Microwave power combining and gracefuldegradation

T.T. Ha, Ph.D.

Indexing terms: Matrix algebra, Microwave components, Power electronics, Microprocessors

Abstract: A scattering matrix approach to the design of microwave power combiners is presented. Both fixedand variable n-way combiners are discussed for power sources of different levels. Recursive expressions ofoutput power in terms of input powers and coupling coefficients of couplers are derived which can be pro-grammed on digital computers to predict the degradation in the event of source failure. For variable /i-waycombiners that are capable of producing truly graceful degradation, phase shift recursive expressions in termsof input powers are also derived which can be implemented on microprocessors to monitor and control thephase shifters.

1 Introduction

Microwave power combining is of particular importance inradar and satellite transmitter applications when manypower sources are combined by various techniques to ob-tain higher power, especially at high-frequency bands wherea large amount of power might not be available from asingle source. For example, the power combining of Impattdiode oscillators at X- and Ku-band1""5 and of power GaAsf.e.t.s.6"8 Also, for economic reasons, it is often desirableto obtain a specific amount of microwave power by com-bining several small power sources than using a large powersource; for example, low-cost helix type t.w.t.s are availableat Ku-band while high-power coupled cavity tubes inthis band are expensive and have poorer characteristicsthan the helix-type tubes. Besides the generation of higherpower, the power-combining techniques can provide agraceful degradation in the case of failure of one or moresources in the combining system. Graceful degradation isreferred to the fact that the output power is reduced butnot completely lost. For example, if an individual amplifierin a balance amplifier9 fails, the output power drops to aquarter of the original or 6 dB below. The purpose of thispaper is to provide a scattering-matrix analysis of existingand successful power-combining techniques such as thenonbinary (chain, serial) structure10'11'14 and the binary(tree, corporate) structure12"14 which combine the outputof n devices in many steps as compared with the directn-way structure which combines the output of n devices ina single step. Combining sources with different levels ofpower will be discussed in detail taking into account theloss in each combining step. Although graceful degradationoccurs in an n-way combining system, the power loss isusually proportional to the square of the ratio of thenumber of failed sources to the number of combiningsources. In order to obtain true graceful degradation ofwhich the output power is proportional to the abovementioned ratio, we propose and study a type of variablecombiner which in principle can achieve the stated goal.

2 Fixed combiners/dividers

A divider is a combiner with the input and output rolesinterchanged; it is not used in power combining of oscil-lators but must be used in conjunction with a combiner forpower transistors or 2-port amplifiers.

Paper 728G, received 28th August 1979Dr. Ha is with GTE Sylvania, Needham, Massachussetts 02062, USA

148

0143-7089/80/030148 + 05 $01-50/0

Nonbinary and binary n-way combiners are shown inFigs. 1—2, respectively, where they are made of 2-waydirectional couplers Q (/ = 1, 2 , . . . ,n — 1), in N com-bining stages. We note that n =N+ I and n — 2N — 1 fornonbinary and binary structures, respectively. Examples of2-way directional couplers are the TEM-line combinerssuch as Wilkinson's,15'16 the branch-line coupler17"19 andthe proximity coupler18 shown in Figs. 3a-c; or the Bethe-hole, Schwinger and magic-T waveguide couplers.19

The direct n-way combiner sums the power of n devicesin one step without proceeding through various combinerssuch as the Wilkinson's n-way combiner shown in Fig. 4a,the Rucker n-way combiner,5 the radial-line combiner6'8

and the promising TEM-line planar n-way combiner20

shown in Fig. 4b.Although the direct n-way combiner offers the possibility

of higher efficiency because the combining power does notgo through various combining stages, it is difficult to buildone to combine sources with different power levels. Further-more it does not offer the prospect of wide-band operationand high isolation from port to port is difficult to obtain.Direct n-way variable combiners for truly graceful degrad-ation might also not be possible to realise while a variable3-way combiner using 2-way variable combiners has beenbuilt and reported.21

In the following discussion we will analyse and designthe nonbinary and binary combiners for different levelsources taking into account the loss of individual couplersQ. The scattering matrix of a lossless 4-port coupler is givenby the following:18"19

S, =

0 0 a ffi

0 0 ;/3 ' a

a /0 0 | 0

t'P o o

0)

where a and |3 are real numbers and a2 + j32 = 1 (/2 = — 1)because of the unitary property of Sc (the superscript Tstands for matrix transposition). If port four is termin-ated by a load with reflection coefficient P L , then the 4-port coupler becomes a 2-way (3-port) combiner. Using acascade-load formula (see p. 59 of Reference 22), thescattering matrix S of a 2-way combiner is given by

ullc ' ^ l c 1 LJ2lc \"*)

