Ch. 2. Transmission Line Theory • field analysis > transmission line theory < basic circuit theory 2.3 The Terminated Lossless Transmission Line • Assumptions o incident wave: π"# π %&'( , generated from a source at z < 0 o characteristic impedance of the line: Z0 § the ratio of voltage to current for such a traveling wave o load impedance: ZL ≠ Z0 • total voltage & current on the line: a sum of incident and reflected wave π (π§) = π"# π %&'( + π"% π #&'( , πΌ(π§) = /01 π %&'( − 23 /05 23 π #&'( , (2.34a) (2.34b) o load impedance ZL at z = 0 π (0) π"# + π"% π7 = = π πΌ(0) π"# − π"% 9 o voltage reflection coefficient, Γ: Γ= /05 /01 = 2; %23 (2.35) 2; #23 § ratio of the reflected to incident voltage wave amplitude at the load (l = 0) o total voltage & current on the line using the voltage reflection coefficient π (π§) = π9# <π %&'( + Γπ #&'( = πΌ(π§) = /31 23 <π %&'( − Γπ #&'( = (2.36a) (2.36a) § superposition of an incident and a reflected wave → “standing waves” § Only when π€ = 0 (ZL = Z0), there is no reflected wave. à Such a load is said to be matched to the line. • Return loss, RL = –20 log |π€| dB o a measure in relative terms of the power of the signal reflected by a discontinuity (= mismatching) in a transmission line or optical fiber § matched load (π€ = 0) à return loss of ∞ dB (no reflected power) § total reflection (|π€| =1) à return loss of 0 dB (all incident power is reflected) • reflection coefficient at arbitrary point (z = -l) Γ(π ) = /35 A 5BCD /31 A 1BCD = Γ(0)π %E&'( (2.42) o π€(0): reflection coefficient at z = 0, as given by (2.35) • The real power flow on the line is a constant (for a lossless line), but that the voltage amplitude, at least for a mismatched line, is oscillatory with position on the line. o The impedance seen looking into the line must vary with position. o input impedance seen looking toward the load at a distance l = -z πFG π(−π ) π9# <π &'H + Γπ %&'H = 1 + Γπ %E&'H = = π = π πΌ (−π ) π9# (π &'H − Γπ %&'H ) 9 1 − Γπ %E&'H 9 § for more usable form πFG = π9 2; #&23 JKG'H 23 #&2; JKG'H à “transmission line impedance equation” (2.43) 2.4 The Smith Chart • It was developed in 1939 by P. Smith at the Bell Telephone Laboratories. • a graphical aid that can be very useful for solving transmission line problems o a useful way of visualizing transmission line phenomenon without the need for detailed numerical calculations o a good intuition about transmission line and impedance-matching problems by learning to think in terms of the Smith chart • “reflection coefficient chart + impedance chart” o 1) a polar plot of the voltage reflection coefficient, π€ = |π€|ejπ § magnitude | π€|: a radius (|π€| ≤1) from the center of the chart § angle π: counterclockwise from the right-hand side of the horizontal diameter o 2) converting from reflection coefficient to normalized impedance (z = Z/Z0) § Reflection coefficient (π€) at the load in terms of the normalized load impedance (zL = ZL / Z0) Γ= π§7 − 1 = |Γ|π &c π§7 + 1 § normalized load impedance (zL) in terms of π€ π§7 = d#|e|A Bf (2.54) d%|e|A Bf § π€ and zL in terms of their real and imaginary pars 1 − ΓjE − ΓFE 2ΓF π7 + ππ₯7 = + π (1 − Γj )E + ΓFE (1 − Γj )E + ΓFE ü Rearranging (2.55) gives lΓj − j; d#j; E m + ΓFE = l (Γj − 1)E + lΓF − E d d#j; m : resistance circle d E d E n; n; m = l m : reactance circle è circles in the π€r and π€i planes . rL = 1 à circle center (π€r = 0.5, π€i = 0), radius of 0.5 . centers of resistance circles: on the horizontal π€i = 0 axis . centers of reactance circles: on the vertical π€r = 1 line . All resistance and reactance circles pass through the π€ = 1 point. • Transmission line impedance equation of (2.43) πFG = π9 1 + Γπ %E&'H 1 − Γπ %E&'H o of the same form as (2.54), differing only by the phase angles of the π€ term o If we have plotted the reflection coefficient | π€ |ejθ at the load, the normalized input impedance seen looking into a length l of transmission line terminated with zL can be found by rotating the point clockwise by an amount 2βl (subtracting 2βl from θ) around the center of the chart. § The radius stays the same since the magnitude of π€ does not change with position along the line (assuming a lossless line). o a line of length λ/2 (or any multiple) à a rotation of 2βl = 2π around the center of the chart à The input impedance of a load seen through a λ/2 line is unchanged. 2.5 The Quarter-Wave Transformer • useful and practical circuit for impedance matching 2.5.1 The Impedance Viewpoint • From (2.44) the input impedance Zin πFG = π9 π7 + ππ9 π‘πππ½π π9 + ππ7 π‘πππ½π o βl = (2π/λ)(λ/4) = π/2 πFG = πdE π 7 o In order for π€ = 0, we must have Zin = Z0. πd = vπ9 π 7 § Z1: geometric mean of the load and source impedance • Length of the matching section: π/4 à A perfect match may be achieved at one frequency, but impedance mismatch will occur at other frequencies. Z0 Z1 RL