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Harmonic Analysis of a Three-Phase Diode BridgeRectifier based on Sampled-Data Model
K. L. Lian Member, IEEE, B. K. Perkins, Member, IEEE, and P. W. Lehn, Senior Member, IEEE
Abstract— This paper presents a time domain method toanalyze the three phase rectifier with capacitor output filter.As demonstrated in the paper, the proposed method analyticallyevaluates harmonics, and obtains exact switching functions byiteratively solving for the switching instants. An analytical Jaco-bian of the mismatch equations is obtained to ensure a quadraticconvergence rate for the iteration process. It is also demonstratedthat a unified approach exists to analyze converters operating inthe continuous conduction mode (CCM) and discontinuous con-duction mode (DCM). One potential application of the proposedmodel is to incorporate it into a harmonic power flow programto yield improved accuracy.
Index Terms— Diode Bridge Rectifier, Harmonics, ContinuousConduction Mode, Discontinuous Conduction Mode, Steady StateAnalysis.
I. INTRODUCTION
THREE phase diode bridge rectifiers are often used inindustry to provide the dc input voltage for motor drives
and dc-to-dc converters. The main drawback of these rectifiersis that they inject significant current harmonics into the powernetwork. These harmonics current injections can detrimentallyaffect the power system by overloading nearby shunt capaci-tors and by distorting the bus voltage at the point of commoncoupling.
Computation of harmonics is routinely accomplishedthrough use of transient time domain simulation. While thisapproach is effective, it is not without challenges. Accuracyof simulation results depends on simulation time step size andsimulation length - quantities that must be estimated based onexperience, or selected using trial and error.
An alternative approach is to employ harmonic domainanalysis methods [1], [2]. By avoiding simulation of circuittransients, these methods yield accurate steady state harmonicspectra in a more computationally efficient manner. Onceagain, user experience is required to obtain accurate harmonicresults, since accuracy depends on the numbers of harmonicsincluded during the calculation process.
In this paper, a time domain sampled-data model is pre-sented to iteratively solve for the current and voltage har-monics injected by a three-phase full bridge rectifier withcapacitive load. The computation time is short comparedto transient time domain simulation because the proposedmethod:
This work was supported by the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC) and by the University of Toronto. B. K. Perkinsis with Hatch Ltd., 2800 Speakman Drive, Mississauga, ON, L5K 2R7. P.W. Lehn and K. L. Lian are with Department of Electrical and ComputerEngineering, University of Toronto, Toronto, ON, M5S 3G4, Canada. (e-mails:[email protected], [email protected], and [email protected]).
1) directly calculates the steady state solution without step-ping through system transients.
2) can employ known waveform symmetry.
In contrast to harmonic domain analysis methods, theproposed time domain sampled-data model can accuratelydetermine harmonics of interest without concern for harmonictruncation error or aliasing effects.
Section II introduces the circuit descriptions of the rectifierbeing analyzed. Section III and IV show how to use theproposed method to analyze discontinuous conduction modes(DCM) and continuous conduction mode(CCM). A computa-tional example is presented in section V to demonstrate thevalidity of the method.
II. CIRCUIT DESCRIPTION
Fig. 1 shows a six-pulse capacitor-filtered diode bridge rec-tifier where the dc load is modeled as an equivalent resistance,Rl [3], [4]. The line harmonics are filtered by the ac chokes,L. This type of rectifier is frequently employed for batterycharger application [5]. It is also used for adjustable speeddrives [6] because it has better drive isolation and lower dccurrent requirements [7], [8] than a conventional inductor-filtered rectifier.
R L
C lR
+
-
dci
dcvai
bi
ci
sav
sbv
scv
1D 3D 5D
4D 6D 2D
n
N
Fig. 1. Six-pulse uncontrolled rectifier with capacitive dc smoothing
Surprisingly, as noted in [9], the literature available oncomprehensive analysis of this rectifier is quite limited. In fact,[3] is the only reference which provides complete analyticalmodels for both DCM and CCM without the aids of transienttime domain simulation. However, the approach in [3] requiresevaluation of a lengthy inverse Laplace Transformation, whichbecomes complicated for the case of CCM analysis. Theproposed model provides a viable alternative to [3]. DCM andCCM are modeled in an efficient and unified fashion withoutthe need for inverse Laplace Transformation.
