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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 2, APRIL 2008
Harmonic Analysis of a Three-Phase Diode Bridge
Rectifier Based on Sampled-Data Model
K. L. Lian, Brian K. Perkins, and P. W. Lehn
Abstract—This paper presents a time domain method to analyze
the three phase rectifier with capacitor output filter. As demonstrated in the paper, the proposed method analytically evaluates
harmonics, and obtains exact switching functions by iteratively
solving for the switching instants. An analytical Jacobian of the
mismatch equations is obtained to ensure a quadratic convergence
rate for the iteration process. It is also demonstrated that a unified
approach exists to analyze converters operating in the continuous
conduction mode and discontinuous conduction mode. One potential application of the proposed model is to incorporate it into a
harmonic power flow program to yield improved accuracy.
Index Terms—Continuous conduction mode (CCM), diode
bridge rectifier, discontinuous conduction mode (DCM), harmonics, steady-state analysis.
I. INTRODUCTION
T
HREE phase diode bridge rectifiers are often used in industry to provide the dc input voltage for motor drives and
dc-to-dc converters. The main drawback of these rectifiers is that
they inject significant current harmonics into the power network.
These harmonics current injections can detrimentally affect the
power system by overloading nearby shunt capacitors and by
distorting the bus voltage at the point of common coupling.
Computation of harmonics is routinely accomplished through
use of transient time domain simulation. While this approach
is effective, it is not without challenges. Accuracy of simulation results depends on simulation time step size and simulation
length—quantities that must be estimated based on experience,
or selected using trial and error.
An alternative approach is to employ harmonic domain
analysis methods [1], [2]. By avoiding simulation of circuit
transients, these methods yield accurate steady-state harmonic
spectra in a more computationally efficient manner. Once again,
user experience is required to obtain accurate harmonic results,
since accuracy depends on the numbers of harmonics included
during the calculation process.
In this paper, a time domain sampled-data model is presented
to iteratively solve for the current and voltage harmonics injected by a three-phase full bridge rectifier with capacitive load.
Manuscript received October 30, 2006. This work was supported in part by
the Natural Sciences and Engineering Research Council of Canada (NSERC)
and in part by the University of Toronto. Paper no. TPWRD-00672-2006.
K. L. Lian and P. W. Lehn are with the Department of Electrical and
Computer Engineering, University of Toronto, Toronto, ON M5S 3G4 Canada
(e-mail: liank@ecf.utoronto.ca; lehn@ecf.utoronto.ca).
B. K. Perkins is with Hatch, Ltd., Mississauga, ON L5K 2R7 Canada (e-mail:
bperkins@hatch.ca).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRD.2008.917671
Fig. 1. Six-pulse uncontrolled rectifier with capacitive dc smoothing.
The computation time is short compared to transient time domain simulation because the proposed method
1) directly calculates the steady-state solution without stepping through system transients;
2) can employ known waveform symmetry.
In contrast to harmonic domain analysis methods, the proposed time domain sampled-data model can accurately determine harmonics of interest without concern for harmonic truncation error or aliasing effects.
Section II introduces the circuit descriptions of the rectifier
being analyzed. Sections III and IV show how to use the
proposed method to analyze discontinuous conduction modes
(DCM) and continuous conduction mode (CCM). A computational example is presented in Section V to demonstrate the
validity of the method.
II. CIRCUIT DESCRIPTION
Fig. 1 shows a six-pulse capacitor-filtered diode bridge rectifier where the dc load is modeled as an equivalent resistance,
[3], [4]. The line harmonics are filtered by the ac chokes, .
This type of rectifier is frequently employed for battery charger
application [5]. It is also used for adjustable speed drives [6] because it has better drive isolation and lower dc current requirements [7], [8] than a conventional inductor-filtered rectifier.
Surprisingly, as noted in [9], the literature available on comprehensive analysis of this rectifier is quite limited. In fact, [3] is
the only reference which provides complete analytical models
for both DCM and CCM without the aids of transient time domain simulation. However, the approach in [3] requires evaluation of a lengthy inverse Laplace Transformation, which becomes complicated for the case of CCM analysis. The proposed
model provides a viable alternative to [3]. DCM and CCM are
modeled in an efficient and unified fashion without the need for
inverse Laplace Transformation.
