new approach to the design of thyristor and transistor switching circuits

New approach to the design of thyristor andtransistor switching circuits

K. Mikotajuk, PhDW.S. Kropacz, PhD

Indexing terms: Power electronics, Transistors, Thyristors, Switches and switching theory

Abstract: The paper gives a new approach to thedesign of power electronic circuits. Time periodicfunctions of excitation and response are requiredas input data for the design. A sequence of thelinear time-invariant circuits is found on the basisof these data. As a result of the synthesis methodnew switching structures are obtained. The syn-thesis method presented can be applied to createnew thyristor and transistor arrangements, partic-ularly thyristor compensators.

1 Introduction

The literature devoted to the analysis of thyristor andtransistor switching circuits is comprehensive. Hithertonew thyristor structures have resulted from thyristornetwork analysis. Hence the theory of this class of cir-cuits can be considered as an analysis composed entirelyof a great number circuits. This has many disadvantages.Among them there are the impediments of teaching ofthis subject. A knowledge of the qualitative properties ofpassive switching circuits is not satisfactory either. Forexample, it is not known what waveforms can be gener-ated in passive switching circuits. For these reasonsworks on a theory of the synthesis of such circuitsstarted.

Methods of synthesising passive switching circuitshave been presented in the papers [1-4]. In the paper [3]identification formulas of state matrices were derived.These formulas require data to be input and outputperiodic time functions. The References are concernedwith the case when an input function is constant. Such acase is classified as inverter operation. This paper is con-cerned with the case when the input is a sine function. Itfollows from Reference 3 that for such an input functionthe case may be classified as a rectifier operation. Thetransformer switched structures for the rectifier operationare the same as for the inverter operation but the identifi-cation formulas of parameters are more complicated. Onthe other hand AC-AC convertors have many applica-tions. An important application of AC-AC convertors is

Paper 5129B (P6), first received 14th January and in revised form 17thNovember 1986Dr. Mikolajuk is with the Institute of Electrical Theory and Measure-ment, Warsaw Technical University, Koszykawa 75, 00-661 Warsaw,Poland.Dr. Kropacz is with the Department of Fundamental Research in Elec-trotechnics, Electrotechnical Institute, Pozaryskiego 28, 04-703Warsaw, Poland.

IEE PROCEEDINGS, Vol. 134, Pt. B, No. 2, MARCH 1987

reactive power control. The arrangements presented inthe paper can work as a reactive power compensators fornonlinear receivers. The synthesis method applied in thepaper is based on the state-space approach. We assumethat input and output functions are port functions of adynamical switched 1-port network. If an input functionis a port current, an output function is a port voltage.Also, if the input function is the port voltage, the outputfunction is the port current. Time functions u(t), y(t), andx(t) are the input function, output function and statevector, respectively. The functions u(t) and y(t) are 1-dimensional as they are 1-port functions. The state vectoris n-dimensional. We also assume that functions u(t) andy(t) are T-periodical and a synthesised circuit is switchedtwo times during the period T at instants t = 0 andt = T/2. During the first half-wave (0 ̂ t ^ T/2) a circuitis described by a minimal realisation {A, B,C, D). Itfollows from Reference 2 that during the second half-wave (T/2 ^ t ^ T) a circuit is described by the minimalrealisation {A, -B, -C, D).

The first synthesis step is concerned with determi-nation of a minimal realisation {A, B, C, />}. In thesecond step parameters of a thyristor structure are calcu-lated.

2 Determination of state matrices

We consider a switched 1-port containing k inductorsand k capacitors. Therefore the size of the state vector isn = 2k. As the input and the output function are 1-dimensional, matrices A, B, C, and D have the form

A =l2k

2k 2k .

