ECE3025_Lecture12

ECE3025_Lecture12 - 22 14‘ U SHEETS 22-142 100 SHEETS...

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Unformatted text preview: 22 14‘! U SHEETS 22-142 100 SHEETS 22-144 200 SHEETS LQAMPA U Iii?) - 50 SHEETS 22-142 100 SHEETS 2-144 200 SHEETS in: ES «2% 4; (""7 22 141 50 SHEETS 22-142 100 SHEETS 2 1M: [)0 SHEFTS ngAMP/JU 2 _ in"?! 545 3025 ism IT‘h‘ cm? mores I” ~ 5 W. V- ,= . r 5 I SMITHL Gil—fifZT May—ES EEC/5:. 3025 The Complete Smith Chart 9 Q \va f. 4. (w . 0. W00 $.0 H- . I. eWDV m @fixhe «w. 1 3 we“. a RADIALLY SEALED PARAMEI'ERS Hi I.! 2 LI LU IA 1.1 T0WARD1LOAD 7) 0.] 0.2 l0 [.2 I.l l l5 | 2 I»! 1.5 H 5 2.5 0 l5 3 IU 20 1% uIIXHD an ‘9‘ r ;_;“ .. C 0o *1.- 2 L5 l.‘ LE [.6 LT |.3 LB 2 [L4 0.6 01 1 0.9 0 0|) 20 0.0l 0.] II H 0.05 785") 6 [I ‘9.» l.3 |.2 0.1 0.2 0.1 l 0.9 0.8 0.? 0.6 0.5 0.1 '46 q. '9’ {IA r 0,5 0.4 '13 D 2 U I U ".7 [LE 03 0.55 0.95 THEN Dam CIRCLE... Plot<\V];-OU SGAr‘dEJ 0 I CENTER 08 0.7 0.6 0.5 0.! 0.3 0.2 0.0 l 0.0 m m R 0 . Test 8. Measurement World, July 2001 How does a Smith chart work? A venerable calculation aid retains its allure in a world of lightning—first computers and graphical user interfaces. By Rick Nelson, Senior Technical Editor The Smith chart appeared in 1939 (Ref. 1) as a graph—based method of simplifying the complex math (that is, calculations involving variables of the form x + jy) needed to describe the characteristics of microwave components. Although Iculators and computers can now make short work of the problems the Smith chart was designed to solve, the Smith chart, like other graphical calculation aids (Ref. 2). remains a valuable tool. Smith chart inventor Philip H. Smith explained in Ref. 1. “From the time I could operate a slide rule, I’ve been interested in graphical representations of mathematical relationships." It's the insights you can derive from the Smith chart’s graphical representations that keep the chart relevant for today's instrumentation and design- . automation applications. On instruments (Ref. 3). Smith chart displays can provide an easy-to- decipher picture of the effect of tweaking the settings in a microwave network; in an EDA program (Figure 1). a Smith chart display can graphically show the effect of altering component values. Although the Smith chart can look imposing. it's nothing more than a special type of 2-D graph. much as polar and semilog and log-log scales - l ' _ constitute special types of 2-D graphs. ,1- ,1" in essence, the Smith chart is a “a - - __A...: ._-.r-"' special plot of the complex 5— parameter 511 (R91 4)- Which is Figure 1. RF eledronic—design-automation programs use Smith equivalent to the complex reflection charts to display the results of operations such as S-parameter coefficient G for single—port microwave simulation. Courtesy of Agilent Technologies. components. Note that in general, - 'ii r= z: + l 1:=l/le’ . and that I G |el‘9 is often expressed as Glu. Note that this latter format omits the absolute—value bars around magnitude G; in complex-notation formats that include the angle sign (I), the preceding variable or constant is http://www.wcb—ce.com/primers/files/SmithCharts/smith_charts.htrn 2/21/2005 —_—-—————_— --——-—-—————_ Page 2 of10 assumed to represent magnitude. Figure 2 shows the specific se of a complex G value 0.6 + j0.3 piotted in rectangular as well as polar coordinates (0.67/26.6°). Why the circles? That’s all well and good, you may say, but _, 1 l __ where do the Smith chart’s familiar circles , “ “ a. (shown in gold in Figure 1) come from? The f “ outer circle (corresponding to the dashed circle in Figure 2) is easy—it corresponds to a / reflection coefficient of magnitude 1. Beuse reflection-coefficient magnitudes must be 1 or less (you can‘t get more reflected energy than the incident energy you apply), regions outside this circle have no significance for the physical systems the Smith chart is designed to represent. It's the other circles (the gold nonccncentric circles and circle segments in Figure 1) that give \. the Smith chart its particular value in solving - , problems and displaying results. As noted -- 1 , .