To simplify the model development and discussion, only
balanced operation of the converter is considered. However,
0885-8977/$25.00 © 2008 IEEE
LIAN et al.: HARMONIC ANALYSIS OF A THREE-PHASE DIODE BRIDGE
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where
and
Fig. 2. Rectifier waveforms in the DCM.
Note that the loop current (i.e., in Fig. 3) and the and
-axis bus voltages, are chosen as state variables to yield a minimum realization for the state space formulation.
A. Diode Constraint Equations
To solve for the conduction time, and nonconduction interval length, , formulation of two diode constraint equations
is required.
must turn on when forward biased, i.e., when
Diode
(3)
Fig. 3. (a) Top: rectifier model during the conduction subinterval. (b) Bottom:
rectifier model during the nonconduction subinterval.
Diode
must turn off at a current zero, i.e., when
the proposed technique can well be extended to the unbalanced
case.
III. DISCONTINUOUS CONDUCTION MODE
(4)
where
For DCM, Fig. 2 shows typical phase currents, together with
the six line-to-line voltages and the dc voltage. As noted in the
figure, two subintervals can be identified in every sixth of a pe, and nonconduction
inriod—conduction
tervals. This repetition pattern allows one to fully describe the
behavior of the rectifier by only considering 1/6th of the period.
The conduction interval with diodes
and
on commences at instant with respect to the zero reference (the intersection point between
and
). Fig. 3(a) shows the circuit
involved during the conduction interval, and (1) gives the corresponding differential equations. In the nonconduction interval,
and
turn off, and the capacitor discharges through the
load resistance [Fig. 3(b)]. The corresponding differential equations are given in (2)
, and
,
which is associated with the change of basis [10] at the transition
instant from nonconduction to conduction interval.
, and
in terms of unknowns ( ,
Expressions for
are defined in the subsequent section.
and ) and input
B. Steady-State Constraint Equations
Under balanced operation, the states
at the
end points of the sixth period interval are related through the
according to
state transition matrix
(5)
where
(1)
, and
, which is associated with
the change of basis from conduction to nonconduction interval.
In addition, the steady-state sixth period symmetry that the
dc voltage (Fig. 2) and ac voltage possess yields the constraint
(2)
(6)
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Fig. 5. AC current space vector with one-sixth period symmetry in the DCM.
based on (8). Then,
is substituted into the diode
constraint mismatch (3) and (4). The mismatch equations then
set the stage for a Newton-type iterative method, which generby (10)
ates the sequence,
(10)
where
and
The iteration terminates when the difference between
,
reaches a required tolerance, .
and
To have quadratic convergence, an analytical Jacobian is constructed. For DCM, the expression of each element in the Jacobian matrix is listed in the Appendix.
Fig. 4. Flow diagram of the proposed numerical iteration process.
where
and
D. Harmonic Analysis
Combining (5) and (6), one gets
(7)
therefore
Once interval lengths and are determined, one can proceed to solve for the current and voltage harmonics.
Since sixth period symmetry also exists for the ac current
space vector (see Fig. 5), the evaluation of the system harmonic
can proceed as follows.
1) The system matrices are augmented with one additional
equation [13] for each harmonic of interest, leading to conduction and nonconduction equations of the form
(8)
In addition, the solution for
is given by
(9)
(11)
Insert (8) and (9) into (3) and (4) yielding two transcendental
equations in terms of unknowns, and . These equations are
solved via numerical iteration, as described in Section III-C.
C. Numerical Iteration for Finding Unknowns
Fig. 4 shows the overall flow diagram of the proposed method.
First,
is initialized, allowing determination of
(12)
LIAN et al.: HARMONIC ANALYSIS OF A THREE-PHASE DIODE BRIDGE
where
and
current space vector harmonic,
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is the th ac
, given as
(13)
where
and
are the three phase ac current harmonics.
represents the th dc voltage harmonics.
in the conduction subinSince
.
terval [Fig. 3(a)], one can express
,
Consequently, based on [13], expressions for
and
are as follows:
Fig. 6. Rectifier waveforms in the CCM.
