A novel series-resonant converter topology

A novel series-resonant converter topology

Citation for published version (APA):

Tilgenkamp, N. V., Haan, de, S. W. H., & Huisman, H. (1987). A novel series-resonant converter topology. IEEE Transactions on Industrial Electronics, 34(2), 240-246. https://doi.org/10.1109/TIE.1987.350960

DOI:

10.1109/TIE.1987.350960

Document status and date: Published: 01/01/1987 Document Version:

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A

Novel

Series-Resonant

Converter

Topology

NICO V. TILGENKAMP, SJOERDW. H. DE HAAN, ANDHENK HUISMAN

Abstract-A converter topology based on the principles of series-resonant (SR) power conversion is described in which the input and output of this converter have one terminal in common, and the transformer is omitted. Both the underlying theory and associated waveforms arepresented.The converter is suitable for dc-dcaswellas

forde-ac conversion.Specialattention isgiventooperationintheregion where theinput voltage isapproximatelyequaltotheoutputvoltage(q

-1).Test results ofa700-Wmultiquadrantdc-dc and/ordc-ac converter areshown.

I. INTRODUCTION

T HE principles of operation of series-resonant power converters (SR converters) have already been discussed bynumerousauthors[1]-[4]. Theadvantagesofconvertersof this type,due to thehighinternalfrequencyandrelatively low switchinglosses are well known. Inmanyapplications, a high-frequency transformer is used for voltage scaling and/or

galvanic isolation. However, quitea few applications do not

need these features and would find a common (ground) terminal for input and output acceptable. In the "standard"

half-bridgeorfull-bridge configurationasdescribedin

[5]

and

[6],

the omission of the transformer will lead to high-frequency common-mode voltages onthe output terminals of

the converter. Therefore the question arises as to whether

there exists a transformerless power network in which the benefits of SRconvertersaremaintained. This paper presents the power network and the associated switching modes and formulates the concepts of series-resonant power conversion ina generalized way, such that both the standard and the new SR convertercan be considered asspecial cases.

It is assumed that the reader is familiar with the operation

modes ofthe standard SRconverter as described in [2], [5],

and [7].

II. PRINCIPLES OF OPERATION

A. Switching Modes

Inthe idealized (half- orfull-bridge) SR power converter, the current waveform consists ofa succession ofsections of sine waves. In betweentwoconsecutive zerocrossingsof the

resonantcurrenttherearegenerally two current segments, the first of which isusually denoted(for historicalreasons) asthe "diode current" and the second as the "thyristor current."

ManuscriptreceivedNovember 8, 1985;revisedApril22, 1986. N. V.TilgenkampwaswiththeDelft UniversityofTechnology.Heisnow

withthe Shellcorporation, Rijswijk, the Netherlands.

S. W. H.deHaan was with theDelftUniversityofTechnology. He isnow with the Department of Electrical Engineering, Eindhoven University of Technology,5600 MBEindhoven, theNetherlands.

H. Huismanis with theDelftUniversity of Technology, 2600GADelft,the Netherlands.

IEEELogNumber8613369.

ii

isj

ioL

_/

(a) W

t-../ / .\ /

(b)

Fig. 1. (a) Standard and (b) new SR converter with associated waveforms. Note that the ratioisrm.s/isa is substantially smaller in the newtopology.

Thesetwocurrentsegmentsaredistributed betweeninputand output ports in such a way as to satisfy the conservation of charge and power.

The distribution of current segments between input and output ports also serves the purpose of damping the state

variables of the resonant circuit. (The word "damping" is used freely here. We will assume the resonant circuit to be

damped aspower is extracted from it). For this purpose, the "diode" current segment is fed back to the input port. This specificimplementation of damping has to be paid forby extra

losses, for thedistortion factor (iSrms/isau) of the input current is higher than is strictly necessary. Damping of the resonant circuit couldalso beprovided by theoutputvoltageonly. The difference between this and a "standard" SR converter is shown inFig. 1.

