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GEORGE E. SODW A Y

Hewlett-Packard Co.Palo Alto, California

If\JTRODUCTlON

There are two requirements for the effective use oftransistors, solid-state devices, and passive components.First, their characteristics must be precisely measured;second, a design capability must exist in terms of themeasured quantities. Scattering (s) parameters satisfythese requirements from both a measurement and designpoint of view. They are particularly useful in themicro'_vave frequency range.

Ordinarily, s-parameters of an active three-terminaldevice are determined by two-port measurements, con-necting the common lead to ground. Unfortunately;--the physical length between the device and the groundplane usually introduces a serious parisitic common-lead inductance, especially if the spacings are made largeenough to obtain a very accurate 50-ohm system. Thesame reason that scattering parameters are measured athigh frequencies (i.e. because accurate shorts and opensare ciifficult to achieve at these frequencies) neces ·itatesmeasuring three-terminal parameters - and thus, reducingconsiderably the errors due to this parasitic common-

- lead inductance.Three-port admittance or impedance transistor para-

meters have been discussed before,1 but they have neverbeen as useful or as desirable as the three-terminal scat-tering parameters at microwave frequencies.- When~making three-port measurements, all three ports areterminated v"ith 50 ohms. Bringing three 50-ohm trans-mission lines up to the device eliminates the common-lead inductance, ensures accurate reference planes, andresults in a very stable measurement system. (Four-portmeasurements can be made in the same way for ICtransistors where the substrate is the fourth terminal.)A 50-ohm termination also approximates the final cir-cuit. environment more closely at microwave frequen-cies than the 0l)en or short terminations required byh, y, or z parameters.

This paper discusses the theory of three-port scatter-ing parameters and shows how previously complicateddesign procedures can be performed very simply inthese terms. For example, all of the two-port para-meters in any common configuration (CB, CE, CC)with any series feedback and any shunt feedback canbe determined by using one single transformation andone matrix transform'l-tion. The t~·o.port parameters,.•.•ith series feedback are related to the 9 measured quan-tities by 12 equations all identical in form, that is, theequations look alike. They only use different variablesand consequently, only one equation has to be solved.Having only one equation to solve has been a tremen-dous help in tying a small desk top computer into themeasurement system for instantaneous device characteri-zations and circuit design.

in IcrovvavewJour~.lal

THREE-PORT SCATTloRING Pft.R/lk~CTERSParameters for the three terminals of a transistor nreshown schematically in Fig. 1, where the three terminalsare all referred to a common ground. The incident nDdreflected power wa"es~ can be represented by the three-port scattering matrix

where ISjj!2iy j- represents the transducer power gain-from port j to port i, and ISiil~ represents the availablegenerator power that is reflected from the device at theith port. *

The measured parameters are referred to the charac-teristic impedance of the three transmission lines thatterminate the device. To be of universal use, the- para-meters for arbitrary termination of the transistor lea-dsare required as a function of these arbitrary termina-tions and the original measured parameters and arbi-trary reference impedances. The expression for the new -scattering parameters is given by

A (1-1'1*) (1 I IO)]/2~. th "thi = ------- - ri - _ IS e 1111-[1 I

element in a diagonal matrix.

Zj -Zl[I = Z/ + Z;*

ri\ = 1'\,

At, rt = rranspose of rhe diagonal matricesA, r, respectively.

The nine new scattering parameters in terms of theoriginal parameters and arbitrary reference impedancesarc given by -

s\;:-= ~~1 { (S11-r1*) 623+r2s12s21(1-r3s33)

+ r2r3 [s23SJ2S31+S2,S,3S32J +

'" For a delai/ed cOlJsideratiolJ of Ihe physical inlerpret.llion ~IJi}' see References 2, 3 find 4.

-~~r::-~:--''-::--_. __ .__.._"- .-------.}'

1 - Incident and reflected waves (a, b rI!3pectivelyl fora transi~tor imbe<lded in a siructuro where all three leadsare t'~rl'l\ii'1ated by the characteristic impedance Zo of a.transmission lino.

-- GC'-!'"~- be

.....c""t-.··-~7..0

fig. 2 - 111cident and reflected waves for.a transistor incommol' emitter configuration with an arbitrary impe-dance ie in the emitter lea·d.

