Smith Chart and its Applications

Transcription

1 International Journal Electronic Electrical Engeerg. ISSN Volume 4, Number 1 (2011), pp International Research Publication House Smith Chart its Applications Arun Kumar Tiwari Head Department, Electronics & Communication Engeerg Branch, Ganeshi Lal Bajaj Institute Technology & Management, Plot No. 2, Knowledge Park-III, Greater NOIDA , U.P., India Abstract The Smith chart one most useful graphical tools for high frequency circuit applications. The chart provides a clever way to vualize functions it contues to endure popularity decades after its origal conception. From a mamatical pot view, Smith chart simply a presentation all possible impedances with respect to coordates defed by reflection coefficient or it can be defed mamatically as one port scatterg parameter S11. The doma defition reflection coefficient a circle radius 1 plane. Th also doma Smith chart. A Smith chart a circular plot with lot terlaced circles on it; when used correctly, matchg impedances with apparent complicated structures can be made without any computations. The only effort required readg followg values along circles. Types Smith Chart There are maly two kds Smith chart, impedance or Z-Smith chart or one admittance or Y-Smith chart. The superposition Z or Y-Smith chart gives Z-Y Smith chart or complete smith chart. In matchg or designg circuits it convenient to overlay impedance Z-Smith Chart admittance Y- Smith Chart called Impedance Admittance Z-Y Smith Chart which basically superimposition dividual Z-Smith Chart Y-Smith Chart. The figure below shows a Z-Y Smith Chart. Impedance admittance charts are used to calculate component values needed for device different parts impedance matchg circuit.

2 76 Arun Kumar Tiwari Figure B1: A Complete Smith chart. Development a Smith Chart A Smith chart developed by examg load where impedance must be matched. Instead considerg its impedance directly, you express its reflection coefficient L, which used to characterize a load (such as admittance, ga, trans conductance). The L more useful when dealg with RF frequencies. We know reflection coefficient defed as ratio between reflected voltage wave cident voltage wave: Figure B2: Impedance at load. The amount reflected signal from load dependent on degree mmatch between source impedance load impedance. Its expression has been defed as follows:

3 Smith Chart its Applications 77 Because impedances are numbers, reflection coefficient willl be a number as well. In order to reduce number unknown parameters, it useful to freeze ones that appear ten are common application. Here Z o ( charactertic impedance) ten a constant a real dustry normalized value, such as 50 impedance by:, 75, 1000, 600. We can n defe a normalized load With th simplification, we can rewrite reflection coefficient formula as: Here we can seee direct relationship between load impedance its reflection coefficient. Unfortunately, nature relation not useful practically, so we can use Smith chart as a type graphical representation above equation. To build chart, equation must be rewritten to extract stard geometrical figures (like circles or stray les). First, equation B.3 reversed to give By settg real parts imagary parts equation B.5 equal, we obta two dependent, new relationships:

4 78 Arun Kumar Tiwari Equation n manipulated by developg equations B.8 through B.13 to fal equation, B. 14. Th equation a relationship form a parametric equation (x-a) 2 + (y-b) 2 = R 2 plane ( r, i) a circle centered at coordates (r/r+ +1, 0) havg a radius 1/ /1+ +r. Figure B3 given below gives depth details. Figure B3: The pots situated on a circle are all impedances characterized by a same real impedance part value. For example, circle, R = 1, centered at coordates (0.5, 0) has a radius 0.5. It cludes pot (0, 0), which reflection zero pot ( load matched with charactertic impedance). A short circuit, as a load, presents a circle centered at coordate (0, 0) has a radius 1. For an open-circuit load, circle degenerates to a sgle pot (centered at 1, 0 with a radius 0). Th corresponds to a maximum reflection coefficient 1, at which entire cident wave reflected totally.