To simplify the model development and discussion, onlybalanced operation of the converter is considered. However,
the proposed technique can well be extended to the unbalancedcase.
III. DISCONTINUOUS CONDUCTION MODE
For DCM, Fig. 2 shows typical phase currents, together withthe six line-to-line voltages and the dc voltage. As noted inthe figure, two subintervals can be identified in every sixth ofa period – conduction (η), and non-conduction (ξ = T/6− η)intervals. This repetition pattern allows one to fully describethe behavior of the rectifier by only considering 1/6 th of theperiod.
Vol
tage
vdc
vsab
vsac
vscb
vsba
vsca
vsbc
0 T/6+ T/2 Tγ
D5
D6
D1
D6
D1
D2
D3
D2
D3
D4
D4
D5
Cur
rent
Time
←γ→
← η →←ξ →← η →←ξ →← η →← ξ →← η →←ξ →← η →←ξ →← η →←ξ →i ai bi c
Fig. 2. Rectifier waveforms in the DCM
n
L
di
1D
C
sav
sbv
R
L6D
di
R
C lR
dcvlR
+
-
dcv+
-
N
N
Fig. 3. (a) Top: rectifier model during the conduction subinterval; (b) Bottom:rectifier model during the non-conduction subinterval
The conduction interval with diodes D1 and D6 on com-mences at instant γ with respect to the zero reference (theintersection point between vsab and vscb). Fig. 3(a) shows thecircuit involved during the conduction interval, and (1) givesthe corresponding differential equations. In the non-conductioninterval, D1 and D6 turn off, and the capacitor dischargesthrough the load resistance (Fig. 3(b)). The correspondingdifferential equations are given in (2).
d
dt
[x1
z
]= Aon
[x1
z
]=
[Ax1 N1
0 Ω
] [x1
z
](1)
d
dt
[x2
z
]= Aoff
[x2
z
]=
[Ax2 N2
0 Ω
] [x2
z
](2)
where x1 =[
id vdc
]T, x2 = [vdc], z =[
vsα vsβ
]T,
[vsα
vsβ
]= Cx
⎡⎣ vsa
vsb
vsc
⎤⎦ ,Cx =
23
[1 −1
2−12
0√
32
−√3
2
],
Ax1 =[ −R
L−12L
1C
−1RlC
],Ax2 =
−1RlC
,Ω =[
0 −ωω 0
],
N1 =[
34L
−√3
4L0 0
], and N2 =
[0 0
].
Note that the loop current (i.e. id in Fig. 3) and the α andβ-axis bus voltages, are chosen as state variables to yield aminimum realization for the state space formulation.
A. Diode Constraint Equations
To solve for the conduction time, γ and non-conduction in-terval length, η, formulation of two diode constraint equationsis required:
Diode D1 must turn on when forward biased, i.e. when:
Md1 = vsab(γ)−vdc(γ) =
[−1 3
2−√
32
] [x2(γ)z(γ)
]= 0
(3)Diode D1 must turn off at a current zero, i.e. when:
Md2 = ia(γ+η) =
[1 0 0 0
]eAonηQd
[x2(γ)z(γ)
]= 0
(4)
where Qd = blkdiag{Qxd, I}, I =[
1 00 1
], and Qxd =[
01
], which is associated with the change of basis [10] at the
transition instant from non-conduction to conduction interval.Expressions for x2(γ), and z(γ) in terms of unknowns (γ,
and η) and input z(0) are defined in the subsequent section.
B. Steady State Constraint Equations
Under balanced operation, the states ([
x2(γ) z(γ)]T
) atthe end points of the sixth period interval are related throughthe state transition matrix Φd according to:[
x2(T6 + γ)
z(T6 + γ)
]= Φd
[x2(γ)z(γ)
](5)
where Φd = eAoff ( T6 −η)PdeAonηQd, Pd =
blkdiag{Pxd, I}, and Pxd =[
0 1], which is associated
with the change of basis from conduction to non-conductioninterval.
