MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science
6.013 -- Electromagnetics and Applications
Problem Set 9 Suggested Reading: Issued: October 29, 2002 Due in Recitation: November 6, 2002 Textbook Sections 5.1-2, 5.5, and 6.5
Problem 9.1 A parallel-plate TEM transmission line is filled with a slightly lossy dielectric having ε = 4εo, µ = µo, and σ = 4 × 10-4 Sm-1 (comparable to distilled water), and crosssection dimensions 10 × 400 microns. The metal plates are perfectly conducting and 400 microns wide (neglect fringing fields). 10 microns a) b) c) µo, 4εo, σ
400 microns What are the inductance L and capacitance C per meter for this line? What is Zo? What is the conductance G per meter [S] for this line? After how many meters will any propagating power drop by 10 dB for this line?
Problem 9.2 The signals on a TEM transmission line can be represented as: v(z,t) = f+(t - z/v) + f-(t + z/v) a) Prove by computation that the total power flowing down this line in the +z direction at any (t,z) equals the power flowing in the +z direction minus the power flowing in the -z direction. Note that powers do not generally superimpose unless the two signals are in some sense orthogonal.
b) Repeat (a) for the case where two equal signals are both propagating in the +z direction only. Do the powers still superimpose?
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Problem 9.3 A line driver at one end of a 20-cm long 200-ohm TEM transmission line triggers a flip-flop at the other end with a step function, as illustrated. The dielectric in the line has ε = 4εo and µ = µo.
50Ω
+
V = 10 u(t) Zo = 200Ω, 4εo, µo 20 cm
ZL = 50Ω Flip-flop
0 z
a) b) c)
Sketch and dimension v(t,z) and i(t,z) on the line at t = 1 ns (10-9 sec). Repeat (a) for t = 2 ns. Sketch quantitatively the load voltage vL(t) until the flip-flop is triggered; its trigger voltage is 4 volts. Note that triggering is excessively delayed. What is the asymptotic value of the load voltage as t→∞? If the line impedance were matched at 50 ohms, would there still be excessive delay? Write a simple equation for v(z,t) valid for 0 < t < 1 ns.
d) e) f)
Problem 9.4 A 100,000-volt DC power transmission line is 10 km long and has a 300-ohm impedance. It is conveying 100 MW to a snug little community when lightning strikes the center of the line at t = 0, putting a 10-ohm resistance across the line at that point for 10 microseconds, after which the line returns to normal. a) b) Sketch and dimension v(z) and i(z) on the line for t = 5 microseconds. Repeat (a) for t = 15 microseconds.
Problem 9.5 The flip-flop of Problem 9.3 is replaced by an inductor L = 10-7 Henries. Again sketch and dimension v(z) and i(z) for t = 2 ns.
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