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 5
Topics:
Smith Chart, transmission line impedance matching, transients on
transmission lines
Reading:
9-5, 9-7
1.
Design a series single-stub tuner to a load consisting of a parallel (shunt)
combination of a 250 Ω resistor and a 3.82 pF capacitor, assuming the section
used to realize d has a characteristic impedance of 50 Ω, and the characteristic
impedance of the transmission line used to realize the stub (l) is 35 Ω. The
operating frequency is 500 MHz.
2.
A shunt double-stub tuner is used to match a load impedance 100 + j100 Ω to a
lossless transmission line of characteristic impedance 300 Ω. The spacing
between the stubs is 3λ/8. Determine the lengths of the stub tuners:
a) If they are both short-circuited.
b) If they are both open-circuited.
3.
Generate a bounce diagram for the voltage v(z, t) for a 1-m–long lossless line
characterized by Z0 = 50 Ω and u = 2c/3 (where c is the velocity of light) if the
line is fed by a step voltage applied at t = 0 by a generator circuit with Vg = 60 V
and Rg =100 Ω. The line is terminated in a load RL = 25 Ω. Use the bounce
diagram to plot v(t) at a point midway along the length of the line from t = 0 to t =
25 ns.
4.
Consider a 5V source with a large internal resistance of Rg = 1000 Ω . The source
is attached to an air-filled 30cm long transmission line of characteristic
impedance Z 0 = 50Ω . The line is terminated with an open-circuit. We are
interested in the transient response of the line to a step function, generated when a
switch connected in series to the 5V source is closed at t=0.
(a) Calculate the half round-trip period on the line (the one-way transit time).
(b) Calculate the voltage at the input of the line at t=0.
(c) Generate and plot the bounce diagram from 0 to 7ns.
(d) Derive a general expression for the voltage at the load as a function of time.
(e) Determine the steady-state voltage at the load as t → ∞ .
(f) Sketch the voltage waveform at the load as a function of time.
(g) Calculate the time required so that the voltage at the load reaches 90% of its
steady-state value found in part (f).
7.
A lossless coaxial cable with Z0 = 100 Ω and of length L =3 m is terminated in a
resistive load RL = 40 Ω as shown in Figure 1. The dielectric between the two
conductors of the cable has a relative dielectric constant of εr =4. At time t=0, a
switch connected to a 15 V battery is closed as shown. The resistor Rg = 50 Ω.
a) Determine the amplitude of the step function initially launched onto the line,
and the one-way transit time of the line.
b) Plot the voltage as a function of time at position z=1 m for 0 ≤ t ≤ 120 ns.
Label your graph clearly and show all relevant voltages and times of
discontinuities.
c) Plot the voltage as a function of time at the load for 0 ≤ t ≤ 120 ns. Label your
graph clearly and show all relevant voltages and times of discontinuities
d) Determine the steady-state value of the voltage observed at i) z=0; and ii) z=L as t →∞.
e) If a 15V pulse generator was used in place of the battery and switch, and the
pulse width is 2 ns, determine the voltage as a function of time at z = 2 m for 0 ≤ t ≤ 120 ns.
f) Repeat part e if the pulse width is 30 ns.
8.
A lossless transmission line of length l = 2 m is connected to a 15 V battery via a
75 Ω source resistance when the switch is closed at t = 0, as shown below. Ignore
the resistor shown in the dashed lines for now. The characteristic impedance of
the line is 50 Ω and the one-way transit time of the line is 10 ns. The load
resistance is RL = 150 Ω.
a) Plot v(z = 0.5 m, t) for 0 ≤ t ≤ 50 ns.
b) Plot v(z, t = 32.5 ns) for 0 ≤ z ≤ 2 m.
c) A shunt resistor shown in the dashed line in Figure 1 is introduced and has a
value of RP =37.5 Ω. Determine the amplitude of the incident step function
launched onto the line at t = 0.
d) Repeat part (b) with the resistor RP in place.