C = [^ ... c2k]

Let the AC input function be

u(t) = Um sin (co01 + $) for — oo < t < oo

and output function

y(t) = Y, Ymi e x P (~ ai t) sin ((ot t + 0,-)

for

where T = 2n/(O0

(1)

(2)

(3)

(4)

(5)

(6a)

(6b)

61

and

y(t) = ~y(t+ T/2) for -oo < t < oo (6c)

We assume

a0 = 0, a,- > 0 for i = 1, . . . , k (la)

Um>0, Ymi>0 fori = 0,...,k (7b)

The example of the waveforms given in eqns. 5-7 isshown in Fig. 1.

u(t)

Fig. 1 Example of the waveforms described by eqn. 5 and 6

Let

Su = t /m[sin <j> - sin (co0 T/2 + <£)] = 2Um sin <t> (8)

Ku = L/m[cos <f) — cos (co0 T/2 + 0)] = 2Um cos <p (9)

G = KJSU (10)

St = yMI.[sin 0, - exp ( - a , T/2) sin (co( T/2 + 0,)]

(11)Ki = Ymilcos <Pi - exp ( - a , T/2) cos (to,- T/2 + 0,)]

(12)

fori = 0, 1, ...,kThe designed circuit should be switched at t = T/2

such that matrices A and Z> remain invariable andmatrices B and C change their sign. It was proved in Ref-erence 3 that a synthesised circuit with the input-outputfunctions given in eqns. 5 and 6 can be classified as arectifier. For rectifier operation the following relationsare valid:

(13)= y(0) - y(T/2)

CAlB[u(Q) - M(T/2)]

for / = (14)

With the use of eqn. 13 and 14 it is possible to determinematrices A, B, C, D. But we are looking for the passiveand reciprocal realisation. Hence it is necessary to regardthe following passivity and reciprocity criterions.

A circuit with the minimal realisation {A, B, C, D) ispassive if there are such real matrices L and Wo that

A*= -LV

a -LW0

(15a)

(156)

(15c)

A circuit with the minimal realisation {A, B, C, D} isreciprocal if the matrices A, B, and C satisfy the follow-ing conditions

(16a)=ArL

= -awhere the matrix £ is diagonal with entries 1 and — 1only.

We choose the state matrix A in a skew-diagonal form

A =

. CO,

- « k -

cok-

- a

(17)

This matrix is chosen a priori but it has many advan-tages. This matrix has a canonical form, which meansthat the matrix contains a minimal number of nonzeroentries. It can be written by inspection of the output func-tion The form of the matrix enables the synthesis of sucha circuit whose parameters are calculated separately foreach frequency cot(i = 1 , . . . , k).

Applying eqn. 13 gives

(18)1 = 0 i = 0

The right-hand side of eqn. 14 for the functions of eqns. 5and 6 can be expressed as follows:for / = 0

yi>(0) - y(l\T/2) - Z)[«(1)(0)

i = 0

for / = 1

y 2 ) (0 ) - / 2 ) ( r / 2 ) - Cfi[«(1)(0)

- Z > [ M ( 2 ) ( 0 ) - M ( 2 ) ( T / 2 ) ]

>tKt - di(o0Ku) (19a)

(196)

As has been mentioned the synthesised circuit contains kinductors and k capacitors, thus the matrix £ in thereciprocity criterion (eqn. 16) has the form

E = diag(/fc, -Ik)

From eqn. 16b the following is obtained:

(20)

bk

bk.— [ ~ C 1 ••• ~CkCk+l • • C2kJ (21)

62 IEE PROCEEDINGS, Vol. 134, Pt. B, No. 2, MARCH 1987

Combining eqns. 17 and 21, the left-hand side of eqn. 14,for / = 0 and / = 1, can be expressed as

xh =CB=

CAB=

(22a)

(22b)

-bk

-b(27c)

2k

Manipulating eqns. 14, 19blk+i-t-bf = Ei for2bib2k_i+1=Hi fort

where

E( = co{ Gi + co0(G0 — d

Hi = d{af - cof + col) •

di = Si/Su

Gi = KJSU

Solutions of eqn. 23 arebi = J(-Ei + yjEf + H?

and 22 givesi=l,...,k

= l,...,k

oG)lk-dioLx

-2aicoiGi-

)/2 for i =

Ei

1, •

co0G)

(O0G

..,k

(23a)

(23b)

(24a)

(24b)

(24c)

(24d)

(25a)

1H

1H

for i=l , . . . , fc (25b)

Thyristor structures with a current input

Fig. 2 Synthesised switched 1-port with a current input

Matrix 27a is realisable as an immittance matrix of aLet the input function defined by eqn. 5 be a current passive (k + l)-port if it is positive semidefinitefunction and the output function defined by eqn. 6 be avoltage function. We synthesise the switched 1-port a ^ &°'shown in Fig. 2.

