Mysteries of the Smith Chart

Transcription

1 Mysteries of the Smith Chart Transmission Lines, Impedance Matching, and Little Known Facts Stephen D. Stearns, K6OIK Chief Technologist TRW Firestorm Wireless Communication Products VG 1 PSA /21/01 Pacificon 01

2 Outline Transmission Line Theory Historical development Heaviside s rewrite of Maxwell s theory, Telegrapher s equations, Impedance, reflection coefficient, SWR, phase constant, and velocity factor Special facts for λ/2, λ/4, and λ/8 lossless lines The Smith Chart Bilinear complex functions Impedance and admittance coordinates (circles, circles, and more circles) Impedance Matching Why match? Impedance matching vs. conjugate impedance matching Single frequency matching Multiple-frequency and broadband matching VG 2 PSA /21/01 Pacificon 01

3 Part 1: Transmission Line Theory VG 3 PSA /21/01 Pacificon 01

4 Key Dates in Electrical Transmission 18s Magnetic telegraphs - Gauss, Henry 1839 Electromagnetic telegraph - Wheatstone & Cook 1844 Telegraph in America - Morse 18s Thousands of miles of telegraph line U.S. and Europe mile cable under English Channel 1855 Distributed analysis of transmission line - Lord Kelvin 1858 Transatlantic cable, project delayed by civil war 1873 Theory of electrodynamics - Maxwell 1876 Invention of telephone - Bell 18s Vectors, vector calculus, reformulation of Maxwell s theory, transmission line theory - Heaviside 1886 Experimental confirmation of Maxwell s Theory - Hertz 1937 Early Smith Chart, published 1939 and Smith VG 4 PSA /21/01 Pacificon 01

5 Numbers to Remember! ,792, VG 5 PSA /21/01 Pacificon 01

6 Heaviside s Vector Formulation of Maxwell s Theory B E = t D H= J+ t D = ρ B= 0 D B = εe = µ H And God said, Let there be light; and there was light. Genesis 1:3 VG 6 PSA /21/01 Pacificon 01

7 Frequency Domain or Phasor Form E= jωµ H H= ( σ + jωε) E E= 0 H= 0 VG 7 PSA /21/01 Pacificon 01

8 Heaviside s Telegrapher s Equations V(x) Uniform transmission line I(x) Equivalent circuit of infinitesimal segment R x L x G x C x dv dx di dx = ( R+ jωl) I( x) = ( G+ jωc) V( x) VG 8 PSA /21/01 Pacificon 01

9 Transmission Line Solution TEM Waves Traveling wave γ Vx ( )=V o e Ix ( ) = x Vx ( ) Z Propagation constant γ = α + jβ = ( R+ jωl)( G+ jωc) o Characteristic impedance Z o = R+ G+ jωl jωc VG 9 PSA /21/01 Pacificon 01

10 Notations Real Parameters R = L = G = C = α = β = λ = v f = X = B = Series resistance per unit length (Ohms/meter) Series inductance per unit length (Henries/meter) Shunt conductance per unit length (Siemens/meter) Shunt capacitance per unit length (Farads/meter) Attenuation constant (nepers/meter) Phase constant (radians/meter) Wavelength (meters) Velocity factor (dimensionless) Reactance (Ohms) Susceptance (Siemens) s = Standing wave radio (dimensionless) VG PSA /21/01 Pacificon 01

11 Notations (Cont d) Complex Parameters Z = R + jx = impedance (Ohms) Z L = Load impedance (Ohms) Z i = Input impedance (Ohms) Z 0 = Characteristic impedance (Ohms) z = Z/Z 0 = r + jx = normalized impedance (dimensionless) Y = G + jb = admittance (Siemens) y = Y/Y 0 = g + jb = normalized admittance (dimensionless) Γ = Γ r + j Γ i = complex reflection coefficient (dimensionless) γ = α + jβ = propagation constant (inverse meters) VG 11 PSA /21/01 Pacificon 01

12 Transmission Line Parameters Physical Dimensions and Material Properties a dielectric (µ, ε, σ) d dielectric (µ, ε, σ) b a c Parameter Coax Twinlead a R Ω/m πδσ c a b 1 πδσ a c VG 12 PSA /21/01 L H/m G S/m C F/m Where skin depth is µ δ 1 1 ln b 2π a + 2 a + b δ = 2πσ ln b a 2πε ln b a 1 πµσ f c For copper µ δ 1 d + π cosh 2a 2a πσ d cosh 1 2a πε d cosh 1 2a σ = c S/m δ = 85. mm at Hz 6.6 µ m at 0 MHz Pacificon 01

