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0. 12 95 Figure 222 Figure 25: Examples of points (Z and reﬂ. coeﬀ.) on Smith chart. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 96 • Input impedance
Recall the expressions for Γ and zL ,
Γ= zL − 1
zL + 1 (180) 1+Γ
(181)
1−Γ
• We know that we can use the Smith chart to get Γ if we are
given zL and we can ﬁnd zL if we are given Γ.
zL = • What about when we connect a transmission line and look along
the line or at the input?
• Recall, when we connect a lossless transmission line this will
only change the phase of Γ along the transmission line. Recall,
the expression we deﬁned for the input impedance at z = −l,
Zin (−l) = Z0 1 + Γe−j 2βl
1 − Γe−j 2βl Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (182) Electromagnetics I: Transmission lines 97 We always normalize by Z0 when using a Smith chart so,
zin (−l) = 1 + Γe−j 2βl
1 − Γe−j 2βl (183) and we have the deﬁnition of the reﬂection coeﬃcient at the load,
Γ = Γejθr . We deﬁne the phase shifted reﬂection coeﬃcient as,
Γl = Γe−j 2βl = Γejθr e−j 2βl (184) so that at the input we have,
Γl = Γej (θr −2βl) (185) What does this correspond to in terms of vectors in complex
plane? Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 98 • Again, the normalized input impedance,
zin = 1 + Γl
1 + Γe−j 2βl
=
1 − Γe−j 2βl
1 − Γl (186) • We see that this has exactly the same form as we had for zL
and Γ. So, we can use the Smith chart in the same way but use
zin and Γl .
• ⇒ ﬁnding the input impedance after the transmission line is
connected to load is easy — rotate Γ clockwise on Smith chart
by 2βl.
• The length of the transmission line that produces one full rotation on the Smith chart is,
2βl = 2π ⇒ 2 2π
λ
l = 2π ⇒ l =
λ
2 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (187) Electromagnetics I: Transmission lines 99 • It is not even necessary to transform phase information into
degrees — it is already done on Smith chart (in terms of wavelength). Look at the Wavelengths towards generator (WTG)
scale.
• Note that the lengths are expressed in fractions of λ . Scale
starts at 180◦ and rotates clockwise.One full circle 2π corresponds to λ/2.
• Most of the time we’ll be interested in diﬀerences between two
points, so the absolute value on WTG will not be important. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 1.4 70 6
1. 8 6 0...
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This note was uploaded on 09/25/2013 for the course ECE 331 taught by Professor Martinsiderious during the Fall '12 term at Portland State.
 Fall '12
 MartinSiderious
 Electromagnet

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