Fromthe currentwaveformsinFig. 1(b)it isobviousthat in this "new" converterno power is fed back to the source. A

practical implementation and a more abstract version of a power networkthat generates the waveforms fromFig. 1(b)is

depicted in Fig. 2(a) and (b), respectively. The network as shown inFig. 2(b), whereall switches are assumed to be built up from pairs of antiparallel thyristors, facilitates multiqua-drantoperationaswell. Because of the symmetry of Fig. 2(b) with respect to the input and the output, the ports can

arbitrarilybelabeledas "input" or "output." Throughout this paper the port with the smallest voltagemagnitude is labeled as

."output" (step-down mode), unless otherwise noted. Fig. 3

gives an overview of the switch operation and associated waveforms for four important switching modes in step-down operation. According to the analysis in Section IIB, these

switchingmodescorrespond to operation in the four respective quadrants in the

UoJ-Io

plane.

0278-0046/87/0500-0240$01

.00 © 1987 IEEE j0

2

"Standard" 0 + New Es_{_[_} Full-Bridge UO Es SR, converter T_S.R. converter ND

PC s -SW, SWz io

-~~~~~~~~SWo

T SWo2 + Ss1s SWs2 O C vC 1 + iso Es input output Uo (a) (b)

Fig. 2. (a) Apracticalimplementationand(b)anabstractrepresentationof the power circuit of the novel SR converter.

three

quadrants, indicating

thatfour-quadrant operationis also

feasiblefor thecaseinwhich the outputvoltageis greaterthan the source

voltage (step-up mode).

B.

Regions of

Operation

For allfourquadrants theenergy balanceover one

positive

half-cycle

canbe stated as

Es(SsdQd+ SsiQt)=UO(SodQd+SotQt) (1)

SWI TCHES CLOSED:swzlisws:l jswz2

swsi

(a)

ioi

SWI TCHES CLOSED.Iswollswol Isws1 swzil

FS2!SWZ2 !SWF2sw02! SIWITCHES CLOSED 1solsw suzl~susl

(a)

\_

X

wX_

iol

l

l\

\J~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ FJ

1

isl I~~~ F: FeI,Fs. r 1F> n

SWITCHES CLOSED iswzl_ISWS215uO2iswzi iswsll_FSWZ2iswoi_SWZ21

(d)

Fig. 3. Switch operation and associated waveforms. In all cases it is supposedthatEs > 0. (a)First quadrant(UO > 0, I, > 0). (b)Second quadrant (UO > 0,It < 0). (c) Third quadrant(U, < 0, IO < 0). (d) Fourthquadrant(UO < 0,I, > 0). Beloweach segmentit is indicated whichswitchesareclosed.

Exchanging the labels "input" and "output" of the networkinFig. 2(b) reveals thatfirst-quadrant operation with the outputvoltage lowerthan the source voltage isequivalent

to second-quadrant operation with the output voltage greater

thanthesourcevoltage. Similarremarksarevalid for the other

where

Qd

and

Qt

represent the magnitude of the

charge

transferred by the diode and thyristor current segments,

respectively.

The coefficients

Sij

denote the direction of

currentflow in the source (i = s) and output (i = o) linesof

the converter for the diode

(j

= d) and thyristor

(j

=

t)

currentintervals, respectively. Note thatSijiseither - 1,0,or +1 andthat

Qd

and

Qt

aredefinedaspositive quantitiesin all

cases. From

(1)

wederive Qt EOSsd- Uo Sod Qd EsSst- UoSot SSd - qS.d st- qS0t

whereq is defined by q = UO/ES.

As Qt/Qd 2 0 by definition, itfollows that

(Ssd-qSod)/(Sst-qSot) C0.

(2)

(3)

FromFig. 3we canobtainthevaluesfor theSi;for thefour

differentquadrants. Filling in the appropriate values of

Sij

in

(3)

yields expressionsfor theallowedrangesof thenormalized

output voltage q, which is displayed in the last column of

Table I. Theallowable range for qdoes indeed comply with

the corresponding quadrant in all cases. This impliesthatthe proposed operation modes, as reflected by Fig. 3, do not

violate theprinciple ofconservation ofenergy.

C. Steady-State Characteristics

All theconverteroperation modesmentioned in SectionIL.B

can be made equivalent to a succession in time of linear networks. Forsteady-state operation it is necessary that state

variables (i.e., capacitor voltage and inductor current) be

continuous atthe boundaries of the successive intervals. Our

analysis will be restricted to steady-state operation modes

where the value ofsuccessive peak capacitorvoltages VCp(k)

and VCp(k + 1) are in accordance with

Vc

(k)

= --

Vcp(k

+

1).

