Chapter2_Notes

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Unformatted text preview: 29 0. 30 —&gt; WAVE LENGT HS T 0.49 OWA RD 0.48 GEN 170 0.47 ERAT OR 160 —&gt; 0. INDUC 46 TIVE 0. REA 15 0 CTA 0 0. 5 NC 45 E CO 0. MP ON 06 14 0 EN 0 .4 T 4 (+ jX /Z o) 110 0. 07 0. 1 4 0. 0.12 0.11 0.39 100 0.1 0.4 09 0. 08 0. 2 4 0 0. 12 95 Figure 2-22 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 = |Γ|ejθr . We deﬁne the phase shifted reﬂection coeﬃcient as, Γl = Γe−j 2βl = |Γ|ejθr e−j 2βl (184) so that at the input we have, Γl = |Γ|ej (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.

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