- "“ above, a graph such as Figure 2's provides for 7 convenient plotting of complex reflection coefficients, but such plots aren't particularly Figure 2. The Smith chart resides in the complex plane useful by themselves. Typically, you’ll want to f fl - - = + I: a: G, At relate reflection coefficients to complex source, 0 re emu" coeffiqentG Gr 6' IGIeI I I”. line. and load impedances. To that end, the mm A' G ' 0'6 +103 ' 067/265 ' Smith chart transforms the rectangular grid of the complex impedance plane into a pattern of circles that can directly overlay the complex reflection caeflicr'ent plane of Figure 2. Ref. 5 provides a Quicktime movie of a rectangular graph of the complex-impedance plane morphing into the polar plot of the typical Smith chart. The following section shows the mathematical derivation that underlies the Smith chart. In effect. the Smith chart performs the algebra embodied in equations 2 through 16. The algebra Recall that nonzero reflection coeflicients arise when a propagating wave encounters an impedance mismatch—for example, when a transmission line having a characteristic impedance 20 = R0 + jXO is terminated with a load impedance ZL = RL + j)(,_ D 20. In that case, the reflection coefficient is Fiure 3. Points of constant resistance form circles on the complex reflection-coefficient plane. Shown here are Z the circles for various values of load resistance. L _ 1 ZL+ZO _Z_L..+]_ Zo "fl Page 3 of 10 In Smith charts, load impedance is olten expressed in the dimensionless normalized form z,_ = rL + xL =ZL/ZO. so Equation 2 becomes f=z_L.—_1 3L+1 Equation 3 is amenable to additional manipulation to obtain zL in terms of G: sz+ f=zL—l sz—zL=—F—-1 zL(l—f)=l+f _1+F _1—r 3L Explic'rtly stating the real and imaginary parts of the complex variables in Equation 4 yields this equation: . _1+r:+jr: which can be rearranged to clearty illustrate its real and imaginary components. The first step is to multiply the numerator and denominator of the right-hand side of Equation 5 by the complex conjugate of its denominator, [(l-ID-j/fl[(1—/;)+j1;] a—arur =1-1f+j2!;'—If =(1-13-1?)+i(21§) (balm (bah/f I ’L +JxL = thereby enabling a form in which real and imaginary parts are readin identifiable and separate: ‘ ....___ ______h Page 4 of 10 l—fi—fl . 2/; (l—nz+zf+’(1—1:)2+If r TL +13% = The real part is then rzl—fi—E L (Ham? and the imaginary part is _ 2/? (1—02“? xL You n further manipulate equations 8 and 9 in the hope of getting them into a form that might suggest a meaningful graphical interpretation. Equation 8, for example, n be altered as follows: _ Eff-1? 1—2/?+/f+1f rL-ZrL!:+rL/f+rL/?=1-Jf-1? rLCZ+jf_2rLj:+T-L+r£fii+fii=l 1:2(rL+1)-—2rL_/;+ [£201 +1)=1—rL 1:2_EE_£+1;Q=1—rL rL+l rL+l I"L The last line of Equation 10 might look familiar. It’s suggestive of this equation you might remember from high school analytic geometry: (ac—Hf +(y-55')2 =r2 Equation 11 represents a circle plotted in an x-y plane with radius rand centered at x = a. y = b. In Equation 10. you can add rL2l(r,_ + 1)2 to each side to convert the G ,terms into a polynomial that you can factor: Page 5 of 10 . 2 2 r + rL+l rL+1 (l—rLXrLH + if (TL + 1)2 (TL + 1y 2 2 (’1 +1)2 (TL +1): You can then arrange Equation 12 into the form of a circle centered at [rLl(rL + 1). 0] and having a radius of 1! (rL + 1): 2 2 1 I: — ’1 +(1: —0i = rL+1 rL+1 . Figure 3 shows the circles for several values of rL. Note that the Q = 0 circle corresponds to the | G | = 1 circle of Figure 2. You can similarly rearrange Equation 9: ._ 2!: 1—2/:+/f+!? xL *ZxL1:+xL/;Q+xL/?