623 = 522533 - S23'S32 (7)

The other seven expressions are obtained by exchangingthe indices on the above equations.

Although the set of equations represented by (4)and (5) can be used for computer analysis, it is un-wieldy to manipulate and does not convey very muchinsight into what is taking place. A far more usefuland rewarding approach has been to leave two of theports terminated by Zo and allow the third port to bearbitrarily terminated. The two-port parameters areobtained in this manner by treating the common leadas arbitrarily terminated in a series impedance differen;than ZOo The maximum available gain, isolation, sta-bility and other characteristics are simply related to thetwo-port parameters and thus, can be evaluated as afunction of this series lead impedance.

To avoid any confusion with indices, an obvious con-vention has been adopted for labeling the three-portscattering parameters for a transistor:

where, for example, Sbb is the driving point reflectioncoefficient of the base with the emitter and collectorboth terminated by Zoo Similarly, Sob is the transducerpower gain for the collector port when driving thebase. Scbis of particular significance for ? device, beingsimilar to h21 when using h-parameters and to S21 whenconsidering two-port s-parameters. The frequency atwhi::h scb goes through 0 dB is defined as f. and repre-sents a minimum value for fma:", The other parametershave similar meanings .

The nine elements of matrix (8) are not all inde-pendent because we are considering a three-terminaldevice. In fact, there are only four independent para-meters; if these are known, the others can be found,being related by the condition:

3

I:j=l

3·Slj::;;,L::. Slj= 1.

i=l

This relation follows from a similar relation for they-parameters where

3

Li=l

3

Y1j =2::j=l

From Equation (4) it is now possible to obtain theexpressions for the two-port parameters, with any feed-back element as a common~ead impedance. See Fig. 2.

Obr'aining Tv/o-Port Parameters From

Three-Port I"torrnatiol1

The two-port parameters for the three possible cOD-figurations are given by three sets of Equations: (12a),(12b), and (12c).

If rl is replaced with -], this is the same as groundingthe common. lead; consequently the above series ofequations give the normal t\yo-port parameters.

Before discussing the properties of Equation (12)series, several interesting observations can be made.First, it has been recognized previously that the gainin the common emitter configuration can be increasedby adding a capacitor in series with the emitter. It canbe shown· from typical d ..t:! for Se<othat when Z. iscapacitive, I(I/re) - seel Cdn be made a minimum, andSfe attains a maximum value. The disadvantage is thatthe other parameters also increase; in fact, Sl~ and S2e(the input and output reflection coefficients in commonemitter configuration) become greater than unity andthe device is very unstable.

It can also be observed from typical data that aninductance ll1 the. common-base lead will usually causeIl/rb) - Sbbl to diminish and the common-base gain toincrease.

SeeSeb S s Another application of the equations is to find aSfe= Seb+ -l----sJ. = Sbb+ ;e eb common-lead impedance which will minimize the re-

___.... __._ . ._. . . __"'- - See- - ..._...-..-' -~-See -------¥-ers~-_transducer power-· ga:n.-For· exam-pie, the valueI. I. of rb which makes Srb = 0 is given by Equation (13)

and a similar expression holds for the other two con-figurations. .SbeS.e _ -I SeeSeeSr. = Sbe+ -1---- S2e- See - 1

- - S - Seere ce re

SebSbe _ + SebSb.Sfb= See+ -1----S1b - See 1- - Sbb - - SbbIb Ib

SebSbeS2b= See+-1---- - SbbIb

Seeseb SeeSeeSfc= Seb+ -1----S2e = See+ -1----- - See - - SeeIe Ie

SbeSe. + SbeSebSre :.-:Sbe+ -1----S1e = Sbb 1- - See - SeoIe Ie

If the magnitude of rb ~ 1, then the element is passiveand a neutralized device can be obtained.

We have touched briefly on some special applicationsof Equations (12). Because of the relative simplicity,a considerable amount of information can be obtainedvery quickly, particularly jf the significance of the two-port parameters, with respect to desired circuit response,is kept in mind. The accuracy of the derived two-portparameters for a given accuracy in the original meas-ured parameters can also be monitored easily.

