UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
The Edward S. Rogers Sr. Department of
Electrical and Computer Engineering
ECE357H1S – ELECTROMAGNETIC FIELDS
PROBLEM SET 1
Topics:
Review of phasors, distributed parameter model
Reading:
Cheng sections 7-7.1, 7-7.2, 9-1, 9-2, 9-3
1.
Consider a transmission line conveying a sinusoidal (harmonic) voltage wave in
the +z direction. The transmission line has a characteristic resistance R0 = 50 Ω.
At some position along the line, the voltage phasor is measured to be V = 30∠67°
V. Write the time domain expression for the current measured at that point.
2.
Consider the time-harmonic vector
B(t ) = 3 cos(ωt + 30°)aˆ x − 4 sin(ωt )aˆ y
a) Find its vector phasor representation B. Also, transform B back to the time
domain, and verify that it equals B(t ) .
b) If B represents a time-harmonic field that is a function of position such that
B(t ) = 3 cos(ωt − βz + 30°)aˆ x − 4 sin(ωt − βz )aˆ y
find the phasor representation B.
3.
Let a(z, t) represent a sinusoidal function where z is the spatial coordinate and t is
time. If A(z) is the phasor representation of a(z, t), show that
∂
a) a ( z , t ) = Re jωA( z )e jωt
∂t
∂2
b) 2 a(z,t) = Re{−ω 2A(z)e jωt }
∂t
{
4.
€
}
Consider two phasors A(z) and B(z). Show that the following is true both using
phasors and time-domain expressions:
a) Re{A( z ) + B( z )} = Re{A( z )}+ Re{B( z )}
b) Re{A( z )}Re{B( z )} =
{
}
Re{A( z )B( z )} Re A( z )B* ( z )
+
2
2
5.
A series RLC circuit is connected to a generator with a voltage
vs(t) = V0cos(ωt + π/3) [V]
a) Write the voltage loop equation in terms of the current i(t), R, L, C, and vs(t).
b) Obtain the corresponding phasor-domain equation.
c) Solve the equation to obtain an express for the phasor current I.
6.
What is the fundamental difference between a transmission line and an ordinary
electric network (circuit)?
Show that the following T-model of a transmission line also yields the telegrapher
equations.
7.
8.
A two-wire line has distributed parameters L = 6.5 µH/m and C = 8.7 pF/m.
a) What is the characteristic impedance (resistance) of the line?
b) What is the velocity of the wave in this transmission line?
c) If a voltage wave with time waveform V − (t ) is traveling along the line in the
–z direction, and has an amplitude of 10 V, write a time-domain expression for
the current wave associated with this voltage wave, using numerical parameters
determined in (a) and (b).