₍₄₎

Therefore we only needto considertwo successive current segments, and add (4) to obtain a set of equations defining steady-stateoperation. Weareespecially interestedhere inthe

relationship between different circuit variables in steady-state operation, for this relationship gives us a tool for evaluating

possibleoperation regions independentof the typeof

control-lerused.

The analysis given in the Appendix yields a general expression for thedelay angle V,. (Fig. 1) inaform similar to

TABLEI

SWITCHOPERATIONANDNORMALIZEDOUTPUT VOLTAGE RANGEFOR THEFOURPROPOSED SWITCHINGMODES

Quadrant Ssd Sst Sod Sot equation (3) range forq 1 0 1 1 '1 -q/(1-q) < 0 0 < q < 1 2 -1 0 -1 -1 (-1+q)/q < 0 0 < q c 1

3 0 1 I -1 0 ql < 0 q < 0

4 -1 0 0 1 -1/-q < 0 q < 0

the expressions found in

[2], [5],

and

[6].

Employing

the notation introduced in Section ILA results in

-2 -t

q

(a) (b)

Fig.4. Regionsin the q - Xplanewheresteady-state operationisfeasible for the four quadrants. (a) First and second, and (b) third and fourth

quadrants.

(-StEs+

Sot Uo

+Vcp)2-(SdE-SodUO +

cp)2-(S

So

UO-SsdEs

+Sod

UO)2

2(SStES

-

Sot

Uo-SSdES

+ SOd

UO)(SSdES

-

SOd

Uo+VCP)

Dividing bothdenominator andnumerator

by

Es

we obtain

(- Sst+qSot+X)2 (Ssd-Sod+X)2-(Sst-qSO-Ssd+qSod)

(6)

2(Sst-qSot-Ssd+qSod)(Ssd-qSod+X)

where X is defined by

Vcp/Es

Expression (6) is a key equation that gives the relationship

between X andq for all operation modes.

For thereaderfamiliar with the full-bridge SRconverterit

may be worthwhiletoevaluate this expression for thatpower

circuit. For the full-bridge SR converterthe

Sij

are

Ssd- -1, SSt=Sod-Sot=1

Thus (6) reduces to the familiarexpression

l+q-qX(7)

1+q-X

which isequivalentto

(I1

+

q)(I

I-cos

lAr)(8

q-cos tlr

The sameexpression maybe found in [6, (41)].

Weproceed with ouranalysis by stating

-I<COS Ar< I. (9)

Substituting (6) in (9) for the four quadrants, while applying

theappropriate Si1 accordingtoTableI,generatesaregionfor every quadrant inwhich steady-state operation is exclusively

possible. These regionscanbemappedontoaplane, in which the horizontal axis denotes q (=U0/Es), the normalized outputvoltage; and the verticalaxisdenotes X(=

Vcp/Es),

the

normalized peak capacitorvoltage. The regionwhere

steady-state operationispossible is shown arcedforeachquadrant in

Fig. 4.

Itisinterestingto notethat inquadrants Iand 2, steady-state

operationisfeasibledown toalower value ofX, i.e., downto

a lower peak capacitor voltage than in the "standard" full-bridge SRconverter, where X is limitedto beinggreater than or equal to 2. This implies that the resonant current remains

continuous over agreateroperating range than in the "stand-ard" SR converter.

III. CONTROL

The controlelectronicscontainsaso-called "

VCpeak-control-ler" which controls the

delay

angle ,r such that the

peak

capacitor voltage

ismaintainedata specifiedvalue Vcpref

(see

(4))

even under

dynamic

conditions. An incorporated

Vcpeak-predictor

generates a real-time prediction of the next

peak

capacitor voltage

based on the actual values of

il(t),

Uj(t),

Es(t),

and

V,(t).

Whenever the predictor indicates that

turnover to the next current segment would render the

specified

valueof Vp, theturnover is actually initiated. The

Vcpeak-predictor

does not account for losses in the resonant

circuit,

so that the actual value of the peak capacitor

voltage

will be

slightly

below thespecified value. Note that the

delay

angle

Prdoesnotservethe control of theaverageloadcurrent.

As shown in

Figs.

1 and

3,

theresonant pulses are

separated

by

azero-currentdwell timeorinterpulsetimetd. Theaverage

loadcurrentiscontrolled

by

adjusting

theinterpulsetime. This control method is basically described in

[7].

