S-Parameters and Smith Charts - thebeekeeper.net

1 S-Parameters and Smith ChartsNick GamrothOct 2004 AbstractThe following are my self-study notes on S-Parameters and SmithCharts. So it s probably all S-ParametersCheck out this two-port network:a1 Vs Zs 2-Port Zl a2 b1 b2 Figure 1: That s hotHow about those arrows? Those are the incident and reflected volt-age waves at each port, normalized to the characteristic impedance, s how that looks:a1=Vi1 Z0(1)a2=Vi2 Z0(2)b1=Vr1 Z0(3)b2=Vr2 Z0(4)1 Nicholas GamrothWhere,Vi1= Voltage wave incident on port 1,Vi2= Voltage wave incident on port 2,Vr1= Voltage wave reflected from port 1,Vr2= Voltage wave reflected from port s OK, but it s not that great. This is though:S11=b1a1 a2=0(5)S22=b2a2 a1=0(6)S21=b2a1 a2=0(7)S12=b1a2 a1=0(8) s not obvious to me why these are useful, but they input and output reflection coefficients, the input reflection coefficient with the output port terminated ina matched load, as in whenZL=Z0,a2 > the outputreflection coefficient when the input port is terminated in a the forward and reverse transmission gains,respectively with both being terminated in matched loads.

2 Reflection CoefficientsReflection coefficients are the same as in transmission havea complex and real part, and represent the quality of the impedancematch between the load and the characteristic impedance of the line. = r+j i=ZL Z0ZL+Z0(9)Sometimes, folks normalizeZL, so you can write: =ZL 1ZL+ 1(10)Not too Gamroth3 Impedance and AdmittanceRemember impedance and admittance? I didn t, so I wrote it s not that important for the purpose of this paper, but I think it ssomething I ought to ImpedanceZ=R+jX(11)WhereZ=impedance,R=re sistance, andX= and inductor and a capacitor,X=j L(12)X=1j C(13) AdmittanceY=G+jB(14)WhereY=admittance,G= conductance, andB= and inductor and a capacitor,X=1j L(15)X=j C(16)4 Smith ChartsNow is when it gets it out. In the Smith chart in figure 2, the blue circle seg-ments are the circles of constant reactance, and the green circles arethe circles of constant resistance.

3 That s the simplest explanation of aSmith chart I can find. You can plot things like a complex impedanceon a Smith chart . Let s try 50 +j100(17)First, normalize this complex impedance to 50 z= 1 +j2(18)That was easy. The point right at the center of a Smith chart istheorigin. Most Charts give more detail that Figure 2, and they ll have3 Nicholas GamrothFigure 2: YOW!things like numbers that you can use to plot things. So, the real partofzis the resistance, and the imaginary part is the reactance. To plotthis complex number on the Smith chart , just go to 1 on the circle ofconstant resistance, and 2 on the circle of constant reactance. Simps!I did this on Figure 3. Point O is the origin, point A is the inter-section of the circles of resistance 1 and reactance 2. Hmm, iwonderwhat the reflection coefficient of this complex resistance is?

4 Ok, it s = + , or in polar, , on most Smith Charts , there sa angle scale printed around the red circle in Figure 3. I m not sureabout the magnitude, but it appears to be the distance between OAand O to the red circle. Yeah, that s it. But I don t know a way toget it by visual inspection (with eyeballs).Here s another tip: to plot something like 10 j50, put that messon the lower half of the chart . That s where theX <0 circles , everything on the top half of a Smith chart represents in-ductance, and everything on the bottom half represents in an inductive or capacitive yeah, and when you plot an impedance, it ll change with fre-quency. So, if you have a RL circuit, it s impedance on the smithchart will look like a line around one of the circles of constant resis-tance (because resistance doesn t change with frequency, natch).

5 Plus,it normally moves clockwise as frequency s about it for plotting things on Smith GamroththetaR = 1X = 2 OAFigure 3: Plot it5 Amplifier StabilityCheck out Figure 4. That s a generalized amplifier setup. TheS isthe amplifier. You can use S-Parameters to figure out if the amplifieris unconditionally stable, and if it s not, you can figure outwhen it is,and how to design source and load circuits to make it Zs S Zl Source Match Load Match s in out L Figure 4: Pump it up!People write S-Parameters in a matrix, like this:S= S11S12S21S22 (19)There are two important numbers that S-Parameters give you,5 Nicholas Gamroththat can determine the stablility of an amplifier. The first one is thedeterminant of the S matrix and is called . The second is K, whichis given by the following formula:K=1 +| |2 |S11|2 |S22|22|S12S21|(20)If| |<1 andK >1, then the amplifier is unconditionally , you can create a stability circle to see when the amplifieris stable and when it s the output, the center of this circle isCLand the radius isRLCL=(S22 S 11) |S22|2 | |2(21)RL= S12S21|S22|2 | |2 (22)For the input, the center isCSand the radius isRSCS=(S11 S 22) |S11|2 | |2(23)RS= S12S21|S11|2 | |2 (24)These circles can be plotted on a Smith chart for added fun.

