting
transformers and passive harmonic filters for the sole purpose of reducing the
waveform distortion produced by the non linear switching characteristics of
power electronic converters. Transformer phase shifting of harmonic currents
from independent converters to achieve harmonic component cancellation is
discussed. Five different types of three phase transformer winding
configurations are explained along with a general mathematical approach for
describing unbalanced harmonic conditions and symmetrical component
harmonic phase shifts through three-phase transformers. Two cases
incorporating transformer phase shifting are explained and actual field
11


measurements are given to help support the theory and justify the applications.
A passive harmonic filter design procedure is devised and outlined using a
generalized transfer function and numerical analysis approach which directly
incorporates the filter design limits based on IEEE-519. Four common types
of harmonic filter systems are presented and two specific applications are
explained.
12


CHAPTER II
PHASE SHIFTING TRANSFORMERS
2.1 Introduction
The application of phase shifting transformers to provide harmonic
component cancellation is not a new technique. When converter loads are
designed and applied to such uses as large drag line mining shovel hoists, large
dc motor drives used in steel mills, large electrolysis rectifiers, and electronic
dc trolley railways for example, multiple pulse configurations above 6-pulse are
commonly encountered. 12-pulse operation is obtained by using two 6-pulse
rectifiers connected to a common dc bus and phase shifting the three-phase ac
input to one of the 6-pulse rectifiers by 30. 18-pulse operation is obtained by
using three 6-pulse rectifiers and phase shifting two of the 6-pulse units by
20 respectively. 24-pulse operation can be obtained by providing a 15
phase shift between two sets of 12-pulse rectifiers or by using four 6-pulse
rectifiers with 0 shift on one unit, +30 on another unit, and 15 on the
remaining two units so that 15 of phase shift exists between each 6-pulse unit.
Higher pulse configurations are possible but they are seldom used since
increasing complexity, control and cost tend to outweigh the benefits of lower
harmonic distortion in the ac source currents and smoother ripple voltages and


currents at the dc bus. The pulse number used in this discussion represents
the total number of dc voltage and/or current pulsations, seen at the dc bus of
the rectifier output, which take place in 360 electrical degrees of the ac
source voltage feeding the rectifier system. In any case, the phase shift angle
required to achieve a certain pulse number can easily be calculated by (2.1).
360
PnL
(2.1)
where
0 Desired phase shift;
p Pulse number of individual rectifier load block (6,12,...);
nL Total number of rectifier load blocks used (1,2,...).
The use of phase shifting to achieve higher pulse numbers for lower dc
ripple and lower ac harmonics is typically used only when each rectifier load
block in a system is controlled. For example, if a variable voltage 12-pulse
rectifier output is desired, two full bridge 6-pulse SCR rectifier groups are
operated such that the gating signals to the individual SCRs are controlled so
that 30 electrical degree symmetry exists and each 6-pulse unit shares half of
the total load. This type of equipment is typically designed and built by
manufacturers who provide complete packages to the users. With lower power
applications, however, higher pulse configurations are not provided and the
levels of dc ripple and ac harmonics are fixed by 6-pulse designs.
14


As mentioned in Chapter I, the number of power electronic motor
drives which use phase controlled rectifiers are increasing rapidly. The
majority of these applications are for motor sizes below 3,000 hp in which case
individual three-phase, 6-pulse, rectifiers are typically used. For these
applications, phase shifting can still be used to reduce the ac current
harmonics flowing in the generating source, but the technique is seldom
considered during the planning stages.
22 Harmonic Component Cancellation [17]
Before the analysis stages of harmonic filtering are pursued, some
attention should be directed toward an alternative solution to controlling
harmonics. If the system is still in the planning stages, serious consideration
should be made concerning the type of transformer connections serving
individual 6-pulse converter loads.
To understand how harmonic cancellation occurs, consider a case where
two 6-pulse drives are being added to a distribution system and are to run
independent of each other as shown in the one-line diagram of figure 2.1.
Here the analysis is simplified by assuming that 1) both motor loads are nearly
the same in torque and speed (i.e., near equal loading), and 2) that the dc link
reactor in both drives is large enough to serve as a constant current source
(i.e., very small dc ripple current) to the invertor sections or dc-load as in the
15


Figure 2.1 ; Two constant current, 6-pulse, motor drives connected
through step-down transformers to achieve harmonic
cancellation
case of a dc motor drive.
As shown in this example, drive #1 is connected to the low voltage
terminals of a delta-delta step-down transformer that has a 0 phase
displacement. Drive #2, on the other hand, is connected to the low voltage
terminals of a delta-wye step-down transformer that is connected as an ANSI
standard 'low-voltage-lags" by 30. The per-unit, phase "A", ac line currents
flowing in the system under these conditions are shown in figure 2.2. The
firing angle a is adjusted to 0 for both motor drives, ia2 lags ial by 30
because the low-voltage line-to-line voltages in the delta-wye lag those in the
16


delta-delta connection. The per-unit line currents on the high voltage sides of
the transformer banks are related to the low voltage side by (2.2) and (2.3).
ial,
-2
iAl,
-2
t 0.0166 0
K
0.0166
Figure 2.2 Time domain current waveforms of the dual converter
connections shown in figure 2.1
iAl = ial
(2.2)
iA2 = ia2 ib2
(2.3)
where
ial Converter #1 line current, a iAl Transformer #1 line current, A ia2 Converter #2 line current, a ib2 Converter #2 line current, b;
iA2 Transformer #2 line current, A.
All of the time domain current waveforms shown in figure 2.2 have the
same relative harmonic magnitude spectrum shown in figure 2.4(a). Note that
17


the harmonic components in figure 2.3 occur at frequencies which follow the
equation 6k 1 (i.e., 5,7, 11,13, 17,19, etc.) where k is a positive integer:
ia(t) = [cos(cot) cos(5cot) + cos(7cot) cos(llcot)
it ( 5 7 11
111
+ cos(13o>t) cos(17cot) + cos(19cot)
(2-4)
where
Id Converter dc current level;
co System radian frequency (377 rad/sec).
Considering that the rectangular waveforms shown in figure 2.2 have six
pulse symmetry as shown in figure 2.3, its Fourier series can be derived as
follows using the form given by [30]:
where
f(t) Time domain function
a Fourier cosine coefficient
bn Fourier sine coefficient
L One half period
n Harmonic index (1,2,3,...)
18


Figure 2.3 Rectangular 6-pulse waveform
Since the waveform of figure 2.3 has even symmetry about the reference
coordinates, the bn terms cancel out since the integral of any odd function over
any even interval equals zero. In addition, the waveform has no net dc
component and the a0 term equals zero as well. Therefore, expanding upon a
yields,
19


and using the angle summation identity,
sin a + sinp = 2s
(2.7)
results in,
a.
n
(2.8)
bn = 0
for
n = 1,2,3,
Expanding f(t) in (2.5) using the result of (2.8), and noting that the
amplitude A equals the converter dc current amplitude, results in the classical
harmonic series for a 6-pulse waveform as given in (2.4).
The total harmonic distortion (THD) for a distorted waveform is a
measure of the RMS equivalent heating the distortion harmonic components
have in relation to the fundamental component and it is given by (2.9).
THD = V IF I2
IFJN&
(2.9)
where
Fn Fourier coefficient of the nth harmonic;
n Harmonic index number.
20


