Midterm 22009

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UNIVERSITY OF TORONTO

FACULTY OF APPLIED SCIENCE AND ENGINEERING

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering

ECE 320H1F

FIELDS AND WAVES

MIDTERM

16 November 2009, 10:10 11:00

Examiner: Prof. Sean V. Hum

NAME:

STUDENT NUMBER:

TOTAL POINTS: 20

Notes:

Include units in your answers.

You can use one double-sided, letter-sized aid sheet.

All non-programmable electronic calculators are allowed.

Only answers that are fully justified will be given full credit.

You can write in pen or pencil, but requests for remarking will be considered for paperswritten in pen only.

GOOD LUCK!

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ECE320 Midterm Page 2

PROBLEM #1. [10 POINTS]

An infinitely long coaxial cable is shown in Figure 1. The radius of the inner conductor is =

a = 1 mm and the radius of the outer conductor is = b = 4.95 mm. The space between the twoconductors is filled with a homogenous, lossless dielectric with a dielectric constant of 3.6. Thecable is excited with a 1 GHz sinusoidal TEM wave travelling in the +z-direction. The electricfield in the dielectric region is given by

E(,,z) =25

exp(j75.4z) a V/m, a < < b.

Figure 1: Infinitely long coaxial cable

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a) Using Maxwells equations, determine a phasor expression for the magnetic field in the

dielectric region a < < b. Refer to page 9 if you need expansions of vector operators incylindrical coordinates. [2 points]

b) Using the magnetic field derived in (a), determine the intrinsic impedance associated with

the dielectric region. How does it compare to the result for plane waves? [2 points]

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c) Write a phasor expression for the current wave on the inner conductor that is associated with

the magnetic field. Assume a reasonable function for the magnetic field if you were unable

to complete part (a). [2 points]

d) Write a phasor expression for the voltage wave between the inner and outer conductors that

is associated with the magnetic field, using the outer conductor as a voltage reference.

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e) Write a phasor expression for the displacement current density in the dielectric region. What

direction does the displacement current flow? [2 points]

f) BONUS: What is the value characteristic impedance associated with this coaxial cable? [1

point]

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PROBLEM #2. [10 POINTS]

A 400 MHz plane wave propagating in a lossless dielectric has an electric field whose amplitude

is 6ax 8ay + 10az V/m at the origin. The wave travels in a direction specified by a unit vector

aN = 0.858ax + 0.515az.

a) Determine the unit vector associated with the direction of the magnetic field. [2 points]

b) The wave travels 4 m in the direction of propagation and is phase delayed by 67.02 radians.Determine the scalar and vector wavenumber associated with plane wave. [2 points]

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c) What is the relative permittivity and conductivity of the medium? Determine the wavelength

of the plane wave. [2 points]

d) Write the time-domain expression for the electric field associated with the plane wave. [2

points]

e) Write the time-domain expression for the magnetic field associated with the plane wave. [2

points]

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USEFUL FORMULAE (YOU MAY DETACH THIS PAGE)

Permittivity of free space: 0 = 8.854

1012 F/m

Permeability of free space: 0 = 4 107 H/m

Speed of light in vacuum: c = 3 108 m/s

Complex permittivity: c = j = j Wavenumber: k =

Complex propagation constant in an unbounded medium: = jk = + j

Phase constant: = /vp = 2/

Intrinsic impedance associated with a plane wave: =

Potential difference VAB = VA VB = AB

E d

Maxwells Equations (time-harmonic form):

Integral form Point form

SD dS= Qencl D = v

S

B

dS= 0

B = 0

CE dL = j

SB dS E= jB

CH dL = Iencl + j

SD dS H= J+ jD

Divergence operator in cylindrical coordinates:

A = 1

(A)

+

1

A

+Azz

Curl operator in cylindrical coordinates:

A = a

1

Az

Az

+ a

Az

Az

+ az

1

(A)

A

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Midterm 22009