In addition, the steady state sixth period symmetry that thedc voltage (Fig. 2) and ac voltage possess yields the constraint[
x2(T6 + γ)
z(T6 + γ)
]= I1/6
d
[x2(γ)z(γ)
](6)
where I1/6d = blkdiag(1,Θ), and Θ =[
cos(π3 ) − sin(π
3 )sin(π
3 ) cos(π3 )
].
Combining (5) and (6), one gets
(I1/6d − Φd)
[x2(γ)z(γ)
]=
[Dd Ed
Fd Kd
] [x2(γ)z(γ)
]= 0
(7)Therefore,
x2(γ) = −Dd−1Edz(γ) (8)
In addition, the solution for z(0) is given by (9)
z(γ) = eΩγz(0) (9)
Insert (8) and (9) into (3) and (4) yielding two transcenden-tal equations in terms of unknowns, γ and η. These equationsare solved via numerical iteration, as described in section C.
C. Numerical Iteration For Finding Unknowns
Fig. 4 shows the overall flow diagram of the proposedmethod. First, ζ =
[γ η
]Tis initialized, allowing deter-
mination of x2(γ) based on (8). Then, x2(γ) is substitutedinto the diode constraint mismatch equations (3) and (4).The mismatch equations then set the stage for a Newton-typeiterative method, which generates the sequence, {ζ}∞
n=0 by(10).
ζ(k+1) = ζ(k) − J−1M (10)
where M =[
Md1
Md2
], and J =
[∂Md
1∂γ
∂Md1
∂η∂Md
2∂γ
∂Md2
∂η
]The
iteration terminates when the difference between ζ (k+1), andζ(k) reaches a required tolerance, ε.
To have quadratic convergence, an analytical Jacobian isconstructed. For DCM, the expression of each element in theJacobian matrix is listed in Appendix I.
D. Harmonic Analysis
Once interval lengths γ and η are determined, one canproceed to solve for the current and voltage harmonics.
Since sixth period symmetry also exists for the ac currentspace vector (see Fig. 5), the evaluation of the system har-monic can proceed as follows:1) The system matrices are augmented with one additionalequation [13] for each harmonic of interest, leading to con-duction and non-conduction equations of the form:
d
dt
⎡⎣ x1
zy1
⎤⎦ =
⎡⎣ Ax1 N1 0
0 Ω 0G1 0 H1
⎤⎦
⎡⎣ x1
zy1
⎤⎦ = Ah
on
⎡⎣ x1
zy1
⎤⎦
(11)
d
dt
⎡⎣ x2
zy2
⎤⎦ =
⎡⎣ Ax2 N2 0
0 Ω 0G2 0 H2
⎤⎦
⎡⎣ x2
zy2
⎤⎦ = Ah
off
⎡⎣ x2
zy2
⎤⎦
(12)where y1 = [ Ih V l
dc ]T , and y2 = [V ldc]. Ih is the hth ac
current space vector harmonic, i(t), given as
i = iα + jiβ (13)
Calculate steady state solutions,
2( )γx based on (8) for DCM
( )kζ
k=k+11
2
d
d
M
M
⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦
MCalculate
No
( 1) ( ) 1k k+ −= −ζ ζ J M where⎡ ⎤∂= ⎢ ⎥∂⎣ ⎦
MJ
ζ
M
( 1) ( )max k k ε+ − <ζ ζ
end
Yes
3( )γx based on (18) for CCM
Set
Initialize( ) ( ),k kγ η
(with k= 0)( ) ( ) ( ) Tk k kγ η⎡ ⎤=
⎣ ⎦ζ
2( )γx or 3( )γx
for DCM
1
2
c
c
M
M
⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦
M for CCMor
Begin
Fig. 4. The flow diagram of the proposed numerical iteration process
Iα (A)
I β (A
)
Fig. 5. Ac current space vector with one-sixth period symmetry in the DCM
where [iαiβ
]= Cx
⎡⎣ ia
ibic
⎤⎦ ,
and ia, ib, ic are the three phase ac current harmonics. V ldc
represents the lth dc voltage harmonics.Since ia = id, ib = −id, ic = 0 in the conduction
subinterval (Fig. 3(a)), one can express i = (1 − j 1√3)id.
