Mysteries of the Smith Chart Transmission Lines, Impedance Matching, and Little Known Facts Stephen D. Stearns, K6OIK Chief Technologist TRW Firestorm Wireless Communication Products [email protected] VG – 1 Pacificon 2001 PSA-00527 — 10/21/2001 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 Pacificon 2001 PSA-00527—10/21/2001 Part 1: Transmission Line Theory VG – 3 Pacificon 2001 PSA-00527 — 10/21/2001 Key Dates in Electrical Transmission 1830s Magnetic telegraphs - Gauss, Henry 1839 Electromagnetic telegraph - Wheatstone & Cook 1844 Telegraph in America - Morse 1850s Thousands of miles of telegraph line U.S. and Europe 1851 40-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 1880s 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 1944 - Smith VG – 4 Pacificon 2001 PSA-00527—10/21/2001 Numbers to Remember! 1.4142135623... 1.7320508075… 1.6180339887... 3.1415926535... 2.718281828459045… 2.54 299,792,458 376.7303134... VG – 5 Pacificon 2001 PSA-00527—10/21/2001 Heaviside’s Vector Formulation of Maxwell’s Theory ∂B ∇×E =− ∂t ∂D ∇×HJ = + ∂t ∇⋅D = ρ ∇⋅B0 = DE= ε BH= µ “And God said, Let there be light; and there was light.” Genesis 1:3 VG – 6 Pacificon 2001 PSA-00527—10/21/2001 Frequency Domain or Phasor Form ∇×EH =−jωµ ∇×HE =()σωε + j ∇⋅E0 = ∇⋅H0 = VG – 7 Pacificon 2001 PSA-00527—10/21/2001 Heaviside’s Telegrapher’s Equations Uniform transmission line Equivalent circuit of infinitesimal segment I(x) R∆xL∆x V(x) G∆xC∆x dV =−()()RjLIx + ω dx dI =−()()GjCVx + ω dx VG – 8 Pacificon 2001 PSA-00527—10/21/2001 Transmission Line Solution TEM Waves Traveling wave γx Vx()=Vo e Vx() Ix()= Zo Propagation constant γα=+jRjLGjC β =()() + ω + ω Characteristic impedance RjL+ ω Z = o GjC+ ω VG – 9 Pacificon 2001 PSA-00527—10/21/2001 Notations Real Parameters R = Series resistance per unit length (Ohms/meter) L = Series inductance per unit length (Henries/meter) G = Shunt conductance per unit length (Siemens/meter) C = Shunt capacitance per unit length (Farads/meter) α = Attenuation constant (nepers/meter) β = Phase constant (radians/meter) λ = Wavelength (meters) vf = Velocity factor (dimensionless) X = Reactance (Ohms) B = Susceptance (Siemens) s = Standing wave radio (dimensionless) VG – 10 Pacificon 2001 PSA-00527—10/21/2001 Notations (Cont’d) Complex Parameters Z = R + jX = impedance (Ohms) ZL = Load impedance (Ohms) Zi = Input impedance (Ohms) Z0 = Characteristic impedance (Ohms) z = Z/Z0 = r + jx = normalized impedance (dimensionless) Y = G + jB = admittance (Siemens) y = Y/Y0 = g + jb = normalized admittance (dimensionless) Γ Γ Γ = r + j i = complex reflection coefficient (dimensionless) γ = α + jβ = propagation constant (inverse meters) VG – 11 Pacificon 2001 PSA-00527—10/21/2001 Transmission Line Parameters Physical Dimensions and Material Properties d dielectric dielectric (µ, ε, σ) (µ, ε, σ) a b a a c Parameter Coax Twinlead 1 11 1 R Ω/m + πδσ πδσ a c 2 c ab µ δ µ b δ 11 −1 d L H/m ln ++ + cosh 22π aab π 22a a πσ πσ G S/m 2 b −1 d ln cosh a 2a C F/m 2πε πε b − d ln cosh 1 a 2a Where skin depth is For copper σ =×58. 