4th International Conference on Power Engineering, Energy and Electrical Drives
Istanbul, Turkey, 13-17 May 2013
Equivalent Circuit Model Containing AC and DC
Side Harmonics of Rectifier Circuits
Ezgi ÜNVERDø 1
Kocaeli University, Engineering Faculty
Departman of Electrical Engineering
Kocaeli -Turkey
1
ezgi.unverdi@kocaeli.edu.tr
Abstract— This paper proposes an equivalent circuit model that
can be used in the presentation of characteristic harmonic
components generated by three-phase diode rectifier circuits. In
conventional equivalent circuits, only the current harmonics on
AC side of converters are considered. For an exact analysis, it is
necessary to deal with both voltage and current harmonics. The
proposed model involves both current harmonics on AC side and
voltage harmonics on DC side of three-phase diode rectifiers.
Keywords- Harmonic model, equivent circuit, three-phase diode
rectifiers
I.
Ali Bekir YILDIZ 2
Kocaeli University, Engineering Faculty
Departman of Electrical Engineering
Kocaeli -Turkey
2
abyildiz@kocaeli.edu.tr
saturated region of the magnetization curve driven by overexcitation [3-4]. AC arc furnaces generate both integer- and
inter-harmonics during their electrode striking, melting and
refining stages of operation. The harmonics generated are
dynamic and random in nature. The fluorescent lamps with
electromagnetic ballasts are among the most important and
efficient lighting devices. The current waveform of a
fluorescent lamp is highly nonsinusoidal due to its luminous
discharge mechanism and due to its magnetic ballast which is
mainly an iron core. Therefore, such lighting loads generate
significant amount of odd harmonics during their operations
[5-6].
INTRODUCTION
The purpose of harmonic studies is to quantify the distortion
in voltage and/or current waveforms at various locations in a
power system. The need for a harmonic study may be
indicated by excessive measured distortion in existing systems
or by installation of harmonic producing equipment. One
important step in harmonic studies is to characterize and to
model harmonic-generating sources. In general, major
harmonic sources can be categorized as (1) Devices that
generate harmonics during their switching processes. The
most commonly seen are power electronic devices; (2)
Devices that generate harmonics due to their nonlinear
voltage-current characteristics; (3) Hybrid devices that include
both types of aforementioned devices; and (4) Devices such as
rotating machines that harmonics are generated because of
nonsinusoidal flux distribution in the stator and the harmonic
interaction between the stator and field windings.
Among those sources of the first category, three-phase power
electronic devices play a significant role in generating
harmonics. Commonly seen three-phase power electronic
devices include static converters that are often used in HVDC
links and motor drives, static VAR compensators, and
cycloconverters [1-2]. These devices are sensitive to supply
voltage distortion and unbalance.
Typical harmonic sources of the second category are
transformers, reactors, AC arc furnaces, and fluorescent lamps
with electromagnetic ballasts. Harmonics generated from
transformers and reactors occur when the core flux enters the
978-1-4673-6392-1/13/$31.00 ©2013 IEEE
POWERENG 2013
For the third category of harmonic sources, it consists DC arc
furnaces and fluorescent lamps with electronic ballasts. The
harmonics produced by a DC arc furnace concentrates at
orders of (6h+1). h = 0, 1, 2, .... However, inter-harmonics
are also generated due to the stochastic nature of the arc. The
electronic ballast for fluorescent lamp employs high frequency
switching to achieve greater flexibility in power conversion.
Due to the large amount of different types of electronic
ballasts, the harmonics produced by a fluorescent lamp with
electronic ballast vary dramatically. In general, these
harmonics are of all orders with even harmonics being of
small magnitudes [7].
