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Electric Power Components and Systems

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Symmetrical Component Analysis of Multi-PulseConverter Systems

M. Abdel-Salam , S. Abdel-Sattar , A. S. Abdallah & H. Ali

To cite this article: M. Abdel-Salam , S. Abdel-Sattar , A. S. Abdallah & H. Ali (2006) SymmetricalComponent Analysis of Multi-Pulse Converter Systems, Electric Power Components and Systems,34:8, 867-888, DOI: 10.1080/15325000600561597

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Electric Power Components and Systems, 34:867–888, 2006Copyright © Taylor & Francis Group, LLCISSN: 1532-5008 print/1532-5016 onlineDOI: 10.1080/15325000600561597

Symmetrical Component Analysis ofMulti-Pulse Converter Systems

M. ABDEL-SALAMS. ABDEL-SATTARA. S. ABDALLAHH. ALI

Electrical Engineering DepartmentAssiut UniversityAssiut, Egypt

This article describes a new method for dynamic simulation of multi-converter sys-tems. This simulation is based on symmetrical components in time domain analysisand general representation of converter transformers to meet Y/�, Y/Y , and Y/Zconnections. The simulation is suitable for harmonic analysis of balanced and unbal-anced AC voltages. The computed currents and voltages agreed reasonably with thosemeasured and reported in the literature for the characteristic and non-characteristicharmonics.

Keywords symmetrical components, multi-pulse converters, harmonic analysis, phaseshift transformers

Introduction

Static power converters have many fields of applications in modern life. These applica-tions are extended from large power converters in electrical and industrial utilities up tosmall power converters in battery chargers and small power electrical devices such aspersonal computers and TV set receivers [1, 2].

Normally, the static power converters use electronic switching devices such as thyris-tors, diodes, power transistors and gate-turn off thyristors to convert bi-directional currentsand voltages into unidirectional currents and voltages or other inverter system [3]. Theincreasing use of these electronic switching devices is due to ease of maintenance, re-ducing their volt-ampere absorption and their high reliability in controlling process [4].However, they have many problems in power systems feeding due to their non-linearityand harmonic content. Such excessive use of these devices can reach 50%–70% of powerload in the nearest future, especially in industrial countries [5–8].

Therefore, study of converter system simulation is an effective tool to evaluate har-monic content with the aim to minimize it to acceptable levels according to standardlimits-IEEE-519-1992. Modeling of phase shift transformer represents difficulties in bothAC and converter power systems [9, 10]. Some mathematical models for dynamic sim-ulation of multi-pulse converters are reported [9, 11] in the literature. A technique was

Manuscript received in final form on 28 November 2005.Address correspondence to Prof. Mazen Abdel-Salam, Electrical Engineering Dept., Faculty

of Engineering, Assiut, 71516, Egypt. E-mail: [email protected]

867

868 M. Abdel-Salam et al.

described to model phase-shifting converter transformers in the harmonic domain [9]. Theresults show that the characteristic DC harmonics for an individual converter in a multi-pulse parallel connection installation increase with pulse number rather than decreaseas with a series connection. Most of these techniques used iterative harmonic analysismethod that have many disadavantages such as uncovergence with strongly interactionbetween AC and DC systems, being modeled separately sepentensery harmonic resultson previous knowledge of injection current harmonics on the AC bus-bar, and also dif-ficulties for representation of phase-shifting transformer, which require many matrices.Other models, such as time–domain simulations, were used with accurate results. How-ever, such models have suffered from long simulation time required to reach steady-statesolutions [12, 13].

The symmetrical component analysis was applied before [14, 15] for power networkcalculations. In this article, a dynamic simulation of multi-converters is proposed based onsymmetrical components, where mathematical equations are predicted for the conductionand commutation system states. This model has the following advantageous:

1. The use of symmetrical component AC analysis has several advantageous such aseasier representations of unbalanced AC voltage and more power for protectionsand network analysis [14, 15].