Consequently, based on [13], expressions for G1, G2, H1,and H2 are as follow:
G1 =[
(1 − j 1√3)λh 0
0 λl
],G2 =
[λl
],
λh =6T
e−jhω( T6 +γ), λl =
3T
e−jlω( T6 +γ),
H1 =[
jhω 00 jlω
],H2 = [jlω] , ω =
2π
T.
2) For characteristic harmonics, (i.e. h = 1,−5, 7,−11, . . .,and l = 0, 6, 12, 18, . . .), the system harmonics can be obtainedby evaluating (14):
[Ihsp
V ldc
]= CheAh
off ( T6 −η)Ph
deAhon(η)Qh
d
⎡⎢⎢⎣
x1(γ)z(γ)
00
⎤⎥⎥⎦ (14)
where Ch =[
0 0 I], Ph
d =[
Pd 0 00 0 I
], and
Qhd =
⎡⎣ Qd 0
0 00 I
⎤⎦ .
Note that the transition matrices, Phd, and Qh
d are neededdue to different basis of Ah
on, and Ahoff .
IV. CONTINUOUS CONDUCTION MODE
Similar to the DCM, two subintervals can also be identifiedin every sixth of the period in CCM (Fig. 6) – conduction (η),and commutation (μ = T/6− η) intervals. However, differentfrom DCM, γ now refers to the time diode (D5) turns off withrespect to the zero reference.
The differential equations describing the conduction interval(Fig. 7(a)) are the same as (1). The differential equationsdescribing the commutation interval (Fig. 7(b)) are given in(15).
d
dt
[x3
z
]= Acom
[x3
z
]=
[Ax3 N3
0 Ω
] [x3
z
](15)
where x3 =[
ie id vdc
], N3 =
⎡⎣ 1
2L−√
32L
1L 00 0
⎤⎦ , and
Ax3 =
⎡⎣ −R
L 0 −13L
0 −RL
−23L
0 1C
−1RlC
⎤⎦ .
Vol
tage
vdc
vsab
vsac
vscb
vsba
vsca
vsbc
0 T/6+ T/2 Tγ
D5
D6
D1D6D5
D1
D6
D1D6D2
D1
D2
D1D3D2
D3
D2
D3D4D2
D3
D4
D3D5D4
D4
D5
D5D4D6
Cur
rent
Time
←γ→
← η →←μ→← η →←μ→← η →←μ→← η →←μ→← η →←μ→←μ→→η ←i ai bi c
Fig. 6. Rectifier waveforms in the CCM
n
L
di
1D
C
sav
sbv
R
L6DR
dcvlR
+
-
N
n
L
di
1D
C
savR
sbv L
6D
R
scv L 2DR
dcv lR+
-
N
ei
Fig. 7. (a) Top: rectifier model during the conduction subinterval; (b) Bottom:rectifier model during the commutation subinterval
A. Diode Constraint Equations
The diode constraint equations for the interval from γ toT/6 + γ in the CCM are now determined. Noting vNn =−vdc/3 in Fig. 7(b), Diode D2 turn-on occurs when
M c1 = −3vc(γ + η) − vdc(γ + η) (16)
=[
0 −1 32
3√
32
]eAonηPc
[x3(γ)z(γ)
]= 0
where Pc = blkdiag(Pxc, I), and Pxc =[ −1
2 1 00 0 1
].
Pc is associated with change of basis at the diode transitioninstant from the commutation to the conduction interval in theCCM.