13 Round Open-Wire Transmission Line Formulas s d Approximate formula Widely published by ARRL and others Accurate only for large spacings: s/d > 3 or large impedances: Z 0 > several hundred Exact formula Accurate for all spacings and impedances VG 13 PSA /21/01 Pacificon 01

14 Comparison of Impedance Formulas Round Open-Wire Line VG 14 PSA /21/01 Pacificon 01

15 K6OIK Square Open-Wire Transmission Line Formula Excellent approximation in the range of practical interest Accurate for small spacings: 1 < s/w < 3 or small impedances: 0 < Z 0 < several hundred s w VG 15 PSA /21/01 Pacificon 01

16 Round vs Square Open-Wire Lines VG 16 PSA /21/01 Pacificon 01

17 Optimal Characteristic Impedances Coax For minimum loss Z o = 77Ω For maximum breakdown voltage For minimum temperature rise Z o = Ω Z o = Ω Z o = Ω has no special significance VG 17 PSA /21/01 Pacificon 01

18 Reflection Coefficient and Impedance Relation at a Terminal Plane Terminal Plane Definition Ζ O Ζ L Γ L Γ= Z Z L L + Z Z o o z = 1 z + 1 Inverse z = Γ Γ For every terminal plane, the complex load impedance and complex reflection coefficient seen to the right give the same information for that terminal plane Question: How do Γ and z change as the terminal plane moves? VG 18 PSA /21/01 Pacificon 01

19 Relations Between Two Terminal Planes Input Terminal Plane Ζ i Γ i Ζ O Output Terminal Plane Ζ L Γ L Impedance relation z i = z + jtan βl L 1 + jz L tan βl Cross relations z i 1 = + Γ 1 Γ L L e e j2βl j2βl Reflection coefficient relation Γ i = Γ L e j2βl z L 1 = + 1 Γ Γ i i e e j2βl j2βl VG 19 PSA /21/01 Pacificon 01

20 Velocity Factor Wavelength λ free space = c f λ Velocity factor actual = v f v f v = = λ c λ actual free space VG PSA /21/01 Pacificon 01

21 How To Measure Velocity Factor of a Line (One Way To Do It) Antenna Analyzer Known length open circuit Antenna Analyzer Same length short circuit v f = 2πfl 1 c Z 1 cot Z open short VG 21 PSA /21/01 Pacificon 01

22 Phase Constant 2π 2πf β = = λ v c actual f radians/meter Phase constant β and velocity factor v f give equivalent information Both can be calculated from line dimensions and material properties Best to measure! β = Im ( R+ jωl)( G+ jωc) VG 22 PSA /21/01 Pacificon 01

23 How to Measure Complex Z o of A Line (One Way to Do It) Antenna Analyzer Unknown or arbitrary length open circuit Antenna Analyzer Same length short circuit Zo = Zopen Zshort Geometric mean of two complex numbers Calculation is trivial in polar form on Smith Chart VG 23 PSA /21/01 Pacificon 01

24 What Special Lengths of Lossless Line Do Half wavelength, l = _/2 Z i = Z L Quarter wavelength, l = _/4 Z i = Z Z 2 o L Eighth wavelength, l = _/8 Zi = Zo if ZL and Zo are real (resistive) VG 24 PSA /21/01 Pacificon 01

25 Standing Wave Ratio Easy to remember from s 1 = + 1 Γ Γ z 1 = + 1 Γ Γ Γ s = 1 s + 1 Γ z = 1 z + 1 VG 25 PSA /21/01 Pacificon 01

26 Part 2: The Smith Chart VG 26 PSA /21/01 Pacificon 01

27 VG 27 PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ± 1 > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Pacificon 01

28 Complex Functions z i w i y (x, y) Complex Number z = x + jy Complex Number w = u + jv v (u, v) x z r f( ) u w r Basic types of complex functions Global Properties Linear lines map to lines Bilinear circles map to circles Local Properties Conformal right angles map to right angles VG 28 PSA /21/01 Pacificon 01

