Agilent AN 154

S-Parameter DesignApplication Note

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The need for new high-frequency, solid-state circuitdesign techniques has been recognized both by micro-wave engineers and circuit designers. These engi-neers are being asked to design solid state circuitsthat will operate at higher and higher frequencies.

The development of microwave transistors and Agilent Technologies’ network analysis instrumen-tation systems that permit complete network char-acterization in the microwave frequency rangehave greatly assisted these engineers in their work.

The Agilent Microwave Division’s lab staff hasdeveloped a high frequency circuit design seminarto assist their counterparts in R&D labs through-out the world. This seminar has been presented in a number of locations in the United States andEurope.

From the experience gained in presenting this orig-inal seminar, we have developed a four-part videotape, S-Parameter Design Seminar. While the tech-nology of high frequency circuit design is everchanging, the concepts upon which this technologyhas been built are relatively invariant.

The content of the S-Parameter Design Seminar isas follows:

A. S-Parameter Design Techniques–Part I(Part No. 90i030A586, VHS; 90030D586, 3/4”)

1. Basic Microwave Review–Part I

This portion of the seminar contains a review of:

a. Transmission line theoryb. S-parametersc. The Smith Chartd. The frequency response of RL-RC-RLC

circuits

2. Basic Microwave Review–Part II

This portion extends the basic concepts to:

a. Scattering-Transfer or T-parametersb. Signal flow graphsc. Voltage and power gain relationshipsd. Stability considerations

B. S-Parameter Design Techniques Part II(Part No. 90030A600, VHS; 90030D600, 3/4”)

1. S-Parameter MeasurementsIn this portion, the characteristics ofmicrowave transistors and the network ana-lyzer instrumentation system used to meas-ure these characteristics are explained.

2. High Frequency Amplifier DesignThe theory of Constant Gain and ConstantNoise Figure Circles is developed in this por-tion of the seminar. This theory is thenapplied in the design of three actual amplifiercircuits.

The style of this application note is somewhatinformal since it is a verbatim transcript of thesevideo tape programs.

Much of the material contained in the seminar, and in this application note, has been developed in greater detail in standard electrical engineeringtextbooks, or in other Agilent application notes.

The value of this application note rests in its bringing together the high frequency circuit designconcepts used today in R&D labs throughout theworld.

We are confident that Application Note 154 andthe video taped S-Parameter Design Seminar willassist you as you continue to develop new high fre-quency circuit designs.

Introduction

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IntroductionThis first portion of Agilent Technologies’ S-Para-meter Design Seminar introduces some fundamen-tal concepts we will use in the analysis and designof high frequency networks.

These concepts are most useful at those frequencieswhere distributed, rather than lumped, parametersmust be considered. We will discuss: (1) scattering or S-parameters, (2) voltage and power gain rela-tionships, (3) stability criteria for two-port net-works in terms of these S-parameters; and we willreview (4) the Smith Chart.

Network CharacterizationS-parameters are basically a means for characteriz-ing n-port networks. By reviewing some traditionalnetwork analysis methods we’ll understand why anadditional method of network characterization isnecessary at higher frequencies.

Figure 1

A two-port device (Fig. 1) can be described by anumber of parameter sets. We’re all familiar withthe H-, Y-, and Z-parameter sets (Fig. 2). All of thesenetwork parameters relate total voltages and totalcurrents at each of the two ports. These are thenetwork variables.

Figure 2

The only difference in the parameter sets is thechoice of independent and dependent variables.The parameters are the constants used to relatethese variables.

To see how parameter sets of this type can bedetermined through measurement, let’s focus onthe H-parameters. H11 is determined by setting V2

equal to zero—applying a short circuit to the outputport of the network. H11 is then the ratio of V1 toI1—the input impedance of the resulting network.H12 is determined by measuring the ratio of V1 toV2—the reverse voltage gain-with the input portopen circuited (Fig. 3). The important thing to notehere is that both open and short circuits are essen-tial for making these measurements.

Figure 3

Moving to higher and higher frequencies, someproblems arise:

1. Equipment is not readily available to measuretotal voltage and total current at the ports of thenetwork.

2. Short and open circuits are difficult to achieveover a broad band of frequencies.

3. Active devices, such as transistors and tunneldiodes, very often will not be short or open circuitstable.

Some method of characterization is necessary toovercome these problems. The logical variables touse at these frequencies are traveling waves ratherthan total voltages and currents.

Chapter 1. Basic Microwave Review I

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Transmission LinesLet’s now investigate the properties of travelingwaves. High frequency systems have a source ofpower. A portion of this power is delivered to aload by means of transmission lines (Fig. 4).

Figure 4

Voltage, current, and power can be considered tobe in the form of waves traveling in both directionsalong this transmission line. A portion of thewaves incident on the load will be reflected. It thenbecomes incident on the source, and in turn re-reflects from the source (if ZS ≠ Zo), resulting in astanding wave on the line.

If this transmission line is uniform in cross sec-tion, it can be thought of as having an equivalentseries impedance and equivalent shunt admittanceper unit length (Fig. 5).

Figure 5.

A lossless line would simply have a series induc-tance and a shunt capacitance. The characteristicimpedance of the lossless line, Zo, is defined as Zo = L/C. At microwave frequencies, most trans-mission lines have a 50-ohm characteristic imped-ance. Other lines of 75-, 90-, and 300-ohm imped-ance are often used.

Although the general techniques developed in thisseminar may be applied for any characteristicimpedance, we will be using lossless 50-ohm trans-mission lines.

We’ve seen that the incident and reflected voltageson a transmission line result in a standing voltagewave on the line.

The value of this total voltage at a given point alongthe length of the transmission line is the sum of theincident and reflected waves at that point (Fig. 6a).

Figure 6

The total current on the line is the differencebetween the incident and reflected voltage wavesdivided by the characteristic impedance of the line(Fig. 6b).

Another very useful relationship is the reflectioncoefficient, Γ. This is a measure of the quality of theimpedance match between the load and the charac-teristic impedance of the line. The reflection coeffi-cient is a complex quantity having a magnitude, rho,and an angle, theta (Fig. 7a). The better the matchbetween the load and the characteristic impedanceof the line, the smaller the reflected voltage waveand the smaller the reflection coefficient.

Figure 7

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This can be seen more clearly if we express thereflection coefficient in terms of load impedanceor load admittance. The reflection coefficient canbe made equal to zero by selecting a load, ZL, equalto the characteristic impedance of the line (Fig. 7b).

To facilitate computations, we will often want tonormalize impedances to the characteristic imped-ance of the transmission line. Expressed in termsof the reflection coefficient, the normalized imped-ance has this form (Fig. 8).

Figure 8

S-ParametersHaving briefly reviewed the properties of transmis-sion lines, let’s insert a two-port network into theline (Fig. 9). We now have additional travelingwaves that are interrelated. Looking at Er2, we seethat it is made up of that portion of Ei2 reflectedfrom the output port of the network as well as thatportion of Ei1 that is transmitted through the net-work. Each of the other waves are similarly madeup of a combination of two waves.

Figure 9

It should be possible to relate these four travelingwaves by some parameter set. While the derivationof this parameter set will be made for two-port net-works, it is applicable for n-ports as well. Let’sstart with the H-parameter set (Fig. 10).

Figure 10

Figure 11

By substituting the expressions for total voltageand total current (Fig. 11) on a transmission lineinto this parameter set, we can rearrange theseequations such that the incident traveling voltagewaves are the independent variables; and thereflected traveling voltage waves are the dependentvariables (Fig. 12).

Figure 12

The functions f11, f21 and f12, f22 represent a new setof network parameters relating traveling voltagewaves rather than total voltages and total currents.In this case these functions are expressed in termsof H-parameters. They could have been derivedfrom any other parameter set.

It is appropriate that we call this new parameterset “scattering parameters,” since they relate thosewaves scattered or reflected from the network tothose waves incident upon the network. Thesescattering parameters will commonly be referred toas S-parameters.

Let’s go one step further. If we divide both sides ofthese equations by Zo, the characteristic imped-ance of the transmission line, the relationship willnot change. It will, however, give us a change invariables (Fig. 13). Let’s now define the new vari-ables:

Figure 13

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Notice that the square of the magnitude of thesenew variables has the dimension of power. |a1|2

can then be thought of as the incident power onport one; |b1|2 as power reflected from port one.These new waves can be called traveling powerwaves rather than traveling voltage waves.Throughout this seminar, we will simply refer tothese waves as traveling waves.

Looking at the new set of equations in a little moredetail, we see that the S-parameters relate thesefour waves in this fashion (Fig. 14):

Figure 14

S-Parameter MeasurementWe saw how the H-parameters are measured. Let’snow see how we go about measuring the S-parame-ters. For S11, we terminate the output port of thenetwork and measure the ratio b1 to a1 (Fig. 15).Terminating the output port in an impedance equalto the characteristic impedance of the transmissionline is equivalent to setting a2 = 0, because a travel-ing wave incident on this load will be totally absorbed.S11 is the input reflection coefficient of the network.Under the same conditions, we can measure S21, theforward transmission through the network. This isthe ratio of b2 to a1 (Fig. 16). This could either bethe gain of an amplifier or the attenuation of a pas-sive network.