IV. OPERATION IN THE q - I REGION

Operation

in the q = 1

region

requires some additional

control with respect to the losses that occur in the resonant

circuit. The

problem

will be pointedout withreferencetothe

idealizedcurrent waveforms inFig. 5.

ThewaveshapeinFig. 5isseentobe changingcontinuously

with

rising

q. For theq * 1 casespowerisextracted fromthe

resonantcircuit in the firstcurrentsegment, andin thesecond

currentsegmentit isdeliveredtotheresonantcircuit. Average

damping

overonehalf-cycle iszero, astheenergy contentof

the resonant circuit is assumedto bethe same atany current

zero. The net energy W that is transferred to the resonant

circuit should bejustenough to compensate for losses.

In the firstquadrant step-downmodethisenergy is equalto

W=

QtEs- (Qt

J

Qd) Uo

.

(10)

In theq = I region no netenergy can be transferred to the

(5)

21

il1

i'sl

x\

isI

O<q<1 q=1 q>1

Fig. 5. Some typical current waveforms in the new SR converter for differentvalues of q.

ioI

SWz1 SWs1 SWs1 SWol SWol SWz1 SWo2 SWo2 SWz2 SWz2 SWs2 SWs2

Fig. 6. Three-segment resonant current waveforms in the first quadrant and associated switch operation for q = 1.

resonant circuit (where Qd 2 0). Due to losses the

Vcpeak-controllerwill thennotbe abletomaintain thepeak capacitor

voltage atthe specified level, causing the oscillation to cease

eventually when the peak capacitor voltage drops belowE,

SeeFig. 4. Twomethods forovercomingthisproblemwill be described.

A. Adding DiscreteAmountsof Energy

Thefirst method isdescribedin[7] and is basedonavirtual shortcircuit of the output, every nowand then. This isdone,

forinstance,by closingswitches

SW,1

and

SW,2

inFig. 2(b).

During this"cycle-stealing" process afixedamountofenergy

is added to the resonant circuit, raising

Vcp

by

2ES.

A

disadvantage of thismethod is that thearray ofcurrentpulses

to the output is interrupted every now and then, thus introducing alow-frequency ripple.

B. Adding anAdjustable Amount ofEnergy

Bythe secondand superior method acontinuousadjustable

amountofenergyissuppliedtotheresonantcircuit such that it

guarantees full static and dynamic stability under all feasible conditions. The approach takesadvantageof the propertiesof the

Vcpak

controller, mentioned previously.

An adjustable amount of energy can be added to the

resonant circuit by a virtual short circuit of the

power-demanding port during part of the thyristor cycle. This

signifies that the resonant current waveform will consist of threecurrentsegments. Thecut-inangle of this third segment is controlled in such a way as to keep the peak capacitor voltage atthe predescribed level. Although this thirdcurrent segmentis only requirednear q = 1, it does noharm if it is implemented for the whole operation range. Because the duration ofthis third segment is typically much shorter than thesecond,theinfluenceonthe formulae isslight in all modes. The proposed resonant waveforms are depicted in Fig. 6,

togetherwithtypicalinputandoutput currentwaveforms. The implementation of such a control mechanism in a

Vcpak-controlled SR converter is straightforward and requires very

little extra hardware.

Notethat thewell-knownSRconverters cantake advantage ofthethree-segmentcurrentwaveformintheq = I regionas

well.

V. VERIFICATION

Theoperation modes ofthe novelconverter topology were

evaluated on a converter that was fully equipped with

antiparallel SCR's. The test converter had the following

specifications:

source voltage

maximum inverterfrequency maximum peak capacitor voltage output voltage range

resonant capacitor

resonant inductor input filtercapacitor

outputfiltercapacitor powerat full load

(1stquadrant) interpulse time

Es

= 100 V

f

= 10kHz

V,p

= 300 V -150 < UO< 150V Ci = 660 nF LI = 260 yH

Cs=

50,tF

CO

= 50

yF

Po

= 750 W td > 10 AS.

Toenablethe SCR'stoturnoff properly, theinterpulse timetd

should exceed the SCR turn-off time (10 its). When MOS-FET's are used, the minimum tdreduces to zero, so that the

powertransfercapability ofthe network increases considera-bly.