2) For characteristic harmonics, (i.e.,
,
), the system harmonics can be
and
obtained by evaluating (14)
Fig. 7. (a) Top: rectifier model during the conduction subinterval. (b) Bottom:
rectifier model during the commutation subinterval.
(14)
where
where
and
and
Note that the transition matrices,
, and
due to different basis of
, and
.
are needed
IV. CONTINUOUS CONDUCTION MODE
Similar to the DCM, two subintervals can also be identified
in every sixth of the period in CCM (Fig. 6)—conduction
,
intervals. However, different
and commutation
turns off with
from DCM, now refers to the time diode
respect to the zero reference.
The differential equations describing the conduction interval
[Fig. 7(a)] are the same as (1). The differential equations describing the commutation interval [Fig. 7(b)] are given in (15)
A. Diode Constraint Equations
The diode constraint equations for the interval from
to
in the CCM are now determined. Noting
in Fig. 7(b), diode
turn-on occurs when
(16)
where
(15)
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and
where
is associated with change of basis at the diode transition instant from the commutation to the conduction interval in
the CCM.
turn-off (refer to Fig. 6) occurs when
Diode
(17)
Similar to the DCM, expressions linking
and
to
are needed to solve for and from (16) and (17).
input
B. Steady-State Constraint Equations
The steady-state constraint for CCM is expressed in (18)
(18)
Fig. 8. AC current space vector with one-sixth period symmetry in the CCM.
where
D. Harmonic Analysis
and
As shown in Fig. 8, the current space vector of CCM also exhibits sixth period symmetry. Consequently, the harmonic analysis can proceed as follows.
1) Similar to the analysis of the DCM, the system matrices of
CCM are augmented with one additional equation for each
harmonic of interest. Consequently, (1) becomes (11), and
(15) becomes (20)
which represents the transition matrix from conduction to commutation interval in the CCM.
Combining sixth period mapping constraint, (19) can be obtained
(20)
where
(19)
where
and
As in the DCM case, constraints (16), and (17) are again transcendental equation that must be solved numerically.
The expression of
and
is obtained based on the fact that
(21)
during the commutation interval [Fig. 7(b)].
,
2) For characteristic harmonics, (i.e.,
), the system harmonics can be
and
found by evaluating (22)
C. Numerical Iteration for Finding Unknowns
For CCM, numerical iteration is carried out just as
outlined in Fig. 4, albeit with new constraint equations
and a new Jacobian. The iteration process yields outputs
, and . The elements of the required Jacobian matrix
are
listed in the Appendix II.
(22)
LIAN et al.: HARMONIC ANALYSIS OF A THREE-PHASE DIODE BRIDGE
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where
and
V. GENERAL COMMENTS
From the above analysis, one can immediately note the strong
similarity between the analysis of DCM and CCM, contrary to
the claims of some authors [5], [14] that the analysis of CCM is
more complex than that of DCM.
In total, there are four different modes [15], [16] of converter
operation: two discontinuous cases (mode 1 and 3), and two
continuous cases (mode 2 and 4).
Mode 1 (also called 2/0 mode) happens when the intervals of
conduction via two diodes alternate with intervals of zero conduction. Mode 2 (also called 3/2 mode) happens when conduction is via alternate 3-and 2-diode paths. Mode 3 (also called
2/3/2/0 mode) occurs when an interval of conduction via two
diodes is followed by a 3-diode interval; this is followed by another 2-diode interval and then by a zero-conduction interval.
In mode 4 (or 3/3 mode), conduction occurs via a sequence of
3-diode paths.
In this paper, only mode 1 and 2 are discussed because:
1) The operating range of mode 3 is very small [16]. Most of
the existing literature [3], [17] only analyze mode 1 for the
DCM.
2) Operation at mode 4 is rare for it is very close to the shortcircuit point [15], [16]. Most of the existing literature [3],
[5], [14] only refer mode 2 as the CCM.
3) The proposed method can be easily extended to mode 3
with slightly added complexity.
4) Constant voltage load can be assumed in mode 4 [9], [18]
to have fairly accurate results, and this greatly simplifies
harmonic analysis.
Also, note that the boundary conditions for each mode has
been identified by [8] and [16] via brute force time domain simulation. Deriving a closed form expression for the boundary condition for each mode is not the objective of this paper.
