Chapter2_Notes

# And open circuit measurements if we measure a

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Unformatted text preview: βl Z0 + jZL tan βl (145) so, ⇒ Zin = ZL for l = nλ/2 (146) • ⇒ Generator connected to ZL via a transmission line that is multiple of λ/2 long induces in the load same currents and voltages as if the transmission line were not there. • Remember that this is valid at only one frequency!. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 74 • Quarter wave transformer When transmission line length l = λ/4 (or = λ/4 + nλ/2), we have another interesting case: • βl = (2π/λ) × (λ/4) = π/2 so tan βl → ∞ • Recall, the input impedance, Zin = Z0 • so that Zin = ZL + jZ0 tan βl Z0 + jZL tan βl 2 Z0 , for l = λ/4 + nλ/2 ZL (147) (148) • How is this useful? Illustrate usefulness of the quarter-wave transformer with an example in ﬁg. 19. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines Feedline Z01 = 50 Ω A Zin 75 λ/4 transformer Z02 A’ ZL = 100 Ω λ/4 Figure 19: Circuit for example 2-9. Figure 2-18 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 76 • Matched line In this case ZL = Z0 • Zin = Z0 • Γ=0 • All incident power at the input is delivered to the load, independent of the line length Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 77 1.9. Power ﬂow So far we’ve only analyzed transmission lines in terms of voltages and currents. There is an alternative point of view, based on power ﬂow on the transmission lines that can be quite useful. • Remember the expressions for voltage and current on transmission line: ˜ V (z ) = V0+ e−jβz + Γejβz (149) V+ ˜ I (z ) = 0 [e−βz − Γeβz ] Z0 (150) • We will substitute in z = −d as we have done before, ˜ V (d) = V0+ ejβd + Γe−jβd (151) V+ ˜ I (d) = 0 [eβd − Γe−βd ] Z0 (152) Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 78 • At the load z = 0 and things simplify and we can write in ˜ ˜ terms of incident and reﬂected waves: V i = V0+ , V r = ΓV0+ , ˜ ˜ I i = V0+ /Z0 and I r = −ΓV0+ /Z0 . • How do we ﬁnd power if we know voltage and current? Two ways to get the answer: in time or phasor domain. • Instantaneous incident power at the load (d = 0) is easy to ﬁnd: P i (0, t) = v i (0, t) · i0,i (t) = |V0+ |2 cos2 (ωt + φ+ ) (W) Z0 (153) • What about the reﬂected power at the load? P r (0, t) = v r (0, t) · ir (0, t) = −|Γ|2 |V0+ |2 cos2 (ωt + φ+ + θr ) (W) Z0 (154) • Note, book includes more general equations for power at location z = −d but we can obtain the same power results by setting d = 0. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 7...
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