2. Simulations of static power converters as a whole part as AC power systemsduring unsymmetrical instantenous faults.

3. The model has a simplified representation of phase-shift transformer which suitsconverter representations at steady-state and during short circuit.

Simplified Multi-Pulse Converter System

Converter System Description

A typical arrangement of a multi-pulse converter system is shown in Figure 1. It consistsof 6-pulse converters connected in series or parallel through phase-shift transformers. Allare connected to a common AC bus-bar, which feeds the converter system by balancedAC voltages [9, 11]. The output of the converter system is connected to a DC load.Normally, such multi-parallel converters system have the following characteristics:

1. Each branch has the same circuit elements of resistance, inductance and capaci-tance.

2. All converter branches have the same rating.3. The number of parallel converter branches N and associated phase-shift angle φ

between converter branches are simply expressed by:

N = P/6.0

φ = 360/P

where P is the pulse number of the multi-pulse converter system.Thus, the number of pulses P of the converter system is determined by the

required smoothness of the system DC output voltage. Subsequently, the numberof branches N and phase shift φ are determined.

4. The converter thyristors are ideally represented by short circuit during conductionand commutation periods and by open circuits during non-conduction periods.

Symmetrical Component Analysis 869

Figure 1. Simplified multi-pulse static power converter system.

5. Accordingly, linear circuit theories can be applied during the conduction andcommutation periods of the system.

General Representation of Converter System Components

The three-phase equivalent circuit for each converter branch of the simplified multi-pulseconverter system is shown in Figure 2. The main elements of the converter branch,

Figure 2. Equivalent circuit of static power converter branch.


as in Figure 2, are:

1. Three-phase balanced AC power supply with emfs eA, eB , and eC and impedanceelements ra1 and La1.

2. Phase-shift transformer with phase-shift angle ψ and impedance elements rb1 andLb1.

3. Three-phase converter bridge and its feeder impedance elements rc1 and Lc1.4. DC load of impedance ZL.

a. Representation of AC Power Supply

The voltage of the supply being assumed unbalanced purely sinusoidal, are:

eA = Ea sin(θ)

eB = Eb sin(θ − 2π/3)

eC = Ec sin(θ + 2π/3)

θ = ωt + α0

(1)

where Ea , Eb, Ec are the voltage amplitudes in p.u., ω is the angular frequency in radiansper second, and α0 is the phase angle which determines the voltage magnitude of phase Aat t = 0.

b. Representation of Phase-Shift Transformer

The transformer windings are Y -connected in the primary side and zigzag connectedin the secondary side with a phase-shift angle ψ of the converter branch, as shown inFigure 2. The primary winding of the phase-shift transformer is assumed to have oneturn while the secondary is considered as having two parts with turns N2 and N3, whichare less than unity. N2 and N3 of the secondary windings are expressed [11] in terms ofthe phase-shift angle � as follows:

N2 = cos(ψ) − sin(ψ)√3

N3 = 2 sin(ψ)√3

(N2)2 + N2N3 + (N3)

2 = 1

(2)

c. Representation of Three-Phase Converter Bridge

The three-phase converter bridge, shown in Figure 2, normally has 6-modes of conduc-tion and commutation states during positive or negative voltage waveforms in sequence12, 123, . . . , 345.

The analysis of conduction and commutation states remain the same irrespectiveof the network topology and pulse number of converter (6, 12, 18, . . . ). Normally, thenetwork topology changes from mode to mode.