Equations (12a, b and c) are all of a single formwhich we can express as

bs=a+-1---,--eI.

where a, band care rcLlted to the measured three-port 'parameters. Equation (14) is a complex equation~ehtll1g the va.riablts sand r; it is a standard' equation111complex vaflable theory. Manipulating Equation (14)shows that the relationship between rand s is a bilin-ear transformation: .

a + db - ac)S = .. 1_.Ie

'.c.

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-~

There are two ways of looking at Equation (14)for s as a function of r. One is similar to that con-sidered for the fwo-port transducer power gain. Inthis case, we Can display circles of constant magnitudeof s on the r plane. For Sre, those are common emitterconstant-gain circles as a funCtion of the common-leadim pedance instead of the load ~r ge.nerator. Th.e radiusand center of the constant-gam clfcles are given by(16) and (17) respectively:

SHUNT FEEDBACK

Not only can the effects of a common lead impedancebe characterized by the set of Equation (12), but alsoshunt feedback can be handled in precisely the same way.

All three leads in the three-port measurement systemare referred to a common ground through a charac-teristic impedance z.,. The parameters measured formthe three-terminal scattering matrix. It is then possibleto make a simple transformation to a new 3 x 3 scat-tering matrix where the ports are referred to one an-other (Fig. 3).

The two-port parameters with any shunt element inany configuration are then given by the same trans-formation as the series case [Equation (12) J. Theseries and shunt feedback transformation can be com-bined resulting in the analysis of very complicatedcircuits.

APPLICATIONS OF THREE-PORTSCATTERING PARi.\MEHRS

An example of the use of the preceding three-portt,ransformation will be described in order to demon-strate the capability and usefulness of the approach. Theexample chosen, because of its wide applications, willshow how the two-port common emitter parametersat 1 and 2 GHz vary with either series or shunt feed-back dements.

Fig. 4 is used as a reference for the mapping ofcircles from the r plane to circles in the s plane. Forexample, Point 1 is a short circuit and the values ofthe transformecl parameters that occur at Point 1 arethose that exist with a short as a series or shunt element.

Fig. 4 - Points on the r plane (r defined by Equation 3)identified for location on the s plane for the series andshunt m~pping. Note the circles which go through 1·6,2-'6, 3-D, 4-6 and 5-6 are constant r ,;ircles, while thosethrough 7-6, 8-6, 9-6 and 10-6 ale constant inductivereactance circles and the corresponding circles through11-6, 12-6, 13-6 and 14-6 are c3pacitive reactanc;,acircles.

g2 = ls12 -- lal2f = cg2 + a'.t b

k2 = :s121c12 - Ib - acl2

The other way to handle Equation (14) is to mapthe r plane onto the s plane. It is well known that,for the bilinear transformation, circles on the r planemap into circles on the s plane. This is significant sinceit means that the Smith Chart for the r plane can bemapped onto tj1e s plane, giving both the magnitudeand phase of s for each complex value of r. Precisiondepends only on how many circles are mapped onto thes plane. This technique gives an exceedingly broadpicture of what is going on.

Fig. 3 -- A three-terminal device iMbedded in a networkwhich can be used to readily evaluate the effects ofshunt feedback.

The graphs 5 through 8 display how the above theorycan be applied to synthesizing the performance oftransistor Clrcuits. Tl1e example given is for a micro·wave small signal transistor with an ft of about 4 GHzand an fmllx of about 6 GHz. The transformation for1 and 2 GHz for series feedback are given by Figs. 5and 6 and for shunt feedback by Figs. 7 and B.

Figs. 5 through 8 have a very general nature, inthat, essentially all high frequency, small signal tran·sistors behave similarly. Some of the information can·tained in Figs. 5 through 8 will be discussed in or.derto provide examples of the meaning and use of thegraphs as well as to point out some of this generalinformation.

Let us see what happens to S1E or the input impedanceas the common lead impedance varies (Figs. 5a and6a). Point 1 represents a short circuit and the result-ing input reflection coefficient is that of the groundedcommon emitter stage. As resistance is added ill theemitter (moving from Points' 1 through 6) S1E movesessentially on a constant resistance line of a few ohmsin the direction of increasing series capacitance. Simi-larly increasing inductance (Points 1, 6, 7, 8, 9, 10)results in essentially an increase in the real part of theinput impedance; the reactance, being relatively con-stant.

