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EEL3472_Spring2011_HO_07

# EEL3472_Spring2011_HO_07 - EEL 3472 Electromagnetic Fields...

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EEL 3472 - Electromagnetic Fields and Applications I Handout #7 Moore 31 January 2011 Spring 2011 Solutions to Homework Assignment #2 All CD Module Portions listed in the questions are optional. 1. Ulaby, p. 121, #2.1 2. Ulaby, p. 121, #2.2 3. Ulaby, p. 121, #2.3 4. Ulaby, p. 122, #2.5 5. Ulaby, p. 122, #2.6 6. Ulaby, p. 122, #2.7 7. Ulaby, p. 122, #2.9 8. Ulaby, p. 122, #2.12 9. Ulaby, p. 122, #2.13 10. Ulaby, p. 122, #2.14 1

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Problem 2.1 A transmission line of length l connects a load to a sinusoidal voltage source with an oscillation frequency f . Assuming the velocity of wave propagation on the line is c , for which of the following situations is it reasonable to ignore the presence of the transmission line in the solution of the circuit: (a) l = 20 cm, f = 20 kHz, (b) l = 50 km, f = 60 Hz, (c) l = 20 cm, f = 600 MHz, (d) l = 1 mm, f = 100 GHz. Solution: A transmission line is negligible when l / λ 0 . 01. (a) l λ = l f u p = ( 20 × 10 2 m ) × ( 20 × 10 3 Hz ) 3 × 10 8 m/s = 1 . 33 × 10 5 (negligible). (b) l λ = l f u p = ( 50 × 10 3 m ) × ( 60 × 10 0 Hz ) 3 × 10 8 m/s = 0 . 01 (borderline) . (c) l λ = l f u p = ( 20 × 10 2 m ) × ( 600 × 10 6 Hz ) 3 × 10 8 m/s = 0 . 40 (nonnegligible) . (d) l λ = l f u p = ( 1 × 10 3 m ) × ( 100 × 10 9 Hz ) 3 × 10 8 m/s = 0 . 33 (nonnegligible) .
Problem 2.2 A two-wire copper transmission line is embedded in a dielectric material with ε r = 2 . 6 and σ = 2 × 10 6 S/m. Its wires are separated by 3 cm and their radii are 1 mm each. (a) Calculate the line parameters R , L , G , and C at 2 GHz. (b) Compare your results with those based on CD Module 2.1. Include a printout of the screen display. Solution: (a) Given: f = 2 × 10 9 Hz , d = 2 × 10 3 m , D = 3 × 10 2 m , σ c = 5 . 8 × 10 7 S/m (copper) , ε r = 2 . 6 , σ = 2 × 10 6 S/m , μ = μ c = μ 0 . From Table 2-1: R s = radicalbig π f μ c / σ c = [ π × 2 × 10 9 × 4 π × 10 7 / 5 . 8 × 10 7 ] 1 / 2 = 1 . 17 × 10 2 , R = 2 R s π d = 2 × 1 . 17 × 10 2 2 π × 10 3 = 3 . 71 / m , L = μ π ln bracketleftbigg ( D / d )+ radicalBig ( D / d ) 2 1 bracketrightbigg = 1 . 36 × 10 6 H/m , G = πσ ln [( D / d )+ radicalbig ( D / d ) 2 1 ] = 1 . 85 × 10 6 S/m , C = G ε σ = 1 . 85 × 10 6 × 8 . 85 × 10 12 × 2 . 6 2 × 10 6 = 2 . 13 × 10 11 F/m . (b) Solution via Module 2.1:

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Problem 2.3 Show that the transmission line model shown in Fig. P2.3 yields the same telegrapher’s equations given by Eqs. (2.14) and (2.16).

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EEL3472_Spring2011_HO_07 - EEL 3472 Electromagnetic Fields...

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