Expanding the coefficient a from (2.8) in (2.9) results in
oo
E W2 =
2fiA
\2/
it
_1_
&
+ + 1
?2 ll2 132 172
\ (2.10)
which resembles a convergent p-series with p = 2 > 1, hence,
A l l l l l l
\ = + + + ++
^ ip 2P 3P 4P "
(2.11)
n=l n*
ip ip 9P ^p 4P np
where the p-series converges for p > 1 and diverges for p £ 1. Evaluating
(2.10) with the fundamental component magnitude results in the following
approximate THD for the 6-pulse waveform:
2y/3A 71
THD n 2y/3A 00 1 SX n=2
* 0.3108
(2.12)
As can be seen in (2.12), the THD for any 6-pulse waveform is independent of
the actual magnitude units since the term A divides out as shown.
For 6-pulse type converters which have small dc ripple current, the total
harmonic current distortion (THDi) is near 31% and the higher order
harmonic components have magnitudes of 1/n relative to the fundamental.
Note that these are only theoretical values of the current harmonic
magnitudes.
21


THDi =31*%
42
Figure 2.4 (a) Theoretical harmonic magnitude spectrum for all 6-
pulse converter waveforms shown in figure 2. (b) Typical
measured, Delta-Wye shifted, 6-pulse current waveform
and harmonic spectrum
22


From a modeling standpoint, assuming a constant current source with
no dc ripple or commutation delay makes life much easier since the time
domain functions can be easily assembled and integrated in a Fourier series to
derive the harmonic components.
For a practical 6-pulse converter, however, the dc current source is not
ideal since large current ripples often exist; for example, discontinuous
conduction at light loads can produce a large dc ripple current, and in some
cases, where negative half-wave symmetry is lost in uncontrolled or half-
controlled rectifiers, even-order harmonic currents can be produced [11]. As a
result, the harmonic spectrum is not equal to that shown in figure 2.4(a).
Rather, the spectrum shown in figure 2.4(b) is more realistic even though the
THDi for both is nearly the same. Note that the rounding of the comers and
the excessive current ripple of the time domain 6-pulse waveform results in a
harmonic spectrum where much more of the distortion current is distributed
into the 5th harmonic component. As shown by [11], as the dc ripple ratio,
defined as
where
I, Peak of dc ripple current;
Id Average dc current,
23


increases toward discontinuous conduction, the 5th harmonic component
approaches, 50% while the higher order harmonics either remain the same, as
in the case of the 11th, or decrease as in the cases of the 7th and 13th
harmonics. In [11], sinusoidal shaped ripple peaks are assumed, the effect of
commutation inductance is ignored, and the following equations are given
which approximate the variation of the harmonic components as a function of
r:
IjC**) = Id(1.102 + 0.014r) (2.14)
w =
6.46r
n 1
7.13r
n
(-i)1
for n = pk 1
(2.15)
W =
6.46 r
n + 1
7.13^
* >
(-1)
for n = pk + 1
(2.16)
where
p Rectifier pulse number (6,12,...)
k Indexing integer (1,2,3,...)
Harmonic cancellation between the two converters in figure 2.1 occur
when the line currents sum together.
24


iAt = iAl + iA2
(2.17)
As shown in figure 2.5, the summation produces a 12-pulse equivalent current
waveform which has the following properties given by (2.18):
1) The harmonic components occur at frequencies according to 12k 1.
2) The THDi is approximately one half of that for a 6-pulse, and it
contains no 5th or 7th harmonics.
3) The remaining harmonic pairs in the 12-pulse have exactly the same
percent magnitudes as the 6-pulse; however, the magnitudes actually
double in size since the fundamental component doubles.
The true 12-pulse harmonic spectrum in figure 2.5 is only valid if the
firing angle a is controlled to be the same for both converters, and the line
currents have equal magnitudes and power factors for both drives. For two
drives operating independently, a and i will vary. To see how these variations
affect the summation of currents, consider the following sequence of
paragraphs.
If the current amplitudes are held constant for both drives and the
i
firing angle: al is delayed with respect to a2 by 5, the resulting time domain
25


<*1 0 -3 y i i i THDi = 15*% Aa = 0
i V 1 _rT i
0 0.0166
,.25 i i i
llAt 1
1 ; nl
llAt, 1
| 11 JL. 0 J Ji m J JLIL JUL
0 n 80
Figure 2.5 12-pulse current summation and harmonic magnitude
spectrum for Act = 0 and equal drive current amplitudes
waveform and harmonic current spectrum shown in figure 2.6 results. Note
that the meaning of Aa used in the following figures refer to the difference
between al and a2:
Aa = a2 al (2.19)
The THDi has increased by 1%, the current iAl lags iA2 by 5, the 11th and
13th harmonics are reduced, and total cancellation of the 5th and 7th
harmonic components is not achieved.
As A a is delayed to 15 shown in figure 2.7, the THDi increases by
~5%, and the 11th and 13th harmonics are almost canceled out completely.
26


0 n 80
Figure 2.6 Same as figure 4 except Aa = 5
At this point, the 5th and 7th harmonics become predominant in the spectrum.
Figure 2.8 shows the classical 6-pulse magnitude spectrum for A a
adjusted to 30. In practice, as A a is delayed further, the average dc link
current decreases and the power transferred to the load decreases. This also
produces a higher current ripple which changes the magnitudes of the
harmonic component distribution. If the firing angles of the two drives are
adjusted so that a2 lags al (i.e., A a < 0), the variation of the THDi is
symmetric to the cases shown in figures 2.6, 2.7 and 2.8. The variation of the
THDi along with the 5th and 7th components are summarized in figure 2.9.
27


THDi = 20 %
Aa = 15
0 ^ 0.0166
,'.2
iAt
, n
IAtj
JL ,
Figure 2.7 A a = 15
Holding A a constant and varying the amplitudes of the converter
current causes the THDi at the summation point to increase. As shown in
figure 2.10, if iAl is held constant, and a is adjusted to be equal for both
drives, then by letting iA2 equal 50% of iAl, the THDi increases by 3%. The
5th and 7th harmonic components increase from zero as do all odd harmonic
pairs which occur at frequencies corresponding to 6(2k+l) 1 for k = 0,1,2...
In the same way, as shown in figure 2.11, if iA2 is fixed at 200%, the THDi
increases by 3% and the odd harmonic pairs appear. For values of iA2
between 0 and 200%, the THDi varies as shown in figure 2.12. The true 12-
pulse operation is achieved only when the THDi is a minimum at iA2 equal to
1 1 1

n n_ Jil J.l. -ILL lH-IL -1LIL -n_n_ .1 ELl IU1
0 n 80
28


3
THDi =31*%
Aa =30
0 n 80
Figure 2.8 A a = 30
100%.
The figures shown have been computed under the conditions when the
loading of the drives are held constant and the firing angles al and a2 are
varied. In a similar manner, the as have been held constant and the loadings
have been varied. This was done to show how the THDi of the current
flowing into the converter terminals varied with Act and loading. In practical
converters, however, the load current varies in magnitude and phase as the
firing angle is changed. They are not independent variables. For inductive
power transfer, the current may increase from zero, reach a maximum near a
= 90, and fall off to 0 as a goes to 180. If the power transfer contains a real
29


-20 0 20 40 60 80
a
Figure 2.9 THDi vs. A a for equal drive currents
and reactive component, the maximum may occur between 0 and 90. In
either case, the current magnitude, power factor, and harmonic content will
vary with a. However, if two drives are connected to the system through
transformers as shown in figure 2.1, and their relative firing angles and
loadings are fairly close to each other as shown in figures 2.9 and 2.12, the
harmonic distortion at the summation point will be less than an equivalent 6-
pulse current flow that would result from transformers having the same type of
connections with no phase shift. From a design aspect, this result can
drastically reduce the size and costs associated with 5th and 7th harmonic
filters if they are applied in conjunction with a 30 transformer phase shift
30