IEE. PROC, Vol. 127, Pt. G, No. 3, JUNE 1980

If the terminating load at port four is matched, i.e., FL = 0,then __

"b 0 a

0 0

<* 70

(2b)

0

0

a

0

0

70

a

70

0_

«i~

a2

0

=

0

0

aal +/0fl2

Mathematically, the 2-way combiner works as follows: letat and b{ (i = 1, 2, 3) be the incident and reflected normal-ised power waves23 at ports of the coupler. For the com-biner, let ax and a2 be the only two incident input waves atports one and two and the output port is port three. Then

(3)

i.e. there is no reflection at the input ports and the outputpower at port three is given by \b3\

2 = \aax + /0a2|2. Let0 = —faa2/a* = ot\a2 \l\ax | (the superscript (*) stands forcomplex conjugate). Since a and 0 are real, this implies thata2/ax must be purely imaginary, i.e. a2 andax must be 90°out of phase. In particular, we let ax = \ax | e ^ 0 + 7r/2> a nda2 = \a2 | e

je, then 0 = a \a2 \l\ax | and, since a2 + 02 = 1, itis necessary that a = ±1/(1 + \a2 \

2 l\ax I2)1/2. This phasetrimming of ax and a2 can be obtained by using an appro-priate length of transmission line and thus, throughout theanalysis, we will assume that the phase of combined sourcescan be adjusted properly. We have

(4«)

= (a2 + 02)2 \ax \2la2 = (a2 + 02)| f l l | 2 /a2

= M 2 / a 2 (4c)

Eqn. 4d states that for given inputs ax and a2 such that#2/^1 = —71̂ 2 l/lfli l> o n e can select real a and 0 and viceversa, such that the output power equals the sum of theinput powers. Using the above results, we can proceed toanalyse the two combining structures. First we considerthe nonbinary combiner in Fig. 1. Let at (i = 1, 2 , . . . , n)be the input incident power wave at port /. From eqn. 4c itis seen that the output power of Q with coupling co-efficients at and 0f is given by

nm=l

i + i

Lm=i

n <4 = L kj 2

where

a, = ± I \am\2

m=\

1/2 ,

(5)

(6)

rr r -out

3 U nFig. 1 Nonbinary (serial, chain) combiner

IEE. PROC, Vol. 127, Pt. G, No. 3, JUNE 1980

0,- = a* \am\m=i

1/2

(7)

with the assumption that ax = \ax \eHd+7T/2) anda,- = |fl,-|

)

Example 1: Let \ax I2 = 5W, \a2 \

2 = 4W, \a3I2 = 6W and

\a4 |2 = 4-5 W with the assumption that ax is 90° out of

phase with ah i = 2, 3, 4. Then upon using eqns. 7 and 8 weobtain at =0-745, 0, =0-666; oc2 =0-775, 02 =0-632;a3 = 0-877 and 03 = 0-480. The coupling in decibels of Qis defined as |20 log |0£||, thus C\ , C2 and C3 are 3-52dB,3-98 dB and 6-38 dB couplers, respectively.

Now, in order to take into account the loss of eachcoupler, let kt > 1 be the coupling loss factor for each Q(i.e. the loss of Q is 10 log k2 (dB), for example, if Q has01 dB loss then kt = 1 -01158). Referred to Fig. 1, the out-put power of Cj , taken into account the loss of Cx, is

l*i I2 = l*i \2I*\OL\ = 7J (I*! I2 + \a2 |2)

Proceeding as before we have the recursive formula

(8)

Uk2ma2

m =^2-m= l

with bo—a1 and

ft = ^

(9)

(10)

(11)

Example 2: Let \at\2 ,i= 1,2,3,4, taking the values given

in Example 1, and assume that Cx, C2 and C3 have anominal 0-3 dB loss each, i.e., kf = 1-0715 (1= 1,2,3).Then Cx is unchanged, oc2 =0-7637, 02 =0-6455, a3 =0-8655 and 03 = 0-5. The output power of C3 is 16-74Was compared to 19-5 W if the Cfs are ideal couplers, and to16-7W if one uses a2 and a3 from Example 1. For manystages of combining of low-power sources, the above pro-cedure can improve the combining efficiency considerably.

