ARRL Radio Designer and the Circles Utility

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Sept/Oct 1998 3

The Smith Chart is a venerable tool for graphic solutionof RCL and transmission-line networks. The Circles Utilityof ARRL’s Radio Designer software provides an “electronic”Smith Chart that is very useful. For those unfamiliar with

the Circles Utility, we begin by exploring basic conceptsand techniques of matching-network design.

By William E. Sabin, WØIYH

1400 Harold Dr SECedar Rapids, IA 52403e-mail sabinw@mwci.net

ARRL Radio Designerand the Circles Utility

Part 1: Smith Chart Basics

One of the interesting anduseful features of the ARRLRadio Designer program is the

Circles Utility. This two-part articlewill look at some of the ways of usingCircles. A brief overview of basic prin-ciples will be followed by some “walk-through” examples that can be used as“templates” or guidance for future ref-erence. The Circles Utility can do thefollowing things:

• Perform Smith Chart operations todesign and analyze transmission-linenetworks and LCR impedance-match-ing networks. Part 1 of this articledeals with these topics.

• Using the Smith Chart, perform

gain-circle operations that are widelyused in active-circuit design, espe-cially amplifiers, using S-parameterequations. Input- and output-match-ing networks can be evaluated. Thestability of an active circuit can beevaluated by plotting stability circles.Noise-figure circles (circles of con-stant noise-figure values) can be plot-ted. Figures of merit are calculated.We can perform plots of certain quan-tities over a frequency range. Theresult is a visual estimate of the per-formance of the circuit. Part 2 of thisarticle will deal with this.

Smith Chart BasicsIt is a good idea to bring up the

Circles Utility and perform the follow-ing operations as you read about them.

First, to enable Circles, we mustenter and Analyze (key F10) a circuitlisting of a two-port network as exem-

plified in Fig 1A, a resonant filter cir-cuit that is also an impedance-match-ing network. The schematic is shownin Fig 1B. Start with the list in Fig 1A,and modify it later as required. I as-sume the reader already knows how touse the Report Editor to get XY rect-angular plots, tables and polar plotsversus frequency of such things asMS11, MS21 etc, and how to set theTerminations. An Optimization of thecircuit is also performed.

Fig 2A shows how various lines ofconstant resistance and constant reac-tance are plotted on the Smith Chart,using the Circles Control Window (wewill henceforth call it CCW). The vari-ous entries that we type into the CCWtext window are just as shown (the“ = “ is optional and a “space” can beused instead). In particular, pure re-actances can only exist on the outercircle of the chart; to move inside, some

4 QEX

resistance must be added. Positive reactance (+X, induc-tance) is in the upper half, negative reactance (–X, capaci-tance) in the lower half. High resistance (R, low conduc-tance G = 1/R) is toward the right.

You can also create an R or X circle at some specific lo-cation. Type “R” or “X” in the CCW, Execute, then place thecursor at the desired location and press the left mouse but-ton. This creates an R or X circle and a label with somespecific value.

To remove a single error in an entry, type “DEL” into theCCW, then Execute. To delete two entries, type “DEL 2,”and so forth. To get the best accuracy in all of the variousoperations, click on Settings/Display/Graphics/Line Widthand set the line width to “1,” the narrowest line.

Fig 2B shows how various lines of constant-conductanceand constant-susceptance values are plotted. This chart isa left-to-right and top-to-bottom reverse image of Fig 2A.Positive susceptance (+B, capacitance) is below and nega-tive susceptance (–B, inductance) above the horizontalaxis. Both charts always show inductance above the axisand capacitance below the axis. High conductance (G, lowresistance, R = 1/G) is toward the left.

G and B circles at some cursor location are created bytyping “G” or “B,” Execute, then place the cursor at thedesired location and press the left mouse button.

The Circles Smith Chart is a YZ chart, which means thatFigs 2A and 2B and all four of the quantities shown therecan be plotted simultaneously on the same chart in two dif-ferent colors. The term “Z” refers to R and X in series and theterm “Y” refers to G and B in parallel. It’s important to keepthis distinction in mind, especially when switching betweenthe two. This YZ capability is a powerful feature that we willuse often. Also, the Z chart is “normalized” to 1 Ω at theorigin. To normalize a 50 Ω system, divide all actual R and±X input values by 50. For other values of Z0, such as 52 Ω,450 Ω etc, use Z0 as a scaling factor. The Y chart is normal-ized to 1 S (Siemens = 1 / Ω) at the origin. For a 0.02 S(1 / (50 Ω)) system, multiply actual G and ±B values by 50,or whatever Z0 is correct. It is very desirable to have a cal-culator available to normalize values; the computations areno problem once you get the “hang of it.” In the interest ofsimplicity, it is best (at least at the beginning) to use resis-tive (not complex) values of Z0. For a transmission line, aresistive Z0 means a line with no attenuation (loss).

Fig 2C shows various values of V (SWR) circles, enteredas shown. At any point on a particular V circle, a value ofRHO, the reflection coefficient, can also be found. For ex-ample, at the intersection of R = 1 and V = 8, the reflectioncoefficient is 0.77 at 39.17° (angle measured counterclock-wise from the horizontal axis). To find this RHO, we use thefollowing procedure:

• Enter the word “RHO” into the CCW and click the Ex-ecute button.

• Place the cursor at the intersection of V = 8 and R = 1.• Press “M” on the keyboard.• This places a mark at this location and also puts the

complex (magnitude and angle) value of RHO in the CCW.The following equations give the relationships involved

in the RHO and V operations:

RHO =+( ) −+( ) +

R jX Z

R jX Z0

0(Eq 1)

where R, X and Z0 are normalized to 1.0 as discussed pre-viously and

VRHO

RHORHO

V

V=

+−

= −+

1

1;

1

1 (Eq 2)

where the vertical bars “| |” denote “magnitude.” From thisequation we see that a V circle is also a circle of constant|RHO|, so the V circle is also called a “constant-reflection”circle. Also of considerable interest is the return loss

RL RHOV

VdB 20 log 20 log

1

1dB( ) = − ⋅ = − ⋅ −

+

(Eq 3)

Return loss is a very sensitive measure of impedancematch that is widely used in test equipment, such as net-work analyzers. This term means “what fraction of thepower that is sent toward the load returns to the genera-tor?” In Radio Designer, RL (dB) is the same as MS11 (dB)and MS22 (dB) at the input and output, respectively, of atwo-port network. The values of RHO and V can be foundfor any combination of R and X or G and B, using the cursormethod as described.

The Smith Chart is basically a reflection-coefficient(RHO) chart. The distance from the center to the outercircle corresponds to |RHO| = 1.0, which is defined as“complete” reflection of a wave, which corresponds to ashort-circuit load (R = 0), an open-circuit load (G = 0) or apurely reactive load. The chart then assigns the R, X, Gand B values in terms of the corresponding complex valuesof RHO according to the equation

Z ZRHO

RHOY

Z= ⋅ +

−=0 0

1

1;

1; RHO and Z possibly complex (Eq 4)

The method described for RHO, steps 1, 2, 3 and 4, can be

* Smith chart example

BLKIND 1 2 L=?1.94198UH? Q=250 F=7.15MHzCAP 2 0 C=?860.327PF? Q=10000 F=7.15MHzIND 2 3 L=?1.83567UH? Q=250 F=7.15MHz

TUNER:2POR 1 3ENDFREQ

STEP 7.0MHZ 7.35MHZ 10 KHZEND

OPTTUNER R1=50 Z2=5 –50 MS11F=7.15MHZ MS11= –50

END* Comments:* Set output load in Report Editor to 5 – j50 ohms* Set generator in Report Editor to 50 ohms* Plot MS11 and MS21

Fig 1—A is a circuit listing for an example two-port networkfor Smith Chart analysis. B is a schematic of the circuit.