=2£ ijf_2ij:+xL-If_2j: =‘xL 2 If—21:+/f——1:=»1 3‘1. xi. adding a constant to make the G iterrns part of a factorable polynomial: 2 2 2 jfu2fl+1+jf——j;+ i =—1+1+ i xL xL 33L . Equation 15 can then be wn'tten as follows: f Page 6 of 10 2 1‘2—21:+1+1;2-—1;+ — =——1+1+ ————— xi; 35L 35L representing a circle of radius “XL centered at [1, 1IxL]. Figure 4 shows several of these circles or circle segments for various values of XL. The segments lying in the top half of the complex-impedance plane represent inductive reactances; those lying in the bottom half represent pacitive readances. Note that the circle centers all lie on the blue G r: 1 vertical line. Only the circles segments that lie within the green | G | =1 circle are relevant for the Smith chart. Note that XL = 0 along the horizontal axis, which represents a circle of infinite radius centered at [1, +y} or [1, —y] in the complex G plane. You can superimpose the circles of Figure 3 and the segments lying within the | G | = 1 circle of Figure4 to get the familiar Smith chart (Figure 5). Note that the Smith chart circles aren’t a replacement for the complex reflection-coefficient plane—in fact, they exist on the plane, which is represented in rectangular form by the gray grid in Figure 5. Now what? The coexistence of complex- impedance and complex—reflection- coefiicient information on a single graph allows you to easily determine how values of one affect the other. Typically, you might want to know what complex reflection coefficient would result from connecting a particular load impedance to a system having a given characteristic impedance. Consider. for example, the normalized load impedance 1 + j2. You n locate the point representing that value on the Smith chart at the intersection of the rL = 1 constant-resistance circle and the xL = 2 constant—reactance circle segment; the intersection is point A in Figure B. erh point A plotted, you can directly read the resulting reflection coefficient: G = 0.5 + j0.5, or G = 0.707!45°. To graphically determine the polar form, simply divide the length of line segment CA by the radius of the rt = 0 {30.25 xlgz Figure 4. Values of constant imaginary load impedances x1- make up circles centered at points along the blue vertil line. The segments lying in the top half of the complex-impedance plane represent inductive reactances; those lying in the bottom half represent capacitive reactances. Only the circle segments within the green circle have meaning for the Smith chart. circle. You can use a protractor to measure the angle; many Smith charts, such as one included in Adobe PDF format on a CD-ROM supplied with Ref. 6, include a protractor scale around the circumference of the rL = 0 circle. Such a scale is suggested in yellow in Figure 6. As another example. the complex- impedance value 1 —j1 is located at point B in Figure 6; at point B, you can read off the corresponding reflection coefficient 6 = 0.2 —j0.4, or G = 0.45!— 63". (Keep in mind here that this example describes dimensionless normalized impedances. For a system characteristic impedance of 50 V, the respective values of load impedances at points A and B would be 50 + j100 V and 10 —j20 V.) Standing wave ratio Smith charts can help you determine input impedances as well as relate load impedances to the reflection coefficient. To understand how that works, first review the operation of standing waves in a transmission line with a mismatched load. Such waves take on a sinusoidal form such as that shown in Figure Ta. In Figure 7, standing waves result when a voltage generator of output voltage VG = (1 V) sin(vt) and source impedance ZG drive a load impedance ZL through a transmission line having characteristic impedance 20, where 25 = 20 b ZL and where angular frequency v corresponds to wavelength t (Figure 7b). The values shown in Figure 7a result from a reflection coefficient of 0.5. i won’t derive the equation that describes the standing wave that appears along the transmission line in Figure 7b; for a derivation, see Ref. 5 or another text covering transmission— line theory. I'm asking you to accept that if you could connect an oscilloscope to various points along the transmission line, you would obtain readings illustrated in Figure 7c. Here, probe A