Figure 15

Figure 16

By terminating the input side of the network, weset a1 = 0. S22, the output reflection coefficient, andS12, the reverse transmission coefficient, can thenbe measured (Fig. 17).

Figure 17

A question often arises about the terminationsused when measuring the S-parameters. Becausethe transmission line is terminated in the charac-teristic impedance of the line, does the networkport have to be matched to that impedance as well?The answer is no!

To see why, let’s look once again at the networkenmeshed in the transmission line (Fig. 18). If theload impedance is equal to the characteristic imped-ance of the line, any wave traveling toward the loadwould be totally absorbed by the load. It would notreflect back to the network. This sets a2 = 0. Thiscondition is completely independent from the net-work’s output impedance.

Figure 18

Multiple-Port NetworksSo far we have just discussed two-port networks.These concepts can be expanded to multiple-portnetworks. To characterize a three-port network, forexample, nine parameters would be required (Fig. 19).S11, the input reflection coefficient at port one, ismeasured by terminating the second and thirdports with an impedance equal to the characteris-tic impedance of the line at these ports. This againensures that a2 = a3 = 0. We could go through theremaining S-parameters and measure them in asimilar way, once the other two ports are properlyterminated.

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Figure 19

What is true for two- and three-port networks issimilarly true for n-port networks (Fig. 20). Thenumber of measurements required for characteriz-ing these more complex networks goes up as thesquare of the number of ports. The concept andmethod of parameter measurement, however, is thesame.

Figure 20

Let’s quickly review what we’ve done up to thispoint. We started off with a familiar networkparameter set relating total voltages and total cur-rents at the ports of the network. We thenreviewed some transmission line concepts. Apply-ing these concepts, we derived a new set of param-eters for a two-port network relating the incidentand reflected traveling waves at the network ports.

The Use of S-ParametersTo gain further insight into the use of S-parame-ters, let’s see how some typical networks can berepresented in terms of S-parameters.

A reciprocal network is defined as having identicaltransmission characteristics from port one to porttwo or from port two to port one (Fig. 21). Thisimplies that the S-parameter matrix is equal to its transpose. In the case of a two-port network, S12 = S21.

Figure 21

A lossless network does not dissipate any power.The power incident on the network must be equalto the power reflected, or ∑|an|2 = ∑|bn|2 (Fig. 22).In the case of a two-port, |a1|2 + |a2|2 = |b1|2 +|b2|2. This implies that the S-matrix is unitary asdefined here. Where: I is the identity matrix and S*is the complex conjugate of the transpose of S. Thisis generally referred to as the hermetian conjugateof S. Typically, we will be using lossless networkswhen we want to place matching networksbetween amplifier stages.

Figure 22

For a lossy network, the net power reflected is lessthan the net incident power (Fig. 23). The differ-ence is the power dissipated in the network. Thisimplies that the statement I – S* S is positive defi-nite, or the eigen-values for this matrix are in theleft half plane so that the impulse response of thenetwork is made up of decaying exponentials.

Figure 23

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Change in Reference PlaneAnother useful relationship is the equation for chang-ing reference planes. We often need this in the meas-urement of transistors and other active deviceswhere, due to device size, it is impractical to attachRF connectors to the actual device terminals.

Imbedding the device in the transmission linestructure, we can then measure the S-parametersat these two planes (Fig. 24). We’ve added a lengthof line, φ1, to port one of the device and anotherlength, φ2, to port two.

Figure 24

The S-parameter matrix, S’, measured at these two planes is related to the S-parameter matrix of the device, S, by this expression. We’ve simplypre-multiplied and post-multiplied the device’s S-parameter matrix by the diagonal matrix, Φ.

To see what’s happening here, let’s carry throughthe multiplication of the S11 term. It will be multi-plied by e –jφ1 twice, since a1 travels through thislength of line, φ1, and gets reflected and then travelsthrough it again (Fig. 25). The transmission term,S’21, would have this form, since the input wave, a1, travels through φ1 and the transmitted wave, b2,through φ2. From the measured S-parameters, S’, wecan then determine the S-parameters of the device,S, with this relationship (Fig. 26).

Figure 25

Figure 26

Analysis of Networks Using S-ParametersLet’s now look at a simple example which willdemonstrate how S-parameters can be determinedanalytically.

Figure 27

Using a shunt admittance, we see the incident andreflected waves at the two ports (Fig. 27). We firstnormalize the admittance and terminate the net-work in the normalized characteristic admittanceof the system (Fig. 28a). This sets a2 = 0. S11, theinput reflection coefficient of the terminated net-work, is then: (Fig. 28b).

To calculate S21, let’s recall that the total voltage atthe input of a shunt element, a1 + b1, is equal to thetotal voltage at the output, a2 + b2 (Fig. 28c). Becausethe network is symmetrical and reciprocal, S22 = S11

and S12 = S21. We have then determined the four S-parameters for a shunt element.

Figure 28

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The Smith ChartAnother basic tool used extensively in amplifierdesign will now be reviewed. Back in the thirties,Phillip Smith, a Bell Lab engineer, devised a graph-ical method for solving the oft-repeated equationsappearing in microwave theory. Equations like theone for reflection coefficient, Γ = (Z – 1)/(Z + 1).Since all the values in this equation are complexnumbers, the tedious task of solving this expres-sion could be reduced by using Smith’s graphicaltechnique. The Smith Chart was a natural name forthis technique.

This chart is essentially a mapping between twoplanes—the Z (or impedance) plane and the Γ (orreflection coefficient) plane. We’re all familiar withthe impedance plane—a rectangular coordinateplane having a real and an imaginary axis. Anyimpedance can be plotted in this plane. For thisdiscussion, we’ll normalize the impedance plane to the characteristic impedance (Fig. 29).

Figure 29

Let’s pick out a few values in this normalized planeand see how they map into the Γ plane. Let z = 1.In a 50-ohm system, this means Z = 50 ohms. Forthis value, |Γ| = 0, the center of the Γ plane.

We now let z be purely imaginary (i.e., z = jx wherex is allowed to vary from minus infinity to plusinfinity). Since Γ = (jx – 1)/(jx + 1), |Γ| = 1 and itsphase angle varies from 0 to 360°. This traces out acircle in the Γ plane (Fig. 29). For positive reac-tance, jx positive, the impedance maps into theupper half circle. For negative reactance, theimpedance maps into the lower half circle. Theupper region is inductive and the lower region iscapacitive.

Now let’s look at some other impedance values. A constant resistance line, going through the pointz = 1 on the real axis, maps into a circle in the Γplane. The upper semicircle represents an imped-ance of 1 + jx, which is inductive; the lower semi-circle, an impedance of 1 – jx or capacitive (Fig. 30).

Figure 30

The constant reactance line, r + j1, also maps intothe Γ plane as a circle. As we approach the imagi-nary axis in the impedance plane, Γ approachesthe unit radius circle. As we cross the imaginaryaxis, the constant reactance circle in the Γ planegoes outside the unit radius circle.

If we now go back and look at z real, we see at z = –1, Γ = ∞. When z is real and less than one, we move out toward the unit radius circle in the Γplane. When the real part of z goes negative, Γ con-tinues along this circle of infinite radius. The entireregion outside the unit radius circle representsimpedances with negative real parts. We will usethis fact later when working with transistors andother active devices, which often have negative realimpedances.

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In the impedance plane, constant resistance andconstant reactance lines intersect. They also crossin the Γ plane. There is a one-to-one correspon-dence between points in the impedance plane andpoints in the Γ plane.

The Smith Chart can be completed by continuingto draw other constant resistance and reactancecircles (Fig. 31).

Figure 31

Applications of the Smith ChartLet’s now try a few examples with the Smith Chartto illustrate its usefulness.

1. Conversion of impedance to admittance: Converting anormalized impedance of 1 + j1 to an admittancecan be accomplished quite easily. Let’s first plotthe point representing the value of z on the SmithChart (Fig. 32). From these relationships, we seethat while the magnitude of admittance is thereciprocal of the magnitude of impedance, the mag-nitude of Γ is the same—but its phase angle ischanged by 180°. On the Smith Chart, the Γ vectorwould rotate through 180°. This point could thenbe read off as an admittance.

Figure 32

We can approach this impedance to admittanceconversion in another way. Rather than rotate theΓ vector by 180°, we could rotate the Smith Chartby 180° (Fig. 33). We can call the rotated chart anadmittance chart and the original an impedancechart. Now we can convert any impedance toadmittance, or vice versa, directly.

Figure 33

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2. Impedances with negative real parts: Let’s now takea look at impedances with negative real parts. Hereagain is a conventional Smith Chart defined by theboundary of the unit radius circle. If we have animpedance that is inductive with a negative realpart, it would map into the Γ plane outside thechart (Fig. 34). One way to bring this point backonto the chart would be to plot the reciprocal of Γ,rather than Γ itself. This would be inconvenientbecause the phase angle would not be preserved.What was a map of an inductive impedanceappears to be capacitive.

Figure 34

If we plot the reciprocal of the complex conjugateof Γ, however, the phase angle is preserved. Thisvalue lies along the same line as the original Γ.Typically in the Agilent Technologies transistordata sheets, impedances of this type are plottedthis way.

There are also compressed Smith Charts availablethat include the unit radius chart plus a great dealof the negative impedance region. This chart has aradius that corresponds to a reflection coefficientwhose magnitude is 3.16 (Fig. 35).