5 Smith Chart its Applications 79 Movg on, we use equations B.15 through B.18 to furr develop equation B.7 to anor parametric equation. Th results equation B.19. Aga, B. 19 a parametric equation type (x-a) 2 + (y-b) 2 = R 2 plane ( r, i) a circle centered at coordates (1, 1/x) havg a radius 1/ /x. Figure B.4 given below gives depth details Figure B4: The pots situated on a circle are all impedances characterized by a same imagary impedance part value x. For example, circle x = 1 centered at coordate (1, 1) has a radius 1. All circles (constant x) clude pot ( 1, 0). Differg with real part circles, x can be positive or negative. Th explas duplicate mirrored circles at bottom side plane. All circle centers are placed on vertical ax, tersectg pot 1. To complete our Smith chart, we superimpose two circles' families. It can n be seen that all circles one family will tersect all circles or family. Knowg impedance, form r + jx, correspondg reflection

6 80 Arun Kumar Tiwari coefficient can be determed. It only necessary to fd tersection pot two circles correspondg to values r x. The reverse operation also possible. Knowg reflection coefficient, fd two circles tersectg at that pot read correspondg values r x on circles. The procedure for th as follows: Determe impedance as a spot on Smith chart. Fd reflection coefficient ( ) for impedance. Havg charactertic impedance, fd impedance. Convert impedance to admittance. Fd equivalent impedance. Fd component values for wanted reflection coefficient. Workg with Admittance The Smith chart built by considerg impedance (restor reactance). Once Smith chart built, it can be used to analyze see parameters both series parallel worlds. Addg elements a series straightforward. New elements can be added ir effects determed by simply movg along circle to ir respective values. However, summg elements parallel anor matter. Th requires considerg additional parameters. Often it easier to work with parallell elements admittance world. We know that, by defition, Y = 1/Z Z = 1/Y. The admittance expressed mhos or -1 ( earlier times it was expressed as Siemens or S). And, as Z, Y must also be. Therefore, Y = G + jb (B. 20), where G called "conductance" B "susceptance" element. It's important to exerce caution, though. By followg logical assumption, we can conclude that G = 1/R B = 1/X. Th, however, not case. If th assumption used, results willl be correct. When workg with admittance, first thg that we must do normalize y = Y/YY o. Th results y = g + jb. So, what happens to reflection coefficient? By workg through followg: It turns out that expression for G opposite, sign, z, (y) = - (z). If we know z, we can vert signs fd a pot situated at same dtance from (0, 0), but opposite direction. Th same result can be obtaed by rotatg an angle 180 around center pot (see Figure B.5).

7 Smith Chart its Applications 81 Figure B5: Results 180 rotation. Of course, while Z 1/ /Z do represent same component, new pot appears as a different impedance ( new value has a different pot Smith chart a different reflection value, so forth). Th occurs because plot an impedance plot. But new pot, fact, an admittance. Therefore, value read on chart has to be read as mhos. Although th method sufficient for makg conversions, it doesn't work for determg circuit resolution when dealg with elements parallel. The Admittance Smith Chart In previous dcussion, we saw thatt every pot on impedance Smith chart can be converted to its admittance counterpart by takg a 180 rotation around orig plane. Thus, an admittance Smith chart can be obtaed by rotatg whole impedance Smith chart by 180. Th extremely convenient, as it elimates necessity buildg anor chart. The tersectg pot all circles (constant conductances constant susceptances) at pot (-1, 0) automatically. With that plot, addg elements parallel also becomes easier. Mamatically, construction admittance Smith chart created by: n, reversg equation:

8 82 Arun Kumar Tiwari Next, by settg real imagary parts equation B.24 equal, we obta two new, dependent relationships: By developg equation B.25, we get followg, which aga a parametric equation type (x-a) 2 + (y-b) 2 = R 2 (equation B.33) plane ( r, i) a circle with its coordates centered at (-g/g+1, 0) havg a radius 1/(1+g). Furrmore, by developg equation B.26, we show that: which aga a parametric equation type (x-a) 2 + (y-b) 2 = R 2 (equation B.38).

9 Smith Chart its Applications 83 Facts about Smith Chart One circuit SMITH chart only half a wavelength: We remember that SMITH chart a polar plot reflection coefficient, which represents ratio amplitudes backward forward waves. Image forward wave gog past you to a load or reflector, n travelg back aga to you as a reflected wave. The total phase shift gog re comg back twice phase shift just gog re. Therefore, re a full 360 degrees or 2 pi radians phase shift for reflections from a load HALF a wavelength away. If you now move reference plane a furr HALF wavelength away from load, re an additional 360 degrees or 2 pi radians phase shift, representg a furr complete circuit reflection (SMITH) chart. Thus for a load a whole wavelength away re a phase shift 720 degrees or 4 pi radians, as round trip 2 whole wavelengths. Thus movg back ONE whole wavelength from load, round trip dtance actually creasg by TWO whole wavelengths, so SMITH chart circumnavigated twice. Smith Chart: graphical representation Mamatical Bas Smith Chart