Symmetrical Component Analysis of Multi-Pulse Converter System

The symmetrical analysis of static power converter elements are evaluated as follows.

a. Symmetrical Component Analysis of AC Power Supply

The positive eA1, negative eA2, and zero eA0 sequence induced emfs of the power supplyare:

eA1 = Em[e−jθ − Une−j (θ+θυ)]/j2.0

eA2 = −Em[e−jθ − Unej (θ+θυ)]j2.0

eA0 = UnEm sin(θ − θu)

Un =√(2Ea − Eb − Ec)2 + 3(Eb − Ec)2

6Em

Em = (Ea + Eb + Ec)/3.0

θu = tan−1√

3(Eb − Ec)

(2Ea − Eb − Ec)

(3)

where Un, Em, and θu are the unbalanced voltage factor, average voltage amplitude andunbalanced voltage angle respctively.

b. Symmetrical Component Analysis of Phase Shift-Transformer

Normally, phase-shift transformer represents difficulties in short circuit and load flowstudies of AC conventional power system [10]. The following symmetrical componentanalysis simplifies the phase-shift transformer representations in power systems.

Y-Zigzag Transformer. The output voltage (va , vb, vc) and input current (iA, iB , iC)equations of the phase-shift transformer, shown in Figure 2, are expressed as:

va = N2vA − N3vC

vb = N2vB − N3vA

vc = N2vC − N3vB

iA = N2ia − N3ib

iB = N2ib − N3ic

iC = N2ic − N3ia

(4)


where vA, vB , vC are the input voltages and ia , ib, ic are the output currents. Thesymmetrical components of these voltages and currents are expressed as:

va1 = (N2 − λN3)vA1

va2 = (N2 − λ2N3)vA2

va0 = (N2 − N3)vA0

iA1 = (N2 − λ2N3)ia1

iA2 = (N2 − λN3)ia2

iA0 = (N2 − N3)ia0

(N2 − λN3)(N2 − λ2N3) = 1.0

λ = ej2π/3.0

λ2 = e−j2π/3.0

(5)

where vA1, vA2, vA0 are the positive, negative and zero sequence components of inputvoltages; iA1, iA2, iA0 are the positive, negative and zero sequence components of inputcurrents; va1, va2, va0 are the positive, negative and zero sequence components of outputvoltages; and ia1, ia2, ia0 are the positive, negative and zero sequence components ofoutput currents.

Zigzag-Y Transformer. The input voltage and output current equations of the phase-shifttransformer can be written in the same way as the (Y/Z) transformer.

c. Symmetrical Component Analysis of Converter during Conduction State

The conduction state equations of Figure 3 are:

var = va

vd = vbr − vcr

vbr − vcr = ZLib

ib = −ic

ia = 0.0

id = Magnitude of (ib)

(6)

where var , vbr , vcr , and vd are the input and DC voltages of three-phase rectifier bridge.The symmetrical equations are:

va1r − va2r = ZLia1

ia1 = −ia2

ia0 = 0.0

(7)


Figure 3. Conduction and commutation states of static power converter branch.

where va1r and va2r are the positive and negative sequence input votage of three phaserectifier bridge.

The sequence network connection during conduction period, Figure 4, gives:

vA1 = eA1 − (La1 + Lb1)(diA1/dt) − (ra1 + rb1)iA1

vA2 = eA2 − (La1 + Lb1)(diA2/dt) − (ra1 + rb1)iA2

(8)

From Eqs. (5), (7), and (8), a set of differential equations is obtained whose solutiondetermines the symmetrical components:

ia1 = N2Em

jZT

[cos(θ − θZT ) − cos(α − θZT )e−(RT /LT )t ]

− N3Em

jZT

[cos(θ ± 2π/3 − θZT ) − cos(α ± 2π/3 − θZT )e−(RT /LT )t ]

− N2UnEm

jZT

[cos(θ + θu − θZT ) − cos(α + θu − θZT )e−(RT /LT )t ]

+ N3UnEm

jZT

[cos(θ + θu ± 2π/3 − θZT ) − cos(α + θu ± 2π/3 − θZT )e−(RT /LT )t]

+ j (√

3/3)I1e−(RT /LT )t

ia2 = −ia1 (9)

Figure 4. Sequence network connection of converter branch during conduction period.


where +2π/3.0 in a positive phase shift angle of Y/Z or Z/Y transformer; −2π/3.0 ina negative phase shift angle of Y/Z or Z/Y transformer and I1 is the initial current atthe beginning of the conduction period.