is located at a point at which peak voltage magnitude is greatest—the peak equals the 1-V peak of the generator output, or incident voltage, plus the in—phase peak reflected voltage of 0.5 V, so on Page 7 of 10 flat fiz-l Figure 5. The circles (green) of Figure 3 and the segments (red) of Figure 4 lying within the | G | = 1 circle combine to form the Smith chart, which lies within the complex reflection-coefficient (G) plane, sh0wn in rectangular form by the gray grid. ISE‘U Figure 6. With a Smith chart. you can plot impedance values using the red and green circles and circle segments and then read reflection-coefficient values from the gray gn'd. Many Smith charts include a scale (yellow) around their circumference that lets you read angle of reflection coefficient. Page 8 of 10 your oscilloscope you would see a time-varying sine wave of 1.5-V peak amplitude (the gray trace in Figure 7c). At point C, however, which is lecated one-quarter of a wavelength (II4) closer to the load. the reflected voltage is 180° out of phase with the incident voltage and subtracts from the incident voltage. so peak magnitude is the 1—V incident voltage minus the 0.5-V reflected voltage, or 0.5 V, and you bl 20 = 26 = would see the red trace. At . intermediate points, you'll see peak values between 0.5 and 1.5 V; at B (offset Ila from the first peak) in Figure To, for example, you'il find a peak magnitude of 1 V. Note that the standing wave repeats every half wavelength (U2) along the transmission line. E. Total voltage {V} O .-' ‘3‘ c: a? a. i" u w r x N U 1.5 sintmt) ail 1.5 v The ratio of the maximum to minimum values of peak voltage amplitude ‘ .1 measured along a standing wave is fit the standing wave ratio SWR. For the K Figure 7 system. SWR = 15105 = 3. Note that 1+|f| 1—|f| it -1, rm not proving Equation 17 here, but Figure 7. (a) Standing waves, which repeat for every half SWR = O .__.‘_m_____w.mm___mm substituting the 0.5 reflection wavelength of the source voltage, arise when (b) a matched coefficient used in the Figure 7 generator and transmission line drive an unmatched load. (c) example into Equation 17 does Time-varying sine waves of different peak magnitudes appear at provide the desired result of 3, different distances along the transmission line as a function of wavelength. The relationship between G and SWR suggests that SWR might have a place within the Smith chart. and indeed it does. In fact. lculations involving SWR first prompted Smith to invent his chart. "By taking advantage of the repetitive nature of the impedance variation along a transmission line and its relation to the standing-wave amplitude ratio and wave position, I devised a rectangular impedance chart in which standing—wave ratios were represented by circles." he explained in Ref. 1. Figure 8 helps to explain how those circles arise. In Figure 8, point L represents a normalized load impedance 2L = 2.5 —j1 = 0.5118” (I chose that particular angle primarily to avoid the need for you to interpolate between resistance and reactance circles to verify the results). The relationship of reflection coefficient and SWR depends only on the reflection coefficient magnitude and not on its phase. If point L corresponds to | G t = 0.5 and SWR = 3. then any point in the complex reflection- coefficient plane equidistant from the origin must also correspond to l G l = Page 9 of 10 0.5 and SWR = 3, and a circle centered at the origin and whose radius is the length of line segment 0L represents a locus of constant-SWR points. (Note that the SWR = 3 circle in Figure 8 shares a tangent line with the rL = 3 circle at the real axis; this relationship between SWR and :1 circles holds for all values of SWR.) Using the standing-wave circle, you can determine input impedanoes looking into any portion of a transmission line such as Figure 7’s if you know the load impedance. Figure 7, for instance, shows an input impedance Z,” to be measured at a distance l0 from the load (toward the generator). Assume that the load impedance is as given by point L in Figure 3. Then, assume that to is 0.139 wavelengths. (Again, I chose this value to avoid interpolation.) One trip - - around the Smith chart is equivalent to f, u _5. .