In the case of S2E (Figs. 5d and 6d) the effect ismore complicated; the magnitude of S2E increases withincreasing L, R or C. \'7ith inductance or resistancein the emitter, the output impedance becomes morecapacitive and, for values of R less than Point 4, thereal part decreases while it increases for inductive loads.

The transducer power gain in a Zo system !S2E!2 de-creases for either a resistor or inductance in the com·man lead. The effect is less at higher frequencies fora given device; for example, a resistance indicated byPoint 4 results in a gain which is the same at both 1and 2 GHz. The very serious effect small inductancescan have at high frequencies could be illustrated by~valuating the drop in gain if, for example, a 100mil lead length were used with this chip. This wouldcorrespond to about p.5 ohms of inductance, or justpast Point 7 at 1 GHz (Fig. 5b), and 25 ohms on Point8 at 2 GHz (Fig. 6b). The drop in gain is significant.The effect is, of course, much more serious as you moveup in frequency to the 4-6 GHz range which is thepresent practical limit for useful transistor operation.A capacitive emitter impedance, in general, increasesthe transducer power gain, but also causes an increasein Sl1 and S22 resulting in instability. Notice also thatthere does not exist a positive real value of impedancewhich will neutralize the device at 1 or 2 GHz.

In this case Point 6 (Figs. 7 and B) or an open circuitcorresponds to the grounded emitter configuration. Thevalues for the parameters obtained with an open shuntimpedance (Point 6, Figs. 6 and B) should, of course,be identical to that for a short circuit emitter seriesimpedance (Point 1, Figs. 5 and 6).

The input impedance S1l-: is relatively independentwith either capacitive or resistive feedback (Figs. 7aand Ba). This is because of the 100v input impedaoceinto the device. The value of S,<, is' 'nueh more sensi·tive to inductive shunt feedback ;~ ind.ic.lted by movingfrom an open circuit Point 6 through Poiots 10, 9,8, 7 and 1 correspondiog to lower values of ind",ctiveimpedance.

IS2112, the transducer power gain, decre-lses 'with re-sistive or capacitive shunt feedback. For example, acollector base capacitance of 1.5 pf causes a drop io,gain from Point 6, Figure 7b, to Point 14 and a drop. to Poiot 13 in Fig. Bb. Also the effect of reducing thecollector base capacitance, for example, by reducingthe base pad size can be. easily ascertained. As induc-tive shunt feedback is added, the gain iocreases to verylarge values until very small v;:l1ues of ioduct2nce arereached when the gaio begins to drop appro:lChingesseotially zero with a short circuit.

The reverse gain S12 incre2ses with any shunt feed·back. It change'" a relatively small amouot for capaci-tive or resistive )eedback, but changes more drasticallyfor inductive feedback.

Point 5, (Figs. 7b and Sb, 100 ohms) gives a gainIS2112 of about 5 dB at 1 and 2 GHz with about 15to 10 ohms of input impedance with 45-60 ohms ofoutput impedance and a low reverse feedback !SI2! < 0.2.More gain could be obtained over this frequency rangeby using inductive peaking in the shunt feedback.

The same gain, about 5 dB, can be obtained at both1 and 2 GHz with about 50 ohms (Point 4) of seriesfeedback (Figs. 5b and 6b).

In this case the input impedance is about 10 oh..rnsbut with about 60 ohms to 30 ohms of capacitive reac-tance (Figs. 5a aod 6a). The output impedance is 10-20 ohms with 60-150 ohms of capacitive reactance. Thereverse feedback goes from 0.2 to 0.4. Additional gaincan be obtained with capacitive series peaking.

This technique has been exceptionally useful in ob-taining a thorough understanding of the behavior ofsmall signal devices in amplifier and oscillator circuitsfrom low frequencies to the very highest frequenciesat which transistors will presently operate. The tech-nique has been used to advant::lge as ao initial or roughsynthesizing tool aod also as a precise and generalanalysis technique for very complex circuits.

Although not illustrated, these transformations areparticularly well suited for coosidering distributed im-pedances. For example, a traosmission line terminatedby a lumped element is represented on the r plane asa circle about the origin with frequc!1cy. This circlealso maps onto the s planes as a .circle.

The three-port measurement system is just an extensionof the two-port system, but what we will describe herein detail is the unique three-port broadbaod system forthe measurement of unbonded transistor chips.