Figure 2.10 With A a = 0, iA2 is adjusted to 50% of iAl
between two sets of 6-pulse type harmonic loads. As mentioned earlier, this is
the main advantage which needs to be considered in the engineering and
system planning stages.
As a concluding note to this section, the development of the waveforms
shown in figures 2.2 through 2.12 was accomplished by building individual data
files, each containing 512 discrete data points in a cycle for approximately two
!
periods of the indicated waveforms. Applying equations (2.2) and (2.3) to the
data files along with two separate indexing variables, al and ct2, used to
account for the relative phase shift, and using a fast fourier transform (FFT),
produced the harmonic spectrums shown. A more rigorous approach to this
31


4
THDi = 18 %
Aa = 0
iA2 = 200 %
0 n 80
Figure 2.11 With A a = 0, iA2 is adjusted to 200% of iAl
problem, of course, would have been to derive separate analytical expressions
for each individual harmonic component for the current summation based on
the Fourier magnitude and phase coefficients. As a result, the functions shown
in figure 2.9 were derived based on the discrete solution sets of the phase
shifting model described above. Actual analytical expressions were not
derived.
32


40
30
THDiL20
10
: 0
0 50 100 150 200
L
Figure 2.12 THDi vs. relative loading between drives from 0 to 200%
at A a = 0
2.3 Topical Transformer Connections
Although a delta-wye connected two winding transformer is the simplest
way to get a 30 phase shift, and by far the most common, it certainly is not
the only possibility especially when other angular phase displacements are
required as in the 18-pulse and 24-pulse cases. As shown in figure 2.13, a
three-phase transformer core can be wound with four winding sets per core
leg: primary set H, and secondary sets X, Y and Z. The H windings are
typically used to connect to the high voltage source of the power system for
the purpose of voltage level reduction to the low voltage winding sets and for
33


providing the exciting flux to the iron core. They can be configured in either
wye or delta connections depending on the 60 Hz zero sequence isolation
requirements. The X, Y and Z winding sets share the same respective iron
core as the H winding set and are, therefore, related by the winding turn ratios
nl9 n2 and n3, as shown. With this core configuration, several secondary
. H.V.
Primary
Windings
L.V.
Secondary
Windings
Figure 2.13 Three-phase transformer core with four winding sets per
core leg
winding configurations are possible. Reference [10] discusses four of the
following five winding connections.
34


2.3.1 Forked-Wye Connection
As shown in figure 2.14, the "forked-wye" phase shifting secondary
winding connection consists of XI, X2 and X3 connected in wye with the "fork"
connection being made using the Y and Z winding sets. Since the Y and Z
winding sets are designed to have the same number of winding turns, they also
produce the same secondary voltage phasor magnitudes and provide 120
angular symmetry about the X windings. The six transformer secondary
A A
Figure 2.14 "Forked-wye" connection
connections make up two three-phase sets of phasor line-to-line and line-to-
neutral voltages indicated by terminals A, B, C, and A', B', C\ The angular
phase shift displacement is achieved by adjusting the winding turn ratio
between n2 and n3 as indicated by the triangle of figure 2.14. Since the angle
0 represents the desired phase shift between the phasor sets, the law a sines
can be used to derive the necessary winding turns ratio:
35


(2.20)
where
n2 Number of X winding turns;
n3 Number of Y and Z winding turns;
0 Desired phase shift in radians.
The primary and secondary voltages are chosen depending on the
source and Hoad interfacing requirements and, given the iron core material
type and flux density design value, the number of turns for each winding set
can be calculated based on the design current carrying capacities, conductor
sizes and insulation thickness to obtain the appropriate window area product.
It is also interesting to note that by simply providing a mid-winding tap
in the X windings, a total of three, three-phase, phasor sets could be obtained
and used to serve three 6-pulse rectifiers forming an equivalent 18-pulse
rectifier with balanced commutating inductance. As shown in figure 2.15, the
mid-winding tap between X and X' for each core leg 1-2-3 would be selected,
with the phase angle set to 20 using (2.20), so that the resulting line-to-line
voltage magnitudes between the A-B-C, A'-B'-C', and A"-B"-C" phasor
sets are equal. It was not known if this transformer connection was ever
invented or used before this thesis was written; perhaps the terminology of a
36


"triple-forked-wye" would best describe it although the "PHIPPS-CONNECTION"
may also apply.
Figure 2.15 "Triple-Forked-Wye" connection
2.32 Extended-Delta Connection
As shown in figure 2.16, the "extended-delta" phase shifting secondary
winding connection consists of XI, X2 and X3 connected in delta with two
opposite Y and Z winding set "extensions" connected to each of the three delta
points. Similar to the "forked-wye" connection, the Y and Z winding sets are
designed to have the same number of winding turns so that balanced symmetry
is maintained between the two phasor sets at terminals A, B, C, and A', B',
C\ The angular phase shift is achieved by adjusting the winding turn ratio
between n2 and n3 similar to the "forked-wye" connection. As shown in figure
37


Figure 2.16 "Extended-delta" connection
2.16, the two triangles share similar sides and are therefore related by the law
of sines applied to each triangle. Using some basic geometry, (2.21) can be
derived.
%
2
(2.21)
where
n2 Number of X winding turns;
n3 Number of Y and Z winding turns;
0 Desired phase shift in radians.
2.3.3 Closed-Polygon Connection
Unlike the "forked-wye" and "extended-delta," the "closed-polygon"
shown in figure 2.17 requires only two secondary winding sets per core leg.
38


Figure 2.17 "Closed-Polygon" connection
The winding connection consists of XI, X2 and X3 configured in a near delta
with opposite Y winding set separation between what would be delta points.
The two, three-phase, phasor sets at terminals A, B, C, and A', B', C' are
made at each polygon point around the loop. The angular phase shift is
achieved by adjusting the winding turn ratio between ri2 and n3 as indicated in
figure 2.17 and it just so happens to be identical to (2.20) for the "forked-wye."
2.3.4 Wye-Delta Two Winding Secondary
As in the "closed-polygon" connection, the "wye-delta" connection
shown in figure 2.18 requires only two secondary winding sets per core leg.
The winding connection consists of XI, X2 and X3 connected in wye and the
Yl, Y2 and Y3 sets connected in delta. The two, three-phase, phasor sets at
terminals A, B, C, and A', B', C' are electrically isolated from each other
39


A
A
Figure 2.18 "Wye-Delta" connection
and it is not possible to vary their angular phase displacements since each set
operates independently.
When two 6-pulse rectifier sets, either phase controlled or straight
diode front-end, are used to form a controlled 12-pulse system, the "wye-delta"
transformer connection is commonly used to provide the required 30 phase
shift between the rectifier sets while at the same time maintaining balanced
commutation inductance between the phasor sets. With this configuration,
harmonic component cancellation occurs in the transformer core legs from flux
cancellation.
2.4 Symmetrical Component Harmonic Phase Shifting
Several authors in the past, including [18], have shown mathematically
how the symmetrical components of an imbalanced set of phasors are phase
40


shifted and/or blocked by the type of transformer winding configuration and
connection used. The concept of symmetrical components was largely applied
to unbalanced phasor sets typically resulting from unbalanced electrical fault
analysis and system studies at the power frequencies (i.e., 50 and 60 Hz
systems). Some technical papers published in IEEE, have mentioned that, for
balanced three-phase systems with balanced voltages and currents, the
waveform distortion harmonics follow the symmetrical component phase
sequence according to Table 2.1 [19]. None of the papers researched before
this thesis was written derived a general equation set, using symmetrical
components, which gave the harmonic component phase shift as a function of
a given transformer phase shift. This section briefly presents the mathematics
of harmonic phase shifting using symmetrical component theory.
The following equations were derived by combining two mathematical
concepts. First, assuming that any distorted waveform that is periodic and
steady state (i.e., all transient terms have decayed to zero and the solution to
the differential equations are particular and not homogeneous) can be
represented by a Fourier series with each harmonic component in the series
being represented by a phasor. Then, for a three-phase system, a harmonic
vector can be defined as (2.22).
41