Fig. 9. Voltage and current waveforms in the DCM produced by PSCAD/
EMTDC.
TABLE I
RESULTS FROM PSCAD/EMTDC AND THE
SAMPLED-DATA MODEL METHOD IN THE DCM
VI. SIMULATION EXAMPLES
In order to demonstrate the validity of the proposed method,
two sets of parameters, extracted from [7], are chosen to result in
continuous and discontinuous conduction modes respectively.
Solutions are compared with those of PSCAD/EMTDC.
A. Discontinuous Mode
To analyze the DCM, the following system parameters are
used:
,
and source voltages
Fig. 10. Complex harmonic spectrum of the capacitor filtered rectifier in the
DCM.
Fig. 9 shows ac currents, and , and dc capacitor voltage,
, predicted by PSCAD/EMTDC.
The values of diode conduction instant, , and conduction
(i.e.,
) in Fig. 9 are listed in Table I
interval, , and
to compare with those predicted by the proposed method.
Figs. 10 and 11 compare ac current space vector and dc
voltage harmonics obtained from PSCAD/EMTDC, and the
proposed sampled-data model method. Fig. 11 are shown in log
scale because the dc voltage harmonic rolls off very rapidly.
As can be seen from Table I, Figs. 10 and 11, excellent agreement exists between the two approaches.
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 2, APRIL 2008
Fig. 11. DC voltage harmonic spectrum of the capacitor filtered rectifier in the
DCM.
Fig. 13. Complex harmonic spectrum of the capacitor filtered rectifier in the
CCM.
Fig. 12. Voltage and current waveforms in the CCM produced by PSCAD/
EMTDC.
TABLE II
RESULTS FROM PSCAD/EMTDC AND THE SAMPLE-DATA
MODEL METHOD IN THE CCM
B. Continuous Mode
As shown in [7], the rectifier operates in the CCM when the
inductance of the ac choke in the DCM is changed to 3 mH and
the rest of the parameters are kept intact.
Fig. 12 shows ac currents, and , and dc capacitor voltage,
, predicted by PSCAD/EMTDC.
Table II lists the extinction time instant, , conduction interval, and the values of the steady-state ac currents and dc
voltages obtained from Fig. 12, together with those obtained
from the sampled-data model.
Fig. 14. DC voltage harmonic spectrum of the capacitor filtered rectifier in the
CCM.
and
obtained from (19) has been conNote that
and
as shown in Table II via (21) so as to
verted to
be comparable with PSCAD/EMTDC results.
The ac current space vector and dc voltage harmonics obtained from PSCAD/EMTDC, and the proposed sampled-data
model method are shown in Figs. 13 and 14, respectively.
Similar to the case of the DCM, excellent agreement exist
between the time domain simulation and the proposed model
for the CCM.
VII. CONCLUSION
A time domain sampled-data model method for the computation of the ac current and dc voltage harmonic generated
by a capacitor filtered three-phase uncontrolled rectifier is presented. The approach employs numerical iteration to determine
the diode’s turn-on and time turn-off times and thereby determine the circuit’s steady steady solution. Harmonics of interest
are solved analytically through a state augmentation method.
LIAN et al.: HARMONIC ANALYSIS OF A THREE-PHASE DIODE BRIDGE
1095
The results have been validated with those of PSCAD/EMTDC
to show the accuracy of the method. One potential application
of the proposed model would be to incorporate it into a harmonic power flow program to improve the accuracy of the existing methods.
Note that the two convolution integrals,
and
can be
easily evaluated by using the matrix augmentation technique
presented in [12]
(27)
APPENDIX
ANALYTICAL JACOBIAN-DCM
where
.
In DCM the Jacobian matrix or the system may be found
analytically in accordance with the following equations.
(28)
(23)
APPENDIX
ANALYTICAL JACOBIAN-CCM
(24)
In CCM the Jacobian matrix or the system may be found analytically in accordance with the following equations.
(25)
(29)
(26)
where
(30)
(31)
(32)
where
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Note that both
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 2, APRIL 2008
and
can be found as follows:
(33)
where
.
(34)
[12] C. F. Van Loan, “Computing integrals involving the matrix exponential,” IEEE Trans. Autom. Control, vol. AC-23, no. 3, pp. 395–404, Jun.