(A)

(B)

Sept/Oct 1998 5

Fig 2—A shows Smith X Chart circles of constant resistance, R, and reactance, X. Positive X is above the horizontal axis. Bshows Smith Y Chart circles of constant conductance, G, and susceptance, B. Positive B is below the axis. C shows SmithChart circles of constant SWR (V) and an example of a reflection-coefficient, RHO, location. D shows Smith Chart arcs ofconstant Q. Positive values of Q apply to inductive reactance or susceptance. Negative values of Q apply to capacitivereactance or susceptance.

used to find values of Z and Y at some location. Type “Y” or“Z” in the CCW, then Execute, place the cursor at a locationand press the “M” key. A marker appears on the chart andthe R and X, or G and B, values appear in the CCW. If amarker is not wanted, press the left mouse button instead.This operation suggests an easy way to transform a series RSand XS to a parallel RP and XP,

because RP = 1 / G and XP =–1 / B at any selected location. Working this backward, wecan change parallel RP to G (= 1 / RP) and XP to B (= –1 / XP),then to RS and XS by doing the Y to Z change.

To place a marker at a specific value of Z = R + jX (or Y= G + jB) do the following:

• Create the appropriate R (or G) circle and an X (or B)circle, as previously described.

• Place the cursor exactly at the intersection and removeyour hand from the mouse.

• Delete the two circles (“DEL 2”) if you do not want themto show.

• Press the “M” key. This places a marker at the Z (or Y)location.

Fig 2D shows arcs of constant Q values. Any values of Rand +X that lie on the Q = +2 line correspond to X / R = Q= 2: for example, the intersection of the R = 1.5 circle andthe X = +3 circle. For capacitive values of X, –Q is plotted

6 QEX

below the horizontal axis. The inter-section of B = –3 and G = 1.5 is also atQ = B / G = 2. Q lines are useful incertain applications that are dis-cussed in Part 2 of this article.

The CCW contents and the completecircle that we create can be saved todisk by typing the SPLT command inthe CCW. A name for the file is re-quested. This same file can be recalledfrom disk by using the RPLT command.The name of the file is requested.

Navigating the ChartAn important topic in Circles opera-

tions concerns the ways that wemodify the Z, Y, R, X, G and B quanti-ties from one chart location to another.Fig 3A shows an initial value of Z atone location. We want to change Z toany of the values Z1 through Z5. Thevalue of Z = R + jX is at the junction ofthe R = 0.18 circle and the +X = 0.26circle, where +X is inductive. The fol-lowing rules are observed in Fig 3A:

The impedance, Z, is at the junctionof a constant-R circle and a constant-X circle. To increase X (make theinductive reactance, therefore the in-ductance, larger), move clockwisealong the R1 circle to point Z1. Thechange in reactance is 0.91 – 0.26 =+0.65. (Inductive reactance increasesin the clockwise direction.)

The Circles Utility has another op-tion, called DX, which works as follows:

• Enter a frequency in the CCW,for example FREQ = 7.15E6, thenExecute.

Fig 3—A shows directions of travel on a Z chart for X and R changes by adding or subtracting series XL, XC or R. B showsdirections of travel on a Y chart for B changes by adding or subtracting shunt BL or BC.

• Type DX into the CCW, thenExecute.

• Place the cursor first at Z andpress the left button.

• Move the cursor to Z1 and pressthe left button again.

• The CCW displays the value of in-ductance, L = DX / (2 π FREQ), thatproduces the change of reactance fromZ to Z1 at the FREQ that was entered.

Instead of moving to Z1, we can movefrom Z to Z2 along the R1 circle. This isa capacitive reactance of value –0.66.(Capacitive reactance increases —Cdecreases—in the counterclockwise di-rection.) The DX option now gives theneeded value of capacitance, providedthat we place the cursor first at Z, thenat Z2. We can also move from Z1 to Z2for a reactance change of –1.31.

To increase the value of R, from loca-tions Z, Z1 or Z2, move counterclock-wise along a line of constant X, asshown, if X is positive (above the hori-zontal axis). If X is negative, the mo-tion is clockwise. The proper directionsto reduce R are obvious in Fig 3A.

Having arrived at Z3, Z4 or Z5, thepreceding operations can be repeated,and in this manner we can travelaround the chart.

Fig 3B shows the travels for the Ychart. The rules are nearly the sameas for the Z chart, and the directionsare as shown. The differences are:

• Inductive susceptance increases(inductance decreases) in the counter-clockwise direction along a line ofconstant conductance.

• Capacitive susceptance increases(capacitance increases) in a clockwisedirection along a line of constant con-ductance.

• Conductance (G) increases in aclockwise direction along a B line for Bless than zero (above the horizontalaxis) and counterclockwise along a Bline for B greater than zero (below thehorizontal axis).

The next task is to move back andforth between the Z chart and the Ychart, using these steps:

• Starting at some Z point, move Zto Z1 (Fig 3A). Then, using the CCW,find the value of Y1 at this point.

• Then move Y1 to Y2 (Fig 3B).• Find the value of Z2 at Y2 and then

move Z2 to Z3, and so forth until thetarget is reached.

Many impedance-matching prob-lems start at a load impedance some-what removed from the center of thechart and work toward the chart cen-ter, R = 1, X = 0 or G = 1, B = 0. (Weassume that the chart center is theimpedance that the generator wantsto “see.” This process models adjust-ment of an antenna tuner. The targetimpedance need not be at the chartcenter. For example, if Z = 1.1 + j0.05or its complex conjugate, place a Zmarker there and make that the tar-get. It’s equally possible to place theload at or near the chart center and thegenerator farther out. In that case, westart at the center and work out to-ward the generator. When we do thisreversal of direction, a series inductor

Sept/Oct 1998 7

Fig 4—A shows the transmission line required totransform Z1 to Z2 along a line of constant SWR,where V = 3.0:1 (RHO = 0.5). Z0 (unnormalized)=49.53 Ω, Z0 (normalized) = 0.99. B (unbalanced)and C (balanced) show the DXD function; D showsthe DBD function.

becomes a series capacitor, and ashunt inductance becomes a shuntcapacitance, and so forth.

For example, a 50 Ω load is to betransformed to some complex imped-ance that a transistor collector (thegenerator) wants to “see.” Similarly (alittle more difficult to visualize butvery important) a transistor inputlooks backward toward a transformed50 Ω (the load that the transistor inputsees, looking back). In order to avoidconfusion and errors the rule is: Startat the load, wherever it is, and worktoward the generator, wherever it is.

The changing of impedance usingtransmission lines and stubs is cov-ered by three additional Circles func-tions as follows:

DT finds the electrical length (indegrees) and characteristic imped-ance [Z0(n) when normalized to 1.0,Z0(u) for a 50 Ω line] of a transmission

line that transforms impedance Z1 toZ2. Z1 and Z2 can lie on the same V(SWR) circle as shown in Fig 4A, orthey may be on different V circles. Usethe following approach for the firstoption: on the same V circle:

• Draw the V circle• Locate Z1 and Z2 on the V circle• Type DT in the CCW, then Execute• Place cursor at Z1 and click left

button, then place cursor at Z2 andclick left button again. The clockwiseelectrical angle is measured. Alterna-tively, do Z2 first then Z1, in whichcase the angle is measured clockwisefrom Z2 to Z1. The complete circle is180° electrical length.

The information appears in the CCWtext window. The physical length is:

metersdegrees velocity factor 0.833

FREQ MHz( ) =

( ) ⋅( ) ⋅( )

(Eq 5)

Note also that the two Z points neednot be on the same V circle. If they areon two different V circles (but notV = ∞) the correct Z0 of the line and itslength (in degrees) will be calculated.Two lines may be required. This valu-able feature is often used, especiallyin microstrip design.