Diode D5 turn-off (refer to Fig. 6) occurs when
M c2 = −ie(γ) =
[ −1 0 0 0 0] [
x3(γ)z(γ)
]= 0
(17)
Similar to the DCM, expressions linking x3(γ) and z(γ)to input z(0) are needed to solve for γ and η from (16), and(17).
B. Steady State Constraint Equations
The steady state constraint for CCM is expressed in (18):[x3(T
6 + γ)z(T
6 + γ)
]= Φc
[x3(γ)z(γ)
](18)
where Φc = eAcom( T6 −η)Qce
AcondηPc, Qc =
blkdiag(Qxc, I), and Qxc =
⎡⎣ 0 0
1 00 1
⎤⎦ , which represents
the transition matrix from conduction to commutation intervalin the CCM.
Combining sixth period mapping constraint, (19) can beobtained.
(I1/6c − Φc)
[x3(γ)z(γ)
]=
[Dc Ec
Fc Kc
] [x3(γ)z(γ)
]= 0
⇒ x3(γ) = −Dc−1Ecz(γ) = −Dc
−1EceΩγz(0) (19)
where
I1/6c = blkdiag(Ied,Θ), and Ied =
⎡⎣ 0 1 0
−1 1 00 0 1
⎤⎦ .
As in the DCM case, constraints (16), and (17) are againtranscendental equation that must be solved numerically.
C. Numerical Iteration For Finding Unknowns
For CCM, numerical iteration is carried out just as outlinedin Fig. 4, albeit with new constraint equations and a newJacobian. The iteration process yields outputs γ, and η. The el-
ements of the required Jacobian matrix, J =
[∂Mc
1∂γ
∂Mc1
∂η∂Mc
2∂γ
∂Mc2
∂η
]are listed in the Appendix II.
D. Harmonic Analysis
As shown in Fig. 8, the current space vector of CCM alsoexhibits sixth period symmetry. Consequently, the harmonicanalysis can proceed as follows:
1) Similar to the analysis of the DCM, the system matricesof CCM are augmented with one additional equation for eachharmonic of interest. Consequently, (1) becomes (11), and (15)becomes (20).
d
dt
⎡⎣ x3
zy1
⎤⎦ =
⎡⎣ Ax3 N3 0
0 Ω 0G3 0 H1
⎤⎦
⎡⎣ x3
zy1
⎤⎦ = Ah
com
⎡⎣ x3
zy1
⎤⎦
(20)where
G3 =[
(j 2√3)λh (1 − j 1√
3)λh
0 λl
].
The expression of G3 is obtained based on the fact thati = iα + jiβ, and[
iαiβ
]=
23
[0 3
2√3 −√
32
] [ieid
](21)
Iα (A)
I β (A
)
Fig. 8. Ac current space vector with one-sixth period symmetry in the CCM
during the commutation interval (Fig. 7(b)).2) For characteristic harmonics, (i.e. h = 1,−5, 7,−11, . . .,
and l = 0, 6, 12, 18, . . .), the system harmonics can be foundby evaluating (22):
[Ihsp
V ldc
]= CheAh
com( T6 −β)Ph
c eAhon(β)Qh
c
⎡⎢⎢⎣
x1(γ)z(γ)
00
⎤⎥⎥⎦ (22)
where Phc =
[Pc 00 I
], and Qh
c =[
Qc 00 I
].
V. GENERAL COMMENTS
From the above analysis, one can immediately note thestrong similarity between the analysis of DCM and CCM,contrary to the claims of some authors [5], [14] that theanalysis of CCM is more complex than that of DCM.
In total, there are four different modes [15], [16] of con-verter operation: two discontinuous cases (mode 1 and 3), andtwo continuous cases (mode 2 and 4) .
Mode 1 (also called 2/0 mode) happens when the intervalsof conduction via two diodes alternate with intervals of zeroconduction. Mode 2 (also called 3/2 mode) happens whenconduction is via alternate 3- and 2- diode paths. Mode 3 (alsocalled 2/3/2/0 mode) occurs when an interval of conduction viatwo diodes is followed by a 3-diode interval; this is followedby another 2-diode interval and then by a zero-conductioninterval. In mode 4 (or 3/3 mode), conduction occurs via asequence 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]. Mostof the existing literature [3], [17] only analyze mode 1for the DCM.