29 Mathematical Basis of the Smith Chart z Γ= z A bilinear conformal complex function u r 1 + jv = ( - )+ ( r+ 1)+ jx jx x v r u Right Half Z Plane Interior Unit Circle Γ Plane VG 29 PSA /21/01 Pacificon 01

30 Smith Chart: Impedance Coordinates VG PSA /21/ r=0 r Series Resistance Γ i x=1 x=0 x= CAPACITIVE REACTANCE COMPONENT (-jx/zo), ORINDUCTIVESUSCEPTANCE(-jB/Yo) x= < WAVELENGTHS TOWARD LOAD < ± 1 r=1 r=2 x Series Reactance 0 > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), ORCAPACITIVESUSCEPTANCE(+jB/Yo) Inductive Reactance (Positive) x=0 RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) Capacitive Reactance (Negative) x= r= r=2 r= Pacificon ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES IN DEGREES ANGLE OF REFLECTION COEFFICIENT Γ r

31 2 8 < WAVELENGTHS TOWARD LOAD < Smith Chart: Admittance Coordinates VG 31 PSA /21/01 Pacificon g=0 Shunt Conductance Γ i b=1 b=0 b= g=1 g=2 Shunt Susceptance b= ± b= INDUCTIVE REACTANCE COMPONENT (+jx/zo), ORCAPACITIVESUSCEPTANCE(+jB/Yo) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES > WAVELENGTHS TOWARD GENERATOR > ANGLE OF REFLECTION COEFFICIENT IN DEGREES 9 RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) CAPACITIVE REACTANCE COMPONENT (-jx/zo), ORINDUCTIVESUSCEPTANCE(-jB/Yo) b g g= Inductive Susceptance (Negative) Capacitive Susceptance (Positive) g=1 b=-1 g= Γ r

32 Smith Chart: Constant SWR Circles VG 32 PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), ORINDUCTIVESUSCEPTANCE(-jB/Yo) -1 9 < WAVELENGTHS TOWARD LOAD < ± 1 > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), ORCAPACITIVESUSCEPTANCE(+jB/Yo) Γi RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES IN DEGREES ANGLE OF REFLECTION COEFFICIENT Γr Pacificon 01

33 Smith Chart: Constant Impedance Magnitude Circles Γ i x 1 2 r VG 33 PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), ORINDUCTIVESUSCEPTANCE(-jB/Yo) -1 9 < WAVELENGTHS TOWARD LOAD < RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ± 1 > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), ORCAPACITIVESUSCEPTANCE(+jB/Yo) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES 8 1 Γ r Pacificon

34 Smith Chart: Constant Impedance Phase Angle Circles Γi x 1 0 VG 34 PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), ORINDUCTIVESUSCEPTANCE(-jB/Yo) -1 9 < WAVELENGTHS TOWARD LOAD < RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ± > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), ORCAPACITIVESUSCEPTANCE(+jB/Yo) Γr Pacificon ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES IN DEGREES ANGLE OF REFLECTION COEFFICIENT

35 Smith Chart: Multiplication, Division, Squares, and Square Roots Unary Operators squares a 2 square roots tangents a tan θ cotangents cot θ inverse tangents tan -1 a Inverse cotangents cot -1 a Binary Operators multiplication X division geometric mean a b c/a ab VG 35 PSA /21/01 Pacificon 01

36 Smith Chart: A Nomogram for Math Calculations Γ i VG 36 PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), ORINDUCTIVESUSCEPTANCE(-jB/Yo) < WAVELENGTHS TOWARD LOAD < 6 4 ± ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES IN DEGREES RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) b= > WAVELENGTHS TOWARD GENERATOR > c=2 ANGLE OF REFLECTION COEFFICIENT INDUCTIVE REACTANCE COMPONENT (+jx/zo), ORCAPACITIVESUSCEPTANCE(+jB/Yo) c=14-2 a=4 Γ r Pacificon 01

37 Part 3: Impedance Matching VG 37 PSA /21/01 Pacificon 01

38 Impedance Matching Categories Single frequency matching Manual synthesis using Smith Chart Eight canonical networks Lumped elements Series and parallel stubs Transmission line sections Multiple frequency matching Ladder networks Multiple stubs Multiple line sections Broadband matching Maximize SWR bandwidth Software for computer-aided manual design Software for network optimization Smith Chart used for visualization only VG 38 PSA /21/01 Pacificon 01