Referring to Fig 2A, the differencebetween two points on a constant-re-sistance circle is a value of reactance.After we mark two reactance points onthis circle (first the initial value, thenthe final value), the DXD operationallows us to get this change of reac-tance with a section of transmissionline connected as a stub in series withthe line, as shown in Fig 4B. Quiteoften two stubs—each having onehalf of the needed reactance—can beinstalled, one in each wire of a bal-anced transmission line as shown inFig 4C. The stub(s) can be:

8 QEX

• A shorted stub (SS) less than λ/4is an inductive reactance.

• An open stub (OS) less than λ/4 isa capacitive reactance.

The CCW asks for one of the follow-ing: I (= Z0, impedance of the transmis-sion line), D ( = degrees of electricallength) or W ( = wavelength). The CCWthen tells which stub to use (OS or SS),its reactance and other parameters.

The DBD operation is just like DXD,except that we put two susceptancemarkers on a constant-conductancecircle as in Fig 2B. We then get a par-allel stub (OS or SS) as shown inFig 4D. The reactance of the stub andother parameters are displayed. Thiskind of stub is widely used.

The combined usage of DT, DXD andDBD in the Circles Utility is a verypowerful way to match impedances us-ing transmission lines. By switchingfrom one to another (and between Z andY) we can travel the chart in transmis-sion-line segments and stubs in a veryelegant way. It is worthwhile to keep inmind that the reactances of transmis-sion-line segments vary over frequencyin somewhat different ways than ordi-nary lumped LC components. This canbe a factor in designing a network.

Problem SolvingThe art and gamesmanship of these

matching exercises are to find theminimum number of components, es-pecially lossy inductors. Always keepthe possibility of using transmission-line segments in mind because of theirhigh Q (low loss) values. Sometimes agreater frequency-response band-width is achieved by using transmis-sion lines than lumped L and C. Wealso want values of L, C and lines thatare realistic, efficient and economical.

Fig 5 (drawn with fine lines to im-prove accuracy) illustrates the manydifferent possibilities for converting aload impedance Zload to R = 1, X = 0. Wewill use, just as one example, the cir-cuit shown in Fig 1B. Zload is at pointA, 0.10 – j1.0. The first component is aseries inductor along the R = 0.10 linefrom A to C. A shunt capacitor movesthe impedance along the G = 0.5 linefrom C to I, and a series inductormoves it along the R = 1.0 line from Ito O (the origin). Fig 1B shows thevalues calculated in this manner. Thiscircuit would have a very good low-pass filtering property, but requiresthree components, and at least two ofthem must be tunable. These valuesare placed in the circuit listing in Fig1A and then optimized for best MS11at 7.15 MHz, using realistic Q values

for the components. MS21 tells thecircuit loss in decibels, about 0.35 dB(when MS11 dB is very large). Theoptimized L and C values are shown.The last inductor “IND 2 3” could havea constant value (remove the questionmarks from the Netlist), and the otherL and C are then optimized.

The first coil from point A could be afixed value if it gets us beyond point B.This inductance could possibly be builtright into the load device (antenna orwhatever). A length of transmissionline (Z0 = 1.0) could get us from, saypoint C, along a constant V circle, overto the R = 1 circle. A simpler approachcould use series inductance up to pointB and shunt capacitance from B to O.Better yet, a series coil from A to C andtwo low-loss capacitors, one in shuntfrom C to F and one in series from F toO. If the series inductive reactancedoesn’t go beyond point B at the lowestfrequency of interest, however, theseschemes won’t work. Another interest-ing idea would be a shunt coil from A toK and a series coil from K to O.

Consider also an inductance from Aup to the horizontal axis (the 0.1 mark)and a 9:1 (impedance ratio) ferrite-core transmission-line transformer

from 0.1 Ω to 0.9 Ω, where a V circlehas the low value 1.11. This transmis-sion line should ideally have a Z0 of0.3 Ω. For a 50-Ω system, this would cor-respond to 15 Ω. Three small, parallel-connected segments of 50 Ω coax wouldbe okay. We can do some fine-tuningwith this method. The resistance addedby the inductor can be considered for amore-accurate graphical answer. Usethe following approach:

• Get a first estimate of the normal-ized inductive reactance (= 1.0) asdescribed above.

• Assume a value of QL for the in-ductor. The RL of the coil is then RL =XL / QL.

• Add RL to the resistance R (= 0.1)of the load and relocate point A on theSmith Chart accordingly.

• On the chart, find a better value ofthe inductor needed to reach the hori-zontal axis.

• Further repetitions of this proce-dure are not necessary.

• The 1:9 transformer (assume it tobe loss-less) now takes us to a newvalue, which will still be close to theorigin if QL is large.

In this example a coil QL of 250 wouldmean a correction of 1.0 / 250 = 0.004 Ω,

Fig 5—Solution paths for the design example.

Sept/Oct 1998 9

which is negligible. In more extremesituations it may be important.

Keep in mind also the discussionregarding the reversal of direction: forexample, starting at the origin (theload) and working outward to the gen-erator impedance or possibly its com-plex conjugate.

The main idea here is to illustratethe power of the Smith Chart in visu-alizing the myriad possible solutions,each of which has possible merits andpossible problems. With such a “shop-ping list” the designer can make thebest decisions. The Circles Utility canthen quickly and easily get values forthe various components.

Also, a candidate circuit of lines/stubs and adjustable L and C should betested over a frequency band for “tun-ing range” of the components. A newDesigner file is created as shown inFig 6 for this purpose. At any arbitraryfrequency that is entered into the OPTblock, the circuit values are optimizedfor maximum MS11. The load imped-ance data is in the DATA block. Thisdata is linked to the ONE circuit ele-ment that has been placed in the maincircuit block. We can now see whatrange of L and C values are needed totune the desired frequency range.

In the example of Fig 6, “IND 2 3” isa fixed value (by assumption, it can’tbe tuned) and it was necessary to in-crease its fixed value (see Fig 1A) sothat we could tune to the low end of the7.0 to 7.3 MHz band with reasonablevalues of the other components. Thisis typical of problems that we mightencounter.

Another interesting exercise is todesign a network that has a certain V(SWR) over a certain frequency band(without retuning). Using Eq 3, a V =2 value corresponds to an MS11 (dB)of –9.5. A V = 1.5 value corresponds toan MS11 (dB) of –14. By observing theplots of MS11 (dB) versus frequencygenerated by Designer, the –9.5 dBand –14 dB frequency ranges can beeasily observed. It is usually neces-sary to “Rescale” the graph, to seethese levels more easily. For furtherinformation on the art and witchcraftof broadband impedance matching,see References 2 and 3.

A Minor BugThe Circles Utility has a problem

when performing the DT (transmis-sion line) operation over a small regionof a V (SWR) circle close to the horizon-tal (X = 0) axis (see Fig 4A). Incorrectresults may appear, due to a minorglitch in the software. The correct pro-

cedure in this situation is to performDT in two segments. The first segmentbegins “away from” the horizontal axisand terminates “at” the horizontalaxis. The second segment begins “at”the horizontal axis and proceeds to thedesired finish point. The total electri-cal angle of the transmission line isthen the sum of the lengths of the twosegments. Some practice will makethis a simple operation. Two RHO op-erations on a V circle also provide theDT electrical angle with no error. Thisproblem does not occur often, but youshould be aware of it.