The harmonic sources of the fourth category mainly include
synchronous machines and induction motors. The mechanism
of harmonic generation in synchronous machines is unique. It
cannot be described by using either machine's nonlinear
voltage-current characteristics or the power electronic
switching model [8]. Only the salient pole synchronous
machines operated under unbalanced conditions can generate
harmonics with sufficient magnitudes. For the cases of
saturation-caused harmonic generation from rotating
machines, the nonlinear voltage-current characteristics can be
used. The interactions between induction motors and harmonic
voltages/currents are more complex than that of a synchronous
machine. Therefore, the modeling of an induction motor can
be very complex. The induction motor is either represented by
equivalent harmonic impedance or by a three-phase coupled
matrix in harmonic analysis [9-10].
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4th International Conference on Power Engineering, Energy and Electrical Drives
Many harmonic models have been proposed for representing
three-phase power electronic devices [11-14]. The most
common model is in the form of a harmonic current source,
which is specified by its magnitude and phase spectrum. More
detailed models become necessary if voltage distortion is
significant or if voltages are unbalanced. Three basic
approaches used to build detailed models include developing
analytical formulae for the Fourier series as a function of
terminal voltage and operating parameters for the harmonic
source, developing analytical models for harmonic source
operation and solving for its current waveform by a suitable
iterative method, and solving for harmonic source steady-state
current waveform with time-domain simulation [15-19].
In this paper, first, the conventional equivalent circuits of the
rectifiers are pointed out, then a new equivalent circuit with
respect to harmonic currents on the AC side and output
voltage harmonics on DC side of three-phase diode rectifiers
is proposed.
II.
CONVENTIONAL EQUIVALENT CIRCUIT OF RECTIFIERS
Converters or rectifiers using semiconductor switching
devices generate harmonics caused by the behavior of
switching. The circuit configuration of a three-phase diode
rectifier, of which the equivalent circuit will be described in
detail, is shown in Fig.1.
isu
r
Istanbul, Turkey, 13-17 May 2013
In Fig.1 and Fig.2, the resistor r represents the source internal
resistance, while no source inductor is connected.
In Fig.1, the exact solution of u-phase source current isu is
given by
ଷ
ଵ
ଷ
ଵ
ξଷ
൅ ቁ ȁ௓ ȁ •‹ሺ߱‫ ݐ‬െ ߮ଵ ሻ ൅ ቀെ
•‹ሺͷ߱‫ݐ‬
ଶగ
ଶగ
ଶȁ௓ఱ ȁ
ଷ
భ
ଵ
ଵ
߮ହ ሻ ൅ •‹ሺ͹߱‫ ݐ‬െ ߮଻ ሻ െ
•‹ሺͳͳ߱‫ ݐ‬െ ߮ଵଵ ሻ ൅
ସȁ௓ళ ȁ
ହȁ௓భభ ȁ
ଵ
•‹ሺͳ͵߱‫ ݐ‬െ ߮ଵ ሻ ǥቁሽ
଻ȁ௓భయ ȁ
݅௦௨ ൌ ξʹܸௌ ሼቀ
െ
(1)
Where ȁ୬ ȁ ൌ ඥሺʹ” ൅ ሻଶ ൅ ሺɘሻଶ ‹ୱ୳ ൌ ‹ୱ୤ ൅ ‹ୱ୦
(2)
Where u-phase voltage;
୳ ൌ ୗ •‹ɘ–
(3)
isf and ish represents the fundamental component, the harmonic
component of the source current, respectively. In Fig.(l), let
the resistance r on the AC side of the rectifier be 1ȍ and the
resistance RL on the DC side be 100ȍ, the inductance L on the
DC side be 50mH and the max. value of the line voltage Vs be
100V. Fig.(3) shows some waveforms calculated from Eq.(l)(3).
idc
+
+
~
Vd(t)
Vw
~
+
RL
Vu
~
Vv
r
+
L
−
r
(a) Source voltage, Vu(t)
Figure 1. Configuration of three phase diode rectifier
The amplitude of harmonic currents generated by a diode
rectifier with a smoothing reactor on the DC side is almost
constant even if a source impedance varies, because the
impedance on the DC side is much larger than the source
impedance on the AC side. Therefore, in a conventional
equivalent circuits, the rectifier has been widely considered as
an ideal current source for harmonics as shown in Fig.2 [1924].