39 Bandwidth Classifications Fractional Bandwidth Narrowband < % Moderate band % to % Broadband > % VG 39 PSA /21/01 Pacificon 01

40 Two Kinds of Matching Conjugate Matching Load Matching Ζ S + - Ζ L Ζ O Ζ L Best use at source (transmitter) Maximizes power delivery to the load Does not minimize reflections unless Z s is real Normally done by the transmitter manufacturer at the circuit design level Ideally Z s (ext) = Ω VG PSA /21/01 Z s = Z L * Z L = Z O Best used at ends of transmission lines Minimizes reflections Does not maximize delivered power unless Z 0 is real Pacificon 01

41 Where Should Matching Network Go? Poor Setup High to Low SWR High SWR Tx Good Setup Antenna Tuner Long Transmission Line Perfect 1:1 Low SWR Low SWR Tx Antenna Tuner Very Low-Loss Line Matching Network Best Setup Insertion Loss x db Low SWR Low SWR Tx Matching Network VG 41 PSA /21/01 Pacificon 01

42 Necessary Test Equipment Antenna analyzer Autek CIA MFJ Noise bridge (less accurate) Network analyzer (more accurate) Hewlett-Packard VG 42 PSA /21/01 Pacificon 01

43 Matching Network Design Recipe 1. Measure transmission line parameters Z o, v f 2. Measure antenna feedpoint impedance across band(s) of interest 3. Measure or calculate impedance across band(s) of interest at network insertion point 4. Narrowband match: Select appropriate lossless L network, 2 or 4 choices Select lumped elements vs stubs Calculate component values Calculate SWR and SWR bandwidth Build and test 5. Broadband match: Use design software - winsmith or equivalent Design n-stage lossless ladder network Select lumped elements vs stubs Calculate component values Calculate SWR and SWR bandwidth VG 43 PSA /21/01 Pacificon 01

44 How to Measure Antenna Feedpoint Impedance Measure impedance through known line Divide measure impedance by Z o Plot impedance point on Smith Chart Move counter clockwise on chart by electrical length of line Read coordinate values from chart Multiply result by Z o VG 44 PSA /21/01 Pacificon 01

45 Smith Chart: Effect of Adding Series Reactances or Shunt Susceptances Γi Γ r VG 45 PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) < WAVELENGTHS TOWARD LOAD < 6 4 RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ± ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) b -x +x +b Pacificon 01

46 VG 46 PSA /21/ Z 0 Z CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) Forbidden Area ± (a) > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Pacificon 01

47 VG 47 PSA /21/ Z 0 Z CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < ± 1 > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) Forbidden Area RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) (b) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Pacificon 01

48 VG 48 PSA /21/ Z 0 Z CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ± (c) > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) Forbidden Area ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Pacificon 01

49 VG 49 PSA /21/ Z 0 Z CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ± (d) > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) Forbidden Area ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Pacificon 01

50 VG PSA /21/ Z 0 Z CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ± Forbidden Area (e) > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Pacificon 01

51 VG 51 PSA /21/ Z 0 Z CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ± 1 > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) Forbidden Area (f) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Pacificon 01

52 VG 52 PSA /21/ Z 0 Z CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) ± Forbidden Area (g) > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Pacificon 01

53 VG 53 PSA /21/ Z 0 Z CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < ± 1 > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) Forbidden Area RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) (h) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Pacificon 01

54 Matching: Four L-Networks Using Lumped Elements Γi VG 54 PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) < WAVELENGTHS TOWARD LOAD < 6 4 ± ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) Z L Z L RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) Finish Start Z L Γ r Z L Pacificon 01

55 Matching: Four L-Networks Using Stubs Γi VG 55 PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) -1 9 < WAVELENGTHS TOWARD LOAD < ± 1 > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) Z L Z L RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) Finish Start ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Z L Γ r Z L Pacificon 01

56 Reactance & Susceptance of Lumped Elements Inductors Capacitors X = 2π L fl B L = 1 2πfL X C = 1 2πfC BC = 2πfC VG 56 PSA /21/01 Pacificon 01

57 Reactance & Susceptance of Stubs Assuming Zo is real, then Shorted Stubs X shorted = Z o lf tan 2π vc f B shorted = Z o 1 2πlf tan vc f Open Stubs X open = Zo lf tan 2π vc f B open = 1 2πlf tan Z vc o f VG 57 PSA /21/01 Pacificon 01