107 S/m δ = 1 c πµσf 85. mm at 60 Hz c δ = VG – 12 6.6 µm at 100 MHz Pacificon 2001 PSA-00527—10/21/2001 Round Open-Wire Transmission Line Formulas d s ❏ Approximate formula ➤ Widely published by ARRL and others ➤ Accurate only for large spacings: s/d > 3 or large impedances: Z0 > several hundred ❏ Exact formula ➤ Accurate for all spacings and impedances VG – 13 Pacificon 2001 PSA-00527—10/21/2001 Comparison of Impedance Formulas Round Open-Wire Line VG – 14 Pacificon 2001 PSA-00527—10/21/2001 K6OIK Square Open-Wire Transmission Line Formula w s ❏ Excellent approximation in the range of practical interest ➤ Accurate for small spacings: 1 < s/w < 3 or small impedances: 0 < Z0 < several hundred VG – 15 Pacificon 2001 PSA-00527—10/21/2001 Round vs Square Open-Wire Lines VG – 16 Pacificon 2001 PSA-00527—10/21/2001 Optimal Characteristic Impedances Coax Ω For minimum loss Zo = 77 Ω For maximum breakdown voltage Zo = 30 Ω For minimum temperature rise Zo = 60 Ω Zo = 50 has no special significance VG – 17 Pacificon 2001 PSA-00527—10/21/2001 Reflection Coefficient and Impedance Relation at a Terminal Plane Terminal Plane Definition Ζ Ζ L O Γ L ZZ− − Γ= Lo= z 1 + + ZZLoz 1 Inverse 1 + Γ z = 1 − 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 Pacificon 2001 PSA-00527—10/21/2001 Relations Between Two Terminal Planes Input Output Terminal Terminal Plane Plane Impedance relation Ζ Ζ L O Γ L zj+ tan β l = L Ζ zi Γi 1 + jztan β l i L Cross relations Reflection coefficient relation 1 + Γ e − jl2β = L zi − jl2β 1 − Γ e − β L ΓΓ= jl2 iL e 1 + Γ e jl2β z = i L − Γ jl2β 1 i e VG – 19 Pacificon 2001 PSA-00527—10/21/2001 Velocity Factor Wavelength λ = c free space f λ = v actual f Velocity factor v λ v == actual f c λ free space VG – 20 Pacificon 2001 PSA-00527—10/21/2001 How To Measure Velocity Factor of a Line (One Way To Do It) Known length Antenna open Analyzer circuit Same length Antenna short Analyzer circuit 21πfl v = f c −Z cot −1 open Zshort VG – 21 Pacificon 2001 PSA-00527—10/21/2001 Phase Constant 22π πf β == radians/meter λ actualvc f ❏ β Phase constant and velocity factor vf give equivalent information ❏ Both can be calculated from line dimensions and material properties βωω=++Im (RjLGjC )( ) ❏ Best to measure! VG – 22 Pacificon 2001 PSA-00527—10/21/2001 How to Measure Complex Zo of A Line (One Way to Do It) Unknown or arbitrary length Antenna open Analyzer circuit Same length Antenna short Analyzer circuit =× ZZZo open short ❏ Geometric mean of two complex numbers ❏ Calculation is trivial in polar form on Smith Chart VG – 23 Pacificon 2001 PSA-00527—10/21/2001 What Special Lengths of Lossless Line Do Half wavelength, l = _/2 = ZZiL Quarter wavelength, l = _/4 Z 2 = o Zi Z L Eighth wavelength, l = _/8 = ZZio if Z L and Z o are real (resistive) VG – 24 Pacificon 2001 PSA-00527—10/21/2001 Standing Wave Ratio Easy to remember from 1 + Γ 1+ Γ s = z = 1 − Γ 1− Γ − z −1 Γ = s 1 Γ = s +1 z +1 VG – 25 Pacificon 2001 PSA-00527—10/21/2001 Part 2: The Smith Chart VG – 26 Pacificon 2001 PSA-00527 — 10/21/2001 PSA-00527 — 10/21/2001 VG – 27 0.0 —> WAV 0.49 ELENGT HS TO 0.48 WA OAD <— 0.0 0.49 RD ARD L GEN TOW ± 0.48 ER THS 180 AT 0.47 NG 170 OR ELE -170 — AV 0.47 > W 0.04 <— 60 -90 90 160 0.46 -1 0.1 0.46 0.1 85 0.04 -85 0.2 150 0.05 0.45 -150 ) IN 0.2 80 -80 /Yo DU 0.45 (-jB CTI 0.05 E VE NC RE TA AC 0.3 EP T 0.06 SC 0.1 AN 75 0 -75 U C 0.3 1 0.44 4 S E 40 0.44 E COMPONENT (G/Yo) COMPONENT (R/Zo), OR CONDUCTANCE RESISTANCE C -1 IV O 0.06 CT M U PO D N N E I N 0.07 0.4 R T 70 -70 O 0.43 ), (+ 0.4 0.43 o jX 0 Z / 1 0.07 3 / 0.2 Z 3 X o 0 -1 -j ) ( , T O N R E C 0.08 0.5 N A 65 -65 O P 0.4 0.42 A 0.5 P 2 0.08 M C O IT 0.3 I C V 120 E E -120 C S N U 0.09 A S 0.41 0.6 T C 0.41 C E 0.6 60 -60 P 0.09 A T E 0.4 A R N E C V I E 1 T 0 ( 1 I 0.1 1 + 0 0.7 C j 0.4 -1 0.5 B 0.4 A 0.7 / 0.1 P Y A o 55 -55 C ) 0.6 0.8 0.11 0.8 0.39 100 0.39 0.11 0.7 -100 0 50 -5 0.9 0.8 0.9 0.12 0.38 0.9 0.38 0.12 1.0 1.0 1.0 90 0.2 -90 0.2 45 0.4 0.2 0.2 -45 0.37 0.4 0.13 0.4 0.4 0.6 0.13 1.2 0.37 0.6 0.6 0.6 0.8 1.4 0.8 0.8 1.2 0.8 1.2 80 0.36 0.14 -80 1.0 1.0 4 0 1.0 0 0.14 -4 1.0 1.6 0.36 1.8 1.4 1.4 0.35 2.0 0.15 7 0.15 0 0 0.35 35 -7 -35 1.6 1.6 0.34 0.16 0.16 0.34 3.0 1.8 1.8 30 60 -60 -30 2.0 0.33 0.17 0.17 2.0 4.0 0.33 5.0 25 5 0 0 0.32 -5 -25 0.18 0.18 0.32 3.0 2 0.31 0 3.0 0 4 0 0.19 0 -2 0.19 -4 10 0.31 4.0 4.0 0.3 0.2 15 5.0 30 20 0.2 -15 -30 0.3 5.0 0.29 10 10 0.21 50 A S 20 N E 50 G E 10 0.21 L -10 R 2 E G 0 0 E O D F -2 T N 20 I R 0.29 A T 50 N N 0.28 E S I M C I I S F S F I E O O N C 0.22 A S N E G E L R E G E O D F R N I 0.22 E F T L N E 0.27 E C I T C I I O F F N E C O 0.23 0.28 0.26 0.24 0.25 0.23 0.27 0.24 0.26 0.25 Pacificon 2001 Complex Functions zi wi Complex Number Complex Number y (x, y) v (u, v) z = x + jy w = u + jv ( ) z f w x r u 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 Pacificon 2001 PSA-00527—10/21/2001 Mathematical Basis of the Smith Chart z −1 Γ= A bilinear conformal + z 1 complex function (-)+rjx1 ujv+= (+)+rjx1 x v r u Right Half Interior Unit Circle Z Plane Γ Plane VG – 29 Pacificon 2001 PSA-00527—10/21/2001 PSA-00527—10/21/2001 VG – 30 Smith Chart: Impedance Coordinates x Series Reactance r=0 r=1 r=2 x=-1 x=0 x=1 r Series Resistance 0 0.0 —> WAV 0.49 ELENGT HS TO 0.48 WA OAD <— 0.0 0.4 RD ARD L 9 GEN TOW ± 0.48 ER THS 180 AT 0.47 NG 170 OR ELE -170 — AV 0.4 > W 7 0.04 <— 160 0.46