Figure 35

In the rest of this seminar, we will see how easilywe can convert measured reflection coefficientdata to impedance information by slipping a SmithChart overlay over the Agilent Technologies net-work analyzer polar display.

3. Frequency response of networks: One final point needsto be covered in this brief review of the Smith Chart,and that is the frequency response for a given net-work. Let’s look at a network having an impedance,z = 0.4 + jx (Fig. 36). As we increase the frequencyof the input signal, the impedance plot for the net-work moves clockwise along a constant resistancecircle whose value is 0.4. This generally clockwisemovement with increasing frequency is typical ofimpedance plots on the Smith Chart for passivenetworks. This is essentially Foster’s ReactanceTheorem.

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Figure 36

If we now look at another circuit having a real partof 0.2 and an imaginary part that is capacitive, theimpedance plot again moves in a clockwise direc-tion with an increase in frequency.

Another circuit that is often encountered is thetank circuit. Here again, the Smith Chart is usefulfor plotting the frequency response (Fig. 37). Forthis circuit at zero frequency, the inductor is ashort circuit. We start our plot at the point, z = 0.As the frequency increases, the inductive reac-tance predominates. We move in a clockwise direc-tion. At resonance, the impedance is purely real,having the value of the resistor. If the resistor hada higher value, the cross-over point at resonancewould be farther to the right on the Smith Chart.As the frequency continues to increase, theresponse moves clockwise into the capacitiveregion of the Smith Chart until we reach infinitefrequency, where the impedance is again zero.

Figure 37

In theory, this complete response for a tank circuitwould be a circle. In practice, since we do not gen-erally have elements that are pure capacitors orpure inductors over the entire frequency range, wewould see other little loops in here that indicateother resonances. These could be due to parasiticinductance in the capacitor or parasitic capaci-tance in the inductor. The diameter of these circlesis somewhat indicative of the Q of the circuit. If wehad an ideal tank circuit, the response would bethe outer circle on the Smith Chart. This wouldindicate an infinite Q.

Agilent Technologies Application Note 117-1describes other possible techniques for measuringthe Q of cavities and YIG spheres using the SmithChart. One of these techniques uses the fact thatwith a tank circuit, the real part of the circuitequals the reactive part at the half-power points.Let’s draw two arcs connecting these points on theSmith Chart (Fig. 38). The centers for these arcsare at ±j1. The radius of the arcs is 2.

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Figure 38

We then increase the frequency and record itsvalue where the response lies on the upper arc.Continuing to increase the frequency, we recordthe resonant frequency and the frequency wherethe response lies on the lower arc. The formula forthe Q of the circuit is simply fo, the resonant fre-quency, divided by the difference in frequencybetween the upper and lower half-power points. Q = fo/∆f.

SummaryLet’s quickly review what we’ve seen with theSmith Chart. It is a mapping of the impedanceplane and the reflection coefficient or Γ plane. Wediscovered that impedances with positive realparts map inside the unit radius circle on theSmith Chart. Impedances with negative real partsmap outside this unit radius circle. Impedanceshaving positive real parts and inductive reactancemap into the upper half of the Smith Chart. Thosewith capacitive reactance map into the lower half.

In the next part of this S-Parameter Design Semi-nar, we will continue our discussion of networkanalysis using S-parameters and flow graph tech-niques.

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This second portion of Agilent Technologies’ BasicMicrowave Review will introduce some additionalconcepts that are used in high frequency amplifierdesign.

Scattering Transfer ParametersLet’s now proceed to a set of network parametersused when cascading networks. We recall that wedeveloped the S-parameters by defining thereflected waves as dependent variables, and inci-dent waves as independent variables (Fig. 39a). Wenow want to rearrange these equations such thatthe input waves a1 and b1 are the dependent vari-ables and the output waves a2 and b2 the independ-ent variables. We’ll call this new parameter setscattering transfer parameters or T-parameters(Fig. 39b).

Figure 39

The T-parameters can be developed by manipulat-ing the S-parameter equations into the appropriateform. Notice that the denominator of each of theseterms is S21 (Fig. 40).

Figure 40

We can also find the S-parameters as a function ofthe T-parameters.

While we defined the T-parameters in a particularway, we could have defined them such that the out-put waves are the dependent variables and the inputwaves are the independent variables. This alter-nate definition can result in some problems whendesigning with active unilateral devices (Fig. 41).

Figure 41

Using the alternate definition for the transferparameters, the denominator in each of theseterms is S12 rather than S21 as we saw earlier.

Working with amplifiers, we often assume thedevice to be unilateral, or S12 = 0. This would causethis alternate T-parameter set to go to infinity.

Both of these definitions for T-parameters can beencountered in practice. In general, we prefer todefine the T-parameters with the output waves asthe dependent variables, and the input waves asthe independent variables.

We use this new set of transfer parameters when wewant to cascade networks—two stages of an ampli-fier, or an amplifier with a matching network forexample (Fig. 42a). From measured S-parameterdata, we can define the T-parameters for the twonetworks. Since the output waves of the first net-work are identical to the input waves of the sec-ond network, we can now simply multiply the twoT-parameter matrices and arrive at a set of equa-tions for the overall network (Fig. 42b).

Figure 42

Chapter 2. Basic Microwave Review II

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Since matrix multiplication is, in general, not com-mutative, these T-parameter matrices must be mul-tiplied in the proper order. When cascading networks,we’ll have to multiply the matrices in the same orderas the networks are connected. Using the alternatedefinition for T-parameters previously described,this matrix multiplication must be done in reverseorder.

This transfer parameter set becomes very usefulwhen using computer-aided design techniqueswhere matrix multiplication is an easy task.

Signal Flow GraphsIf we design manually, however, we can use stillanother technique—signal flow graphs—to followincident and reflected waves through the networks.This is a comparatively new technique for microwavenetwork analysis.

A. RulesWe’ll follow certain rules when we build up a net-work flow graph.

1. Each variable, a1, a2, b1, and b2 will be desig-nated as a node.

2. Each of the S-parameters will be a branch.

3. Branches enter dependent variable nodes, andemanate from the independent variable nodes.

4. In our S-parameter equations, the reflectedwaves b1 and b2 are the dependent variables andthe incident waves a1 and a2 are the independentvariables.

5. Each node is equal to the sum of the branchesentering it.

Let’s now apply these rules to the two S-parameterequations (Fig. 43a). The first equation has threenodes: bl, a1, and a2. b1 is a dependent node and isconnected to a1 through the branch S11 and to nodea2 through the branch S12. The second equation isconstructed similarly. We can now overlay these tohave a complete flow graph for a two-port network(Fig. 43b).

Figure 43

The relationship between the traveling waves isnow easily seen. We have a1 incident on the net-work. Part of it transmits through the network tobecome part of b2. Part of it is reflected to becomepart of b1. Meanwhile, the a2 wave entering porttwo is transmitted through the network to becomepart of b1 as well as being reflected from port twoas part of b2. By merely following the arrows, wecan tell what’s going on in the network.

This technique will be all the more useful as wecascade networks or add feedback paths.

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B. Application of Flow GraphsLet’s now look at several typical networks we willencounter in amplifier designs. A generator withsome internal voltage source and an internalimpedance will have a wave emanating from it. Theflow graph for the generator introduces a newterm, bS (Fig. 44). It’s defined by the impedance ofthe generator. The units in this equation look pecu-liar, but we have to remember that we originallynormalized the traveling waves to Zo. The magni-tude of bS squared then has the dimension ofpower.

Figure 44

For a load, the flow graph is simply ΓL, the complexreflection coefficient of the load (Fig. 45).

Figure 45

When the load is connected to the generator, wesee a wave emanating from the generator incidenton the load, and a wave reflected back to the gen-erator from the load (Fig. 46).

Figure 46

To demonstrate the utility of flow graphs, let’sembed a two-port network between a source and a load. Combining the examples we have seen, wecan now draw a flow graph for the system (Fig. 47).

Figure 47

We can now apply the rule known as Mason’s rule(or as it is often called, the non-touching loop rule)to solve for the value of any node in this network.Before applying the rule, however, we must firstdefine several additional terms.

A first order loop is defined as the product of thebranches encountered in a journey starting from anode and moving in the direction of the arrowsback to that original node. To illustrate this, let’sstart at node a1. One first order loop is S11ΓS.Another first order loop is S21ΓLS12ΓS. If we nowstart at node a2, we find a third first order loop,S22ΓL. Any of the other loops we encounter is oneof these three first order loops.

A second order loop is defined as the product of anytwo non-touching first order loops. Of the threefirst order loops just found, only S11ΓS and S22ΓL donot touch in any way. The product of these twoloops establishes the second order loop for thisnetwork. More complicated networks, involvingfeedback paths for example, might have severalsecond order loops.

A third order loop is the product of any three non-touching first order loops. This example does nothave any third order loops, but again, more compli-cated networks would have third order loops andeven higher order loops.

Let’s now suppose that we are interested in thevalue of bl. In this example, bS is the only independ-ent variable because its value will determine thevalue of each of the other variables in the network.B1, therefore, will be a function of bS. To determineb1, we first have to identify the paths leading frombS to b1. Following the arrows, we see two paths—(1) S11 and (2) S2lΓLSl2.