2) Operation at mode 4 is rare for it is very close tothe short-circuit point [15], [16]. Most of the existingliterature [3], [5], [14] only refer mode 2 as the CCM.
3) The proposed method can be easily extended to mode 3with slightly added complexity.
4) Constant voltage load can be assumed in mode 4 [9],[18] to have fairly accurate results, and this greatlysimplifies harmonic analysis.
Also, note that the boundary conditions for each mode hasbeen identified by [8], [16] via brute force time domain sim-ulation. Deriving a closed form expression for the boundarycondition for each mode is not the objective of this paper.
VI. SIMULATION EXAMPLES
In order to demonstrate the validity of the proposed method,two sets of parameters, extracted from [7], are chosen to resultin continuous and discontinuous conduction modes respec-tively. Solutions are compared with those of PSCAD/EMTDC.
A. Discontinuous Mode
To analyze the DCM, the following system parameters areused: R = 0.001Ω, L = 0.1mH , C = 1000μF , Rl = 25Ω,and source voltages:
vsa = 120√
2 sin(377t)vsb = 120
√2 sin(377t− 2π
3 )vsc = 120
√2 sin(377t + 2π
3 )
Fig. 9 shows ac currents, iα and iβ , and dc capacitor voltage,vdc, predicted by PSCAD/EMTDC.
−400
0
284.7908
400
Vol
tage
(V
)
vsab
vscb
vdc
0 0.0083 0.0167 −50
0
50
Time (Sec)
Cur
rent
(A
)
γ←→ η← →
iα iβ
Fig. 9. Voltage and current waveforms in the DCM produced byPSCAD/EMTDC
The values of diode conduction instant, γ, and conductioninterval, η, and x2(γ) (i.e. vdc(γ)) in Fig. 9 are listed in TableI to compare with those predicted by the proposed method.
TABLE I
RESULTS FROM PSCAD/EMTDC AND THE SAMPLED-DATA MODEL
METHOD IN THE DCM
PSCAD/EMTDC Sampled-Data Modelγ 7.3927 × 10−4s 7.2553 × 10−4sη 1.6355 × 10−3s 1.6030 × 10−3svdc(γ) 284.7908 V 284.7951V
Fig. 10 and 11 compare ac current space vector and dcvoltage harmonics obtained from PSCAD/EMTDC, and theproposed sampled-data model method. Fig. 11 are shownin log scale because the dc voltage harmonic rolls off veryrapidly.
As can be seen from Table I, Fig. 10 and 11, excellentagreement exists between the two approaches.
−17 −11 −5 1 7 13 190
2
4
6
8
10
12
14
harm
onic
am
plitu
de (
A)
harmonic number
PSCADSampled−data Model
Fig. 10. Complex harmonic spectrum of the capacitor filtered rectifier in theDCM
0 6 12 180.01
0.1
1
10
100
200 300
harm
onic
am
plitu
de (
V)
in lo
g sc
ale
harmonic number
PSCADSampled−data Model
Fig. 11. Dc voltage harmonic spectrum of the capacitor filtered rectifier inthe DCM
B. Continuous Mode
As shown in [7], the rectifier operates in the CCM when theinductance of the ac choke in the DCM is changed to 3mHand the rest of the parameters are kept intact.
Fig. 12 shows ac currents, iα and iβ , and dc capacitorvoltage, vdc, predicted by PSCAD/EMTDC.
Table II lists the extinction time instant, γ, conductioninterval, η and the values of the steady state ac currents and dc
−600
0
267.6064
600
Vol
tage
(V
)
vsab
vscb
−3vsc
vdc
0 0.0083 0.0167 −20
−4.4297
0
7.6719
20
Time (sec)
Cur
rent
(A
)
γ← → η← →
iαiβ
Fig. 12. Voltage and current waveforms in the CCM produced byPSCAD/EMTDC
voltages obtained from Fig. 12, together with those obtainedfrom the sampled-data model.