Additional ReadingR. Dean Straw, Editor, The ARRL Antenna

Book, 18th edition, Chapter 28, has an ex-cellent general discussion of the SmithChart. ARRL Order No. 6133. ARRL publi-

cations are available from your local ARRLdealer or directly from the ARRL. Mail or-ders to Pub Sales Dept, ARRL, 225 MainSt, Newington, CT 06111-1494. You cancall us toll-free at tel 888-277-5289; faxyour order to 860-594-0303; or send e-mailto pubsales@arrl.or g. Check out the fullARRL publications line on the World WideWeb at http://www.arrl.org/catalo g.

Wilfred Caron, Antenna Impedance Match-ing (Newington: 1989, ARRL), Order No.2200. This book has very thorough cover-age of matching methods using transmis-sion lines and LC components.

W. E. Sabin, WØIYH, “Broadband HF An-tenna Matching With ARRL Radio De-signer,” QST, August 1995, p 33.

W. E. Sabin, WØIYH, “Computer Modeling ofCoax Cable Circuits,” QEX, August 1996,pp 3.

W. E. Sabin, WØIYH, “Understanding the T-tuner (C-L-C) Transmatch,” QEX, Decem-ber 1997, pp 13.

* Band tuning example

BLKIND 1 2 L=?3.92257UH? Q=250 F=7.15MHZCAP 2 0 C=?414.698PF? Q=1000 F=7.15MHZIND 2 3 L=2.3UH Q=250 F=7.15MHZ

* Note that this L is held constantONE 3 0 ZDAT

TUNER:1POR 1 0END

FREQSTEP 6.9MHZ 7.4MHZ 10KHZ

END

OPTTUNER R1=50 MS11F=7.25MHZ MS11= -50

END

DATAZDAT: Z RI

* Freq Real Imag7.00MHZ 3 -707.10MHZ 4 -607.20MHZ 6 -407.30MHZ 8 -30

END

* Comments:* Put the ONE element in the circuit block.* Change TUNER to a 1POR 1 0 (one port) as shown.* Set Report Editor to MS11 with TERM=50+j0* Put load impedance data in the DATA block, real thenimaginary.* DATA is interpolated between freq entries.* In the OPT block, insert any freq value in the range,one at a time.* Analyze, then Optimize at the freq that is in the OPTblock.* Check to see the min and max values of tuning Ls and Csthat are needed.* Check to see that MS11 reaches a very large value.

Fig 6—Net list for band tuning example.

Nov/Dec 1998 3

Have you been intimidated by amplifier design? Maybeyou want to step up to ARRL Radio Designer for the task.

Either way, this will help you grasp this powerful design tool.

By William E. Sabin, WØIYH

1400 Harold Dr SECedar Rapids, IA 52403e-mail sabinw@mwci.net

ARRL Radio Designer andthe Circles Utility, Part 2:

Small-Signal Amplifier Design

Part 1 of this article was in-tended to help the reader toacquire greater familiarity

with the Circles Utility of Radio De-signer. We looked at passive networkdesign using transmission lines andLC components, using the variousmethods of navigating the SmithChart. In this Part 2 we will considerthe interesting and very useful meth-ods of designing small-signal linearamplifiers using gain circles, stabilitycircles, noise-figure circles and vari-ous figures of merit. These techniquesare well known and widely used in in-dustry, and we will focus on the par-

ticular features of the Radio Designerprogram in this regard.

S-Parameter BasicsWe begin with a brief overview of

S-parameters, especially as they re-late to Radio Designer. Fig 1 shows atwo-port network (we will call it “N”)that is connected to a generator and aload. The generator has a resistanceRG and is connected to the input portthrough a transmission line having acharacteristic impedance Z01. The out-put is connected to a load RL througha transmission line Z02. If the genera-tor sends a voltage wave V1

+ toward N,a current wave, I V /Z1 1 01

+ += , accompa-nies it. The power in this forward wave

travels back to the generator ( –I1– is

correct because I1– moves in phase

with V1– in a direction opposite to I1

+ ).If we square both sides of this equa-tion and divide both sides by Z01we get

V

ZI Z P

12

011

201 1

––( )

= −( ) ⋅ = − (Eq 1)

This is a power wave that is re-flected from N back to the generator.We see that V1

+ and I1+ are in-phase

(Z01 is a resistance) and V1– and –I1

–

are also in-phase, but V1+ and V1

– (andI1

+ and –I1– ) may not be in-phase with

each other. Both waves are in facttraveling simultaneously in oppositedirections (they are easily measuredindependently using a directional cou-pler) but at any point the net voltage(or current) is the vector sum of the

is P V I V /Z I Z1 1 1 12

01 12

01+ + + + += = ( ) = ( ) . A

reflected voltage wave, V –I Z1–

1–

01= ,

4 QEX

two waves. The difference between P1+ and P1

– is the powerthat is accepted by N, and it is also the power that is deliv-ered by the generator. At the output of N, Fig 1 shows apower wave traveling from the load toward N and a powerwave from N to the load, and the same rules apply here.

The square roots of these four power waves are voltagewaves that are said to be normalized with respect to Z0.

aV

Zb

V

Za

V

Zb

V

Z1

1

011

1

012

2

022

2

02

= = = =+ − + −

; ; ;

(Eq 2a)They can also be expressed as normalized current waves

a I Z b I Z a I Z b I Z1 1 01 1 1 01 2 2 02 2 2 02= ⋅ = ⋅ = ⋅ = ⋅+ − + − ; ; ;

(Eq 2b)Although Z01 and Z02 are often the same, especially in

network analyzers, they need not be the same. In RadioDesigner the Report Editor allows us to specify the Termi-nations (Z0) separately. For example, if N is a loss-less 25:1impedance-ratio matching network from a 200 + j0 Ω gen-erator to an 8 + j0 Ω load we can set the Input Terminationat 200 + j0 Ω and the Output Termination at 8 + j0 Ω. RadioDesigner then tells us that this particular N has no mis-match loss (the impedance matches at the input and out-put are perfect).

If the input impedance of N in this example is not 200 Ω,the power available from the 200 Ω generator will not beaccepted by N. There is a mismatch loss. An exactly iden-tical statement is that the power not accepted by N is re-flected and returns to the generator. The same is true atthe output. If we move the voltage generator to the rightside (leave RG = Z01 and RL = Z02 where they are) the samesituation exists at the right-hand side of N. If the outputimpedance of N is not 8 Ω, some power is reflected at theoutput terminals. In practice, we routinely make thisswitch of the generator so that we can measure the trans-mission and reflection properties of N at the output side.

Fig 1 shows a1 and a2 as normalized voltage waves rep-resenting the signals arriving at N and b1 and b2 as nor-malized voltage waves leaving N. The other thing we see isthat the portion of a1 accepted by N (from the generator onthe left) is transmitted, after impedance transformation,to the output terminals. A portion of a2 is transmitted (fromgenerator on the right), after impedance transformation,from output back to input.

We need one other thing to complete the picture. If theload RL = Z02 then a wave b2 will be completely absorbedby RL. If RG = Z01 a wave b1 will be absorbed by RG. If thisis not true, re-reflections occur at RL or RG, and the situa-tion becomes a little more complicated. In test equipment,

the effect of these re-reflections is that they create errorsin our measurements of the S-parameters (unless we com-pensate for them). In Radio Designer, we will always as-sume that RL = Z02. An identical situation occurs on theinput side and we make RG = Z01. Because RL = Z02, it istrue that a2 = 0 when the generator is connected at theinput. Also, because RG = Z01, a1 = 0 when the generator isconnected to the output. Radio Designer takes care of this.