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The next step is to find the non-touching loopswith respect to the paths just found. Here, the pathS11 and the first order loop, S22ΓL have no nodes orbranches in common. With this condition met, wecan call S22ΓL a non-touching loop with respect tothe path S11.

The other path, S21ΓLS12, touches all of the net-work’s first order loops, hence there are no non-touching loops with respect to this path. Again, inmore complex networks, there would be higherorder non-touching loops.

Let’s now look at the non-touching loop rule itself(Fig. 48). This equation appears to be rather omi-nous at first glance, but once we go through it termby term, it will be less awesome. This rule deter-mines the ratio of two variables, a dependent to anindependent variable. (In our example, we areinterested in the ratio b1 to bS.)

Figure 48

Pl, P2, etc., are the various paths connecting thesevariables.

This term, ∑L(l)(1), is the sum of all first orderloops that do not touch the first path between thevariables.

This term, ∑L(2)(1), is the sum of all second orderloops that do not touch that path, and so on downthe line.

Now, this term, ∑L(1)(2), is the sum of all first orderloops that do not touch the second path.

The denominator in this expression is a function ofthe network geometry. It is simply one minus thesum of all first order loops, plus the sum of all sec-ond order loops, minus the third order loops, andso on.

Let’s now apply this non-touching loop rule to ournetwork (Fig. 49). The ratio of bl, the dependentvariable, to bS, the independent variable, is equalto the first path, S1l, multiplied by one minus thenon-touching first order loop with respect to thispath, ΓLS22.

Figure 49

The second path, S21ΓLS12, is simply multiplied byone because there are no non-touching loops withrespect to this path.

The denominator is one minus the sum of firstorder loops plus the second order loop.

This concludes our example. With a little experi-ence drawing flow graphs of complex networks,you can see how this technique will facilitate yournetwork analysis. In fact, using the flow graphtechnique, we can now derive several expressionsfor power and gain that are of interest to the cir-cuit designer.

First of all, we need to know the power delivered to aload. Remember that the square of the magnitudesof the incident and reflected waves has the dimen-sion of power. The power delivered to a load isthen the difference between the incident powerand the reflected power, Pdel = |a|2 –|b|2.

The power available from a source is that power deliv-ered to a conjugately matched load. This impliesthat the reflection coefficient of the load is the con-jugate of the source reflection coefficient, ΓS* = ΓL.

Figure 50

Looking at the flow graph describing these condi-tions (Fig. 50), we see that the power availablefrom the source is:

Pavs = |b|2 –|a|2

Using the flow graph techniques previouslydescribed, we see that:

18

The power available from the source reduces to(Fig. 51):

Figure 51

We can also develop voltage and power gain equa-tions that will be useful in our amplifier designsusing these flow graph techniques. For a two-portnetwork, the voltage gain is equal to the total volt-age at the output divided by the total voltage at theinput,

a2 + b2Av =a1 + b1

If we divide the numerator and denominator by bs,we can relate each of the dependent variables of thesystem to the one independent variable (Fig. 52a).Now we have four expressions or four ratios thatwe can determine from the non-touching loop rule.

Figure 52

We can simplify this derivation by rememberingthat the denominator in the expression for thenon-touching loop rule is a function of the networkgeometry. It will be the same for each of theseratios, and will cancel out in the end. So we onlyneed to be concerned with the numerators of theseratios.

Let’s trace through a couple of these expressions tofirm up our understanding of the process (Fig. 52b).A2 is connected to bS through the path S21ΓL. Allfirst order loops touch this path, so this path issimply multiplied by one. b2 is connected to bS

through the path S21. All first order loops also

touch this path. a1 is connected to bS directly, andthere is one non-touching loop, S22ΓL. We havealready determined the ratio of b1 to bS, so we cansimply write down the numerator of that expres-sion. We have now derived the voltage gain of thetwo-port network.

The last expression we wish to develop is that fortransducer power gain. This will be very important in the amplifier design examples contained in thefinal section of this seminar. Transducer powergain is defined as the power delivered to a loaddivided by the power available from a source.

PdelGt =Pavs

We have already derived these two expressions.

What remains is to solve the ratio b2 to bS (Fig. 53a).The only path connecting bS and b2 is S21. There areno non-touching loops with respect to this path.The denominator is the same as in the previousexample: one minus the first order loops plus thesecond order loop. Taking the magnitude of thisratio, squaring, and substituting the result yieldsthe expression for transducer power gain of a two-port network (Fig. 53b).

Figure 53

Needless to say, this is not a simple relationship, asthe terms are generally complex quantities. Calcu-lator or computer routines will greatly facilitate thecircuit designer’s task.

Later, when designing amplifiers, we will see thatwe can simplify this expression by assuming thatthe amplifier is a unilateral device, or S12 = 0. Ingeneral, however, this assumption cannot be made,and we will be forced to deal with this expression.

19

One of the things you might want to do is to opti-mize or maximize the transducer gain of the net-work. Since the S-parameters at one frequency areconstants depending on the device selected andthe bias conditions, we have to turn our attentionto the source and load reflection coefficients.

Stability ConsiderationsTo maximize the transducer gain, we must conju-gately match the input and the output. Before wedo this, we will have to look at what might happento the network in terms of stability—will the ampli-fier oscillate with certain values of impedanceused in the matching process?

There are two traditional expressions used whenspeaking of stability: conditional and uncondi-tional stability.

A network is conditionally stable if the real part of Zin

and Zout is greater than zero for some positive realsource and load impedances at a specific frequency.

A network is unconditionally stable if the real part ofZin and Zout is greater than zero for all positive realsource and load impedances at a specific frequency.

It is important to note that these two conditionsapply only at one specific frequency. The condi-tions we will now discuss will have to be investi-gated at many frequencies to ensure broadbandstability. Going back to our Smith Chart discus-sion, we recall that positive real source and loadimpedances imply: |ΓS|and |ΓL| ≤1.

Figure 54

Let’s look first at the condition where we want toconjugately match the network to the load andsource to achieve maximum transducer gain (Fig. 54).Under these conditions, we can say that the networkwill be stable if this factor, K, is greater than one(Fig. 55). Conjugately matched conditions mean

that the reflection coefficient of the source, ΓS, isequal to the conjugate of the input reflection coef-ficient, Γin.

ΓS = Γin*

ΓL is equal to the conjugate of the output reflectioncoefficient, Γout.

ΓL = Γout*

Figure 55

A precaution must be mentioned here. The K factormust not be considered alone. If we were operatingunder matched conditions in order to achieve max-imum gain, we would have to consider the following:(1) What effect would temperature changes or drift-ing S-parameters of the transistor have on the sta-bility of the amplifier? (2) We would also have tobe concerned with the effect on stability as we sub-stitute transistors into the circuit. (3) Would thesechanging conditions affect the conjugate match orthe stability of the amplifier? Therefore, we reallyneed to consider these other conditions in additionto the K factor.

Looking at the input and output reflection coeffi-cient equations, we see a similarity of form (Fig. 56).The only difference is that S11 replaces S22, and ΓL

replaces ΓS.

Figure 56

If we set this equation, |Γin|, equal to one, aboundary would be established. On one side of theboundary, we would expect |Γin| <1. On the otherside, we would expect |Γin| >1.

Let’s first find the boundary by solving this expres-sion (Fig. 57). We insert the real and imaginaryvalues for the S-parameters and solve for ΓL.

Figure 57

20

The solutions for ΓL will lie on a circle. The radiusof the circle will be given by this expression as afunction of S-parameters (Fig. 58a).

Figure 58

The center of the circle will have this form (Fig. 58b).Having measured the S-parameters of a two-portdevice at one frequency, we can calculate thesequantities.

If we now plot these values on a Smith Chart, wecan determine the locus of all values of ΓL thatmake |Γin| = 1.

This circle then represents the boundary (Fig. 59).The area either inside or outside the circle willrepresent a stable operating condition.

Figure 59

To determine which area represents this stable oper-ating condition, let’s make ZL = 50 ohms, or ΓL = 0.This represents the point at the center of the SmithChart. Under these conditions, |Γin| = |S11|.

Let’s now assume that S11 has been measured andits magnitude is less than one. Γin’s magnitude isalso less than one. This means that this point, ΓL = 0,represents a stable operating condition. This region(Fig. 60) then represents the stable operating con-dition for the entire network.

Figure 60

If we select another value of ΓL that falls inside thestability circle, we would have an input reflectioncoefficient that would be greater than one, and thenetwork would be potentially unstable.

If we only deal with passive loads, that is, loadshaving a reflection coefficient less than or equal toone, then we only have to stay away from thoseΓL’s that are in this region (Fig. 61) to ensure sta-ble operation for the amplifier we are designing.Chances are, we should also stay away from imped-ances in the border region, since the other factorslike changing temperature, the aging of the transis-tors, or the replacement of transistors might causethe center or radius of the stability circle to shift.The impedance of the load could then fall in theexpanded unstable region, and we would again bein trouble.

21

Figure 61

If, on the other hand, |S11| >1, with ZL = 50 Ω, thenthis area would be the stable region and thisregion the unstable area (Fig. 62).