Table 2.1
Harmonic Frequency Index (n) Symmetrical Component Phase Sequence (0,1,2)
1 1
2 2
3 0
4 1
5 2
6 0
7 1
8 2
9 0
10 1
11 2
12 0
13 1
-.abc
Fnaej[l9'
F.V*
FV"6-
(2.22)
where
Fnabc Harmonic vector of index n;
Fna a

Fnb b

Fnc c(f> phasor magnitude of harmonic index n.
Second, assuming that any unbalanced set of three-phase phasors may be
transformed into three balanced sets of symmetrical components and that the
42


superposition theorem holds for linear systems with individual harmonic
components acting one at a time, a symmetrical component harmonic vector
can be defined:
(2.23)
where
(2.24)
with
Fn012 Symmetrical component a

Fn(0) Zero sequence phasor of harmonic index n;
Fn(1) Positive sequence phasor of harmonic index n;
Fn(2) Negative sequence phasor of harmonic index n;
and, with the symmetrical component transformation matrix A defined as
1 1 1
A = 1 a2 a
(2.25)
1 a a2
using the phase shift operator a equal to
a = e
(2.26)
Solving (2.23) for Fn012 and expanding yields
43


(2.27)
T3012
*n
= a_1f;
-lijabc
1 1 1 FV<
1 1 a a2 Fy>a.
3
1 a2 a FV6-
(2.28)
Equation (2.28), in general, can be used for any set of unbalanced three-phase
phasors: voltage, current, power, etc. For example, the three-phase line
currents, shown in figure 2.19, serving a 1000 hp, 6-pulse, phase controlled
rectifier motor drive, running near 50% of full load at 600 volts, were
measured using a three-channel digital oscillograph. As can be seen from the
graph, the line currents have imbalance. If each phase current is decomposed
into a Fourier series starting at any arbitrary reference point common to all
three, each: harmonic phasor will contain a distind set of harmonic
symmetrical components according to (2.28). It should be obvious at this point
that if the system of phasors is unbalanced, each harmonic component can
contain all of the symmetrical components. For example, the third harmonic
component of the waveforms in figure 2.19 is composed of approximately 24
amps positive sequence, 13 amps negative sequence, and negligible zero
sequence. The fifth harmonic component consists of 30 amps positive
sequence, 201 amps negative sequence, and 22 amps zero sequence.
44


Another example is shown in figure 2.20. Here, the line currents to a
500 hp, variable speed, induction motor drive were measured while operating
at approximately 80% of full load at 480 volts. While each harmonic contains
three symmetrical components, the dominant phasor of each set tends to
follow Table 2.1 for 6k 1 harmonics. The triplen order harmonics which
follow 3k, are composed of mostly positive and negative sequence symmetrical
components.
The small zero sequence contributions in figures 2.19 and 2.20 exist,
most likely, as a result of measurement error. Since three-phase full bridge
rectifiers do not have a neutral return path, no zero sequence orders should
exist. However, in grounded three-phase systems, the drive chassis is typically
grounded and the rectifier, in some applications, has line-to-ground surge
arresters and surge capacitors for transient protection. While this provides a
path for zero sequence harmonic current flow, it is unlikely that magnitudes on
the order of several amps would flow under normal conditions.
Split-core current transducers placed around the phase conductors
feeding the drive rectifiers were used for both measurements. In the case of
the 1000 hp dc motor drive, six 500 MCM copper cables are paralleled per
phase and the window area of the clamp on transducer is not large enough to
accommodate all six conductors; in this particular case, only two conductors
could be fitted through the transducer window and the results were multiplied
45


by 3. In the case of figure 2.20, the current transducers were placed around
2,000 amp copper bus work in metal clad swicthgear and the transducer
windows could not be centered exactly the same for each phase. Therefore,
the measurement errors are likely to be partially due to proximity flux
unbalances in each set of measurements between phases.
46


*k
ODD Harmonics:
EVEN Harmonics:
od
FO
od
FI
odl
F2
odl
1 8 483.7 30.5
3 2 24.4 12.6
5 21.6 30.4 201.3
7 4.7 49.6 6.9
9 1.1 4.6 2.8
11 8 11.1 45.2
13 2.2 14 4.3
15 1.2 2.9 2.3
17 5.1 7.3 20.9
19 1.4 5.1 3.2
21 1.2 2.3 1.5
23 3.8 5.3 12.6
25 1.1 2.6 2.5
27 1.3 1.5 1.3
29 3.3 4.6 9.7
31 1 1.4 2.4
33 1.2 1.6 1.2
35 3 4.1 7.5
37 0.8 0.9 2.4
39 1.2 1.2 0.9
41 2.6 3.5 5.7
43 0.6 0.5 1.8
45 1.1 1.2 0.8
47 2.4 3.2 4.9
49 0.5 0.5 1.8
ev Ifo 1 sv\ I FI ] ev| |F2 1
2 0.4 3.4 0.9
4 0.4 0.6 1.2
6 0.6 1.4 0.8
8 0.4 1.4 0.6
10 0.5 1.1 0.7
12 0.6 0.4 0.5
14 0.4 1.4 0.5
16 0.6 0.8 1.2
18 0.4 0.7 0.2
20 0.5 1.1 0.6
22 0.8 0.7 1.3
24 0.4 0.4 0.7
26 0.5 1.3 0.5
28 0.8 0.9 1.3
30 0.4 0.4 0.4
32 0.6 1 0.7
34 0.7 0.7 1.4
36 0.4 0.1 0.5
38 0.5 1 0.6
40 0.7 0.8 1.2
42 0.5 0.3 0.5
44 0.4 1 0.7
46 0.8 0.8 1.3
48 0.6 0.2 0.6
50 0.5 1.2 0.9
Figure 2.19
Unbalanced 6-pulse rectifier, line current waveforms of a
1000 hp dc motor drive running near 50% of full load
47


1000
T
T
T
ODD Harmonics:
od iF0J |F1od| |F2od|
i 13.2 497.1 16
3 0.2 1.3 2.3
5 4.1 4 130.8
7 0.9 25.9 1.5
9 0.2 0.9 0.8
11 2.4 2.5 39.7
13 1.2 18.3 1.5
15 0.2 0.7 0.6
17 1.9 2.1 19.9
19 1.2 12.9 1.5
21 0.2 0.4 0.5
23 1.5 1.6 11.7
25 1.1 9.2 1.3
27 0.2 0.4 0.4
29 1.1 1.3 6.9
31 1 6.3 0.9
33 0.1 0.3 0.4
35 0.6 0.9 3.8
37 0.7 4.3 0.7
39 0 0.2 0.4
41 0.2 0.6 1.8
43 0.5 2.6 0.4
45 0 0.1 0.3
47 0 0.2 0.4
49 0.3 1.4 0.2
k
EVEN Harmonics:
ev F0 ev| FI ev| |F2
2 0.1 1 0.4
4 0.1 0.4 0.4
6 0.1 0.6 0.2
8 0.3 0.5 0.4
10 0.2 0.2 0.1
12 0.1 0.1 0.2
14 0.1 0.3 0.1
16 0.1 0.2 0.1
18 0.1 0.1 0.1
20 0.1 0.2 0.1
22 0.1 0.2 0.1
24 0.1 0.2 0.1
26 0.1 0.1 0.1
28 0.1 0.3 0.2
30 0.1 0.1 0.1
32 0.2 0.1 0.1
34 0.1 0.1 0.1
36 0.1 0.1 0.1
38 0.2 0.2 0.1
40 0 0.1 0.2
42 0.1 0.1 0.1
44 0.2 0.2 0.1
46 0.1 0.1 0.1
48 0.1 0.1 0.1
50 0.2 0.3 0.1
Figure 2.20
Nearly balanced rectifier line current waveforms of a 500
hp induction motor drive running near 80% of full load
48