RT = 2(ra1 + rb1 + rc1) + RL

LT = 2(La1 + Lb1 + Lc1) + LL

ZT =√(RT )2 + (XT )2

XT = ωLT

θZT = tan−1(XT /RT )

Thus, the conduction current equations are expressed as:

ia = 0.0

ib = λ2ia1 + λia2 = (λ2 − λ)ia1 = −j√

3ia1

ic = −ib

(10)

d. Symmetrical Component Analysis of Converter during Commutation State

The commutation state equations of Figure 3 are:

var − vbr = vd

var − vbr = ZLia

vbr = vcr

ia + ib + ic = 0.0

id = Magnitude of (ia)

(11)

The symmetrical voltages and currents equations are:

va1r = va2r

va1r = ZL(ia1 + ia2)/3.0

ia0 = 0.0

(12)

The sequence network connection during commutation period, shown in Figure 5, gives:

va1r = va2r = (LL/3.0)(dia1/dt + dia2/dt) + (RL/3.0)(ia1 + ia2) (13)


Figure 5. Sequence network connection of converter branch during commutation period.

Solution of this differential Eq. (13) gives the symmetrical components:

ia11 = N2Em

2Z3[sin(θ − θZ3) − sin(α − θZ3)e

−(R3/L3)t ]

− N3Em

2Z3[sin(θ ± 2π/3 − θZ3) − sin(α ± 2π/3 − θZ3)e

−(R3/L3)t ]

+ N2UnEm

2Z3[sin(θ + θu − θZ3) − sin(α + θu − θZ3)e

−(R3/L3)t ]

− N3UnEm

2Z3[sin(θ + θu ± 2π/3 − θZ3) − sin(α + θu ± 2π/3 − θZ3)e

−(R3/L3)t ]

ia12 = N2Em

j2Z1[cos(θ − θZ1) − cos(α − θZ1)e

−(R1/L1)t ]

− N3Em

j2Z1[cos(θ ± 2π/3 − θZ1) − cos(α ± 2π/3 − θZ1)e

−(R1/L1)t ]

− N2UnEm

j2Z1[cos(θ + θu − θZ1) − cos(α + θu − θZ1)e

−(R1/L1)t ]

+ N3UnEm

j2Z1[cos(θ + θu ± 2π/3 − θZ1) − cos(α + θu ± 2π/3 − θZ1)e

−(R1/L1)t ]

ia1 = ia11 + ia12 + 0.5I2e−(R3/L3)t − j (

√3/6)I2e

−(R1/L1)t

ia2 = ia11 − ia12 + 0.5I2e−(R3/L3)t + j (

√3/6)I2e

−(R1/L1)t (14)


where I2 is the initial current value at the beginning of commutation state,

R1 = ra1 + rb1 + rc1

L1 = La1 + Lb1 + Lc1

R3 = R1 + (2/3)RL

L3 = L1 + (2/3)LL

Z1 =√(R1)2 + (X1)2

Z3 =√(R3)2 + (X3)2

X1 = ωL1

X3 = ωL3

θZ1 = tan−1(x1/R1)

θZ3 = tan−1(x3/R3)

Thus, the commutation current equations are expressed as:

ia = ia1 +ia2

ia = 2ia11 + I2e−(R3/L3)t

ib = λ2ia1 + λia2

ib = −ia11 − j√

3ia12 − 0.5I2e−(R1/L1)t − 0.5I2e

−(R3/L3)t

ic = λia1 + λ2ia2

ic = −ia11 + j√

3ia12 + 0.5I2e−(R1/L1)t − 0.5I2e

−(R3/L3)t

(15)

e. Initial and Boundary Conditions of Converter Branch

The initial and boundary conditions between subsequent states (either conduction orcommutation state) of converter branch are taken from the final values of the precedingstate in a manner as to ensure magnetic flux and electric charge continuity [16]. Thismeans continuity of currents through inductors and charges across capacitors duringchange of converter states from conduction to commutation and vise versa.

f. Bus-Bar Current and Voltage Equations of the Multi-Pulse Converter System

The temporal variation of currents and voltages of the multi-pulse converter system,shown in Figure 1, are obtained as follows:

1. Determine the currents and voltages of each converter branch at a given instantaccording to its states (either conduction or commutation states).