58 Broadband Matching Network Design Recipe Using 4-Element π-resonant and T-Resonant Networks Putting network insertion point close to load (antenna) gives greatest SWR bandwidth Step 1: Using an L-network, move the midband impedance point to prime point A or B. Bandwidth will be maximized if the minimum reactance or susceptance L-network is chosen in this step. Step 2: Wrap the impedance locus into the SWR circle by adding a series or parallel resonant circuit as required to complete the π-resonant or T-resonant network VG 58 PSA /21/01 Pacificon 01

59 VG 59 PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVESUSCEPTANCE(-jB/Yo) -1 9 < WAVELENGTHS TOWARD LOAD < ± 1 > WAVELENGTHS TOWARD GENERATOR > INDUCTIVE REACTANCE COMPONENT (+jx/zo), ORCAPACITIVESUSCEPTANCE(+jB/Yo) Resonant and T-Resonant Network For Moderate and Broadband Matching Γ i Z L Z L Z L Z L RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) SWR = 1.5 f l A B f u f mid ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES ANGLE OF REFLECTION COEFFICIENT IN DEGREES Γ r Z L Z L Z L Z L Pacificon 01

60 Caron s Example : VHF Folded Blade Antenna VG PSA /21/ CAPACITIVE REACTANCE COMPONENT (-jx/zo), ORINDUCTIVESUSCEPTANCE(-jB/Yo) < WAVELENGTHS TOWARD LOAD < 6 4 ± ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES IN DEGREES RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) > WAVELENGTHS TOWARD GENERATOR > ANGLE OF REFLECTION COEFFICIENT INDUCTIVE REACTANCE COMPONENT (+jx/zo), ORCAPACITIVESUSCEPTANCE(+jB/Yo) Pacificon 01

61 Caron s Matching Solution VG 61 PSA /21/ CAPACITIVE REACTANCECOMPONENT (-jx/zo), ORINDUCTIVESUSCEPTANCE(-jB/Yo) -1 9 < WAVELENGTHS TOWARD LOAD < ± 1 > WAVELENGTHS TOWARD GENERATOR > 1 RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) INDUCTIVE REACTANCE COMPONENT (+jx/zo), ORCAPACITIVESUSCEPTANCE(+jB/Yo) Pacificon ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES IN DEGREES ANGLE OF REFLECTION COEFFICIENT Max VSWR=7

62 WinSmith Display Max VSWR=1.59 VG 62 PSA /21/01 Pacificon 01

63 ARRL Radio Designer VG 63 PSA /21/01 Pacificon 01

64 ARRL Radio Designer VG 64 PSA /21/01 Pacificon 01

65 Caron s Example 11: Long Wire Receiving Antenna VG 65 PSA /21/01 Pacificon 01

66 Caron s Solution Max VSWR=6.2 VG 66 PSA /21/01 Pacificon 01

67 Caron s Solution Max VSWR=6.2 VG 67 PSA /21/01 Pacificon 01

68 Eagleware s Solution 1 Max VSWR=5 VG 68 PSA /21/01 Pacificon 01

69 Eagleware s Solution 2 Max VSWR=4.77 VG 69 PSA /21/01 Pacificon 01

70 Eagleware s Solution 3 Max VSWR=6.59 VG PSA /21/01 Pacificon 01

71 Eagleware s Solution 4: Fano Limit Max VSWR=3.42 VG 71 PSA /21/01 Pacificon 01

72 GM3HAT 4-Band Dipole of Delight VG 72 PSA /21/01 Pacificon 01

73 GM3HAT Feedpoint Impedance At Five Harmonic Frequencies VSWR=12 VSWR=2 VG 73 PSA /21/01 Pacificon 01

74 K6OIK m Suboctave Band Matching Network 1Ω 3.5 MHz 5.9 µh 25Ω 3.5 MHz 18.8 µh Z ant VG 74 PSA /21/01 Pacificon 01

75 K6OIK s Multiband Match Performance VSWR=6 VG 75 PSA /21/01 Pacificon 01

76 Software for Smith Charting and Network Design Smith Chart Analysis & Display MicroSmith 2.3, ARRL, 1992, $39 A primitive DOS program. winsmith, Noble Publishing, 1995, $79 Written by Eagleware. Easy to use. Restricted to ladder networks. Doesn t have series stubs. Lacks an optimizer. Matching Network Optimization & Synthesis ARRL Radio Designer 1.5, ARRL, 1995, $1 No longer sold. ARD s optimizer works with Serenade netlists and handles more variables than Serenade SV s optimizer. Serenade SV (student version), Ansoft, 00, $0 (free) Download (about Mbytes) from: Advanced Automated Smith Chart, Artech House, 1998, $395 =MATCH=, Eagleware, $699 (requires GENESYS Basic, $1997) Harmonica Linear Design Suite, Ansoft, 00, $60 VG 76 PSA /21/01 Pacificon 01