Figure 62

To ensure that we have an unconditionally stable con-dition at a given frequency in our amplifier design,we must be able to place any passive load on thenetwork and drive it with any source impedancewithout moving into an unstable condition.

From a graphical point of view, we want to be surethat the stability circle falls completely outside theSmith Chart, and we want to make sure that theinside of the stability circle represents the unstableregion (Fig. 63). The area outside the stability cir-cle, including the Smith Chart, would then repre-sent the stable operating region.

Figure 63

To satisfy this requirement, we must ensure thatthe magnitude of the vector, CL, the distance fromthe center of the Smith Chart to the center of thestability circle, minus the radius of the stabilitycircle, rL, is greater than one. This means that theclosest point on the stability circle would be out-side the unit radius circle or Smith Chart.

To ensure that the region inside the Smith Chartrepresents the stable operating condition, theinput or output impedance of the network musthave a real part greater than zero when the net-work is terminated in 50 ohms. For completeness,we must also add the output stability circle to gaina better understanding of this concept. This meansthat the magnitude of S11 and S22 must be less thanone.

One word of caution about stability.

S-parameters are typically measured at some par-ticular frequency. The stability circles are drawnfor that frequency. We can be sure that the ampli-fier will be stable at that frequency, but will itoscillate at some other frequency either inside oroutside the frequency range of the amplifier?

22

Typically, we want to investigate stability over abroad range of frequencies and construct stabilitycircles wherever we might suspect a problem.Shown here are the stability circles drawn forthree different frequencies (Fig. 64). To ensure stability between f1 and f3, we stay away fromimpedances in this (shaded) area. While thisprocess may sound tedious, we do have somenotion based on experience where something may get us into trouble.

Figure 64

Stability is strongly dependent on the |S12| |S21|product (Fig. 65). |S21| is a generally decreasingfunction of frequency from fβ on.

Figure 65

|S12| is an increasing function.

Looking at the |S12| |S21| product, we see that itincreases below fβ, flattens out, then decreases athigher frequencies.

It is in this flat region that we must worry aboutinstability.

On the other hand, if we synthesize elements suchas inductors by using high impedance transmissionlines, we might have capacitance rather thaninductance at higher frequencies, as seen here onthe Impedance Phase plot (Fig. 66). If we suspectthat this might cause oscillation, we would investi-gate stability in the region where the inductor iscapacitive. Using tunnel diodes having negativeimpedance all the way down to dc, we would haveto investigate stability right on down in frequencyto make sure that oscillations did not occur outsidethe band in which we are working.

Figure 66

23

The material presented in this program is a contin-uation of Agilent Technologies’ video tape S-Para-meter Design Seminar.

S-ParametersA. Their ImportanceMicrowave transistor technology is continuallypushing maximum operating frequencies everupward. As a result, manufacturers of transistorsare specifying their transistors in terms of S-parameters. This affects two groups of design engineers. Transistor circuit designers must nowswitch their thinking from the well-known H-, Y-,and Z-parameters in their circuit designs to the S or scattering parameters. Microwave engineers,because transistor technology is moving into theirfrequency domain, must now become conversantwith transistor terminology and begin to think ofapplying transistors to the circuits they work with.

In this tape we will:

1. Review the S-parameter concept.

2. Show the results of typical S-parameter meas-urements of a 12-GHz transistor.

3. Demonstrate the network analyzer system usedin these measurements.

B. Review of S-ParametersThe function of network analysis is to completelycharacterize or describe a network so we’ll knowhow it will behave when stimulated by some signal.For a two-port device, such as a transistor, we cancompletely describe or characterize it by establish-ing a set of equations that relate the voltages andcurrents at the two ports (Fig. 67).

Figure 67

In low frequency transistor work, one such set ofequations relates total voltages across the portsand total current into or out of these ports interms of H-parameters. For example,

V1h11 =I1 V2 = 0

These parameters are obtained under either openor short circuit conditions.

At higher frequencies, especially in the microwavedomain, these operating conditions present a prob-lem since a short circuit looks like an inductor andan open circuit has some leakage capacitance.Often, if the network is an active device such as atransistor, it will oscillate when terminated with areactive load.

It is imperative that some new method for charac-terizing these devices at high frequencies is used.

Figure 68

If we embed our two-port device into a transmis-sion line, and terminate the transmission line in itscharacteristic impedance, we can think of the stim-ulus signal as a traveling wave incident on thedevice, and the response signal as a wave reflect-ing from the device or being transmitted throughthe device (Fig. 68). We can then establish this newset of equations relating these incident and “scat-tered” waves (Fig. 69a): E1r and E2r are the voltagesreflected from the 1st and 2nd ports, E1i and E2i

are the voltages incident upon the 1st and 2ndports. By dividing through by Zo, where Zo is thecharacteristic impedance of the transmission line,we can alter these equations to a more recogniza-ble form (Fig. 69b). Where, for example, |b1|2 =Power reflected from the 1st port and|a1|2 = Powerincident on the 1st port.

Chapter 3. S-Parameter Measurements

24

Figure 69

S11 is then equal to b1/a1 with a2 = 0 or no incidentwave on port 2. This is accomplished by terminat-ing the output of the two-port in an impedanceequal to Zo.

C. SummaryS11 = input reflection coefficient with the outputmatched.

S21 = forward transmission coefficient with theoutput matched.

This is the gain or attenuation of the network.

S22 = output reflection coefficient with the inputmatched.

S12 = reverse transmission coefficient with theinput matched.

To the question “Why are S-parameters impor-tant?” you can now give several answers:

1. S-parameters are determined with resistive ter-minations. This obviates the difficulties involved inobtaining the broadband open and short circuitconditions required for the H-, Y-, and Z-parameters.

2. Parasitic oscillations in active devices are mini-mized when these devices are terminated in resis-tive loads.

3. Equipment is available for determining S-param-eters since only incident and reflected voltagesneed to be measured.

Characterization of Microwave TransistorsNow that we’ve briefly reviewed S-parameter the-ory, let’s look at some typical transistor parame-ters. There are three terms often used by transistorcircuit designers (Fig. 70):

1. ft or the frequency at which the short circuitcurrent gain is equal to one;

2. fS or the frequency where |S21| =1 or the powergain of the device, |S21|2, expressed in dB is zero;

3. fmax or the frequency where the maximum avail-able power gain, Gamax, of the device is equal toone. Fmax is also referred to as the maximum fre-quency of oscillation.

Figure 70

To determine fs of a transistor connected in a com-mon emitter configuration, we drive the base witha 50-ohm voltage source and terminate the collec-tor in the 50-ohm characteristic impedance. Thisresults in a gain versus frequency plot that decaysat about 6 dB per octave at higher frequencies.

Due to the problems involved in obtaining trueshort circuits at high frequencies, the short circuitcurrent gain |hfe| cannot be measured directly, butcan be derived from measured S-parameter data.The shape of this gain versus frequency curve issimilar to that of |S21|2 and, for this example, ft isslightly less than fs.

25

Fmax is determined after conjugately matching thevoltage source to the transistor input, and the tran-sistor output to the characteristic impedance ofthe line. The resulting gain is the maximum avail-able power gain as a function of frequency. It ishigher than |S21|2 because of impedance matchingat the input and output. With proper impedancematching techniques, the transistor is usable abovefs in actual circuit design.

S-Parameters of TransistorsA. IntroductionLet’s now shift our attention to the actual S-param-eters of a transistor. We’ll look at transistors inchip form and after the chips have been packaged.The advantage of characterizing the chip is thatyou will get a better qualitative understanding ofthe transistor. However, fixtures to hold these chipsare not readily available. Most engineers will beusing packaged transistors in their R&D work.There are fixtures available for characterizingpackaged transistors, and we will demonstratethese later on (Fig. 71). The bias conditions usedwhen obtaining these transistors’ parameters con-nected as common emitter: Vcb = 15 V and Ic = 15 mA.

Figure 71

B. S11 of Common EmitterThe input reflection characteristic, S11, of the chiptransistor seems to be following a constant resist-ance circle on the Smith Chart (Fig. 72). At lowerfrequencies, the capacitive reactance is clearly visi-ble, and as the frequency increases, this reactancedecreases and the resistance becomes more evident.A small inductance is also evident, which for thisexample, resonates with the capacitance at 10 GHz.

Figure 72

An equivalent circuit can be drawn that exhibitssuch characteristics (Fig. 73). The resistance comesfrom the bulk resistivities in the transistor’s baseregion plus any contact resistance resulting frommaking connections to the device. The capacitanceis due mainly to the base-emitter junction. Theinductance results from the emitter resistancebeing referred back to the input by a complex β atthese high frequencies.

Figure 73

If you characterize the same chip transistor afterpackaging, the input reflection characteristic againstarts in the capacitive reactance region at lowerfrequencies and then moves into the inductivereactance region at higher frequencies (Fig. 72).Another equivalent circuit explaining this charac-teristic can be drawn (Fig. 74). Package inductanceand capacitance contribute to the radial shiftinward as well as to the extension of the S11 char-acteristic into the upper portion of the Smith Chart.

Figure 74

26

C. S22 of Common EmitterThe output reflection coefficient, S22, is again inthe capacitive reactance portion of the SmithChart (Fig. 75). If you overlay an admittance SmithChart, you can see that this characteristic roughlyfollows a constant conductance circle. This type ofcharacteristic represents a shunt RC type of equiv-alent circuit where the angle spanned would becontrolled by capacitive elements, and the radialdistance from the center of the Smith Chart wouldbe a function of the real parts (Fig. 76).