Although a high efficiency is potentially an important feature ofthe noveltopology,theconverterwas notlaidout to prove this. The test converter was primarily intended to

evaluate the operation modes and the control electronics. Nevertheless, a maximum efficiency of 88 percent was

obtained.

Fig. 7(a)-(d) shows thesource-andload-current waveforms for all four quadrants for

lql

< 1. The basic two-segment

waveshapes are in agreement with the waveforms of Fig. 1. The overshoot in

il

at the end of each pulse is caused by the reverserecoveryof the SCR's. The waveforms of iS and io are

generatedbythe control

electronics,

which clips the overshoot mentioned. Waveforms for

Iql

> 1 are not shown, because

they are essentially equal to the

lql

< I waveforms (see

Section II.A).

The significance ofthree-segment current waveforms fol-lows fromFig. 8. The converter isable to operate in a

steady-state mode at q 1, because the third current segment

maintains the capacitor peak voltageatthe predescribed level of 300 V. The first current segment, which is basically a

dampingsegment, is asshortasthecontrolelectronicsallows

it tobe. The current

i,

in the common (ground) line shows a

tiny negative firstsegment. Due to converter losses the second segment is effectively damping. Only the third segment

delivers energy to the resonant circuit.

All fourquadrant operation modes, both

IqI

< 1 and

IqI

>

1, arebrought together inFig. 9. Theconverteris fed from a

dc source

E,

= 100 Vand generates a 20-Hz output voltage

U0

= 140 sin (40 7rt)at a capacitive load of 50 ,uF. From the

-1

Fig. 8. Three-segment current waveforms. Upper trace: il, 20 A/div. Middle traces: i, 20 A/div, i, 20 A/div. Lower trace: i4 20 A/div; 20 tss/div.

(c)

(d)

Fig. 7. Current waveforms for the four quadrants. E, = 100 V. Upper

trace: i,, 20 A/div. Middletrace: i, 20 A/div. Lowertrace: i_ 20 A/div;

20pts/div. (a)U. = +50 V,I, = +3.2A. (b)U, = +50V, I, = -3.4

A. (c) U, = -50V, I,, = -1.9A. (d) U, = -50V,IO = +2.5A.

I

2

I

1

lIdOl

1

4

1

;'

I

le

I

Fig. 9. Waveforms atdc toacpowerconversion. A 20-Hz sinoid (100V rms)isgeneratedatacapacitiveload.E, = 100V, CO = 50/AF. Upper

trace: ii, 20 A/div. Middle traces: i., 20 A/div; V_, 365 V/div. Lower trace: U,, 145 V/div, 5 ms/div. The quadrants are indicated below the picture.

signsof

UO

andI it follows in which quadrantthe converter

operates.Above Fig. 9 a timescale isindicated. From20to25

mstheconverter is idle because the output capacitor is loaded to avalue which is close enoughtothe referencevoltage. The

three-segment current waveform guaranteesthat at t = 25 ms the q I barrier is passed withoutconsiderable loss ofpeak

capacitorvoltage.The continuouschangeofUOandq causes a

corresponding change in Qd and

Q,

(see (1)), which is

reflected in the envelopes of io and

il.

The theoretical limitations of the converter in the

U0-I,

plane are shown in Fig. 10. The maximum current in the 3rd and4th quadrant (trajectories e

andf

) aresubstantially below

the

1st

and 2nd quadrant currents (trajectories a andd). Inthe

3rd and4th quadrant operations the thyristor current is either flowing through the source or the load (Fig. 3(c) and (d)), while in the

1st

and 2nd quadrants the thyristor current is simultaneously flowing through the source and load (Fig. 3(a) and (b)). The average output current can be calculated from the average resonant current. From the definition of

Sij

(see

Section

II.B)

it follows that

to

SodQd

+

Sot

Qt

(11)

Fig. 10. Maximum-currentandmaximum-voltage trajectories of the novel converter. Thepeakcapacitorvoltageismaintainedat300 V whileEs =

100 V. Solidline: calculated trajectories. Squares: measureddata.

Eliminating theratio Qd/Qtfrom (11)

using

(1) leadsto

IO'_

ES(SotSsd-SstSodS)

II Es(Ssd-Sst)+

Uo(Sot-Sod)

(12)

where

Si;

follows from Table I. Forinstanceattrajectoryathe ratio IO/II equals 1, whileattrajectory

f

it equals -E

/(E5

+

UO)