77 References: Articles Robert L. Thomas, Broadband Impedance Matching in High-Q Networks, EDN, pp , December, Neal C. Silence, The Smith Chart and Its Usage in RF Design, RF Design, pp , April Thomas R. Cuthbert, Jr., Broadband Impedance Matching Methods, RF Design, pp , August Thomas R. Cuthbert, Jr., Broadband Impedance Matching - Fast and Simple, RF Design, pp. 38, November William E. Sabin, Broadband HF Matching with ARRL Radio Designer, QST, pp , August William E. Sabin, ARRL Radio Designer and the Circles Utility, Part 1: Smith Chart Basics, QEX, pp. 3 9, Sept/Oct William E. Sabin, ARRL Radio Designer and the Circles Utility, Part 2: Small-Signal Amplifier Design, QEX, pp. 3 11, Nov/Dec VG 77 PSA /21/01 Pacificon 01

78 References: Articles (Cont d) Steve Sparks, A Practical Amateur Application of the Smith Chart, Communications Quarterly, pp. 2 6, Summer This article contains serious Smith charting errors. See the comments by Garry Shapiro, NI6T, Communications Quarterly, p. 3, Fall K.C. Chan and A. Harter, Impedance Matching and the Smith Chart The Fundamentals, RF Design, pp , July 00. VG 78 PSA /21/01 Pacificon 01

79 References: Books Robert A. Chipman, Transmission Lines, Schaum Outline Series, McGraw-Hill, Basic, mathematical. A classic, but out-of-print. Robert L. Thomas, A Practical Introduction to Impedance Matching, Artech House, 1976, ISBN Intermediate, mathematical. A nice treatment of 4-element networks for wideband matching. Pieter L. D. Abrie, The Design of Impedance-Matching Networks for Radio-Frequency and Microwave Amplifiers, Artech House, 1985, ISBN Advanced, mathematical. Wilfred Caron, Antenna Impedance Matching, ARRL, 1989, ISBN Intermediate, non-mathematical. Caron illustrates manual Smith chart methods at their best, but which nonetheless have been completely replaced by computeraided network design. Brian C. Wadell, Transmission Line Design Handbook, Artech House, 1991, ISBN Analysis and formulas for many transmission lines. Comparable to M.A.R. Gunston or K.C. Gupta, et al., below. VG 79 PSA /21/01 Pacificon 01

80 References: Books (Cont d) Phillip H. Smith, Electronic Applications of the Smith Chart in Waveguide Circuit, and Component Analysis, 2nd edition, Noble Publishing, 1995, ISBN Originally published by McGraw-Hill, 1969, and reprinted by Krieger, Intermediate, mathematical. K.C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines, 2nd ed., Artech House, 1996, ISBN X. Analysis and formulas for many transmission lines. M.A.R. Gunston, Microwave Transmission-Line Impedance Data, Noble Publishing, 1997, ISBN Originally published by Van Nostrand Reinhold, Analysis and formulas for many transmission lines. Comparable to B. Wadell above. ARRL Antenna Book, 18th edition, chapters 24-28, ARRL, 1997, ISBN Elementary, non-mathematical. M. Walter Maxwell, W2DU, Reflections II: Transmission Lines and Antennas, 2nd ed., Worldradio Books, 01, ISBN A non-mathematical treatment of transmission lines and matching that examines and corrects common misunderstandings. VG PSA /21/01 Pacificon 01

81 References: Application Notes, Videos, and Web Sites Application Notes Times Microwave s Complete Coaxial Cable Catalog & Handbook Download from: Videos Glenn Parker, Introduction to the Smith Chart, Noble Publishing, minutes, $99. Useful Web Sites This Presentation is at VG 81 PSA /21/01 Pacificon 01