Note that ie(γ) and id(γ) obtained from (19) has beenconverted to iα(γ) and iβ(γ) as shown in Table II via (21) soas to be comparable with PSCAD/EMTDC results.
TABLE II
RESULTS FROM PSCAD/EMTDC AND THE SAMPLE-DATA MODEL
METHOD IN THE CCM
PSCAD/EMTDC Sampled-Data Modelγ 1.0631 × 10−3s 1.0441 × 10−3sη 1.8406 × 10−3s 1.8159 × 10−3siα(γ) 7.6719A 7.6593Aiβ(γ) −4.4297A −4.4221Avdc(γ) 267.6064V 267.6159V
The ac current space vector and dc voltage harmonicsobtained from PSCAD/EMTDC, and the proposed sampled-data model method are shown in Fig. 13 and 14, respectively.
Similar to the case of the DCM, excellent agreement existbetween the time domain simulation and the proposed modelfor the CCM.
VII. CONCLUSIONS
A time domain sampled-data model method for the com-putation of the ac current and dc voltage harmonic generatedby a capacitor filtered three-phase uncontrolled rectifier is pre-sented. The approach employs numerical iteration to determinethe diode’s turn-on and time turn-off times and thereby deter-mine the circuit’s steady steady solution. Harmonics of interestare solved analytically through a state augmentation method.The results have been validated with those of PSCAD/EMTDCto show the accuracy of the method. One potential applicationof the proposed model would be to incorporate it into aharmonic power flow program to improve the accuracy of theexisting methods.
APPENDIX IANALYTICAL JACOBIAN - DCM
In DCM the Jacobian matrix or the system may be foundanalytically in accordance with the following equations.
−17 −11 −5 1 7 13 190
2
4
6
8
10
12
harm
onic
am
plitu
de (
A)
harmonic number
PSCADSampled−data Model
Fig. 13. Complex harmonic spectrum of the capacitor filtered rectifier in theCCM
0 6 12 180.01
0.1
1
10
100
200 300
harm
onic
am
plitu
de (
V)
in lo
g sc
ale
harmonic number
PSCADSampled−data Model
Fig. 14. Dc voltage harmonic spectrum of the capacitor filtered rectifier inthe CCM
∂Md1
∂γ=
[32
−√3
2
]eΩγΩz(0) − ∂x2(γ)
∂γ(23)
∂Md1
∂η= −∂x2(γ)
∂η(24)
∂Md2
∂γ=
[1 0
] {eAx1ηQxd∂x2(γ)
∂γ+ Γ1 + N1eΩηz(γ)−
eAx1ηN1z(γ)} (25)
∂Md2
∂η=
[1 0
] {Ax1eAx1ηQxdx2(γ)+eAx1ηQxd∂x2(γ)
∂η+
Γ1 + N1eΩηz(γ)} (26)
where
∂x2(γ)∂γ
= (1 − Φxd)−1eAx2ξPxd{Γ1 + N1eΩηz(γ) −eAx1ηN1z(γ)},
∂x2(γ)∂η
= −(1−Φxd)−2(Ax2Φxd−eAx2ξPxdeAx1ηAx1Qxd)·
eAx2ξPxdΓ2 − (1 − Φxd)−1eAx2ξAx2PxdΓ2+
(1 − Φxd)−1eAx2ξPxd{Γ1 + N1eΩηz(γ)}.
Φxd = eAx2(T6 −η)PxdeAx1ηQxd,
Γ1 =
γ+η∫γ
eAx1(γ+η−τ)Ax1N1z(τ)dτ,
Γ2 =
γ+η∫γ
eAx1(γ+η−τ)N1z(τ)dτ.
Note that the two convolution integrals, Γ1 and Γ2 can beeasily evaluated by using the matrix augmentation techniquepresented in [12]:
Γ1 =[
1 0 0 00 1 0 0
]eAonGη
⎡⎣ 0
0z(γ)
⎤⎦ (27)
where AonG =[
Ax1 Ax1N1
0 Ω
].