For the binary combiner in Fig. 2, it is easily seen thatthe output power of Q x (i = 1,2, . . . , 2N~1) is given by

= ka-.P/of., ( 1 - 1 , 2 , . .

where

V2

(12)

(13)

C12

n=2

Fig. 2

cM2.1

ClN-1

. .= (*

-1.N -out

CN-22T

stage 1 stage 2 stage N-1 stage NBinary (tree, corporate) combiner

149

and hence the output of Cim (i = 1, 2 , . . . , 2N~m andm= 1,2,3, ... ,N) is given by the following recursiveexpression letting bio = at:

and m= 1 , 2 , 3 , . . . , TV) ( 1 4 )

where

= ± \bv.m- (15)

If A:̂ m is the loss factor of Ct m, then ait m will be replacedbyk'i,m<Xi,m ineqns. 12-15. '

It is noted that if ax = \a\eJid + n/2) and a2 = . . .an =|a | eJ0 then af = ft = ± l/\/2 for all i, for the binary com-biner, and at = ± {i/(i + 1)}1/2 and ft = ± {1/(1 + /)}1/2 forall /, for the nonbinary combiner.

Eqns. 10 and 14 also provide a convenient way to analysethe degradation of one or more sources in the two com-bining structures. Using these two equations, the output ofQ and Cim in Figs. 1 and 2 can be rewritten recursively asfollows:

(16)

where bo = ax and i = 1,2,.... ,n = N + 1, for the non-binary combiner and

7Ki

(I7)i,m

where /n = 1 , 2 , 3 , . . . ,7V and / = 1, 2 , . . . , 2N~m and&,->o

= ai> for the binary combiner.Eqns. 16 and 17 can be easily programmed on a digital

computer to predict the degradation of the combiningsystems should one or more a,s fail completely or partially

out

2 in

out

in 1

out

Fig. 3 Two-way combinera Wilkinsonb Branch linec Proximity

out

Fig. 4 n-way combiner

a Wilkinsonb Planar

150

with different phase and magnitude change. In the case thatax = \a\eM+n/2> and a2 = .. . = an = \a\eiQ then, if msources fail completely, the output power for both com-bining structures are (n —m)2 \a\2/n as seen from eqns. 16and 17 as compared to (n — ni)\a\2 of the input power(assuming ideal couplers). For example, if n = 4 and m = 1,then (n -rri)2 \a\2/n = 2-25\a\2 (a loss of 2-5 dB in power)as compared to (« — rri) \a\2 = 3 \a\2 (a loss of 1-25 dB inpower) for the input power. If we define the degradationcombining efficiency 77 as the ratio of the output power tothe input power, then r] — (n —m)/n for the equal ampli-tude sources.

3 Variable combiners/dividers

As seen from the above analysis, fixed combiners cannotprovide a degradation combining efficiency 77 = 1, i.e. thetheoretical efficiency. In order to achieve 77= 1, variablecombiners must be used. Consider the network TV shown inFig. 5 a phase shifter, with phase 0, is cascaded in betweentwo identical 3 dB couplers Cx and C2 (a = 0 = l/\/2) withport four of the second coupler C2 terminated in a matchedload. We will show that this is a variable 2-way combinerthat can produce a truly graceful degradation in the case offailure of either ax otai.\jz\ax = \ai\eid andfl2 = \a2\e

id

be the two input incident power waves at ports one andtwo of N, with the assumption that the in-phase relation-ship of ax and a2 can be adjusted properly at the inputs.Then upon using eqns. 1 and 3 with a = j3 = l/\/2, notingthat the phase shifter will change the phase of the outputreflected power wave at port three of Cx by 0, we have thefollowing output reflected wave b3 at port three of TV:

\a2 |

where

0, = 0+tan-1( |fl1

62 = 0 - 0 +tan"1

»« +e»>) (18)

(19)

ki l ) + 7r/2 (20)

In order for \b3 \2 = \ax \

2 + \a2 \2 in normal operation, it is

sufficient that 6X = 02 , i.e.

b3 = Vlfli I2 + k 2 | 2 e;<?1 (216)

In the case of failure, suppose at = yt \at | eJd ,(i= 1,2) and

0 < 7,- < 1 (again we assume the phase shift of a,- can beadjusted at the inputs), then in order for \b31

2 = \y1a112 +

l72c2 |2, i.e., for 77 = 1, 0 in eqn. 21a must be

(22)In particular, if \cti\ = 0 then 0 = + IT and if \a2\ = 0 then0 = 0. If the roles of inputs and outputs are interchanged,TV becomes a variable 2-way divider. If the variable 2-waycombiners are used in Figs. 1 and 2, we obtain a variable«-way combiner. For the nonbinary combiner, the phaseshift <f>i and the output wave b{ at the variable 2-way com-