Figure 75

Figure 76

The output reflection coefficient of the packagedtransistor is again shifted radially inward and theangle spanned is extended. From an equivalent cir-cuit standpoint (Fig. 77), you can see that we haveadded the package inductance and changed thecapacitance. This added inductance causes thisparameter to shift away from a constant conduc-tance circle.

Figure 77

D. S21 of Common EmitterThe forward transmission coefficient, S21, that wehave seen before when discussing fs, exhibits avoltage gain value slightly greater than 4 or 12 dBat 1 GHz and crosses the unity gain circle between4 and 5 GHz (Fig. 78). The packaged transistorexhibits slightly less gain and a unity gaincrossover at around 4 GHz.

Figure 78

27

In an equivalent circuit, we could add a currentsource as the element giving gain to the transistor(Fig. 79).

Figure 79

E. S12 of Common EmitterSince a transistor is not a unilateral device, thereverse transmission characteristic, S12, will havesome finite value in chip form. On a polar plot, theS12 characteristic approximates a circular path(Fig. 80).

Figure 80

If you were to plot |S12| on a Bode Diagram, youwould see a gradual buildup at about 6 dB/octaveat low frequencies, a leveling off, and then ultimatelya decay at the higher frequencies. Let’s now super-impose a Bode Plot of |S21|. It is constant at frequen-cies below fβ and then decays at about 6 dB/octave.Therefore, the product of these two characteristicswould increase up to fβ, around 100 to 200 MHz,and remain relatively flat until a break point ataround the fs of the transistor (Fig. 81).

Figure 81

This |S12||S21|product is significant since it bothrepresents a figure of merit of the feedback or sta-bility term of the device and it also appears in thecomplete equations for input and output reflectioncoefficients.

F. Combined Equivalent CircuitIf you were to now combine the equivalent circuitsdrawn up to this point, you could arrive at a quali-tative model that describes the transistor’s opera-tion (Fig. 82).

Figure 82

Measurement DemonstrationNow that you’ve seen some typical transistor char-acteristics, let’s actually make several measure-ments to see how simply and accurately you canmake the measurements that will provide you withthe necessary data for designing your circuits.

The S-parameter characteristics we have seen arethose of a Model 35821E Transistor. In these meas-urements we will measure the transistor in a K-disc common emitter package.

28

The standard bias conditions are:

Vcb = 15 volts Ic = 15 mA

On the polar display with the Smith Chart overlayinserted, the input impedance can be read offdirectly. To ensure that we are in the linear regionof the transistor, we can measure S11 at two inputpower levels to the transistor. If these readings do not change, we know that we are driving thetransistor at an optimum power level and the S-parameters are truly the small signal characteris-tics. If we now vary the collector current bias level,we note very little difference (Fig. 90).

Figure 90

Returning the current level to the original value,we now decrease Vcb and note a shift of the origi-nal characteristic. Decreasing Vcb causes the epi-taxial layer to be less depleted so you would expectless capacitive reactance in the input equivalentcircuit.

Figure 91

Now you can measure the output reflection coeffi-cient (Fig. 91). Let’s now reduce the collector cur-rent and note the effect on this characteristic. Theradial shift outward indicates an increase in thereal part of the output impedance. This shift is dueto the real part being inversely proportional to thegm of the transistor, while the collector current isdirectly proportional to gm (Fig. 92).

Figure 92

Let’s now return Ic to 15 mA and decrease Vcb. Younote a radial shift inward. This shift is againrelated to the depletion of the epitaxial layer.

29

Let’s now turn our attention to the gain of thetransistor and depress S21 with the bias conditionsback at their original values. The forward gain ofthe device, S21, is now visible. This characteristic isalso affected by varying the bias conditions (Fig. 93).

Figure 93

Let’s look now at the reverse transmission charac-teristic, S12. This value is much smaller than theforward gain, so we will have to introduce moretest channel gain into the system to enable us tohave a reasonable display. This characteristic isrelatively invariant to bias changes.

One characteristic that often appears on transistordata sheets is the relation of power gain |S21|2 ver-sus collector current at one frequency. This charac-teristic curve was determined at 1 GHz (Fig. 94).

Figure 94

At low current levels, there is a gradual increase,then the power gain characteristic flattens out anddecreases at higher current levels. The base-collectortransit time determines the flat plateau. The highcurrent roll-off is due to two effects: (1) thermaleffects on the transistor; (2) if we try to pump morecurrent into the device than it can handle, the baseof the transistor stretches electrically. Since theelectrons move across the base-collector junctionat a finite rate, the current density increases as we try to pump more current in, until, at the limit,the base has stretched to the width of the epitaxiallayer and this will account for the gain goingtoward zero.

SummaryThis tape has presented an overview of S-parame-ter theory and has related this theory to actualtransistor characterization.

The remaining tapes in this S-Parameter DesignSeminar are devoted to high frequency circuitdesign techniques using S-parameters. Constantgain and noise figure circles will be discussed andthen used in designing unilateral narrow andbroadband amplifiers.

This amplifier (Fig. 95), for example, was designedwith S-parameter data, and operates from 100 MHzto 2 GHz with a typical gain of 40 dB and flat towithin 3 dB across the band. A similar amplifierwill be designed in the next tape.

Figure 95

The use of these design techniques and measure-ment equipment will also prove valuable to you inyour device development.

30

IntroductionIn this tape, the practical application of S-parameterswill be discussed. They will specifically be appliedto unilateral amplifier design. This tape is a contin-uation of the Agilent Technologies Microwave Divi-sion’s S-Parameter Design Seminar. We will discuss:Transducer Power Gain, Constant Gain, and Con-stant Noise Figure Circles; and then use these con-cepts with S-parameter data in the design of ampli-fiers for the case where the transistor can beassumed to be unilateral, or S12 = 0.

S-Parameter ReviewBefore introducing these concepts, let’s brieflyreview S-parameters.

Figure 96

As opposed to the more conventional parametersets which relate total voltages and total currentsat the network ports, S-parameters relate travelingwaves (Fig. 96). The incident waves, a1 and a2, arethe independent variables, and the reflected waves,b1 and b2, are the dependent variables. The networkis assumed to be embedded in a transmission linesystem of known characteristic impedance whichshall be designated Zo. The S-parameters are thenmeasured with Zo terminations on each of the portsof the network. Under these conditions, S11 and S22,the input and output reflection coefficients, andS21 and S12, the forward and reverse transmissioncoefficients, can be measured (Fig. 97).

Figure 97

Transducer Power GainIn the design of amplifiers, we are most interestedin the transducer power gain. An expression canbe derived for transducer power gain if we firstredraw the two-port network using flow graphtechniques (Fig. 98).

The transducer power gain is defined as the powerdelivered to the load divided by the power avail-able from the source. The ratio of b2 to bs can befound by applying the non-touching loop rule forflow graphs resulting in this expression for trans-ducer power gain (Fig. 99).

Figure 98

Figure 99

If we now assume the network to be unilateral,that is, S12 is equal to zero, this term (S21S12ΓLΓS)drops out and the resulting expression can be sep-arated into three distinct parts. This expressionwill be referred to as the unilateral transducerpower gain (Fig. 100).

Figure 100

The first term is related to the transistor or otheractive device being used. Once the device and itsbias conditions are established, S21 is determinedand remains invariant throughout the design.

Chapter 4. High Frequency Amplifier Design

31

The other two terms, however, are not only relatedto the remaining S-parameters of the two-portdevice, S11 and S22, but also to the source and loadreflection coefficients. It is these latter two quanti-ties which we will be able to control in the designof the amplifier. We will employ lossless impedancetransforming networks at the input and outputports of the network. We can then think of the uni-lateral transducer power gain as being made up ofthree distinct and independent gain terms and theamplifier as three distinct gain blocks (Fig. 101).

Figure 101

The Gs term affects the degree of mismatch betweenthe characteristic impedance of the source and theinput reflection coefficient of the two-port device.Even though the Gs block is made up of passivecomponents, it can have a gain contribution greaterthan unity. This is true because an intrinsic mismatchloss exists between Zo and S11, and the impedancetransforming elements can be employed to improvethis match, thus decreasing the mismatch loss, andcan, therefore, be thought of as providing gain.

The Go term is, as we said before, related to thedevice and its bias conditions and is simply equalto |S21|2.

The third term in the expression, GL, serves thesame function as the Gs term, but affects thematching at the output rather than the input.

Maximum unilateral transducer gain can be accom-plished by choosing impedance matching networkssuch that Γs = S11* and ΓL = S22* (Fig. 102).

Figure 102

Constant Gain Circles

Let’s look at the Gs term now in a little more detail.We have just seen that for Γs = S11*, Gs is equal to amaximum. It is also clear that for | Γs| = 1, Gs hasa value of zero. For any arbitrary value of Gs

between these extremes of zero and Gs max, solu-tions for Γs lie on a circle (Fig. 103).

Figure 103

It is convenient to plot these circles on a SmithChart. The circles have their centers located on thevector drawn from the center of the Smith Chart tothe point S11* (Fig. 104).