Γ2 =[
1 0 0 00 1 0 0
]eAonη
⎡⎣ 0
0z(γ)
⎤⎦ (28)
APPENDIX IIANALYTICAL JACOBIAN - CCM
In CCM the Jacobian matrix or the system may be foundanalytically in accordance with the following equations.
∂M c1
∂γ=
[32
3√
32
]eΩ(γ+η)Ωz(0) − [
0 1] ·
eAx1(η)pxc∂x3(γ)
∂γ(29)
∂M c1
∂η=
[32
3√
32
]eΩ(γ+η)Ωz(0) − [
0 1] ·
{eAx1(η)Ax1pxcx3(γ) + eAx1(η)pxc∂x3(γ)
∂η}
(30)
∂M c2
∂γ= − [
1 0 0] ∂x3(γ)
∂γ(31)
∂M c2
∂η= − [
1 0 0] ∂x3(γ)
∂η(32)
where
∂x3(γ)∂γ
= [Ied − Φxc]−1
eAx3μQxc{Γ1 + N1eΩηz(γ) −eAx1ηN1z(γ)} + Γ3 + N3eΩ(T/6)z(γ) −eAx3μN3eΩηz(γ),
∂x3(γ)∂η
= − [Ied − Φxc]−2 (Ax3Φxc − eAx3μQxce
Ax1η ·Ax1Pxc) · (eAx3μQxcΓ2 + Γ4 + [I6 − Φxc]−1 ·{−Ax3eAx3μQxcΓ2 + eAx3μQxc[Γ1 +N1eΩμz(γ)] − eAx3μN3eΩηz(γ)},
Γ3 =
T/6+γ∫γ+η
eAx3(T/6+γ−τ)Ax3N3z(τ)dτ,
Γ4 =
T/6+γ∫γ+η
eAx3(T/6+γ−τ)N3z(τ)dτ.
Note that both Γ3 and Γ4 can be found as follows:
Γ3 =
⎡⎣ 1 0 0 0 0
0 1 0 0 00 0 1 0 0
⎤⎦ eAcomGμ
⎡⎢⎢⎣
000
z(γ + μ)
⎤⎥⎥⎦ (33)
where AcomG =[
Ax3 N3
0 Ω
].
Γ4 =
⎡⎣ 1 0 0 0 0
0 1 0 0 00 0 1 0 0
⎤⎦ eAcomμ
⎡⎢⎢⎣
000
z(γ + μ)
⎤⎥⎥⎦ (34)
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K. L. Lian received the B.A.Sc.(Hons.), M.A.Sc,Ph. D. degrees in electrical engineering in 2001,2003, and 2007, respectively, all from the Univer-sity of Toronto. He is currently a visiting researchscientist at the Central Research Institute of ElectricPower Industry (CRIEPI) in Japan. His research in-terests include mathematical modeling and analysisof nonlinear and power electronic converters and realtime simulations of power systems.
Brian K. Perkians has been involved in a broadrange of industrial projects ranging from industrialpower distribution to smelting furnace applicationssince joining Hatch in 2000. Prior to joining Hatch,Brian acquired a broad range of experience in bothacademic and industrial milieus. After completinghis Ph.D. in Power Systems at the University ofToronto (1997), he served as a post-doctoral internwith Siemens AG in Erlangen, Germany where hecontributed to active filter development (the SIPCONproduct line) and developed software for the design
and evaluation of rectifier harmonic compensation filters. This software hasbeen used for the design and verification of compensation schemes for rectifierload associated with electrolysis and aluminum smelting applications.
P. W. Lehn received the B.Sc. and M.Sc. degreesin electrical engineering from the University ofManitoba in 1990 and 1992, respectively, and thePh.D. degree from the University of Toronto in1999. From 1992 to 1994, he was with the NetworkPlanning Group of Siemens AG, Erlangen, Germany.Currently, he is an Associate Professor at the Uni-versity of Toronto. His research interests includemodelling and control of converters, and integrationof renewable energy source into the power grid.