6 Takea gander at Figure 5. Those got plotted using the equations just s a couple of conditions to determining whether or nottheamplifier is stable. First, there s a couple of parameters :D1=|S11|2 | |2(25)D2=|S22|2 | |2(26)D1is associated with the source, andD2is associated with theload. Depending on the sign of theDparameter, the stable region ofthe amplifier is either outside or inside the stability circle. ForDx>0,the amplifier is stable outside of the stability circle, and forDx<0,the amplifier is stable inside the stability you look at Figure 5, and assume thatD2>0, then we can saythat for the load, the amplifier is stable for all regions thataren t tothe upper right of the arc labeled Load Stability things go well, you ll have a unconditionally stable this happens, depending on theDxparameter, the stability circlewill either be outside of the Smith chart , or completely encompass6 Nicholas GamrothLoad Stability CircleSource StabilityCircleFigure 5: Feet shoulder width apart for better stabilityit.

7 If you see this, give yourself a pat on the back for making such awonderful amplifier. Otherwise, don t act so all that mess would be a hassle to calculate and draw, lookat the Orfanidi reference to download a whole gang of MATLAB scripts to do it for you. That s what I made the Smith chart graphicswith, and it s pretty easy to use. Mostly, you just have to putin theS- parameters and it does the rest for Unilateral Figure of MeritHere s a little gem called the Unilateral Figure of |S 11S12S21S 22| (1 |S11|2) (1 |S22|2) (27)A lot of times, you don t get a lot of reverse transmission, aka|S12|<<|S21|. When this happens, you can call the two-port unilat-eral. Here sgu:gu=1|1 U|2(28)7 Nicholas Gamrothguis called the relative gain ratio, and if it s near 1, you can probablytreat the two-port as unilateral.

8 A lot of things simplify somewhat inpower calculations because you can setS12== Power GainsWhen you make an amplifier, you want it to plump up the volume as much as possible. That doesn t always happen if you just put inloads and sources all willy-nilly. You gotta think about that first step in thinking about that stuff is figuring out how muchgain you can get, and things like three of these gains that tell you things like that. They reall sort of messes, so I m not going to write them down. I will listthem though. They are: transducer power gain,GT, available powergain,Ga, and operating power gain, ,GTseems to be the most important, because it accounts forthe effects of both the source and load impedances. What s mostimportant is that the transducer power gain is maximized when thetwo-port is simultaneously conjugate matched.

9 That happens when in= Sand L= out. This also makes the three gains given aboveequal. This gain is called the maximum available gain. LookslikeK 1 for this to m mostly condensing this stuff from the Sophocles Orfanidis(what an awesome name) book listed in the references, so for moredetail, take a look Designing Matching NetworksNow things might actually get useful, at least for amplifiers. We ll the big deal here is figuring out Sand Lto conjugate matchthe source and load. The next two equations are the long way togetthis done: S=S11+S12S21 L1 S22 L(29) L=S22+S12S21 S1 S22 S(30)That s not too bad, but if you look at it close, you get excitedwhen you see what happens in the unilateral case (S12= 0): S=S 11(31) L=S 22(32)8 Nicholas GamrothWhat could be easier? It s possible to solve it in the bilateral case(bilateral is the opposite of unilateral, dude), but it involves someparameters I didn t mention.

10 They re easy to get though, andtheindustrious reader will have no trouble figuring out how to solve thebilateral case. (Hint, they re simultaneous equations.)Vs Zs S Zl Load Match L1 C1 Figure 6: It s a matchOK. Enough fooling around. Let s design a matching networkfor a two-port. Look at figure 4. We re going to design a sourcematching network that looks like Figure 6. There are other waysto do impedance matching, but I like circuits, so we re goingto usecircuits. So, first we find the simultaneous conjugate match by usingthe equations above. Next, convert to an impedance (we re justdoing the source side, so convert S). Now we haveZSwhich weconvert toZinby taking its complex conjugate. Finally, we just plugthisZin, along with ourZSinto a couple of equations, and we re equations are: (Note:Z=R+jX)Q= RSRL 1 +X2 SRSRL(33)X1=XS RSQRSRL 1(34)X2= (XL RLQ)(35)OK, we re not really done.