A schematic of the system is shown in Fig. 9 andphotographs of the actual setup in Figs. 10, 11, 12,13 and 14. The signal is directed inciden.ton.onep.(),rt ..._ >"aod measured reBected from this port and transmitted ..::~::::,~;~"i

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In order to accomplish this, a new set of parameters,the three-terminal scattering parameters for a transistor,are utilized. Not only can the conventional two-portparameters be obtained simply from the measured quan-tities, but also the paper shows how the effect of addinga series or shunt impedance to the device can be obtained

,mathematically by using a simple extension of the basicequation involved.

3. Bodway, George E., "Two-Port Power Flow Analysis UsingThe da~a for a conventional microwave transistor is Generalized Scattering Parameters," the mht"OUJave journal,

utilized for sho\\'-iiig--h0\v--a--mapping -technique-cal1-._b~_.\'?l:_!_O~ ~o. 6, May 1967.applied which shows visibly at a single ghnce, at a 4. Kurokawa, i-., IEEE Tra;1Saclions~MTTJMarch 1965, p; 194. ,_

out- the rther two ports. Next the .second port is drivenand then the third, resulting in the measurement of the9 scattering parameters. The switching of the signaland measurement ports is controlled by electrical sig-nals triggered either manually or by a computer. Thesigli:ds are 'detected by a s<lmpler and compared againsta referel1ce. The resulting output is displayed or. apolar chart, oscilloscope, etc., or can be fed directly toa computer. .

Transistor chips are presently being measured on aproduction basis for use on hybrid integrated circuitson this equipment. A chip can be measured from 0.1to 12.4 GHz on this equipment. l\lmost any informa-tion about the device can be obtained; fm.x' \S21!2, etc.,or performance in an amplifier or oscillator. This in-formation can also be obtained as a function of thedc bias conditions. The loading, testing, calculating,unloading and sorting can be done routinely in lessthan 2 minutes pcr device. The device is then readyto be bonded down on a microcircuit. It is' assurednot only that the device will ,vork but that the circuitwill perform as required \vith a very high yield evenwith many devices per circuit.

A practical and accurate technique for measuring un-bonded transistor chips from 0.1 to 12.4 GHz has beendescribed. ,

'particular frequency, the effect of adding any series orshunt feedback element. The data and genera! effectsshown are typical of any microwave sma]] signal tran-sistors and the many figur~s shown are therefore ofgeneral LIsefor reference information.

The equipment used to 'accomplish the measurementof transistor chips is described including a descriptionand pictures of the techniques used to. make contact tothe transistor chips.

In this paper and one previou;;;]y published, the foun-dation has been laid for the precise measurements oftransistor chips in terms of useful microwave p:uametersas we]] as describing powerful design tools particularlybut not limited to the precise but simple design of micro-wave hybrid thin film circuitry. The utilization of thismaterial in designing microwave circuits such as oscil-lators and amplifiers wi]] be described in forthcomingarticles. •

The author wishes to express his appreciation to every-one who assisted in preparing and editing this manu-script. Particularly to Mee Chow for the considerableeffort required in preparing the artwork, Joan McClungand Roseanne Calc!we]] for preparing the manuscriptand to Larry Rayher for editing the paper.

1. linville and Gibbons, "Transistors and Active Circuits," Mc-Graw-Hili Book Co., Inc., New York, 1961.

2. Bodway,George E., "Two-Port Power Flow Analysis of linearActive Circuits Using the Generalized Scattering Parameters,"Hewlett-Packard Internal Report, April 1966.

GEORGE E. BODWAY received a B.S. in 1960, an M.S. in 1964and a Ph,D, in Engineering Physics in 1967 from the Universityof California at Berkeley. He is presently the Section Managerresponsible for the Microcircuits and Solid State Program for theMicrowave Division of the Hewlett-Pai=ka,d Company.

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. Figs. 5a, b, c and d _. Common emitter series feedback impedal1ce mapped ontothe s-parameter planes at 1 GHz. The shaded regions correspond toinductive impedance while the colored areas are capacitive.

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Figs. 6a, h, c and d - Common emitter series feedback impeda:.ce mapped ontothe s-parameter planes at 2 ·GHz. The shaded regions correspond toinductive impedances while the colored areas are capacitive.

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