Another application of (2.28) is to determine the harmonic phase shift
through three-phase transformers of arbitrary phase shift. For balanced
conditions
F = |Fnl = |Fnl
- IF,
(2.29)
and
Therefore,
and,
(2.30)
Fn 1 '
2jrn
Fnb ft" ii e-'
. 2im
Fn. [eTj
(2.31)
hi [l 1 l] 1 '
F ejn0 . 2nn
Fi _ n 1 a a2 e 6
3
Fn 1 a2 a . 2nn e 3 .
(2.32)
49


Equation (2.32) can be expanded for Fn, Fn1 and Fn2 with n indexed over the
harmonic range, n = 1,2,3 for any arbitrary transformer phase shift 0 to
determine the symmetrical component phase shift.
When symmetrical components flow through a three-phase transformer,
a new operator can be defined to account for the direction of rotation; hence,

(2.33)
such that negative sequence harmonic components rotate clockwise (i|r = +1),
and positive sequence components rotate counterclockwise (t|r = -1).
Combining (2.32) and (2.33) results in (2.34) which is a generalized equation
describing the symmetrical component phase shift through a three-phase
transformer, assuming balanced phase conditions exist.
n
n
,2
F ej8<*n +
1 1 1
1 a a2
1 a2 a
-j
. 2n
J
, 2nn
(2.34)
To see how (2.34) works, consider the 18-pulse rectifier of figure 2.21.
If three, 6-pulse, full bridge, controlled rectifiers, A, B and C, are connected to
the secondary of a "triple-forked-wye" transformer with 20 phase shifts as
shown, (2.34) can be used to predict positive and negative sequence harmonic
symmetrical component phase shifts. For example, at the 11th harmonic in
50


rectifier A, with 011A = +20, I11A2 = I A.-120; in rectifier B, with 011B = +0,
I11B2 = IA_0; and, in rectifier C, with 011C = -20, I11C2 = IA.+120. Hence, the
resultant 11th harmonic, I11T2 = 0. This same general 120 symmetric
cancellation occurs for all positive and negative sequence harmonic
components except for the harmonic pairs in the 18-pulse spectrum (i.e., 18k
1 for k = 1,2,3...) in which case, these pairs have a net 0 phase shift and add
together arithmetically with a resultant magnitude of 31^
>
Figure 2.21 An 18-Pulse application using a "triple-forked-wye"
connected three winding secondary
51


CHAPTER III
PHASE SHIFTING APPLICATIONS
3.1 An Application for Distributed Converters
Consider applying harmonic cancellation transformer connections to a
distributed set of variable frequency drives (VFDs) used in an oil field as
shown in the one-line diagram of figure 3.1. Three of the four 25 kV
distribution feeders serve several down-hole pumping motors through VFDs
connected to the secondary of 480 V step-down transformer banks. The VFD
output voltages are stepped up to 2,400 volts for the down-hole cable run to
the pump motors which are located at the bottom of the wells-the lengths of
these cable sections are on the order of 3.66 km (12,000 feet) straight down!
A small cogenerator is connected to the fourth feeder.
All of the transformers have been installed with the same connection.
What will the 69 kV source see as it "looks" into the oil field distribution
system? If all of the VFD as are fairly coincident, the system will look like a
giant 6-pulse load with approximately 20%, if not more, of the total VFD
current being composed of the 5th harmonic. Under these conditions, the
harmonic voltage distortion on the 25 kV system will be significant if the
parallel resonant frequency of the system and the capacitor banks is near the


5th harmonic. For this example, figures 3.2 and 3.3 show the measured line to
neutral voltage waveforms at the 25 kV bus of figure 3.1 with and without the
three feeder capacitor banks on-line. The voltage waveform of figure 3.2 has
approximately 15% THDv with the 5th harmonic being the predominant
component in the spectrum [17], [20]. At the design and planning stages of
the development of this distribution system, half of the VFD loads could have
been connected with delta-delta step-down transformers, and the other half
with delta-wyes at approximately the same cost as installing all with the same
type connection. While the odd harmonic pairs would not be canceled out
completely, as in a true 12-pulse configuration where the as are controlled,
they could have been reduced significantly. This would greatly reduce the
amount of 5th harmonic voltage distortion on both the 25 kV and 69 kV
systems.
53


Figure 3.1 One-line diagram of a distributed set of VFD loads used
in an oil field for down-hole pumps
54


0
n
42
Figure 3.2 25 kV bus line-to-neutral voltage waveform and harmonic
spectrum with capacitor banks on-line
55


THDv = 7 %
= 14.75103-volt
vpeak
= 21.6 103-volt
0 ty. 0.0166
I 1 1 1-


L flQn ihflnnnnnni^nnnfl -nlkllnnn 1
0.15
0 n 42
Figure 3.3 25 kV bus line-to-neutral voltage waveform and harmonic
spectrum with capacitor banks off-line
32 An Application for Ski Lift Converters [21]
As shown in figure 3.4, a wye-delta phase shifting transformer was
applied to a new motor drive converter on a detachable quad ski lift for
harmonic cancellation with an existing ski lift converter.
56


UTILITY
25kV DISTRIBUTION
&
___pcq
=" 2500kVA
LIFT #2
LIFT #1
Figure 3.4 One-line diagram of independent 6-pulse ac/dc converter
motor drives used for two detachable quad ski lifts
incorporating a phase shifting transformer
57


Figure 3.5 shows the measured phase "A" time domain current and
harmonic spectrum for lift #1 running at approximately 50% of rated full-load.
As shown, the current waveform has a dc ripple ratio very close to 1.5 and
contains a 5th harmonic component of nearly 46%. The 7th and 11th
harmonics are small by comparison.
0 n 80
THDi = 49 %
IAlj
=462*amp
Figure 3.5 Lift #1 phase "A" time domain current waveform and
harmonic spectrum
58


Figure 3.6 shows the phase "A" current for lift #2 running at
approximately 90% of rated full-load. The 5th harmonic component
is approximately 32% and the dc ripple ratio is between 0.7 and 0.8.
Figure 3.6 Lift #2 phase "A" time domain current waveform and
harmonic spectrum
Note that the predominant harmonics occur at frequencies corresponding to 6k
- 1 for k = 1,2,3..., and that the upper harmonics which occur at frequencies
corresponding to 6k + 1 are very small. As shown in [11], the upper
components of each harmonic pair cross zero for ripple ratios between 0.7 and
59


1.4 while the lower components of each harmonic pair remain nearly constant
for ripple ratios between 0 and 1.5.
With both converters running and adjusting the firing angles to be
nearly equal, the phase "A" current, quasi 12-pulse, waveform and harmonic
spectrum shown in figure 3.7 were measured at the secondary of the main
power transformer as shown in figure 3.4. Note the significant reduction in
the THDi and 5th harmonic component. The 5th harmonic component was
reduced below 4% without using any harmonic filtering!
Under typical operating conditions, the current waveform and spectrum
shown in figure 3.26 were measured when both ski lifts were running close to
full-speed and the firing angles of the converters were not controlled. The 5th
harmonic component was approximately 13% and the THDi was only 18%.
While these harmonic current levels may seem large compared to IEEE-519
limits, they are much less than what they would have been if a phase shifting
transformer had not been used. In this case the THDi could have been as
high as 40%.
When filters were designed to further reduce the harmonic current
injected into the utility, the 5th harmonic filter size was significantly reduced
from what would normally be required if a phase shifting transformer was not
used. Since transformer costs are typically on the order of $15/kVA and low
voltage filter costs are on the order of $70/kVAr, it is both economical and
60