1978.
[13] P. W. Lehn, “Direct harmonic analysis of the voltage source converter,”
IEEE Trans. Power Del., vol. 18, no. 3, pp. 1024–1042, Jul. 2003.
[14] S. Hansen, L. Asiminoaei, and F. Blaabjerg, “Simple and advanced
methods for calculating six-pulse diode rectifier line-side harmonics,”
in Proc. 38th Ind. Applicat. Conf., Oct. 2003, vol. 3, pp. 2056–2062.
[15] M. Hancock, “Rectifier action with constant load voltage: Infinite-Capacitance condition,” in Proc. Inst. Elect. Eng. London, 1973, vol. 120,
no. 12, pp. 1529–1530.
[16] W. F. Ray, “The effect of supply reactance on regulation and power
factor for an unctrolled 3-phase bridge rectifier with a capactive load,”
in Proc. Int. Conf. Power Electron. Variable-Speed Drives, May 1–4,
1984, pp. 111–114.
[17] M. Grotzbach and B. Draxler, “Line side behaviour of uncontrolled
rectifier bridges with capacitive dc smoothing,” in Proc. 3rd EPE Conf.,
Aachen, Germany, 1989, pp. 761–764.
[18] V. Caliskan, D. Perreault, T. Jahns, and J. Kassakian, “Analysis of
three-phase rectifiers with constant-voltage loads,” IEEE Trans Circuits
Syst., vol. 50, no. 9, pp. 1220–1225, Sep. 2003.
K. L. Lian received the B.A.Sc. (Hons.), M.A.Sc.,
and Ph.D. degrees in electrical engineering in 2001,
2003, and 2007, respectively, all from the University
of Toronto, Toronto, ON, Canada.
He is currently a Visiting Research Scientist at the
Central Research Institute of Electric Power Industry
(CRIEPI), Japan. His research interests include mathematical modeling and analysis of nonlinear components and power-electronic converters, and real-time
simulations for power systems.
REFERENCES
[1] M. Chen, Z. Qian, and X. Yuan, “Frequency-domain analysis of uncontrolled rectifiers,” in Proc. Appl. Power Electron. Conf. Expo., 2004,
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[2] M. Sakuoi, H. Fujita, and M. Shioya, “A method for calculating harmonic currents of a three-phase bridge uncontrolled rectifier with DC
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[3] G. Carpinelli, F. Iacovone, A. Russo, P. Varilone, and P. Verde, “Analytical modeling for harmonic analysis of line current of vsi-fed drives,”
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[5] J. Schaefer, Rectifier Circuits: Theory and Design. New York: Wiley,
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[6] A. Emanuel and J. Orr, “Six-pulse converter atypical harmonics caused
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[7] B. Pilvelait, T. Ortmeyer, and M. Grizer, “Harmonic evaluation of inductor location in A variable speed drive,” in Proc. ICHPS V Int. Conf.
Harmonics Power Systems., Sep. 22–25, 1992, pp. 267–271.
[8] M. Grotzbach and R. Reiner, “Line current harmonics of vsi-fed
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Brian K. Perkians received the Ph.D. degree in
power systems from the University of Toronto,
Toronto, ON, Canada, in 1997.
He has been involved in a broad range of industrial
projects ranging from industrial power distribution
to smelting furnace applications since joining
Hatch, Ltd., Mississauga, ON, Canada, in 2000.
Prior to joining Hatch, he acquired a broad range
of experience in academic and industrial milieus.
He was a Postdoctoral Intern with Siemens AG,
Erlangen, Germany, where he contributed to active
filter development (the SIPCON product line) and developed software for the
design and evaluation of rectifier harmonic compensation filters. This software
has been used for the design and verification of compensation schemes for
rectifier load associated with electrolysis and aluminum smelting applications.
P. W. Lehn received the B.Sc. and M.Sc. degrees in
electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 1990 and 1992, respectively, and the Ph.D. degree from the University
of Toronto, Toronto, ON, Canada, in 1999.
From 1992 to 1994, he was with the Network Planning Group, Siemens AG, Erlangen, Germany. Currently, he is an Associate Professor with the University of Toronto. His research interests include modeling and control of converters and integration of renewable energy source into the power grid.