2. Determine the bus-bar currents (iAS , iBS , and iCS) of the multi-pulse converterpower supply by adding the currents of all the converter branches at the sameinstant.

3. Determine the bus-bar current derivatives (diAS/dt , diBS/dt , and diCS/dt) ofmulti-pulse converter power supply by adding the current derivatives of eachconverter branch at the same instant.

4. Determine the bus-bar voltages of the multi-pulse converter system as:

vAB = eA − La1(diAS/dt) − ra1iAS

vBC = eB − La1(diBS/dt) − ra1iBS

vCB = eC − La1(diCS/dt) − ra1iCS

(16)

5. Subsequently, the input voltages of each phase shift transformer as:

vA = vAB − Lb1(diA/dt) − rb1iA

vB = vBB − Lb1(diB/dt) − rb1iB

vC = vCB − Lb1(diC/dt) − rb1iC

(17)

g. DC Side Calculations

i. DC Connected Bridges in Series.

iDC = id

vDC =N∑

i=1

(vd)i

(18)

ii. DC Connected Bridges in Parallel.

vDC = vd

iDC =N∑

i=1

(id)i

(19)

Computer Algorithm

The flow chart of computer program is shown in Figure 6. The following steps de-scribe the procedure followed in building a computer program for solving the abovedescribing differential equations. At the instant of switching on, the converter bus-barvoltages are equal to the supply emfs, and the following values are computed after a timeincrement:

1. The rectifier currents of each converter branch and their derivatives.


Figure 6. Flow chart of computation steps for multi-pulse power converters.

2. The input currents of each converter branch and their derivatives.3. The total currents and their derivatives of all converter branches.4. The bus-bar voltage of the multi-pulse converter system.5. The input voltage of each phase-shift transformer.6. The output voltage of each phase-shift transformer.

With further time increments, steps 1–6 are repeated until a predetermined period oftime is elapsed. The harmonic content of such a system is calculated according to digitalfourier analysis.


Results and Discussion

6-Pulse Converter

To indicate the capability of the proposed simulation method, the computed results arecompared with those measured before [8] for 6-pulse converter system with �/Y phase-shift transformer. The system parameter calculations are given in Appendix A and theirestimated values in per-unit are: Em = 1, ra1 = 0.02, rb1 = 0.05, rc1 = 0.02, RL = 2.03,La1 = 8.750e−05, Lb1 = 2.600e−5, Lc1 = 1.590e−04, LL = 1.00e−03, and conductingangle = 15 degrees.

The base values of the system are:

1. Base values at point of common coupling side are Em = 460√

2, kVABase = 1.4,and base current = 2.485 amp.

2. Base values at load bus side are Em = 190√

2, kVABase = 1.4, and base current =6.0156 amp.

The current and voltage waveforms of 6-pulse converter at the input and outputof �/Y transformer are compared with those measured experimentally [8] (Figures 7and 8).

Tables 1 and 2 give the computed and measured values of the total harmonic distor-tion voltage THDV and current THDI distortion. The computed values of the characteris-tic harmonics agree reasonably with those measured experimentally. Also, the computed

Figure 7. Current and voltage waveforms of 6-pulse converter at the input of �/Y transformer(a) computed results, (b) measured results of reference [8].


Figure 8. Current and voltage waveforms of 6-pulse converter at the output of �/Y transformer(a) computed results, (b) measured results of reference [8].

