Ch. 2. Transmission Line Theory
• field analysis > transmission line theory < basic circuit theory
2.3 The Terminated Lossless Transmission Line
• Assumptions
o incident wave: 𝑉"# 𝑒 %&'( , 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
𝑉 (𝑧) = 𝑉"# 𝑒 %&'( + 𝑉"% 𝑒 #&'( ,
𝐼(𝑧) =
/01
𝑒 %&'( −
23
/05
23
𝑒 #&'( ,
(2.34a)
(2.34b)
o load impedance ZL at z = 0
𝑉 (0) 𝑉"# + 𝑉"%
𝑍7 =
=
𝑍
𝐼(0) 𝑉"# − 𝑉"% 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
𝑉 (𝑧) = 𝑉9# <𝑒 %&'( + Γ𝑒 #&'( =
𝐼(𝑧) =
/31
23
<𝑒 %&'( − Γ𝑒 #&'( =
(2.36a)
(2.36a)
§ superposition of an incident and a reflected wave → “standing waves”
§ Only when 𝛤 = 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 |𝛤| 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 (𝛤 = 0) à return loss of ∞ dB (no reflected power)
§
total reflection (|𝛤| =1) à return loss of 0 dB (all incident power is reflected)
• reflection coefficient at arbitrary point (z = -l)
Γ(𝑙 ) =
/35 A 5BCD
/31 A 1BCD
= Γ(0)𝑒 %E&'(
(2.42)
o 𝛤(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
𝑍FG
𝑉(−𝑙 ) 𝑉9# <𝑒 &'H + Γ𝑒 %&'H =
1 + Γ𝑒 %E&'H
=
=
𝑍 =
𝑍
𝐼 (−𝑙 ) 𝑉9# (𝑒 &'H − Γ𝑒 %&'H ) 9 1 − Γ𝑒 %E&'H 9
§ for more usable form
𝑍FG = 𝑍9
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, 𝛤 = |𝛤|ej𝜃
§ magnitude | 𝛤|: a radius (|𝛤| ≤1) from the center of the chart
§ angle 𝜃: counterclockwise from the right-hand side of the horizontal
diameter
o 2) converting from reflection coefficient to normalized impedance (z = Z/Z0)
§ Reflection coefficient (𝛤) at the load in terms of the normalized load
impedance (zL = ZL / Z0)
Γ=
𝑧7 − 1
= |Γ|𝑒 &c
𝑧7 + 1
§ normalized load impedance (zL) in terms of 𝛤
𝑧7 =
d#|e|A Bf
(2.54)
d%|e|A Bf
§ 𝛤 and zL in terms of their real and imaginary pars
1 − ΓjE − ΓFE
2ΓF
𝑟7 + 𝑗𝑥7 =
+
𝑗
(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 𝛤r and 𝛤i planes
. rL = 1 à circle center (𝛤r = 0.5, 𝛤i = 0), radius of 0.5
. centers of resistance circles: on the horizontal 𝛤i = 0 axis
. centers of reactance circles: on the vertical 𝛤r = 1 line
. All resistance and reactance circles pass through the 𝛤 = 1 point.
• Transmission line impedance equation of (2.43)
𝑍FG = 𝑍9
1 + Γ𝑒 %E&'H
1 − Γ𝑒 %E&'H
o of the same form as (2.54), differing only by the phase angles of the 𝛤 term
o If we have plotted the reflection coefficient | 𝛤 |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 𝛤 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
𝑍FG = 𝑍9
𝑍7 + 𝑗𝑍9 𝑡𝑎𝑛𝛽𝑙
𝑍9 + 𝑗𝑍7 𝑡𝑎𝑛𝛽𝑙
o βl = (2π/λ)(λ/4) = π/2
𝑍FG =
𝑍dE
𝑅7
o In order for 𝛤 = 0, we must have Zin = Z0.
𝑍d = v𝑍9 𝑅7
§ Z1: geometric mean of the load and source impedance
• Length of the matching section: 𝜆/4
à A perfect match may be achieved at one frequency, but impedance mismatch will
occur at other frequencies.
Z0 Z1 RL