We can now define the S-parameters, which we “custom-arily,” although not necessarily, do in terms of voltagewaves as follows:

•S

V

Z

V

Z

V

V

b

aa11 0

1

01

1

01

1

1

1

12= = = =

−

+

−

+ ;(Eq 3)

This is the complex ratio of a reflected wave leaving port1 to a wave arriving at port 1 with no wave arriving at port2 (no reflection from RL, and the generator is at the left).It is also known as the reflection coefficient ΓIN and inRadio Designer Circles Utility as RHO. Input SWR can befound from |S11| [SWR = (1 + |S11|) / (1 – |S11|)—Ed.].S11 can be plotted on a Smith Chart.

•S

V

Z

V

Z

V

V

b

aa22 0

2

02

2

02

2

2

2

21= = = =

−

+

−

+ ;(Eq 4)

This is the complex ratio of a reflected wave leaving port2 to a wave arriving at port 2 with no wave arriving at port1 (no reflection from RG and the generator is at the right).Also known as the reflection coefficient ΓOUT and in RadioDesigner’s Circles Utility as RHO. Output SWR can befound from |S22|. S22 can be plotted on a Smith Chart.

•S

V

Z

V

Z

b

aa21 0

2

02

1

01

2

12= = =

−

+ ;(Eq 5)

This is the complex ratio (gain or attenuation) of a waveleaving port 2 to a wave arriving at port 1 with no wavearriving at port 2 (no reflection from RL). S21 is not animpedance or a reflection factor, so it is plotted on a polargraph. The quantity |S21|2 is referred to in transistor datasheets as the “intrinsic” (same as “transducer”) power gain,

Fig 1—Definitions of terms used in S-parameter analysis and design.

Nov/Dec 1998 5

when measured with a network ana-lyzer. “Available” and “actual” powergain are, in most cases, differentfrom this measurement (see laterdiscussion).

•S

V

Z

V

Z

b

aa12 0

1

01

2

02

1

21= = =

−

+ ;

(Eq 6)This is the complex ratio of a wave

leaving port 1 to a wave arriving atport 2 with no wave arriving at port 1(no reflection from RG).• Using these definitions, we canwrite the two-port equations that areused in S-parameter analysis anddesign:

b S a S a

b S a S a

1 1 2

2 1 2

11 12

21 22

= ⋅ + ⋅

= ⋅ + ⋅ (Eq 7)

The wave leaving port 1 (b1) is thesum of a portion of a wave (a1) that isreflected by port 1 (S11) and a wave(a2) that is fed-through from port 2(S12). Similarly for b2.• Radio Designer uses the followingforms of S-parameters, using S11 asan example:

MS11 magnitude of S11

PS11 phase of S11

RS11 = real part of S11

IS11 = imaginary part of S11

==

(Eq 8)

S-parameters versus frequency canbe plotted in rectangular form, usingthe Rectangular option in Report Edi-tor. The Polar option plots magnitudeand angle of S11 and S22 on a SmithChart and S12 and S21 on a polarchart.

Gain DefinitionsBefore proceeding, we should say

something about the various kinds ofpower gain that are used in Designer.Assume that Designer correctly termi-nates the device in Fig 2A (no reflec-tions from source or load). The trans-ducer or insertion power gain was pre-viously defined as |S21|2. Designeruses “available” gain GA and “actual”(aka “operating”) gain GP. These aregiven by the equations

GS

SG

S

SA P=

− −=

21

1 22

21

1 11

2

2

2

2and

(Eq 9)If S22 or S11 is not zero, the discrep-

ancies with transducer gain need to beappreciated. We will encounter thesedifferences in this article.

The Small-Signal AmplifierFig 2A shows an amplifier gain block

with a set of transistor S-parametersavailable as data in a Data Block of aDesigner Circuit File. We want to lookat some important properties of theamplifier. In Fig 2A the amplifier, asdefined by its 50 Ω S-parameters, isterminated at the input and outputwith 50 Ω. Its forward “transducer”power gain, also called “insertiongain,” is |S21|2, and its reverse power“gain” or attenuation is |S12|2. Itsinput and output reflection coeffi-cients are S11 and S22. But suppose,in Fig 2B, that ZIN = ZS = 100 Ω andZOUT = ZL = 200 Ω. The impedance-matching networks at the input andoutput would increase the gain. Theinput network would produce 3 dBmore gain and the output networkwould increase the gain 6 dB, for atotal increase of 9 dB.

For this reason, we say that the in-put network has (is responsible for) a“gain” of 3 dB and the output networkhas a “gain” of 6 dB. Other input/output networks can be designed thatproduce smaller values of gain. Theimportant thing for the Circles Utilityis that on a Smith Chart a particularvalue of total amplifier gain, includingthe added gain that is due to thematching networks, is associated withcircles. These circles are called gaincircles. Fig 2B shows the input/outputmatching networks added to theamplifier.

It might be thought that we wouldalways want to simultaneously imped-ance match the amplifier’s input/out-put to get the greatest possible gain.But for maximum linear power outputcapability, a wider bandwidth or for

stability reasons the required sourceimpedance or load impedance may be alot different. Gain, bandwidth and sig-nal handling always involve trade-offs.The choices of gain circle values andinput/output impedances are the toolsthat we use. Conventional or transmis-sion-line transformer coupling is animportant approach at frequencieswhere they can be used. The best noisefigure is often obtained by correctlymismatching the input.

When working with the input/out-put matching networks, we will beusing reflection coefficients ratherthan impedance values. We recallfrom Part 1 that an impedance on theSmith Chart corresponds to a certainvalue of complex reflection coefficient,RHO. We will also use such symbolsas ΓS, ΓL, ΓIN and ΓOUT for source,load, input and output values, respec-tively. Fig 2B shows this usage, whichis compatible with the usage of S11and S22, which are also reflectioncoefficients.

When working with gain circles, wemust be aware of the effects of inter-nal feedback (S12) in the amplifier.This feedback can cause instabilityand oscillation if we are not careful.For this reason a different kind ofcircle, called a stability circle, is alsoplotted on the Smith Chart. There aretwo kinds of stability circles, inputstability circles and output stabilitycircles. The input stability circle is as-sociated with values of the amplifiersource impedance ZS (or ΓS) thatmight cause instability. The outputstability circle is associated with val-ues of load impedance ZL (or ΓL) thatmight cause instability. The main ideais to select points on the input and

Fig 2—Amplifier with input and output matching networks. (A) without matchingnetworks (B) with matching networks added.

6 QEX

output gain circles that are suffi-ciently far away from the input andoutput stability circles that there areno stability problems due to tempera-ture or statistical variations or fre-quency changes.

Due to feedback, the input and out-put gain circles interact with eachother. Designer uses a special proce-dure to plot these simultaneously inthe presence of feedback. Because ofthis feature we need not be concernedabout the so-called Unilateral Gainconcept, which in times past assumedthe S12 = 0 approximation in order tosimplify the calculations. Another fea-ture, called RHOL or RHOS, locatesthe source and load reflection coeffi-cients that provide some desired gainvalue. As you can see, this entire pro-cedure is a real can of worms whenperformed manually, but the CirclesUtility removes nearly all of the pain.

To repeat, if S12 is zero, the feedbackproblem disappears and the amplifieris said to be unilateral. A transistor canpossibly be neutralized (unilater-alized) to reduce the effects of feedback.Designer tells us the maximum avail-able gain AMAG possible or, if feedbackis a problem, the maximum stable gainMSG. AMAG is the gain when both in-put and output of a stable transistorare conjugately matched to source andload. Designer also gives us a stabilityfigure of merit, K, that gives us guid-ance regarding conditional stability. IfK > 1.0, the stability circles are outsidethe chart and the entire chart is astable region, there is no problem. Con-ditional stability (K < 1) means that ev-erything may still be okay if we designthe input/output networks carefullyand stay far away from the stabilitycircles. If S12 is small enough, the cir-cuit is said to be unconditionally stable,which means that no combination ofreactances at input and output cancause oscillation or excessive regenera-tion. If S12 is significant and we try toincrease the gain-circle value to thepoint where an unstable region is en-tered, Circles denies entry and printsthe message “MSG = xx.xx dB.” MSG isthe maximum stable gain when K = 1.0.If we move into an unstable region, thegain would increase very rapidly be-yond the MSG, and we get this mes-sage. A very important way to improvestability and increase K beyond the 1.0mark is to modify the circuit in a waythat reduces the product of forwardgain and feedback (ie, loop gain). Wewill study one example of how to dothis, but there are many ways.