ish
r
(b) Source current, isu(t)
Rectifier
J(t)
Figure 2. Conventional equivalent circuit to harmonics on AC side
POWERENG 2013
(c) Output voltage, Vd(t)
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4th International Conference on Power Engineering, Energy and Electrical Drives
Istanbul, Turkey, 13-17 May 2013
RL
Vd(t)
+
~
L
DC side
Figure 4 Equivalent voltage source, Vd(t), applied to the load
(d) Fundamental component of the source current, isf(t)
The equivalent voltage source, Vd(t), contains both the
fundamental component and harmonic components of the
voltage. The equivalent source waveform, accordingly the
output voltage of the rectifier circuit, is given in Fig.5.
(e) Harmonic component of the source current, ish(t)
Figure 3 Calculated waveforms based on Eq.(l)-(3).
In Fig.1 and Eq.(l), a harmonic current magnitude iSh on the
AC side is determined by only the resistor RL on the DC side,
if RL >> r. In this case, an equivalent circuit of Fig.1 on a per
phase base can be considered as an ideal current source even if
the resistor r varies. Fig.2 shows the conventional equivalent
circuit with respect to harmonic currents. The current source,
J(t), presents the harmonic currents.
III.
PROPOSED EQUIVALENT CIRCUIT FOR RECTIFIERS
In conventional equivalent circuits of rectifiers, only the
current harmonics on AC side are considered. Whereas, in
rectifier circuits, there are both current harmonics on AC side
and voltage harmonics on DC side. For an exact analysis, it is
necessary to deal with both harmonics. For this purpose, a new
equivalent circuit model containing both current harmonics
and voltage harmonics is proposed.
First, Let’s consider the output voltage harmonics in DC side
of the rectifier circuit in Fig.1. In the rectifier circuits, an AC
source is processed through a set of switches to create a welldefined waveform. We can represent the combined action of
an actual source (AC source) and a set of switches by an
equivalent source. The equivalent sources provide a very
strong advantage: The new circuits are linear, and avoid the
nonlinearity and complication of switches. We can use
superposition, Laplace transforms, or other techniques from
linear network analysis to analyze rectifiers. Based on
superposition, a term-by-term for the Fourier series of the
current and voltage in the rectifier circuits can be solved.
Equivalent voltage source, Vd(t), applied to the load in Fig.1 is
presented in Fig.4.
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Figure 5. Equivalent source waveform of rectifier
The Fourier series for the voltage Vd(t) can be expressed in
trigonometric form as
∞
∞
n =1
n =1
Vd (t ) = Vo + ¦ a n cos nωt + ¦ b n sin nωt
(4.a)
∞
Vd ( t ) = Vo + ¦ c n sin(nωt + ϕ n )
(4.b)
n =1
Every component of Fourier series corresponds to a voltage
source. They are symbolized as Vo, v1(ω1t), v2(ω2t),… in Fig.6,
where Vo is average load voltage as DC source, v1(ωt) is a AC
source having ω1 frequency, v2(ωt) is a AC source having ω2
frequency,….
Vo
i(t)
+
RL
+
v1(ω1t)
Vd(t)
~
+
v2(ω2t)
~
L
+
vn(ωnt)
~
Figure 6 Equivalent voltage sources corresponding to Fourier series
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4th International Conference on Power Engineering, Energy and Electrical Drives
Istanbul, Turkey, 13-17 May 2013
The Fourier series expression relating to Fig.5 is given by
‫ݒ‬ௗ ሺ‫ݐ‬ሻ ൌ
ଷξଷ௏ೠ
గ
൅ σ’௡ୀ଺ǡଵଶǡଵ଼ǥ
଺ξଷ௏ೠ
గሺ௡మ ିଵሻ
‘•ሺ݊߱‫ ݐ‬൅ ߨሻ
(5)
Second, Let’s deal with the current harmonics in AC side of
the rectifier circuit in Fig.1. The source currents, as shown in
Fig.3b, are dependent on the load current. Therefore, these
currents are modeled by a current controlled current source,
which is controlled with load current. On a per phase base, the
dependent harmonic current source, Js(t), is shown in Fig.7.