The hybrid matrices M (for 0 ^ t ^ T/2) and M (forT/2 ^ t ^ T) of the nondynamical (2k + l)-port shownin Fig. 2 are

For example, for k = 1 condition 28 means that

a(d0 + d1)^[-E + V(£2 + H2j]/2 (29)

- B :3-

-b k + l -b

-: -3] •

bk+i ~cok

btk -co!

d -b^ ••• -bk i uk+1

— bt, ak i COL

k + l-b

-b2k

-cok

(26a)

(26b)

The transformer realisation of the hybrid matrix M witha current input function is performed according to thegeneral structure presented in Reference 3. The matricesresulting from the division of the matrix M into imped-ance and admittance parts as shown in eqn. 26a are

For reactance circuits (a, = 0, i = 1, ..., fe) inequality 28leads to the condition

Hi, = 0 for i = 1,..., k (30)

Z =

Y =

(27a)

(21b)

Let us assume that condition 28 is fulfilled. After the con-gruent factorisation of matrix 27a the turns ratio matrixof the multiport transformer is obtained:

ho 0

ho h

0

fcO

(31)

IEE PROCEEDINGS, Vol. 134, Pt. B, No. 2, MARCH 1987 63

where 4 Thyristor structures with a voltage input

W for i = 1, ..., fch =

The congruent factorisation of matrix 27ft gives

T, =

(32)

(33)

(34)

Let the input function defined by eqn. 5 be a voltagefunction and the output function defined by eqn. 6 be acurrent function. We synthesise the switched 1-port

The matrices Z'a, Y'c, and x'b follow from the division ofthe hybrid matrix M as shown in eqn. 26ft. After the con-gruent factorisation of matrix Za, 22fc + 1 variants ofmatrix t'a can be obtained. Two of the principal variantsare

-t00

0 0

— t.

and

Xa2 =

Ok

too 0hi ~h

(35a)

-U

Fig. 4 General thyristor structures followed from variant 2

The congruent factorisation of the matrix Yc gives

(35ft)

(36)

The turns-ratio matrix x'b results directly from eqn. 26ft:

(37)

Variant 1 leads to the thyristor circuit shown in Fig. 3and variant 2 leads to the thyristor circuit shown in Fig.4. Fig. 5 Synthesised switched 1-port with a voltage input

network shown in Fig. 5. The hybrid matrices M and Mof the nondynamical (2k + l)-port shown in Fig. 5 arethe same as defined by eqn. 26. But the division thehybrid matrices into admittance and impedance partsgives different results:

1 [JIQ 51H ••• 51H

m

ffll IS

«, His

Y =

Z =

dbl

'k+l 2k

— CO,,

(38a)

(38ft)

(38c)


64

The passivity condition now is similar as in the precedingcase

Ya > 0 (39)

Let us assume that condition 39 is fulfilled. The con-gruent factorisation of matrix 38a gives

IEE PROCEEDINGS, Vol. 134, Pt. B, No. 2, MARCH 1987

xn =

^00 0̂1

0 h<-0k

o tk j

where

U = <*i, hi = b,/y/(a^ for i = 1, . . . , k

= lid- X i

The congruent factorisation of matrix 38b gives

T , =

(40)

(41a)

(41b)

(42)

The matrices F̂ > Z'c> a n d TJ, follow from the division ofthe hybrid matrix M (eqn. 26b). After the congruent fac-torisation of matrices Y'a, 2

2k+i variants of matrix x'a canbe obtained. Two of the principal variants are

(43a)

(43b)

< = xc (44)

The turns-ratio matrix x'b results directly from the matrix

<1 =

1

• j

0

. 0

0

0

The congruent

- h i •••

toi

factorisation

— tok

tk -

hk~

-tk .

of matrix Z'c gives

— cob

(45)

Variant 1 leads to the thyristor circuit shown in Fig. 6and variant 2 leads to the thyristor circuit shown in Fig.7.