Another issue is tunability, which is

*UHF AMPLIFIER DESIGN USING THE CIRCLES UTILITY

BLK

IND 1 2 L=?16.45NH?

CAP 2 0 C=?4.39PF?

INNET:2POR 1 2

END

******************************

BLK

TWO 1 2 0 MRF901

* RES 2 0 R=150

XISTOR:2POR 1 2

END

******************************

BLK

CAP 1 0 C=?1.68PF?

IND 1 2 L=?23.12NH?

OUTNET:2POR 1 2

END

******************************

BLK

INNET 1 2

XISTOR 2 3

OUTNET 3 4

AMP:2POR 1 4

END

*****************************

FREQ

435MHZ

* STEP 420MHZ 450MHZ 5MHZ

END

*****************************

OPT

AMP R1=50 R2=50

MS11

MS22

END

*******************************************************************

DATA

MRF901: S MP RREF=50

* MRF901 0.4GHZ 0.5GHZ 2 50.0 1 VCE=10V IC=5MA

* FREQ S11 S21 S12 S22

0.2GHZ 0.63 -69 9.69 135 0.05 57 0.79 -25

0.5GHZ 0.53 -135 5.65 101 0.07 43 0.54 -38

* NOI RN

* FREQ FMIN_DB MGOPT GOPT RN

* 0.5GHZ 1.80 0.63 7 0.96

END

*Notes:

*In DATA block, S means S-parameters, MP means magnitude and phase,

* RREF means reference Z0 for S-parameters.

Fig 3—Amplifier circuit description.

Nov/Dec 1998 7

the ability to “tune” the input/outputmatching networks. Tunability can bepoor if adjusting one circuit causes theother to become detuned because of S12feedback. Designer helps to find bothnetworks simultaneously. Variationsin S12 over frequency and tempera-ture, etc, can aggravate this problem.

Another consideration is the noisefigure of the amplifier. If we put tran-sistor noise data in a Data Block wecan plot noise figure circles on theSmith Chart, each of which corre-sponds to a particular value of noisefigure. Losses (L and C component Q)in the input matching network canalso be included in the Designer circuitfile to improve the accuracy of gainand noise-figure simulations.

To summarize, the main task forDesigner is to find locations on accept-able gain circles that are sufficentlyremoved from instability regions andin regions of acceptable noise figure (acompromise). We then design input/output matching networks that imple-ment the compromises, using the pro-cedures discussed in Part 1 of thisarticle. The L network is a simple oneto implement, where it is adequate,but T, π and other multielement typeshave greater capabilities where selec-tivity and control of impedance versusfrequency, in addition to impedanceratio, are important. In particular,selectivity at the input is often impor-tant in order to reduce interference. Insome power amplifiers or second-stageamplifiers, noise figure may be a sec-ondary consideration.

When designing the networks wemay want to control the selectivity (Q),for example in a wideband amplifier.If we draw Q lines, for example +2 and–2, on the Smith Chart and confine allimpedance (DX) and admittance (DB)paths to lie within these lines, therewill be some limitation of selectivity.The networks may need more thantwo components, perhaps a T or π, toachieve the goal. We can also use mul-tiple components to increase selectiv-ity (use higher Q circles).

This entire procedure applies only toclass-A linear amplifiers in which thevalues of S-parameters do not changewith signal level. If these changes oc-cur, special large-signal S-parameterprocedures are used that are beyondthe scope of this article.

The final result is a set of S-param-eter and noise-figure data for the com-plete circuit of Fig 2B, repeated at eachfrequency of interest and using a spe-cific set of L and C values (possiblytransmission-line or microstrip seg-

ments) for the matching networks. Thedesign must be tested over a frequencyrange to make sure that oscillation (ornear-oscillation) cannot occur at somefrequency that may possibly be far re-moved from the intended frequencyband. In other words, we do not get tooclose to a stability circle. This design isthen available as a 2POR (two-portdevice) for some larger system. Oneexample of this would be a push-pullamplifier using hybrid splitters andcombiners. Or, the final S-parameterand noise data can be used in a De-signer Data Block that defines a new“device.” We can also easily get theS-parameters of a modified transistor.Stray Ls and Cs of the various com-ponents may also improve accuracy ofthe circuit description.

The Amplifier to be SimulatedFig 3 is the Designer circuit file for

the amplifier in Fig 2B, an MRF901bipolar transistor (about $1.60 insmall quantities) with input and out-put matching networks that are to bedesigned for the 420-450 MHz (70 cm)band. The input network is INNET andthe output network is OUTNET. Thetransistor is XISTOR, which containsa TWO circuit element that referencesthe MRF901. The DATA block containsthe S-parameter data for the transis-tor. This kind of information is foundin data manuals (and Designer’s built-in data files) over frequency and atvarious values of Vce and Ic. Designerwill interpolate between 200 MHz dataand 500 MHz data to get data for the70 cm band. The complete circuit is theAMP block. Each of these blocks can beedited and analyzed individually. Thespecific design frequency of interest is435 MHz. After the networks are de-signed the frequency can be swept from420 to 450 MHz to get the overallS-parameters versus FREQ for AMPas rectangular graph plots, polar plotsor as data tables.

If we wish to, we can see what thetransistor’s S-parameters look like at435 MHz or at several frequencies. Wecan do this by analyzing the XISTORblock separately. Use Report Editor toset up a table that displays XISTOR’sfour S-parameters, magnitude (notdecibels) and angle. The Terminationsfor this should be 50 Ω. Ordinarily,Circles will calculate these automati-cally as it does its work. We do have tospecify the various frequencies. If wemodify the transistor by adding resis-tors or parasitic Ls and Cs we can, ifwe wish, easily get the new S-param-eters and use them in our design work.

There are some problems about noisesimulations. Many fine transistorssuch as the MRF901 do not explicitlygive all of the NOI parameters versusfrequency that Designer needs to createnoise-figure circles (it may be possibleto estimate them from the data sheets).It is more common to see noise-figuredata or circles presented as tables orgraphs in data manuals (eg, MotorolaRF Device Data, DL110/D REV 7). Theyalso indicate the complex source-impedance or reflection-coefficient ΓSneeded to get those noise figures. In thissituation it is probably best to workback and forth between Designer andthe data manual to get the best result.I will illustrate the method in this ar-ticle. Fig 3 gives an example of hownoise data might be placed in a DATAblock, when it is available. See theNoise Figure sidebar.

Putting the CirclesUtility to Work

Create the circuit file 2-port networkin Fig 3 and Analyze it at the singleFREQ 435MHZ. Put an asterisk beforethe 420 to 450 frequency sweep toeliminate it from the analysis. The 150Ω resistor should also be “commentedout” with an asterisk. Use Report Edi-tor to create a graph of MS11, MS22,MS12 and MS21, all in decibels, for theAMP block (the complete circuit). Bringup the Circles Utility. Fig 4 shows theappearance of the chart as we performthe following steps.1. In the CCW, type R 1 and X 0 to geta couple of red reference lines on thechart.2. Type CKT XISTOR. This selects theMRF901 transistor to be analyzed allby itself at 435 MHz. The rest of thecircuit is ignored.3. See Fig 4A. Type S1, then S2. Theseare the input stability circle (blue)and output stability circle (grey) forthe MRF901. They fall inside theSmith Chart and the CCW says thatthe regions outside the circles arestable. So in future steps we will oper-ate only in those regions. Regions in-side the circles produce conditionsthat are prone to oscillation. Some-times only the regions inside arestable and the CCW will tell us so.4. We also see in Fig 4A that XISTOR,at this point, is terminated only with50 Ω (the center of the Smith Chart).Therefore, XISTOR by itself is stable.This points out a major reason for us-ing S-parameters to characterize theactive device. We should always verifythis, especially for multiple-transistor“devices” that we create.