These circles are interpreted as follows:

Any Γs along a 2 dB circle would result in a Gs = 2 dB.

Any Γs along the 0 dB circle would result in a Gs =0 dB, and so on.

For points in this region (within the 0 dB circle),the impedance transforming network is such as toimprove the input impedance match and for pointsin this region (area outside the 0 dB circle), thedevice is further mismatched. These circles arecalled constant gain circles.

32

Figure 104

Since the expression for the output gain term, GL,has the same form as that of Gs, a similar set ofconstant gain circles can be drawn for this term.These circles can be located precisely on the SmithChart by applying these formulas (Fig. 105):

Figure 105

1. di being the distance from the center of theSmith Chart to the center of the constant gaincircle along the vector S11*

2. Ri is the radius of the circle

3. gi is the normalized gain value for the gain circle Gi.

Constant Noise-Figure CircuitsAnother important aspect of amplifier design isnoise figure, which is defined as the ratio of theS/N ratio at the input to the S/N ratio at the out-put.

S/N inNF =S/N out

In general, the noise figure for a linear two-porthas this form (Fig. 106a), where rn is the equivalentinput noise resistance of the two-port. Gs and bs rep-resent the real and imaginary parts of the sourceadmittance, and go and bo represent the real andimaginary parts of that source admittance whichresults in the minimum noise figure, Fmin.

If we now express Ys and Yo in terms of reflectioncoefficients and substitute these equations in thenoise figure expression, we see once again that theresulting equation has the form of a circle (Fig. 106B).For a given noise figure, F, the solutions for Γs willlie on a circle. The equations for these circles canbe found given the parameters Γo, Fmin, and rn.Unless accurately specified on the data sheet forthe device being used, these quantities must befound experimentally.

Generally, the source reflection coefficient wouldbe varied by means of a slide screw tuner or stubtuners to obtain a minimum noise figure as readon a noise figure meter. Fmin can then be read offthe meter and the source reflection coefficient canbe determined on a network analyzer.

Figure 106

33

The equivalent noise resistance, rn, can be foundby making one additional noise figure reading witha known source reflection coefficient. If a 50-ohmsource were used, for example, Γs = 0 and thisexpression could be used to calculate rn (Fig. 107).

Figure 107

To determine a family of noise figure circles, let’sfirst define a noise figure parameter, Ni:

Here, Fi is the value of the desired noise figure cir-cle and Γo, Fmin, and rn are as previously defined.With a value for Ni determined, the center andradius of the circle can be found by these expres-sions (Fig. 108).

Figure 108

From these equations, we see that Ni = 0, where Fi

= Fmin; and the center of the Fmin circle with zeroradius is located at Γo on the Smith Chart. The cen-ters of the other noise figure circles lie along the Γo

vector.

This plot then gives the noise figure for a particu-lar device for any arbitrary source impedance at aparticular frequency (Fig. 109). For example, givena source impedance of 40 + j 50 ohms, the noisefigure would be 5 dB. Likewise, a source of 50 ohmswould result in a noise figure of approximately 3.5 dB.

Figure 109

Constant gain circles can now be overlaid on thesenoise figure circles (Fig. 110). The resulting plotclearly indicates the tradeoffs between gain andnoise figure that have to be made in the design oflow noise stages. In general, maximum gain andminimum noise figure cannot be obtained simulta-neously. In this example, designing for maximumgain results in a noise figure of about 6 dB, whiledesigning for minimum noise figure results inapproximately 2 dB less than maximum gain.

Figure 110

34

The relative importance of the two design objec-tives, gain and noise figure, will dictate the com-promise that must be made in the design.

It is also important to remember that the contribu-tions of the second stage to the overall amplifiernoise figure can be significant, especially if thefirst stage gain is low (Fig. 111). It is, therefore, notalways wise to minimize first stage noise figure ifthe cost in gain is too great. Very often there is abetter compromise between first stage gain andnoise figure which results in a lower overall ampli-fier noise figure.

Figure 111

Design ExamplesWith the concepts of constant gain and constantnoise figure circles well in hand, let’s now embarkon some actual design examples.

Shown here is a typical single stage amplifier withthe device enmeshed between the input and outputmatching networks (Fig. 112). The device we willbe using for the design examples is an Agilent-2112 GHz transistor.

Figure 112

The forward gain characteristic, S21, of this partic-ular transistor was measured in the previous videotape of this seminar, and we noted that |S21|2 is adecreasing function with frequency having a slopeof approximately 6 dB per octave. Gumax, which isthe maximum unilateral transducer gain, is essen-tially parallel to the forward gain curve (Fig. 113).This is not necessarily true in general, but in thiscase, as we can see on this Smith Chart plot, themagnitudes of S11 and S22 for this device are essen-tially constant over the frequency band in question(Fig. 114). Thus, the maximum values of the inputand output matching terms are also relatively con-stant over this frequency range.

Figure 113

Figure 114

35

To illustrate the various considerations in thedesign of unilateral amplifier stages, let’s selectthree design examples (Fig. 115). In the first exam-ple, we want to design an amplifier stage at 1 GHzhaving a gain equal to Gumax, which in this case is18.3 dB. No consideration will be given in thisdesign to noise figure. In the second example, wewill aim for minimum noise figure with a gain of16 dB. The third example will be the design of anamplifier covering the frequency band from 1 to 2 GHz with a minimum gain of 10 dB and a noisefigure less than 4.5 dB.

Figure 115

In these examples, we will assume a source imped-ance and a load impedance of 50 ohms. In general,however, the source impedance could be complexsuch as the output impedance of the previousstage. Likewise, the output load is quite often theinput impedance of a following stage.

A. Design for Gumax

Now, in this first example, since we will be design-ing for Gumax at 1 GHz, the input matching networkwill be designed to conjugate match the inputimpedance of the transistor. This will provide a netgain contribution of 3 dB (Fig. 116).

Figure 116

The output matching network will be used to con-jugately match the 50-ohm load impedance to theoutput impedance of the transistor. From themeasured data for S22 at 1 GHz, we find that thismatching network will provide a gain contributionof 1.3 dB at the output.

Since the gain of the transistor at 1 GHz with 50-ohmsource and load termination is 14 dB, the overallgain of this single stage amplifier will be 18.3 dB.The matching elements used can be any routineelement, including inductors, capacitors, andtransmission lines.

In general, to transform one impedance to anyother impedance at one frequency requires twovariable elements. A transmission line does, byitself, comprise two variables in that both itsimpedance and its length can be varied. In ourexample, however, we will use only inductors andcapacitors for the matching elements.

36

The next step in the design process is to plot on aSmith Chart the input and output constant gaincircles. If noise figure was a design consideration,it would be necessary to plot the noise figure cir-cles as well. In most cases it is not necessary toplot an entire family of constant gain circles. Forthis example, only the two circles representingmaximum gain are needed. These circles have zeroradius and are located at S11* and S22* (Fig. 117).

Figure 117

To facilitate the design of the matching networks,let’s first overlay another Smith Chart on the onewe now have. This added Smith Chart is orientedat 180° angle with respect to the original chart.The original chart can then be used to read imped-ances and the overlaid chart to read admittances.

To determine the matching network for the output,we start from our load impedance of 50 ohms at thecenter of the chart and proceed along a constantresistance circle until we arrive at the constantconductance circle that intersects the point repre-senting S22.* This represents a negative reactanceof 75 ohms. Hence, the first element is a seriescapacitor.

We now add an inductive susceptance along theconstant conductance circle so that the impedancelooking into the matching network will be equal to S22.*

The same procedure can now be applied at theinput, resulting in a shunt capacitor and a seriesinductor (Fig. 118).

Figure 118

There are, of course, other networks that wouldaccomplish the purpose of matching the device tothe 50-ohm load and source. We could, for exam-ple, have added an inductive reactance and then ashunt capacitor for the output matching.

Choosing which matching network to use is gener-ally a practical choice. Notice in this example, thefirst choice we made provides us with a convenientmeans of biasing the transistor without addingadditional parasitics to the network other than thebypass capacitor. Another consideration might bethe realizability of the elements. One configurationmight give element values that are more realizablethan the other.

Along these same lines, if the element valuesobtained in this process are too large, smaller val-ues can generally be obtained by adding more cir-cuit elements, but as you can see, at the cost ofadded complexity.

In any case, our design example is essentially com-plete with the final circuit looking like this.

37

So far we have not considered noise figure in thisdesign example. By plotting the noise figure circlesfor the device being used, we can readily determinethe noise figure of the final circuit, which in thiscase is approximately 6 dB (Fig. 119).

Figure 119

B. Design for Minimum Noise FigureLet’s now proceed with the second design examplein which low noise figure is the design objective.This single stage amplifier will be designed to haveminimum noise figure and 16 dB gain at 1 GHz(Fig. 120).

Figure 120

To accomplish the design, we first determine theinput matching necessary to achieve minimumnoise figure. Then, using the constant gain circles,G1, the gain contribution at the input can be deter-mined. Knowing the gain of the device at 1 GHz,the desired value of G2, the gain contribution of theoutput matching network can then be found. Theappropriate output matching network can be deter-mined by using the constant gain circles for theoutput.