-The maximum average resonant current follows from

Io=4CVcpf1- ₍₁₃₎

Thetrajectories atofarecalculated from(12) and (13) where

V,p

ismaintainedat300V.Thecalculationsarecarriedoutfor

aninterpulsetimeoftd = 10,us. The measured data isplotted

in Fig. 10as well. At first sight the measured data is in fair agreement withthecalculatedtrajectories. However,asshown in [5] and [6], the converter losses can be represented by a

raise in the output voltage of

Ploss/I,

Ifweassume that this correspondstoshiftingthecalculated trajectoriesover

approx-imately +15 Vinthe UOdirection in the 1stand4th quadrant andover -15Vinthe2nd and 3rdquadrant,thanaverygood agreement can by noticed.

VI. CONCLUSIONS

The proposed power network and control techniques make it possible to construct a transformerless SR converter with

commongroundforinput and output ports, with thefollowing

properties.

1) The converter is capable of both "step up" and "step down" operation.

2)A single-quadrantconverter usesonly sixsemiconductor

switches for "full-bridge" operation.

3) The converterhas an inherent high efficiency due to: a) the factthatonly twosemiconductor switches are in the

currentpath;

b) the absence of a power transformer; and

c) the virtual absence ofswitching lossesas a result ofthe resonant character ofthe operation.

4) The converter operatesas adcautotransformerwithouta

transformer being present.

5) The converter is inherently short-circuit proof.

Pk Pk'+r(k) T k ,0 t(k) W.t

(a)

sSiE SOjUo

(b)

Fig. 11. (a) GeneralizedSR power circuit. (b) Positivehalf-cycle ofthe

resonantcurrentin thecircuitof(a).

6)

Theconverteroffers the

possibility

togenerate a

single-phase

ac

voltage

with lowdistortion froma dcsource.

7)

By adding

branches to thepower network and

applying

adequate

control

techniques

converters with any number of

input

and output lines can be constructed using only one

resonantcircuit.

APPENDIX

We will

analyze

the circuit of

Fig.

11 with respect to the

waveform

depicted

in

Fig.

11.

For

brevity

we introduce the

following

notations:

Y1=C1/L1t I£1

/dIE

Y,

=--l

Zi

= LI/C Usd=SsdEs Ust=SstEs (14) (15) Uod=SodUo Uot=Sot Uo VCA=

VC(03k)

VCB= Vc(3k+irk) VcC = Vc(fk+lrk+Ctk) iB=

i(3k

+

O4rk).

(16) (17) (18)

(19)

(20)

We recall thecircuitequationsvalid for (k < ( < (3k + i/lrk

i(/3)= Yl(Usd-

Uod- VCA)

sin

((3-13k)

(21)

Vc(3)=

Vc([k)+(Usd-Uod- VCA){1-cos ((-(3k)} (22) and thecircuit

equations

valid forSk + i/irk < ( < Sk + 1/rk

+ 'tk

i($)

=

i((

+

1/irk)

COS

((-Ok

-rk)

+

Y1

(

Ust-Uot-VCB)

sin

((3

-k-

-i/rk)

(23)

VC()= VcB+i(13k+74rk) sin ((-(kV-'rk)

The resonant current will be zero at the end of the

thyristor

which aftersome

manipulating

reduces to

(-

VA-

Ust+Uot) 2-(Us5d- Uod-VCA) -(Uot-Usd+Uod)2

COS rK 2

-2(Ust-Uot-Usd+

Ust

Uod)(Usd-Uod

-VcA

)

(31)

current segment, thus

i(Ik

+ Prk+

t<k)

=O.

(25)

From (23) and (25) wederive

tan (Itk)

ZI- UOk

- -

}k

(26)

forphysical reasons

0<

r(k)C

-7r (27)

substituting (27) in (26):

-

VCA-

Ust+Uot>0 (28)

K(-VCA-USt+UO)-Z3i+(UIst-Uot VcB)2. (29)

Using (29), (22) and (21) weobtain

(- cA-Ust+UOt)2 (USd-sdUd- VA) sin2 Vrk

+{Ust Uot VcA (Usd Uod-

VcA)(l-OCOS

rk)}2.

(30)

By substitution of(15) to (16) in (31) we turn back to our original notation and (5) is obtained.

REFERENCES

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