THDi = 12 %
IAtJ =1212amp
Figure 3.7 Phase "A" current summation and harmonic spectrum of
Lifts #1 and #2 with nearly equal firing angles
practical to use phase shifting transformers to reduce the filtering costs.
61


0 n 80
Figure 3.8 Typical operating conditions: phase "A" current
summation and harmonic spectrum of Lifts #1 and #2
with uncontrolled firing angles
62


CHAPTER IV
FILTER TRANSFER FUNCTIONS and CONFIGURATIONS
4.1 Justification for Harmonic Control Measures
When power factor correction capacitors are added to a power system,
a natural resonance is formed at some critical frequency which can be excited
into parallel resonance by waveform distortion produced by rectifier/converter
loads (i.e., see figures 3.2 and 3.3). In such cases, severe voltage distortion can
be produced which can affect the performance of other, voltage sensitive,
loads connected to the electric power system at points of common coupling. It
is, therefore, often required that the flow of harmonic current produced by
rectifier/converter loads be controlled so that the systems natural parallel
resonances are not "kicked" into forced oscillation. It is also very important to
reduce the flow of harmonic currents and control the level of the resulting
voltage distortion through passive power system apparatus so that a high
degree of power quality can be maintained and the system heating losses and
insulation life degradation can be minimized among other potential problems.
One method for answering the question of what degree of harmonic
control measures are necessary is to look at how voltage and current harmonic


distortion affect critical equipment on the power system. The main power
system components which need to be considered are 1) power transformers
and reactors, 2) power capacitors, 3) power cables, 4) protective relaying and
metering, 5) generators and 6) synchronous and induction motors.
4.1.1 Power Transformers and Reactors
Power transformers and iron core reactors are affected by both voltage
and current distortion. Current distortion flowing through the windings
produce additional stray eddy-current losses which rise with frequency from
skin-effect and induce harmonic fluxes which increase core heating. Voltage
distortion on the exciting terminals of the transformer or reactor can produce
increased excitation losses in the core resulting from hysteresis and increased
losses in the interwinding capacitances at high frequencies. Power
transformers which serve large rectifier loads require either special core,
winding and cooling designs better known as F-Factor rated transformers, or
they require derating based on the harmonic current spectrum and the eddy
current loss factor at the hot spot under rated conditions (Peer).
4.1.2 Power Capacitors
Since power factor correction capacitors are used extensively for both
reactive power and voltage support on distribution feeders, they are very
64


abundant. ;The flow of harmonic current on the power system can excite the
natural parallel resonance of a capacitor bank which may result in severe
voltage distortion and amplified current flows at the resonant frequency. The
voltage distortion can cause additional reactive power flow, increased dielectric
losses and heating. Harmonic frequencies above the natural parallel resonant
frequency flow through shunt capacitors and can cause additional plate heating
and dielectric loss. In may instances, resonant conditions and/or high
frequency current flow through capacitors can cause premature operation of
protective fuses or breakers. This, many times, can be a first sign that
harmonic distortion problems exist in a power system.
4.1.3 Power Cables
High-voltage power cables have significant capacitive reactance which
acts as a low impedance to high frequency harmonics. In addition, similar to
shunt capacitor banks, a natural resonant frequency is formed which can
produce amplified harmonic currents and voltages. These effects can increase
the dielectric heating and degrade the life of the insulation system. In
addition, skin-effect heating increases with frequency and leads to higher
temperature rises.
65


4.1.4 Protective Relaying and Metering
The presence of harmonic voltages and/or currents can affect the
performance of electro-mechanical relays in timing and pickup depending on
the individual characteristics and applications. In electro-mechanical induction
disc type meters, positive and negative errors can result depending on the
phase rotation direction of the harmonics and direction of harmonic power
flow. For microprocessor based sampling type relays and meters, harmonic
frequencies above the Nyquest sampling rate can cause additional errors or
adversely affect performance.
4.1.5 Generators and Motors
Harmonic voltage and current affect both generators and motors by
increasing winding losses and core losses. In addition, negative sequence
harmonic voltages can produce counter rotating magnetic fluxes in the air-gap
which produce opposing shaft torques and increased rotor heating.
In all cases, increased heating in the power system apparatus leads to
higher temperature rises and reduced insulation life. Reference [22] discusses
the analytical techniques for estimating the lifetime reduction for insulation
systems based on harmonic distortion heating in transformers and rotating
machines. The derivation used is based both on theoretical loss analysis and
measured temperature rise data. Reference [23] develops a procedure for
66


estimating induction motor derating factors based on the level of harmonic
voltage present on the system. While both papers use a harmonic voltage
factor approach in their analysis, [22] presents a much more detailed
examination of transformers, induction motors and universal machines as well
as makes recommendations and suggests possible limits for harmonic voltage
levels. As shown in [22], the insulation lifetime reduction is far greater for
induction machines than it is for transformers. As an approximate rule, the
insulation lift time on motors is halved for every 10 C rise in temperature
above its rated rise. The adverse effects distortion have on power system
apparatus is well documented and help justify the reasons for controlling and
reducing distortion levels.
One method for controlling the flow of harmonic currents produced by
rectifier/converter loads is to provide shunt current paths through series
resonant tuned filters connected in parallel with the power system impedances
so that it remains localized and cannot contribute significant pollution to the
system. Currently, passive harmonic filter application is the method practiced
most often and is readily available to power system engineers and designers
for reducing harmonic voltage and current distortion through alternate circuit
path operation. Several IEEE transaction papers have been written and
published which introduce the theory and implementation of advanced
techniques for controlling harmonic current flow such as magnetic flux
67


compensation, harmonic current injection, dc ripple injection, series and shunt
active filter systems, and pulse width modulated static var harmonic
compensators. However, practical systems have not been extensively
developed and are not yet available on the market. It may still be sometime
before these advanced techniques are fully developed and are readily available
so as to successfully compete with passive harmonic filter systems; by the time
they can compete, advanced rectifier/converter designs which use active line
current shaping techniques will reduce the need for large scale harmonic
filtering systems in new installations assuming industry implements the
technology. Hence, the life expectancy and success of these advanced
harmonic control techniques is questionable and quite possibly short-lived at
best. Until that time arrives, if it even does, passive harmonic filters can be
designed and applied alone or in combination with transformer phase shifting
and/or higher pulse number rectifier configurations to control waveform
distortion on the power system.
This chapter deals exclusively with passive harmonic filter design. Four
common filter configurations are presented. Possibly for the first time, a
transfer function approach to filter design and system modeling performance is
presented which can be set up to account for harmonic distortion control limits
directly in the filter design process. The current version of ANSI/IEEE-519
harmonic distortion limits and guidelines are discussed. A practical filter
68


design applied to the ski lift converter loads discussed in Chapter in is shown,
the filter configuration is presented, the design steps are discussed, and the
actual measured results are provided and analyzed.
42 Filter Placement
As shown in figure 5.1, there are two practical locations where passive
harmonic filters may be effectively applied. Similar to power factor correction
capacitor placement, the optimum location results in maximized harmonic
reduction performance, minimized equipment costs and system losses. Many
times, placing the filter system as near to the nonlinear load as possible (i.e.,
at the same voltage level, preferably at the motor drive terminals) results in
the greatest attenuation of harmonic distortion for a given filter reactive power
rating. There are both advantages and disadvantages for either filter location
as well as several economic alternatives to consider. In any case, however, the
analysis begins with the generalized frequency dependant filter system
impedance representation Zf. This impedance can take on several forms
L
depending on the desired response and it is the basic building block on which
several useful filter system design transfer functions can be defined.
69