8 QEX

Fig 4—(A) shows stability circles for the example. (B) input and output-gain circles, input and output matching-networkreflection coefficients. (C) designing the input matching L-network. (D) designing output matching L -network. (E) amplifierwith 150 Ω loading resistor.

Nov/Dec 1998 9

5. Type K. This is the stability factorand it is 0.52. A number less than 1.0indicates conditional stability (pos-sible instability). The stability circlesbeing inside the Smith Chart also sug-gests that K is less than 1.6. Type GMAX and get the resultMSG = 19.97 dB. MSG (maximumstable gain) is the largest possible gainthat will not result in instability. Themessage GAMAX (maximum avail-able gain) corresponds to uncondi-tional stability. Circles will not allowa gain greater than GAMAX, or MSGif that is smaller than GAMAX.7. We want to decide a value of gain forthe amplifier that is less than MSGand is “sufficiently” (judgment call)removed from the stability circles. Seethe Gain Circles sidebar. In this ex-ample, we need to use bilateral gaincircles because of the internal feed-back of this transistor (as discussedearlier). G1B refers to “availablepower gain GA,” which is used in manyapplications. The “B” stands for “bilat-eral.” We could also use G2B, whichrefers to “actual power gain GP,” whichis also frequently used. We will useG1B for this example.8. We will pick a GA gain value of16 dB. See Fig 4B. Type G1B 16 in theCCW. We get two gain circles, a solidyellow line for the input and a dashedyellow line for the output. For eachpoint on one circle there is a corre-sponding point on the other circlesuch that the 16 dB of available gain isachieved. The transistor itself has anavailable gain GA = 18.2 dB or an ac-tual gain GP = 17.9 dB at 435 MHz,but because we are concerned aboutstability, we will design for GA = 16 dB.The question occurs, “If the transistoritself has close to the required gainand is stable, why bother with net-works?” The answers are (a) in mostsituations we will want some selectiv-ity in the amplifier, and (b) we maywant to improve noise figure. We alsohave to consider that the source/loadmay not be the ideal 50 Ω at frequen-cies outside the 420 to 450 MHz band,and this might cause stability prob-lems if there is no selectivity.9. Now we want to pick a value of re-flection coefficient on each of thesecircles that will (a) give us the gainthat we want, (b) have no stabilityproblems and (c) will result in an ac-ceptable noise figure for the amplifier.10. See Fig 4B. Type RHOL 16, thenExecute. The number16 specifies theavailable decibel gain that we want.Now place the cursor at the point indi-cated on the solid yellow (input) circle

and press the left mouse button. Amarker appears on the input linewhere we placed the cursor and amarker also appears on the output(dashed) line. The CCW now showstwo values, RHO S mag = 0.30, ang =–16.57°, and RHO L mag = 0.48, ang =32.26°. The first is the reflection coef-ficient ΓS (looking back from the tran-sistor base toward the generator)needed for the input matching net-work (INNET) and the second is thereflection coefficient ΓL (looking fromthe collector toward the load) for theoutput matching network (OUTNET).These networks will provide the over-all gain of 16 dB at 435 MHz with verygood stability. Several trials are al-ways needed to get the best placementof these markers with respect to thestability circles. DEL the undesiredtrial points and re-enter the desiredpoints, when you know where youwant them to be.11. We must also consider the noisefigure. I referred to the Motorola datasheets for the MRF901 and saw thatat 500 MHz, Vce = 10, Ic = 5 mA and forthe location of my marker on the inputyellow circle, I could expect a noisefigure of about 2 dB. This was closeenough to 435 MHz that I was satis-fied. My choice of S-parameters alsowas listed at these same values of Vceand Ic. See the Noise Figure sidebar.12. We must now obtain values for theinput-matching network. See Fig 4C.Type G in the CCW, then Execute.Place the cursor at the marker on thesolid yellow (input) circle and press

the left button. This creates a G = 0.55circle. We are now ready to get thevalues. Type DX, then Execute. Placethe cursor at A, the origin point, andpress the left button. Move the cursorto B, the junction of R = 1, G = 0.55 andpress the left button. The series induc-tance value in the CCW is 16.45 nH.Now use DB from the junction B downto C, the marker on the solid yellowcircle, to get the shunt capacitancevalue, which is 4.41 pF. Enter thesevalues in the INNET circuit block asshown in Fig 3. See Fig 4D. Create theG = 0.38 line at point C. Repeat theprocedure DX from A to B and DB fromB to C to get the values for OUTNETwhich are 23.4 nH and 1.68 pF. Enterthese values in OUTNET as shown. Besure to get the connections in the rightorder. The schematic and node num-bers are shown in Fig 5A.13. To see what has been accom-plished, change the FREQ block tosweep from 420 to 450 MHz and viewthe plot of AMP. MS21 dB is nearly 16(the exact GA from Eq 9 is 16.02 dB).The output impedance is close to 50 Ω,as shown by MS22 dB. MS12 dB isabout –24, which is not too bad but onlyabout 8 dB greater in magnitude, inthe negative direction, than MS21.MS11 dB is very poor, only about –1.5,meaning that the input to the ampli-fier is not well matched to the 50 Ωgenerator. This is how Circles has as-sured the amplifier stability and con-trolled GA. Nevertheless, the designapproach that we have used tells usthat we have a noise figure of not more

Fig 5—Amplifier diagrams. (A) no resistor loading, (B) 150 Ω resistor loading.

10 QEX

Gain CirclesThere are some options regarding gain circles that should be appreciated

for successful design. They are as follows:1. The maximum available generator power is delivered to a ΓΙN that is the con-jugate of its ΓS (see Fig 2). The maximum device available power is deliveredto a ΓL that is the conjugate of its ΓOUT. These two define the maximum avail-able gain (GAMAX). On the other hand, available gain is the ratio of the powerthat is actually available from the output, when the input and output are mis-matched, to the maximum power that is available from the generator. Avail-able gain is very often less than maximum available gain. It is widely used incalculating the gain and noise figure of cascaded devices.2. Power gain (aka actual gain or operating gain) is the ratio of the power thatis delivered to the load to the power that is delivered to the input.3. G1 refers to the input gain circle. For example, the G1 15 circle gives allvalues of the input ΓS that can be used to get 15 dB of available gain (if wealso use the right value of load ΓL). It doesn’t tell us what ΓL is. RHO, at anypoint on the circle, applies only to the input network. G1B, on the other hand,does the same thing as G1 and also gives a circle for the output. The pair ofpoints, one at input, one at output, defined by the RHOL 15 command, tell uswhat to do at both ends of the amplifier. With G1B, we select a point on theinput circle. If the stage is almost unilateral, the output circle is very small andmay even be difficult to see on the screen. I like G1B a lot better.4. G2 15 is similar to G1 15, but it refers to actual or operating power gain andonly to the output of the amplifier. G2B 15 gives us two circles, one at the outputand one at the input. RHOS 15 gives a point on each circle that provide 15 dB ofactual power gain. With G2B, we select a point on the output circle. G2B is alsopreferable, I believe. When selecting the point on the output circle, the resultinginput point should occur at a location of good noise figure, if possible. It is forthis reason that the G1B approach is usually preferred for noise matching.