10 84 Arun Kumar Tiwari Smith Chart: Impedance Coordates Smith Chart: Admittance Coordates Smith Chart: Constant Impedance Phase Angle Circles

11 Smith Chart its Applications 85 Smith Chart: Constant VSWR circles Smith Chart: Constant Impedance Magnitude Circles Smith Chart: for Multiplication, Divion, Squares, Square Roots Unary Operators 2 squares a square roots a tangents tan Ө cotangents cot Ө verse tangents tan-1 a verse cotangents cot-1 a

12 86 Arun Kumar Tiwari Bary Operators multiplication a b divion c/a geometric mean ab Smith Chart: A Nomogram for Mat th Calculations The stereographic representations plane are obtaed by real constructions. Instead, analogue representation trigonometric functions could be used, which tangent an angle pot tersection radius unit circle prolonged to tersect vertical tangent at x= =1. The basic representation holds that x-projection radius, that y-projection. The equivalent th mappg needs a angle to work with, ought to correspond to polar stereographic projection rar than central stereographic projection. In former case, modulus projection rar tan Θ / 2 r than tan Θ, so suggested mappg W = tan Θ / 2. But Constant after troducg abbreviation.

13 Smith Chart its Applications 87 Figure: contours generate a useful nomogram, Smith Chart. Figure B6: Contour plots for th mappg constitute nomograms which, after havg been labeled drawn arttically, are known as Smith Charts. They are considerable use transmsion le ory, are used without factor i. One great advantage th representation thatt whole right half-plane, circle, imagary ax takg up residence on its circumference. The real ax maps to real ax, but given that fity maps to 1, whole coordate grid les parallel to real to imagary ax ends up as two families mutually orthogonal circles, alll passg through one whose numbers have positive real parts, mapped to unit 1. Movg along Smith chart

14 88 Arun Kumar Tiwari Figure B 7: Movement along Smith chart. Problem solvg usg Smith chart Given below are basic Smith Chart techniques for loss-less transmsion les: Given Z(, Fd Γ( Given Γ(, Fd Z( Given ΓR ZR, Fd Γ( Z( Given ΓR ZR, Fd Voltage Stg Wave Ratio (VSWR) Given Z(, Fd Y( Its use for solvg le admittances Its use fdg Q-factor Given Z(, Fd Γ( 1. Normalize impedance Z( R X z ( = = + j = r + Zo Zo Zo jx 2. Fd circle constant normalized restance r 3. Fd arc constant normalized reactance x 4. The tersection two curves dicates reflection coefficient plane. The chart provides directly magnitude phase angle Γ(. Example: Fd Γ(, Given Z d =25 + j100 with Z0 =50 Ω

15 Smith Chart its Applications 89 Given Γ(, Fd Z( 1. Determe pot representg given reflection coefficient G( on chart. 2. Read values normalized restance r normalized reactance x that correspond to reflection coefficient pot. 3. The normalized impedance z( = r + j x actual impedance Z( = Zo. z( = Zo.(r + jx) = Zo.r + j Zo.x Given ΓR ZR, Fd Γ( Z( The magnitude reflection coefficient constant along a loss-less transmsion le termated by a specified load, sce Γ( = Γ exp( j2β = Γ R R Therefore, on plane, a circle with center at orig radius R represents all possible reflection coefficients found along transmsion le. When circle constant magnitude reflection coefficient drawn on Smith chart, one can determe values le impedance at any location. The graphical step-by-step procedure : 1. Identify load reflection coefficient R normalized load impedance ZR on Smith chart. 2. Draw The magnitude reflection coefficient constant along a circle constant reflection coefficient amplitude Γ( = ΓR.