(b) Load voltage, Vd(t)
r
Vs(t)
+
Figure 9. Simulation results relating to the equivalent circuit
From the point of harmonic studies, the equivalent circuit can
be divided into two circuits: one is an equivalent circuit to the
fundamental component and the other is an equivalent circuit
to harmonics.
Js(t)
~
AC side
Figure 7 Dependent harmonic current source
Now, to derive the new equivalent circuit containing harmonic
currents on the AC side and harmonic voltages on the DC side
of rectifiers, we discuss an equivalent circuit on a per phase
base shown in Fig.8. This model, primarily, is a combination
of Fig.4 and Fig.7.
r
isu
First, the equivalent circuit to the fundamental components on
both the AC side and the DC side of the rectifier is discussed.
In this case, the first harmonic components of current source,
Js(t), and voltage source Vd(t) in the model are considered:
J1(ωt), v1(ω1t). The source voltage, Vs(t) remains unchanged.
The equivalent circuit to the fundamental components is
obtained as in Fig.10.
isf
RL
idc
Vs(t)
RL
Vs(t)
+
Js(t)
~
Vd(t)
r
+
+
~
J1(ω1t)
v1(ω1t)
+
~
L
~
L
AC side
DC side
Figure 8. Exact harmonic model of rectifier circuit
Since the equivalent circuit shown in Fig.8 contains both AC
side and DC side of the rectifier circuit, it meets the need for
exact analysis of the rectifier circuit. The waveforms relating
to the exact equivalent circuit are given in Fig.9. It can be seen
that the waveforms obtained from the proposed equivalent
circuit model are the same as the waveforms of Fig.1.
.
AC side
DC side
Figure 10. Equivalent circuit to fundamental components on the DC and AC
side.
The fundamental component of source current, isf(t), obtained
from Fig.10 is given in Fig.11. It is the same as Fig.3.d.
Figure 11. Simulation result relating to fundamental component, isf(t)
(a) Source current , is(t)
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Next, the equivalent circuit to harmonic components on both
the AC side and the DC side of the rectifier is discussed. This
equivalent circuit can be obtained under the condition that the
source voltage Vs(t)= 0, the first component of harmonic
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4th International Conference on Power Engineering, Energy and Electrical Drives
current source J1(ω1t)= 0 and the first component of harmonic
voltage source v1(ω1t)= 0 in Fig.8. The equivalent circuit to
harmonics is shown in Fig.12.
ish
r
[4]
[5]
[6]
RL
Jh(t)
Uh
+
[7]
~
L
[8]
[9]
Fig. 12 Equivalent circuit to harmonic components on the DC and AC side
[10]
The harmonic component of source current, ish(t), obtained
from Fig.12 is given in Fig.13. It is the same as Fig.3.e.
[11]
[12]
[13]
[14]
Figure 13. Simulation result relating to harmonic component, ish(t)
IV.
CONCLUSION
For an accurate harmonic analysis of converter circuits, the
adequate models of harmonic sources are required. For this
purpose, the paper describes the exact equivalent circuit
containing current harmonics on the AC side and voltage
harmonics on DC side of three-phase diode rectifiers. In the
study, first, the conventional equivalent circuit, which is
considered as an ideal current source for harmonics, is pointed
out. Next, the new equivalent circuit relating to current and
voltage harmonics, which meet the lacks of the conventional
equivalent circuit, is proposed. Simulation results show that
the proposed equivalent circuit models properly current and
voltage harmonics of rectifier circuits
.
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