Fig. 5 Variable 2-way combiner

IEE. PROC, Vol. 127, Pt. G, No. 3, JUNE 1980

biner Q in the chain are given by the following recursiveexpressions with bo = ax, for i = 1, 2, . . . , « = N + 1:

4>t = - t a n - H l ^ - i l / k + iD + tan-Hk. + J / l^-xD + Tr^(23)

bi = ' l l « m | V * ' (24)m=l

where 0O = 0 and

h = 0,-,+tan"1 / £ km|2/kf+1| (25)m = l

ai+l = \ai + l\eJdi-> (26)

Eqn. 26 indicates that a phase shifter </>,- is required for eachport / (f = 2, 3, . . . , « = TV + 1) to adjust the phase of at tothat of bt_x which is 0 , - j . For truly graceful degradation(7? = 1), all a,-s in eqn. 23 are replaced by ytat, 0 < 7,- < 1,in the case of failures of any numbers of fl,s. For the binarycombiner we have, for ai = \at \eje,

+ tan"1 (|Z>2l#m -M\b2i.XtTn _ , | ) + TT/2 (27)

Km = V l ^ - i . m - i P + lftaf.™-!!2 <^ '" 'm (28)

o = af (w(29)

where 0 = Oandfy o = af (w = 1 , 2 , . . . ,N;i= 1 ,2 , . . . ,

Eqn. 29 indicates that a phase shifter ipim is needed atthe output of each Q m to ensure that the output waves arein phase. For both variable «-way combining structures,there are a total of 2n — 3 phase shifters, which can be con-trolled by a microprocessor that receives control inputsfrom fault detection circuits in each combined source at.The programmable instructions are given by eqns. 23-29.In practice, the variable 2-way combiners are more lossythan the fixed 2-way combiners and therefore variable/?-way combiners are probably less practical for n > 6 whenthe combining loss becomes unacceptable.

Example 3: It is desired to combine three sources \a{ \2 =

50W, (/= 1,2,3) to obtain truly graceful degradation,(77 = 1) in the case of failure. Using eqns. 23-26 we obtain0i = + 7r/2 and 6X = 0 + n/4 where 0 is the phase of a!, a2

and a3. Thus the phase of a3 must be shifted by n/4 duringnormal operation, i.e. <p3 = ir/4, and also 02 = + 70-5°. Ifdi = 0 then 0j = + 7T, 0! = 0 and the phase of a3 must beshifted back to its original, i.e. </?3 = 0 and 02 = + 7r/2.Since ax = a2, the operation remains unchanged for a2 = 0.If a3 = 0, 0! = + TT/2 and <p2 = 0.

4 Conclusion

We have presented design principles for microwave powercombiners of sources of different levels taking into accountthe coupler losses. It is seen that the nonbinary combineroffers the advantage of the ease of changing the number ofports. To add a port to an existing structure, a coupler withthe coupling coefficient given by eqn. 6 is added to thechain and the other couplers remain unchanged. The dis-advantage of this type of combiner is the difficulty to buildcouplers with high coupling coefficients when large numbersof sources are combined; some couplings are too weak forbranch-line couplers and some are too strong for proximitycouplers. The loss in couplers is also determined by the

IEE. PROC, Vol. 127, Pt. G, No. 3, JUNE 1980

coupling coefficients and the bandwidth is determined bythe choice of couplers. The binary combiner offers less lossand hence higher efficiency than the nonbinary combineras the number of combining sources increases.24 A com-bination of these two structures can offer a compromisebetween flexibility and loss.

For these two types of structures, recursive expressionsare derived that can be easily programmed on a digitalcomputer to predict the degradation of the output powerin the event of failure of any combination of combinedsources. One problem in the case of failure is the mismatchat the inputs of the combiner caused by the shift in v.s.w.r.of the failed sources. This can be prevented by using inputcirculators for isolations; the use of circulators does notnecessarily increase system cost of complexity, since it isoften desirable to have them in the modules for faultprotection, improvement of port-to-port isolation andinput v.s.w.r. — although it can add loss to the combiningstructure. In satellite—earth terminals, a 'fail-soft' high-power amplifier in the transmitter is often more desirablethan the switched redundant configuration. The primaryadvantage of using a 'fail-soft' amplifier is that the failure ofeither amplifier does not disrupt transmission eveninstantaneously. The power combiners offer such anadvantage.