-‘"- 5W," o-‘g‘ " 0-37 = W39 traversing one-half wavelength along a \\ Eve "'9‘ ‘ mm" genera”! standing wave, and Smith charts often include 0- to 0.5-wavelength scales around their circumferences (usually tying outside the reflection-coefficient angle scale previously discussed). Figure 8. Constant SWR circles are centered at the origin of the complex reflection-coefficient plane. The yellow scale represents per—unit wavelength movements away from the load toward a generator along a transmission line. Such a scale is show in yellow in Figure 8, where clockwise movement corresponds to movement away from the load and toward the generator (some charts also include a c0unter-clockwise scale for movement toward the load). Using that Scale, you can rotate the red vector intersecting point L clockwise for 0.139 wavelengths, ending up at the blue vector. That vector intersects the SWR = 3 circle at point I, at which you can read Figure 7’s input impedance 2,". Point | lies at the intersection of the 0.45 resistance circle and —0.5 reactanoe circle, so 2m = 0.45 -j0.5. Still going strong The Smith chart remains an invaluable aid for a variety of applications, from the design of impedance- matching networks to the determination of the feed-point impedance of an antenna based on a measurement taken at the input of a random length of transmission line (Ref. 7). Whether you are using it as a computational tool—as its inventor intended—or as the graphical interface to instrumentation or EDA software, it provides insights to circuit operation not available from the raw data that number crunching machines can produce from microwave component measurements and simulations. T&llllw References 1. “Philip Smith, Electrical Engineer," an oral history conducted in 1973 by Frank A. Polkinghorn, IEEE History Center, Rutgers University, New Brunswick, NJ. www.ieee.orglorganizationslhistory_centerl oral_histofiesltranscriptslsmith3.htrnl. 2. See, for example, "Parallel Resistance Nomograph," www.trrmorld.comlreferencelnom_ oddsendshtm. 3. Nelson, Rick, “Vector network analyzers grade microwave components,” Test & Measurement Wortd, April 2001. p. 16. www.tmwor1d.coml Page 10 oflO articles/2001m_VNAs.htm. 4. Nelson. Rick, “What are S-parameters, anyway?” Test & Measurement World. February 2001. p. 23. www.trnwond.comlartidesl2001l02_sparameters.htm. 5. Agilent Technologies provides an interactive demonstration you can download. www.tm.agilent. comldatalstatidengttmolNotesfinteradivel an-95-1Iclassesfimatch.html. 6. Ulaby, Fawwa T., Fundamentals of Applied Electromagnetics, 1999 ed.. Prentice-Hall. Upper Saddle River, NJ. 1999. 7. Straw, R. Dean. ed., "Smith Chart Calculations," Chapter 28. The ARRL Antenna Book, 18th ed., The American Radio Relay League. Newington. CT, 1997—98. Rick Nelson received a BSEE degree from Penn State University. He has six years experience designing electronic industrial-control systems. A member of the IEEE, he has served as the managing editor of EDN, and he became a senior technical editor at T&MW in 1‘ 998. E-mail: [email protected] Copyright 2001, Tesl 3. Measurement World. Published by Cahners Business Information. Newton, MA. Download a Smith chart from WebEE by clicking here 3D: I +3» LY The Complete Smith Chart Black Magic Design RADIAUX SEALED PARAMEI'ERS TDWARDTLOAD 7) l0 2 La 1.6 M l.2 LII IS 15 ' «IMHO 20 -I '6 1.2 LE 14 LB 1.8 2 0.1 0.2 -4030 D Q 15 0.4 0.6 0.0 I 0 00 20 0.0! 0. l 1 ! 9I0 II III 0.2 IJ 1.5 [.6 IT LI [.92 I 2 IJ [.1 0.05 0| 0.?! 0.4 l 0.9 0.! 0.1 0.6 0.5 0. 55 0.99 0l 0.9 0.0 0.? 0.6 I15 0.‘ 0.3 0.2 CENTER ORIGIN ...
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ECE3025_Lecture12 - 22 14‘ U SHEETS 22-142 100 SHEETS...

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