ps9 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science Problem Set No. 9 6.630 Electromagnetics Issued: Week 10 Fall Term 2006 Due: Week 11 -----------------------------------------------------------------------------Reading assignment: Section 2.6; J. A. Kong, “Electromagnetic Wave Theory”. Problem P9.1 1 Zo Y1 Zo ZL Y2 2 Use the Smith chart to match the following load impedence, ZL to the line impedance, Zo = 50Ω by ﬁnding a suitable 1 and 2 . (a) ZL = (100 + 25j )Ω (b) ZL = (100 − 25j )Ω (c) ZL = (25 + 50j )Ω (d) ZL = (25 − 50j )Ω Problem P9.2 (a) Determine the propagation constant θ(ω ) , for the lumped transmission line shown in following ﬁgure . Show that the result can be placed in the form sin2 ω 2 − ω0 2 θ = 2 2 ω1 2 2 where ω0 and ω1 are appropriate constants. L0 L0 C0 C0 I2 I1 L0 C0 + Vk − Ik L0 C0 + Ik+1 + V − + V1 − + C1 V2 − C1 C1 Vk+1 − C1 (b) For what range of ω will the line support a propagating wave, i.e., admit real θ solutions? Problem P9.3 I0 a C0 a� + V1 − I1 + L0 V2 − I2 + V3 − + Vn − In (a) Determine the dispersion relation for the backward wave line shown in following ﬁgure. That is, assume Vn = Ae−jnθ , In = B e−jnθ and determine θ(ω ) . Sketch θ(ω ) for ω > 1 ω0 = 2√L0 C0 . (b) For a given ω > ω0 , the result in part (a) yields two real real values of θ (excluding n values which diﬀer by 2nπ ). Determine the impedance, Z = Vn , for each of these I modes and show that the time-averaged power ﬂow is in the direction opposite the phase velocity. (c) A voltage source vs (t) = Vs sin ω0 t is connected to terminal pair a − a . Determine the steady-state vn (t) . (d) Suppose the source connected to a − a is given by vs (t) = Vs where ω1 ω0 . Determine vn (t) . sin ω1 t sin ω0 t ω1 t ...
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ps9 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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