A fixed 4-way combiner can offer a degradation of2-5 dB losses in power in case of failure of either oneidentical amplifier in the chain. But in many cases trulygraceful degradation is required, then variable combinersmust be employed. The time required for the full outputpower to be achieved depends upon the switching speed ofthe phase shifters, which can be in the range of millisecondseven when using rotating half-wave plates in waveguidecombiners.25 For earth-station transmitters at Ku- andKa-band, where the power required is about 500—600 W,25

the use of variable combiners can save cost and increasereliability; furthermore the loss in a variable 3-way wave-guide combiner at Ku-band is under 0-4 dB.21

5 References

1 KUROKAWA, K., and MAGALHAES, F.M.: 'An X-band 10Wmultiple-diode oscillator', Proc. IEEE, 1971, 59, p. 102

2 KUROKAWA, K.: The single cavity multiple-diode oscillator',IEEE Trans., 1971, MTT-19, p. 793

3 MAGALHAES, F.M., and KUROKAWA, K.: 'A single-tunedoscillator for IMPATT characterisations', Proc. IEEE, 1970,58,p .83l

4 RUSSEL, K.J., and HARP, R.S.: 'A multistage high-power solid-.'state X-band amplifier'. IEEE International solid-state circuitsconference, Digest of Technical Papers, 1978, p. 166

5 RUCKER, C.T.: 'A multiple-diode high-average power avalanche-diode oscillator', IEEE Trans., 1969, MTT-17, p. 1156

6 SCHELLENBERG, J.M., and COHN, M.: 'A wideband radialpower combiner for FET amplifiers'. IEEE International solid-state circuits conference, Digest of Technical Papers, 1978

7 SUYAMA, K., FUKUTA, M., and KOMIZO, H.: 'DesignMESFET amps for more power', Microwave Syst. News, 1978,8, p. 61

8 COHN, M., GILLER, B.D., and SCHELLENBERG, J.M.: 'A10W broadband FET combiner/amplifier', IEEE MTT-S Int.Microwave Symp. Dig., 1979, p. 292

9 KUROKAWA, K.: 'Design theory of balanced transistor ampli-fiers', Bell Syst. Tech. J., 1965, p. 1675

10 MOHR, R.J.: 'A microwave power divider', IRE Trans., 1961,MTT-9, p.573

11 NAKAJIMA, M.: 'A proposed multistage microwave powercombiner', Proc. IEEE, 1973, 61, p. 242

12 MIZUSHIMA, S.: '2 n oscillators combined with 3dB directionalcouplers for output power summing', ibid., 1967, 55, p. 2166

13 FUKUI, H.: 'Frequency locking and modulation of microwavesilicon avalanche diode oscillators', ibid., 1966, 54, p. 1475

151

147514 MORSE, A.W.: 'Modify combiner designs to term high power

amps', Microwaves, 1978, 17, p. 7015 WILKINSON, E.J.: 'An N-way hybrid power divider', IRE

Trans., I960, MTT-8, p. 11616 PARAD, L., and MOYNIHAN, R.: 'Split-tee power divider',

IEEE Trans., 1965, MTT-13, p. 9117 PON, C.Y.: 'Hybrid-ring directional coupler for arbitrary power

divisions', IRE Trans., 1961, MTT-9, p. 52918 LEVY, R.: 'Directional couplers', in YOUNG, L. (Ed.):

'Advances in microwaves', 1966, l ,p . 11519 COLLIN, R.E.: 'Foundations for microwave engineering'

(McGraw-Hill, 1966)

20 GALANI, Z., and TEMPLE, S.J.: 'A broadband planar N-waycombiner/divider', IEEE MTT-S Int. Microwave Symp. Dig.1977, p. 499

21 WILKINSON, E.J., and SOMMERS, D.J.: 'Variable multiportpower combiners', Microwave J., 1978, 21, p. 59

22 NEWCOMB, R.W.: 'Linear multiport synthesis' (McGraw-Hill,1966)

23 KUROKAWA, K.: 'Power waves and the scattering matrix',IEEE Trans., 1965, MTT-13, p. 194

24 RUSSELL, K.J.: 'Microwave power combining techniques', ibid.,1979,MTT-27,p.472

25 WEISCHADLE, G.M., and KOURY, A.: 'SBS terminals demandadvanced design', Microwave Syst. News, 1979, 9, p. 70

152 IEE. PROC, Vol. 127, Pt. G, No. 3, JUNE 1980