Table 1Harmonic content of 6-pulse converter at the input of

�/Y transformer

Computed values Measured values in %in %a of reference [8]b

Harmonicorder Current Voltage Current Voltage

1 100 100 100 1005 21.7 2.7 20.5 2.77 10.1 2.2 9.8 2.4

11 6.6 2 6.9 1.713 5.1 1.7 5.1 1.617 2.7 1.3 3.5 1.219 2.6 1.1 2.9 1.2

aTHDI = 25.68%, THDV = 4.94%.bTHDI = 25.00%, THDV = 5.2%.


Table 2Harmonic content of 6-pulse converter at the output of

�/Y transformer

Computed values Measured values in %in %a of reference [8]b

Harmonicorder Current Voltage Current Voltage

1 100 100 100 1005 22 4.1 22.3 37 10.0 2.9 10.3 2.6

11 6.8 2.7 7.4 2.013 5.1 2.4 5.3 1.817 2.9 1.7 3.9 1.419 2.7 1.6 3.1 1.6

aTHDI = 26.04%, THDV = 6.9%.bTHDI = 27.00%, THDV = 5.7%.

total harmonic distortion values of the current and voltage also agree resonably withthose measured experimentally.

Morever, the computed results are compared with Kimbark equation [6] during com-mutation periods.

Id = (V/√

2Xc)[cos(α) − cos(α + µ)] (20)

where V is the rms line-line voltage, α is the conduction angle, µ is the overlap angle,and Xc is the commutation reactance.

According to Eq. (20), Id = 4.344 amp for α = 15 and µ = 12.97 and the computedcurrent of proposed simulation method = 4.251 amp. The results are fairly acceptableand the difference is small due to the fact that Eq. (20) assumes that the DC current Idis ripple free and neglects circuit resistance.

The difference between computed and measured results may be attributed to ne-glecting no-load current which contains large harmonic content, using estimated systemparameters, little firing error or little unbalance in voltage and system parameters, andlimited accuracy measuring instruments.

12-Pulse Converter

For 12-pulse converter with Y/Y and Y/� converter transformers of phase-shift angleψ of 0 and 30 degrees, the phase shift φ between converter branches is 30 degrees. Thepower supply of the system has unbalaced factor 5.4% and their unbalanced emfs in %are:

Ea = 88 0 , Eb = 106 240 , and Ec = 100 120

The estimated impedance parameters of the system are La1 = 5e−05, Lb1 = 1.00e−05,LL = 8.50e−02, ra1 = 0.001, rb1 = 0.001, and RL = 2.03.

The AC input currents of 12-pulse converter and their harmonic contents for un-balanced power supply are compared with those obtaind by EMTP computer simulation


Figure 9. AC input currents of 12-pulse converter feeding from unbalanced power supply (a) com-puted results, (b) previous results of reference [17].

package, as shown in Figures 9 and 10 [17]. The results are fairly acceptable and thelittle differences are due to the use of estimated system parameters. Also, the DC out-put votage of the system is comparing by EMTP results of a previous reference [17](Figure 11). The results are fairly reconizably acceptable.

Unbalanced AC Voltages on Multi-Pulse Converter

Percentage of unbalance in AC voltage Un was varied from 5% to 40%. The results ofzero sequence harmonics of order 3, 6, 9, 15 for both 6-pulse and 12-pulse convertersare shown in Figure 12. The values of the 3rd and 9th harmonics increase sharply withthe percentage of unbalance voltage when compared with the 6th and 15th harmonics(Figure 12). Also, the values of zero sequence harmonics are reduced with increasingpulse-number at the same level of unbalance in AC voltage.

The DC output voltage harmonic values are plotted in Figure 13 against the percent-age unbalanced in AC voltage for both 6-pulse and 12-pulse converters. The unbalancein the AC voltages of multi-pulse converters increases the even harmonics of the DCoutput voltage, such as 2nd, 4th, and 6th. The values of 2nd harmonic are increasedsharply as compared to other even harmonics, and also, the even harmonic become lesswith increasing converter pulse number.