Figure 4.1 Practical filter locations
Harmonic
'-j
o


4.3 Filter Transfer Functions
There are a number of important transfer functions which can be
derived for filter design and system modeling purposes. As in all three-phase
power system analysis, one of the main design goals is to maintain balanced
voltages and currents in the network and to maintain reliability and efficiency.
With these in mind, the following transfer functions are based on single-phase
equivalent circuits assuming that the system is designed and operated under
balanced conditions. Although it is beyond the scope of this thesis to discuss
unbalanced harmonic systems, if the system was to operate unbalanced,
symmetrical component harmonic models and transfer functions could, with
moderate effort, be derived and used in addition to a complete three-phase
computer simulated model to approximate the unbalanced conditions and
system responses. When filter systems are used in unbalanced loading
conditions, the basic single-phase design relations can still be used to calculate
the initial component values. Those values can then be inserted into a full
three-phase unbalanced model for further analysis and design refinement.
Fortunately, the application of single-phase rectifier loads which cause
unbalanced harmonic systems are of the low power type and, therefore, only
represent a small fraction of harmonic producing loads in general. Therefore,
the need for unbalanced harmonic analysis and filter operation is much less
than it is for balanced operation.
71


4.3.1 Filter Impedance Transfer Function
This transfer function is the basic building block on which the modeling
begins. It is defined to be the complex impedance frequency response of the
filter system expressed in the s-domain using Laplace transforms of the
individual filter circuit elements. As shown in figure 4.2, if a general filter
branch is defined at its terminals, (4.1) can be defined as follows:
I f (a)
----1--------o +
ZfOOG vf(s)
Figure 4.2 Filter impedance representation
Hf(s) = Z/s)
Vf(s)
W
(4.1)
where
Hf(s) Filter impedance s-domain transfer function;
Zf(s) Filter s-domain complex impedance;
Vf(s) Single-phase equivalent filter s-domain voltage;
Ij(s) Filter branch s-domain current.
As mentioned above, Zj(s) is considered to be a single phase equivalent
impedance when used to model a three-phase system. If the actual filter is
configured in a delta connection, care must be taken not to confuse line-to-line
72


and line-to-neutral impedances when using (4.1). For balanced three-phase
systems, (4.2) is used as the wye-delta transform.
(4.2)
where
Zy Line-to-neutral, Wye, impedance;
ZA Line-to-line, Delta, impedance.
Hf(s) can be used to design and tune the filter as a separate system before it is
modeled in the power system network. Depending on the type and complexity
of the filter configuration, Zf(s) can be factored into a combination of
denominators (poles) and numerators (zeros) substituting
to derive the particular tuning and damping relations between component
variables.
4.3.2 Filter/System Impedance Transfer Function
After the filter system is configured and Zf(s) is known, this impedance
can be connected to the power system network to derive the filter/system
impedance transfer function Hfs(s) where Zs(s) is represented as the system
Thevenin equivalent system network impedance.
s = jo
(4.3)
73


I (s)
------O +
Zs(s)Q
Zf (s)
V(s)
o
Figure 4.3 Filter/System impedance representation
1
1 + 1 (4.4)
Zf(s) + Zs(s)
If other energy storage elements such as capacitances and/or
inductances exist in the power system network, they will affect the overall
performance of the filter system when it is installed. Hfs(s) is a powerful tool
which can be used to gain insight into the combined frequency response of the
filter connected to the system.
- T7^
I(s) If(s) + Is(s)
4.3.3 Filter/System Current Divider Transfer Function
There are two types of the current divider transfer functions which can
be derived for the filter system connected to the power system network.
Referring to figure 4.3, Hcds(s) is the ratio of system current to injected
current; and Hcdf(s) is the ratio of filter current to injected current.
74


(4.5)
H^Cs) -
W
Z,(s)
I(s) Z/s) + Za(s)
*
. Vs)
Z,(s)
I(s) Zj(s) + Zs(s)
Z (s)
H^S)=^H-(S)
Using (4.4) in (4.5) and (4.6), results in
1
H*(s) =
1
1
Zf(s)Zs(s)
Z/s) + Z8(s)
Z/s) Zs(s)

(4.6)
(4.7)
(4.8)
= Z1(s)Hcds(s)
= Z^H^s)
H-(S) = Z(s) ^ (4.9)
h^) z;s) iw (4.10)
Transfer functions (4.5) and (4.6) are very important because they serve
two roles. When the filter system is being designed, the impedances of the
transfer functions can be used to assess the overall system performance. After
the filter system is installed and operational, the harmonic current flows can
be measured and the appropriate current divider ratios can be computed and
75


plotted on the same graph for a filter performance comparison of designed vs.
measured response. Equation (4.5) is useful for designing and determining
harmonic current distortion limit compliance with IEEE-519 standards.
As discussed later, the IEEE-519 current distortion limits can be used
in conjunction with the expected harmonic current injection from the converter
rectifier to compute the minimum current divider ratios required to meet the
imposed limits. These ratios can then be used to define the minimum filter
attenuations for which Hcds(s) is based.
4.4 lst-Order High-Pass Filter
A first-order filter consists of a capacitor bank connected directly to the
power system bus and it is typically intended to filter high frequency harmonics
from the system as shown in figure 4.4 with the system impedance assumed to
be a simple inductance, Zs(s) = sLs. The "order" of the filter, in this case the
1st, is taken as the highest exponent of the characteristic s-domain polynomial
of Hf(s).
i <4U)
When used intentionally, the primary application of this type filter is to
attenuate high frequency harmonic current components which cause telephone
interference and reduce the voltage notching caused by SCR rectifier
76


-20dB/Dec
Figure 4.4 lst-Order high-pass
commutation as well as provide partial power factor correction of the
fundamental load current. When used unintentionally, as in the case of 60 Hz
power factor correction capacitor banks, the system natural parallel resonant
frequency may fall near one or more critical driving harmonic current
frequencies and significant voltage distortion may result, as discussed in
Chapter HI.
For the plot shown in figure 4.4, the construction of Hcds(jto) consists of
evaluating the function at both low and high frequencies. Starting with
and considering low frequencies,
77


H^O'u) =
1
joC
(4.12)
1
joC
+ jti)Ls
0) <<
1
joC
|joLs
(4.13)
and,
H*(j)
1
j(QC _
1
jo>C
For high frequencies,
o >>
1
jwC
|jLs
(4.14)
(4.15)
and,

1
jtoC
jLs
-1
g>2LsC
(4.16)
As shown by (4.16), the roll-off of the high frequency components is l/o>2 or
78


-40 dB per decade when (4.12) is expressed in dB form. If the system
impedance is assumed to have some resistance, Rs, then the maximum can be
found as follows:
HcasM co)u=_l_
v/L^
JC
££+r8 + ^
iC
, iCRs jc J
i + - + *-
jLa
^LSC JhsC
1
jCR.
1 + ---i 1
= _ij_
jCRa JRsN
c
(4.17)
and,
^ lHcds(i)L_L_ (4.18)
with,
Qs
_L is
RSN C
(4.19)
Therefore,
79


H. Q, (4-20)
When the transfer functions contain a few simple terms, as in the case
shown above, the algebra required to derive analytical closed-form expressions
for the asymptotes and local maxima and minima is fairly straight forward.
4.5 lst-Order Damped High-Pass Filter
As shown in figure 4.5, a series connected resistance is sometimes used
to provide a damping characteristic to the high-pass filter.
H^s) = -L + R (4.21)
Figure 4.5 lst-Order damped high-pass
80