Noise FigureThe following notes apply to noise-figure analysis using Circles:The format for noise data in the DATA block is:NOI RN (or) NC x (first line)* FREQ FMIN_DB MGOPT GOPT RN (or) NC (comments)420MHZ 1.83 0.320 35.7 0.25 (data)435MHZ 1.88 0.758 65.2 0.36 (data)etcRN signifies that the RN noise data is the equivalent noise resistance, normalized with respect to x. If the data is given

in ohms, it is divided by x to get the normalized RN. For example if x = 50 Ω and RN = 60 Ω then RN (normalized) =60 Ω / 50 Ω = 1.2. If the data is already normalized, omit x.

NC signifies that the noise data is noise figure in decibels (either RN or NC may be available from data sheets).FMIN_DB is the minimum value of noise figure in decibels at FREQ. This is the value of noise figure when the match-

ing networks are optimized for lowest noise figure. It is the best that can be achieved at FREQ.MGOPT is the magnitude and GOPT is the angle of the input reflection coefficient ΓS = ΓOPT that results in FMIN_DB.The noise situation is perhaps best appreciated by studying the equation:

F F4 RN

1 1 MG GMIN

S OPT2

S2

OPT2 OPT OPT OPT= +

⋅ ⋅ −

−( ) ⋅ += ∠

Γ Γ

Γ ΓΓ;

where F and FMIN are noise factors and ΓS is the input reflection coefficient. If ΓS = ΓOPT then F = FMIN. If ΓS = 0 (nomatching network) the intrinsic value of F for XISTOR itself is observed. The values of ΓS that define a particular value ofF lie on a circle in the Smith Chart. Noise figure NF(dB) = 10 log (F).

Type N 3 in the CCW to get a 3 dB noise-figure circle, and so forth. Type N 0 to see the minimum noise figure.

than 2 dB, according to the data sheetfor the MRF901. Also, if we optimizeonly the components in the output net-work the output match to 50 Ω can beimproved. In other words, the transis-tor output is conjugate matched to theload. At 420 MHz, GA increases about0.5 dB and at 450 MHz decreases about0.5 dB.14. To get some additional perspective,set up the Circles chart for CKT AMP,the complete amplifier, at FREQ =435 MHz. Plot S1, S2 and G1B 16. S1and S2 are well inside the chart. Thetwo gain circles intersect at the origin,just as they should (the load and sourceare both 50 Ω) and the amplifier is quitestable, but only conditionally so (K =0.52). If the amplifier is not terminatedproperly, if a combination of small pa-rameter changes occur or if we try toincrease gain, oscillation might occur.15. The Sweep feature deserves atten-tion. Page 12-9 of the Designer manualdescribes how S1, S2, G1[B], G2[B] orN circles of a circuit or subcircuit canbe plotted at each of the frequenciesdefined in the FREQ block, using theSW FREQ command. Or, a plot ofG1[B], G2[B] or N circles over somedecibel range, minimum to maximum,at some specified FREQ can be plotted,using the SW DB command. Whenyou’re through with the Sweep mode,be sure to enter SW END.

The MRF901 transistor appears to

be a slightly difficult one to use in itsnative condition. An interesting thingcan be observed about this example.If we use G2B 16 and RHOS 16, instead

of G1B 16 and RHOL 16 and redesignthe networks, we will find that the in-put side is much better matched to the50 Ω generator and the output side is

Nov/Dec 1998 11

poorly matched, as compared with theexample that we worked out. Thereader is encouraged to try this. How-ever, there are better ways to improvethe design and we will look at one ex-ample of how we might do this.1. Remove the asterisk in front of the150Ω resistor from collector to ground.2. Analyze the circuit and bring upCircles. Use CKT XISTOR, FREQ =435 MHz.3. Plot S1 and S2, then type K andGMAX. We see that K = 1.17, GMAX =17.46 dB and S1 and S2 are well out-side the Smith Chart, as shown in Fig4E. We now have an unconditionallystable amplifier, which gives us morelatitude in the design options. Wehave achieved this by modifying theXISTOR block with a resistor. Thequantity |S21•S12| of XISTOR hasbeen reduced from 0.46 to 0.26 (checkit out).4. Use bilateral G1B 17.4 and RHOL17.4 to get the gain circles and themarker points. Note that the twocircles are very small, which meansthat both ends are very close to a con-jugate match and therefore maximumgain. Use DX and DB to get the inputand output-network values, as shownin Fig 5B. Place these values inINNET and OUTNET.5. Looking at the graph of AMP’sS-parameters we see that MS11 andMS22 are better than –15 dB at theband edges and better than –25 dB atthe center. We now have a very stableamplifier with 17.4 dB of gain that is“cascadable,” meaning that it matchesa 50 Ω source and load within a 2:1SWR. The noise figure is about thesame, or slightly better, according tothe MRF901 datasheet. It may be pos-sible to optimize both matching net-works over the frequency band. Wemay also be able to adjust the inputnetwork to a value of better noise fig-ure, with a small loss of gain and asmall loss of input impedance match.

Here’s another educational experi-ment to try:1. In the DATA block, reduce the S12magnitude data from 0.05 to 0.005 andfrom 0.07 to 0.007. This is a 20 dB re-duction in internal feedback. Also com-ment out (asterisk) the 150 Ω resistor.2. In Circles, GAMAX is 19.97 dB. UseG1B 19.5 and RHOL 19.5 (almostmaximum gain). The stability circlesare far removed from the chart.3. Get the input and output networks.This is also a “cascadable” amplifierwith a noise figure in the 2 dB range.This is what a unilateral transistorcan accomplish if we can find one, or if

we can neutralize the MRF901. Thecascode connection or the differentialpair should be considered as goodways to reduce S12, although the noisefigure may take a hit (two noise mak-ing devices instead of one).

Optimization is a very nice way toimprove input and output matching,but it has to be used carefully. My ex-perience has been that it works muchbetter with unconditionally stabletransistors. Otherwise, the amplifiercan wind up in an oscillating region.This happened with the first example.The method shown in Fig 3 is simpleand effective. Some experience withthe optimizer will suggest strategiesthat help to avoid pitfalls. In addition,we should always confirm the resultsby checking the K value and the stabil-ity circles of AMP. The origin point ofthis chart (the generator and load im-pedances) should be far removed fromthe stability circles.

If you perform the various exercisesthat we have presented here and studythe two sidebars, skill will develop asyou get familiar with the various key-

board and graphical operations. Itsoon becomes a simple enough routinethat “what if?” experiments can beexplored, with confidence in the re-sults. Data sheets contain the neces-sary information so that you can de-sign and analyze scientifically.

Recommended ReadingG. Gonzales, Microwave Transistor Amplifi-

ers, 2nd ed, Crestone Technical Books,Noble Publishing Co, Tucker, GA; tel 770-908-2320; URL www.noblepub.co m .

Hewlett-Packard “S-Parameter Techniques”Application Note 95-1 available from http://www.tmo.hp.com/tmo/Notes/interac-tive/an-95-1/index.htm l as an 80-page.PDF document. A network design docu-ment is also available.

M. Wilson, et al, Ed, The ARRL UHF/Micro-wave Experimenter’s Manual (Newington:ARRL, 1990), ARRL Order No. 3126.ARRL publications are available fromyour local ARRL dealer or directly from theARRL. Mail orders to Pub Sales Dept,ARRL, 225 Main St, Newington, CT 06111-1494. You can call us toll-free at tel 888-277-5289; fax your order to 860-594-0303; or send e-mail to pubsales@arrl.or g . Check out the full ARRL publi-cations line on the World Wide Web athttp://www.arrl.org/catalog .