5 Examples

5.1 1 -port with a current input functionWe will design a passive reciprocal switched RLCT1-port network generating a T-periodic voltage

2

y(t)= X Vmk exp {-<xkt) sin (o)kt + <f>k) (46)k = 0

T ( T\for 0 < t < — and y(t) = -yi t + — 1 where

Vm0 = Vml = Vm2 = 20 V

oc0 = 0, ax = 30, a2 = 2

4>0 = 1.570796632 rad,

4>x = -0.0715298872 rad

4>2 = -2.62740875 x 10 ~3 rad

coo = 314.159265 rad/s,

(Ol = 942.477795 rad/s

co2 = 1570.79632 rad/s

for t G <0, TIT), (T = 0.02 s) if the input function is

u{t) = Im sin (cot + 4>) (47)

where

Im = 20 A, w = co 0 , 0 = <t>0.

" •

J1—

*0k

i[Ik> *



The input/output functions are shown in Fig. 8. Now, thetask is to find the canonical realization {A, B, C, D).From eqns. 17, 18, 21, 24 and 25 we have

A =

- 3 0 0 0 -942.4780 - 2 0 -1570.796 00 1570.796 - 2 0

. 942.478 0 0 - 3 0

0.5842157130.74049071539.44353630028.643516100 J

C = [-0.584215713 -0.740490715

39.4435363 28.6435161]

D = [0.93519078]

(48a)

(48ft)

(48c)

(4M)

In =

0.806009054 0 00.106662708 5.477225570 00.523606007 0 1.41421356 J

(49)

The congruent factorisation of matrices yc and y'c givesxc = x'c where

tc = diag [1.41421356 5.47722557] (50)

The final structure of thyristor circuit is shown in Fig. 9.

52 1 -port network with a voltage input functionLet the AC input voltage function be

u(t) = Vm sin (cot + 4>)

where Km = 5V, co = 314.159265 rad/s,1.57079632 rad and the output current function is

y(t) = exp (-cckt) sin (cok t + <f)k)

(51)

<t> =

(52)

JljT 1


'. t!T/2

\

bk . i "

-^k

W

b2k

h

uin

51H ••• i i H

\

Fig. 8 Response voltage function of example 5.1u{t)

M

Applying eqns. 26 and 27 after the congruent factor-isation of matrix xa the turns-ratio matrix of the multi-port transformer is obtained

T ( T\for 0 < t < — and y{t) = — y\ t + — I where

2 \ 1J

Lo = 10 A, Iml = 20 A, Im2 = 7 A, /m3 = 8 A

a0 = 0, <x1 = 50, a2 = 10, a3 = 100

co0 = Q),col= 942.477795 rad/s,

a>2 = 2199.11485 rad/s

co3 = 1570.79632 rad/s

0O = (f)i(f)l = -0.118999721 rad,

02 = -9.23997838 x 10"3 rad

03 = -0.0532447333 rad

The data functions are shown in Fig. 10. From eqns.17-25 we obtain the minimal realisation


A =

-500000

+ 942.4777

0-10

00

2199.11490

00

-1001570.7965

00

00

-1570.7963-100

00

Fig. 9 Circuit followed from variant 2 of example 5.1t00 = 0.806009054t10 = 0.106662708t20 = 0.523606007

t, = 5.477225570t2 = 1.414213560

63 = 39.44353630064 = 28.643516100

— - — . ^

p

\ /\ /\y

\\/p

/!T/2

j

Fig. 10 Response current function of example 5.2u(t)

02199.1149

00

-100

-942.47770000

-50

(53a)

B =

4.107476172.53484921.65912416

41.638981754.2097119

. 55.1612800

(53*)

C= [-4.10747617 -2.5348492 -1.65912416

41.6389817 54.2097119 55.16128] (53c)

D = [1.4606228] (53d)

The turns-ratio matrix of the 7-port transformer is

0.673143833 0.580884850 7.071067810 0

L0 0

!„ =

(54)

0.801589699 0.1659124160 03.16227766 00 10

The final circuit is shown in Fig. 11.