As shown in the previous section, Hmax can be determined analytically by
evaluating the transfer function Hcds(j) at the resonant frequency. Assuming
the source impedance is a simple inductance, the following equations can be
written:
Za(s) = sLs (4.22)
y/QZ
JLC
+ R
jC
jc
1 +
jCR
i + JCR ic .
^LjC ^L,C
i.JCR
i + JCR
ffi
- i
jCR
1 "jRN
K
c
(4.23)
81


and,

= \/Q2 + 1 *Q
where,
q4n
The high frequency asymptote is found by considering
G> >>
1
jC
|jLs
FWH -
1
jcoC
+ R
1
jwC
R
R + jcoL
+ R + jcoL
1 +
jLs

l2 +
/ T
(oL,
\2
coL
R
(4.24)
(4.25)
(4.26)
(4.27)
82


since,
/
\
Ci>Ls
"F
\2
>> 1
(4.28)
As shown by (4.27), the roll-off of the high frequency components above the
frequency 1/(RC) is only l/w or -20 dB per decade. Hence, the application of
the series damping resistance significantly limits high frequency performance
over that of the undamped lst-order high-pass. This tends to make it less
desirable for telephone interference reduction applications.
In certain instances, when the parallel resonant frequency resulting
from the cancellation of the system inductive reactance by the high-pass filter
capacitive reactance falls on or near a critical harmonic frequency, a high-pass
damping resistance can be connected in series with the capacitance to control
and reduce the amplification. Such a resistance, however, increases the
fundamental frequency power loss and reduces the effectiveness of the high-
pass attenuation above the frequency 1/(RC) as shown by Hcds(s). In
applications involving several 2nd-order series resonant filter sections
connected in parallel and tuned to increasing discrete harmonic frequencies, a
lst-order damped high-pass filter is commonly used as the last filter section in
the bunch. The damping resistance then serves to reduce the highest order
parallel resonance formed between the high-pass filter capacitance and the
83


effective equivalent inductive reactance resulting from the parallel
combination of the utility system inductance and the 2nd-order filter branch
inductances.
In certain power factor correction applications, when a capacitor bank
is connected behind a series inductance from a harmonic generating source, a
series resonant circuit will be formed between the filter capacitance and the
series inductance (i.e., the inductive leakage reactance of a power transformer,
for example) as shown in figure 4.6. If this resonance falls on a driving
harmonic, unwanted harmonic current may flow through the capacitance and
cause overheating or excessive system voltage distortion. A damping
resistance can then be added to limit the harmonic current flow.
4.6 2nd-Order Series Resonant Filter
The single most used harmonic filter topology is, perhaps, the 2nd-order
series resonant type. It consists of a series combination of a capacitance,
inductance and small damping resistance as shown in figure 4.7 and is typically
used to filter a single discrete harmonic frequency such as the 3rd, 5th, 7th,
11th, etc. The damping resistance is usually set by the physical limitations of
the equivalent series inductor resistance and the equivalent series capacitor
resistance and no external resistance is added to the circuit.
84


co (Lg +L)
Figure 4.6 lst-Order damped high-pass filter behind a series
reactance
Zs(s)M
Figure 4.7 2nd-Order series resonant filter
The representation of the 2nd-order series resonant filter can be
expressed in an equation form as given by (4.29) and (4.30), or it can be
expressed graphically as shown in figure 4.8. Both forms show the classical
85


series resonant circuit. At low frequencies, the filter is dominantly capacitive
and, therefore, provides reactive power to the system. At high frequencies, the
filter is inductive and provides little attenuation for high frequency distortion.
At the resonant frequency, the capacitive reactance cancels the inductive
reactance and the filter is entirely resistive. Obviously, the lower the filter
resistance, the more attenuation. However, there are practical limits to R.
Hfs) = R + sL +
sC
_ 1 + RCs + LCs2
Cs
A i + ! Is] ( \2
* +
s [ Q col ^ / J
A
c
1
G> = ------
v/LC
Q
1 J
R\ C
(4.29)
(4.30)
where
A Gain coefficient;
6>0 Series resonant frequency;
Q Filter quality factor.
86


lHf(j)ldB
Figure 4.8 2nd-order filter impedance transfer function
For high voltage applications with a given filter reactive power rating, the
current flowing in the filter is typically of low RMS magnitude and does not
require large current carrying conductors or gapped inductor cores with an
abundance of iron. In fact, air-core inductors are regularly used. In these
cases, filter quality factors of 50 £ Q £ 150 are typical [11]. Low voltage
applications, on the other hand, usually require gapped iron inductor cores and
large current carrying conductors. The result is higher I2R and core losses
with increased heating and a higher temperature rise. These factors tend to
raise the effective series resistance and lower the quality factor. Hence, for
low voltage applications, filter quality factors of 10 £ Q £ 50 are typical.
The 2nd-order series resonant type filters are commonly applied at
practically all power system voltage levels: transmission, distribution, and
utilization. For example, large HVDC transmission converters typically use
87


several 2nd-order filter sections tuned to discrete frequencies and connected in
parallel at the ac terminals of each converter to provide both reactive power
and the desired harmonic filtering. On distribution systems, 2nd-order filters
are typically used to provide filtering for applications where several harmonic
producing loads share a common coupling; or, they are typically employed
where power factor capacitor banks require detuning to control the natural
system parallel resonant frequencies from landing on a critical driving
harmonic in which case the capacitor bank is configured into a detuned filter.
The most common location, however, is mostly at the utilization voltage level
since individual converter loads often exist alone and economic and operating
constraints do not justify filter applications or costs at the higher voltage levels.
The filter/system current divider transfer function Hcds(s) can be
evaluated at high and low frequencies to determine the functions asymptotes
similar to the above discussion for the lst-order filters: (4.13) and (4.15).
Those expressions are indicated in figure 4.9 for the case where the system is
considered to be a simple inductance and Ls > L. Figure 4.10 shows the case
for Ls < L. To find the local maxima of | Hcds(j) | or | Hfs(s) | requires much
more work, however. Because of the interaction between the capacitive
reactance and the filter inductive reactance, does not occur at the
parallel resonant frequency formed by the filter capacitance and the system
inductance.
88


Figure 4.9 2nd-Order transfer functions with L* > L
H-x |Hcds(i>)L = _L_
_L_
(4.31)
A numerical method is, therefore, recommended for this estimate. Two
possible solutions exist: 1) the transfer functions can be evaluated numerically
and plotted graphically to determine the approximate maxima; or 2) the
derivatives of the functions can be evaluated numerically to determine
frequencies where the maxima occurs. These frequencies can then be back
89


Figure 4.10 2nd-Order transfer functions with Ls < L
substituted and a short iterative procedure can be used to gain any numerical
degree of precision desired. It should be noted that finding a closed form
expression for these functions involves finding (4.32).
(4.32)
£p*0->I-0
While (4.32) may look innocent as shown above, the evaluation proves to be
quite laborious, indeed! Hence, numerical methods become very attractive,
especially when the system and filter impedances increase in complexity.
Fortunately, the minima of the transfer functions is very close to the series
90


resonant frequency and typically only tends to increase by a few percent worst
case when Ls < L.
4.7 2nd-Order Damped Series Resonant Filter
Another popular topology of the 2nd-order series resonant filter is to
provide an inductor bypass resistance as shown in figure 4.11. The bypass
resistance provides a high-pass characteristic as shown in figures 5.12 and 5.13.
Using a little algebra, the filter impedance transfer function can be expressed
in normalized form by (4.33).
Hf(s) =
sC
R,
bp
R + sL
RR^-C + L R. .
1 + --£------s + ---LCs2
R + R,
bp
R + R,
bp
Cs
1 +
R + R,
1 + -2-
<>
bp y
/ \
1 +
£bp
Ci)
V J
( \2
(0
\ J
V/
(4.33)
As defined in (4.30), the normalized polynomial coefficients of (4.33) can be
equated to find the gain coefficient A, the series resonant frequency g>0, the
quality factor Qbp, and the simple pole frequency o>p. Also, noting the typical
case where R < < Rbp, the approximations given in (4.34) can be made.
91