Conclusions

A generalized modeling technique to represent multi-pulse converter systems is pre-sented in this article. This technique is based on symmetrical network analysis during


Figure 10. Harmonic currents of 12-pulse converter feeding from unbalanced power supply(a) computed results, (b) results of reference [17].

conduction and commutation states by using general equations of multi-pulse converterssystem.

The computed results are fairly acceptable with respect to previous experimental andtheoretical investigations for 6-, 12-pulse converters. This system simulation is simplerand has a capability to represent in time-domain multi-pulse converters, taking into ac-count the unbalanced power supply, source impedance and the smoothing DC reactor.The main merit of such systems is the reduction of harmonic currents and voltages withthe increase of pulse number. The 3rd and 9th harmonics of the AC input currents and2nd harmonic of the DC output voltage are very sensitive for increasing of the unbalancedAC power voltages system.

Appendix A

Estimated Parameters of Six-Pulse Converter

The parameters calculation are based on the following equations [5]:

P = √2V I cos(φ) = VdId (A.1)

Id = 1.22I (A.2)


Figure 11. DC output voltage of 12-pulse converter with unbalanced power supply (a) computedresults, (b) previous results of reference [17].

where V is the RMS line-to-line voltage, I is the RMS line current, cos(φ) is the powerfactor, Vd is the DC voltage, and Id is the DC current.

Load Bus Parameters (Vd , Id , RL)

From reference [8], one obtains:

Base VA = 1400 VABase voltage = 190 voltsBase impedance = ZB = (190)2/1400 = 25.786 ohm

From reference [8], the following measuring data are given:

S = 1302 VA at load busIL = 1302

190√

3= 3.956 amp

Power factor = 0.936 at load busId = 1.22 × 3.956 = 4.826 ampVd = S × Power factor/Id = 1302 × 0.936/4.826 = 252.522 voltsRL = Vd/Id = 252.522/4.826 = 52.325 ohmRL = RL/ZB = 52.522/25.786 = 2.03 p.u.


Figure 12. Zero sequence harmonics against unbalanced AC voltage (a) 6-pulse converter, (b) 12-pulse converter.

Feeder-to-Bridge Parameters (Lc1, rc1)

From reference [5]:

Power Factor = 0.5[cos(α) + cos(α + µ)] (A.3)

where α is the conduction angle (= 15 degrees), µ is the overlap angle.

cos(α + µ) = 2 cos(φ) − cos(α) (A.4)

Id = (V/√

2Xc)[cos(α) − cos(α + µ)] (A.5)


Figure 13. DC output voltage harmonics against unbalanced AC voltage power supply (a) 6-pulseconverter, (b) 12-pulse converter.

where cos(α + µ) = 2.0 × 0.963 − 0.966 = 0.906

Xc1 = V√2Id

[cos(α) − cos(α + µ)] (A.6)

Xc1 = 1901.414×4.826 [0.966 − 0.906] = 1.67 ohm

Xc1 = Xc1ZB

= 1.6725.786 = 0.06 p.u.

Lc1 = Xc1/ω, ω = 2πfLc1 = 1.59e−04 for f = 60 HZrc1 = 0.02


�/Y Transformer Parameters (Lb1, rb1)


S = 1375 VAPower factor = 0.931Transformer losses = 3I 2rb1 = 1375 × 0.931 − 1302 × 0.936 = 61 wattsrb1 = 61/3(3.956)2 = 1.3 ohmrb1 = 1.3/25.786 = 0.05 p.u.cos(α + µ) = 2 × 0.931 − 0.966 = 0.896XCT = 190

1.414×4.826 [0.966 − 0.896] = 1.949 ohm

XCT = XCT

ZB= 1.949

25.786 = 0.07 p.u.Xb1 = XCT − Xc1 = 0.07 − 0.06 = 0.01 p.u.Lb1 = Xb1/ω, ω = 2πfLb1 = 2.6 × 10−5 p.u.

Source Impedance (La1, ra1)


Xa1 = 2.0/60 = 0.033 p.u.La1 = 8.753 × 10−5

ra1 = 0.02

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