6 ConclusionsThe feature of the method presented is universality,although it is restricted to circuits which are switchedtwice during a period. Input and output time functionsare required as data for the synthesis method. Such dataare often available when a circuit is being designed. But itdoes not mean that the synthesis in the form presented inthis paper can be directly used in a design. There are

Fig. 11 Circuit followed from variant 1 of example 5.2t00 = 0.673143833 t03 = 0.165912416 b5 = 54.2097119f01 = 0.58088485 b4 = 41.6389817 b6 = 55.1612800tn7 = 0.801589699


restrictions. The structures presented contain a greatnumber of transformers. It ensued from the fact that syn-thesis of resistance n-port networks with the use of trans-formers is easy, whereas the problem of transformerlesssynthesis of passive resistance networks is still unsolved.On the other hand, for practical arrangements trans-formerless construction is desired because of the highcost and weight of transformers. In the structures pre-sented some transformers can be easily eliminated. But inthe synthesis it was assumed that transformers andswitches are ideal. Hence, it is necessary to check how thestructures obtained will behave when real elements areapplied.

The analysis of commutation processes is the nextproblem which needs further investigation, as not all timeinstants are equally convenient for switching. In spite ofthe limitations, the structures presented present a newclass of thyristor arrangements. They can be consideredas starting structures for AC-AC thyristor convertors. Itseems possible to apply the results obtained to the designof reactive power compensators. It was seen in Section 2that the circuits obtained can derive a desired non-sinusoidal current when a voltage source is sinusoidal. Ina particular case the assumed waveform can be con-sidered as a function expanded in a Fourier series. Hence,this waveform consists of the required harmonic spec-trum.

7 References1 KROPACZ, W.S., MIKOLAJUK, K., and OGONOWSKI, J.: 'Syn-

thesis of passive switching networks', Proceedings of the IEEE Inter-national Symposium on Circuits and systems, Rome, Italy, 1982, pp.329-332

2 KROPACZ, W.S., MIKOLAJUK, K., and TOBOfcA, A.: 'Synthesisof passive networks containing periodically operated thyristors', Int.J. Circuit Theory & AppL, 1984,12, pp. 375-393

3 MIKOfcAJUK, K., and KROPACZ, W.S.: 'Canonical structures ofDC-AC thyristors convenors', ibid., 1986,14, pp. 1-33

4 MIKOLAJUK, K., KROPACZ, W.S., and TOBOfcA, A.: 'Realisabil-ity of passive switching circuits', Proceedings of the European Con-ference on Circuit theory and design, Prague, Czechoslovakia, 1985,pp. 757-760

5 ANDERSON, B.D.O., and VONGPANITLERD, S.: 'Networkanalysis and synthesis. A modern systems theory approach' (PrenticeHall, 1973)

6 DARLINGTON, S.: 'A history of networks synthesis and filtertheory for circuit composed of resistors, inductors and capacitors',IEEE Trans., 1984, CAS-31, pp. 3-13

7 ALESINA, A., and VENTURINI, M.G.B.: 'Solid-state power conver-sion. A Fourier analysis approach to generalized transformer synthe-sis', ibid., 1981, CAS-28, pp. 318-330

8 ANDERSON, B.D.O., and MOYLAN, P.J.: 'Synthesis of linear time-varying passive network', ibid., 1974, CAS-21, pp. 678-687

9 SHIEH, L.S., and TAJVARI, A.: 'Analysis and synthesis of matrixtransfer functions using the new block-state equations in block-tridiagonal forms', IEE Proc. D, Control Theory & AppL, 1980, 127,pp. 19-31


new approach to the design of thyristor and transistor switching circuits

Documents