In this example a shunt capacitor and series induc-tor can be used to achieve the desired impedancefor minimum noise figure. Referring once again tothe Smith Chart and the mapping techniques usedpreviously, we follow the constant conductance cir-cle from the center of the Smith Chart and thenproceed along a constant resistance circle (Fig. 121).Sometimes this requires several trials until theexact constant resistance circle that intersects the minimum noise figure point is found.

Figure 121

Since we now know that the minimum noise figurecircle on the Smith Chart represents a specificsource reflection coefficient, we can insert thisvalue of Γs into the formula for G1 to determine the value of the input constant gain circle passingthrough this point. In this case, it is the 1.22 dBgain circle. This is 1.8 dB less than the maximumgain attainable by matching the input.

38

We can now calculate the output gain circle as fol-lows. The desired gain is 16 dB. The gain due tothe input matching networks is 1.22 dB and theforward gain of the device with 50-ohm source andload terminations is 14 dB. The gain desired fromthe matching network at the output is, therefore,0.78 dB.

The output matching can again be accomplished by using a series capacitor with a shunt inductor.Notice that for the output matching there are aninfinite number of points which would result in again of 0.78 dB. Essentially, any point on the 0.78 dBcircle would give us the desired amount of gain.

There is, however, a unique feature about the com-bination of matching elements just selected. Thevalue of capacitance was chosen such that this pointfell on the constant conductance circle that passesthrough the maximum gain circle represented byS22* (Fig. 122).

Figure 122

This combination of elements would assure an out-put gain-frequency response that would be sym-metric with frequency.

If, for example, the frequency were increasedslightly, the capacitive reactance and the inductivesusceptance would both decrease, and the resultingimpedance would be at this point.

Similarly, decreasing the frequency would result inthis impedance. Both of these points fall on a con-stant gain circle of larger radius, and hence, lowergain (Fig. 123).

Figure 123

The gain response for the output matching, there-fore, is more or less symmetric around the centerfrequency (Fig. 124).

Figure 124

39

If we look at the input side, however, we have a fardifferent situation. With an increase in frequency,both the capacitive susceptance and inductivereactance increase, resulting in an increase in gain.When the frequency is reduced, these quantitiesdecrease, resulting in a lower value of gain. Thegain contribution at the input is, therefore, unsym-metric with respect to frequency.

Since the overall gain as a function of frequency isthe combination of the G1, Go, and G2 terms, theresulting gain would be reasonably symmetricabout the center frequency. (If another point onthe 0.78 dB gain circle at the output were chosen,the final overall gain characteristic would be asym-metric with frequency.)

The important point is that the selection of thematching elements for the output, in this case, isnot as arbitrary as it first appears. The final selec-tion must be based not only on the gain at a spe-cific frequency but also on the desired frequencyresponse. The element values can now be calcu-lated resulting in the circuit shown (Fig. 125).

Figure 125

From a practical standpoint, one would not achievethe noise figure design objective for a number ofreasons. For one thing, the circuit elements usedfor matching purposes would have a certain lossassociated with them. This resistive loss would adddirectly to the noise figure. To keep this loss to aminimum, it is desirable to use high-Q circuit ele-ments and to use the minimum number of elementsnecessary to obtain the desired source impedance.Second, there will be some contribution to theoverall noise figure from the second stage. Third,additional degradation in noise figure would occurbecause of device and element variations from unitto unit.

In practice, typically 1/2 to 1 dB would be addedfrom these sources to the predicted theoreticalnoise figure for a narrow band design. As much as2 dB could be added in the case of an octave banddesign such as in our next example.

C. Broadband Design for Specific Gain and Noise Figure Here, the design objective will be 10 dB unilateraltransducer gain from 1 to 2 GHz with a noise fig-ure less than 4.5 dB (Fig. 126).

In this example, the input and output matchingnetworks will be designed to have a gain of 10 dBat the band edges only, The gain at 1.5 GHz willthen be calculated. The response will be found tolook similar to this curve. If a greater degree offlatness were necessary, additional matching ele-ments would have to be added. We could thendesign for 10 dB gain at three different frequen-cies, or more if necessary. Three frequencies wouldgenerally be the practical limit to the graphicaldesign approach we have been using.

Figure 126

When matching is required at more than three fre-quencies, computer-aided design techniques aregenerally employed.

40

The schematic again looks similar to that which wehad in the previous two examples (Fig. 127).

Figure 127

The design approach is to:

1. Match the input for the best compromise betweenlow and high frequency noise figure. In this case, itis important to keep the number of matching elementsto a minimum for the reasons cited earlier. It mightbe possible to get an additional 0.2 dB improvementin noise figure with one additional element, but theextra element might, in turn, add an additional inser-tion loss at 0.2 dB or more.

2. The next step is to determine the gain contribu-tion at the input as a result of the noise figure match-ing. This then allows us to calculate the desiredgain at the output from the design objective.

3. The output matching elements are then selected,completing the design.

Let’s first plot S11* for 1 and 2 GHz and then plotthe points resulting in minimum noise figure forthese frequencies (Fig. 128).

Figure 128

To match the input for minimum noise figure wemust first choose a combination of elements thatgives the best compromise at the two design fre-quencies. On the plot we see that a shunt capacitorfollowed by a series inductor will provide a sourceimpedance such that the noise figure will be lessthan 3.5 dB at 1 GHz and less than 4.5 dB at 2 GHz1

(Fig. 129). At both frequencies we are about asclose as is practical to the theoretical minimumnoise figure for the device.

Figure 129

1. For the particular transistor measured. We want to emphasize that the methodologyfollowed in these design examples is more important than the specific numbers.

41

The constant gain circles, which intersect thepoints established by the input matching network,are calculated and found to have the values 0.3 dBat 1 GHz and 1.5 dB at 2 GHz.

The desired gain due to output matching can nowbe calculated and found to be –4.3 dB at 1 GHz and+0.5 dB at 2 GHz (Fig. 130).

Figure 130

The constant gain circles having these gain valuesare then plotted as shown (Fig. 131). A trial-and-error process is followed to find the proper match-ing elements to provide the required output matchat the two frequencies. Let’s start from the 50-ohmpoint on the Smith Chart and add an arbitrary nega-tive series reactance and the appropriate negativeshunt susceptance to land on the 0.5 dB gain circleat 2 GHz. We then determine where these matchingelements will bring us at 1 GHz, and in this case,we fall short of reaching the proper gain circle(Fig. 132). By increasing the capacitive reactancewe find a combination that lands on both circlesand the design is complete (Fig. 133).

Figure 131

Figure 132

Figure 133

So far, by our design procedure we have forced thegain to be 10 dB at 1 and 2 GHz. The question nat-urally arises: what happens between these two fre-quencies? From the matching element values alreadydetermined, we can calculate the matching networkimpedances at 1.5 GHz and then determine the con-stant gain circles that intersect those points.

For this case, we find that the output gain circlehas a value of –0.25 dB and the input gain circle, avalue of +1 dB. The gain of the device at 1.5 GHz is10.5 dB. Hence the overall gain of the amplifier at1.5 GHz is 11.25 dB.

42

In summary form, the contribution of the threeamplifier gain blocks at the three frequencies canbe seen (Fig. 134). If the resulting gain characteris-tic was not sufficiently flat, we would add an addi-tional matching element at the output and selectvalues for this added element such that we landedon the original gain circles for 1 and 2 GHz, but onthe –1.5 dB rather than –0.25 dB circle at 1.5 GHz.This would give us a gain for the amplifier of 10 dBat 1, 1.5, and 2 GHz with some ripple in between.

Figure 134

If, however, we were satisfied with the first gain-frequency characteristic, our final schematic wouldlook like this (Fig. 135).

Figure 135

D. Multistage DesignWhile the examples we have just completed are sin-gle stage amplifiers, the techniques presented areequally applicable to multistage amplifier design.The difference in a multistage design is that thesource and load impedances for a given stage ofthe amplifier are, in general, not 50 ohms but arerather an arbitrary complex impedance. In certaincases, this impedance might even have a negativereal part.

Multistage design can be handled by simply shift-ing the reference impedance to the appropriatepoint on the Smith Chart. This is illustrated in thefollowing example.

Let’s now concentrate on the matching networkdesign between two identical stages (Fig. 136). Forthe first stage, as we have seen, there is a gain of1.3 dB attainable by matching the output to 50 ohms.Similarly, there is a gain of 3 dB attainable by match-ing 50 ohms to the input of the second stage. Wecan then think of a gain of 4.3 dB being attainableby matching the output impedance of the firststage to the input impedance of the second stage.

Figure 136

The constant gain circles for the first stage outputwould then be plotted on the Smith Chart (Fig. 137).The maximum gain is now 4.3 dB. The gain circlethat intersects the point represented by the secondstage’s S11 has a value of 0 dB.

Figure 137

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To design for a specific interstage gain, we could,as before, add a series capacitor followed by ashunt inductor—except in this case we start fromthe input impedance of the second stage, S11,rather than the 50-ohm point.

The resulting interstage matching network lookslike this. As you can see, the design of multistageamplifiers is handled as easily as single stagedesigns (Fig. 138).

Figure 138

SummaryThe measurement and design techniques demon-strated in this video-tape seminar are presentlybeing used by engineers in advanced R&D labsthroughout the world. When coupled with designand optimization computer programs, engineerswill have at their disposal the most powerful andrapid design tools available.

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