16 90 Arun Kumar Tiwari 3. Startg from pot representg load, travel on circle 2Π Θ = 2 βd = 2 d λ The new location on chart corresponds to location d on transmsion le. Here, values Γ( Z( can be read from chart as before. Example: Given Z R = 25 + j100ω with Zo = 50Ω fd Z( Γ( for d = 0.18λ Given ΓR ZR, Fd Voltage Stg Wave Ratio (VSWR) The Voltage stg Wave Ratio or VSWR defed as VSWR = V V max m 1+ Γ = 1 Γ R R The normalized impedance at a maximum location stg wave pattern given by 1+ Γ( d max ) 1+ ΓR z( d max ) = = = VSWR! 1 Γ( d ) 1 Γ max Th quantity always real 1. The VSWR simply obtaed on Smith chart, by readg value (real) normalized impedance, at location dmax where real positive. R

17 Smith Chart its Applications 91 The graphical step-by-step procedure : 1. Identify load reflection coefficient R normalized load impedance ZR on Smith chart. 2. Draw circle constant reflection coefficient amplitude Γ( = ΓR. 3. Fd tersection th circle with real positive ax for reflection coefficient (correspondg to transmsion le location dmax). 4. A circle constant normalized restance will also tersect th pot. Read or terpolate value normalized restance to determe VSWR. Example Fd VSWR for ZR1 = 25 + j100 Ω ; ZR2 = 25 j100 Ω (Zo = 50 Ω) Given Z(, Fd Y( The normalized impedance admittance are defed as 1+ Γ( z( = 1 Γ( 1 Γ( y( = 1+ Γ( s ce, λ Γ( d + ) = Γ( 4 λ 1+ Γ( d + ) λ 4 1 Γ( z( d + ) = = = y( 4 λ 1+ Γ( 1 Γ( d + ) 4

18 92 Arun Kumar Tiwari It important to note equality z ( d + λ ) = y( which valid for normalized 4 impedance admittance. The actual values are given by λ λ Z( d + ) = Zo. z( d + ) 4 4 y( Y ( = Yo. y( = Zo Where, Y0=1 /Z0 charactertic admittance transmsion le. The graphical step-by-step procedure : 1. Identify load reflection coefficient R normalized load impedance ZR on Smith chart. 2. Draw circle constant reflection coefficient amplitude Γ( = ΓR. 3. The normalized admittance located at a pot on circle constant Γ which diametrically opposite to normalized impedance Example Given ZR = 25 + j100 Ω with Zo = 50 Ω, fd YR Calculation le admittances By shiftg space reference to admittance location, one can move on chart just readg numerical values as representg admittances. Let s review impedance-admittance termology: Impedance = Restance + j Reactance, Z = R + jx Admittance = Conductance + j Susceptance, Y = G + jb

19 Smith Chart its Applications 93 On impedance chart, correct reflection coefficient always represented by vector correspondg to normalized impedance. Charts specifically prepared for admittances are modified to give correct reflection coefficient correspondence admittance. Sce related impedance admittance are on opposite sides same Smith chart, imagary parts always have different sign. Therefore, a positive (ductive) reactance corresponds to a negative (ductive) susceptance, while a negative (capacitive)reactance corresponds to a positive (capacitive) susceptance. Numerically, we have z = r + jx y = g + r jx y = ( r + jx)( r g = r 2 1 jb = r + jx r + x 2, b = r r jx = 2 jx) r + x x + x 2 2 Calculation Q-factor After havg located impedance Q-factor can be directly read f from Smith Chart as mentioned below: If usg an impedance Z-Smith Chart Q n = x /r, where, Q n nodal quality factor Z=r+jx normalized impedance. 2

20 94 Arun Kumar Tiwari In order to derive th consider, So by dividg se two equations we get Q n. If usg admittance Y-Smith Chart Q n = b /g where, Q n nodal quality factor Y=g+jb normalized admittance. In order to derive th consider, So by dividg se two equations we get Q n. References [1] Mysteries Smith Chart, Stephen D. Stearns, K6OIK,Chief Technologt, TRW Firestorm Wireless Communication Products, stearns@ieee.org [2] Electronic Applications Smith Chart: In Waveguide, Circuit, Component Analys, by Phillip H. Smith. Hardcover pages 2nd edition (October 2000 [3] Advanced Automated Smith Chart, Version 3.0: Stware User's Manual, by Leonard Schwab (March 1998). [4] Electronic Applications Of The Smith Chart: In Waveguide, Circuit, And Component Analys by Phillip H. Smith [5] "Smith chart Tutorial," by Dr. Russell P. Jedlicka, Klipsch School Electrical Computer Engeerg, New Mexico State University, September 2002 Internet URL s [6] [7] [8]