INF5481 RF Circuit, Theory and Design
Assignment #1
Problem 1
Show that the input impedance of a lossless transmission line repeats itself
every λ/2, i.e.
for m ∈ N and l, λ ∈ R
Zin (l) = Zin (l + mλ/2)
Problem 2
Given a lossless transmission line with a 100 Ω load and SWR = 1.5, find the
two possible values for the characteristic impedance Z0 .
Problem 3
We have a 75 Ω coaxial transmission line with length l = 20 cm, load ZL =
37.5 + j75 Ω and a dielectric with εr = 2.56 at f = 3 GHz.
(a)
Find the input impedance Zin
(b)
Find the reflection coefficient at the load ΓL = Γ0
(c)
Find the reflection coefficient at the input Γin
(d)
Calculate the SWR.
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INF5481: Assignment #1
Problem 4
Z0 =?
l =?
ZL = 80 + j20 Ω
Zin = 100 Ω
Figure 1: Diagram for problem 4, where Z0 and l is unknown.
You have a load ZL = 80 + j20 Ω which you want to match to 100 Ω, see
figure 1. Find the required Z0 (real) and length l (real).
Assume all transmission lines are lossless and air filled, calculate the
length in both λ and in meters for f = 3 GHz.
TIPS: Express the input impedance Zin as two equations, one for the real
part and another for the complex part. Treating tan(βl) as an unknown, you
now have 2 equations and 2 unkowns.
Problem 5
In this exercise we will explore the voltage at different locations along the
“Quarter-Wave” transmission line in figure 2. We will assume a lossless
transmission line Z0 . Plot the voltage magnitude V (z) for −l ≤ z ≤ 0.
ZG = 100 Ω
Z0 = 100 Ω
ZL = 80 − j40 Ω
VG = 10 V
l = 3λ/2
d=l
z = −l
d=0
z=0
Figure 2: Diagram for problem 5, with generator and load. Also shown are
the two coordinate systems used by the book.
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INF5481: Assignment #1
Problem 6
We now study a “lambda-quarter transformer” shown in figure 3.
ZGs = 50 Ω
Z0 = 50 Ω
ZGp = 25 Ω +
VG = 10 V
ZL = 100 Ω
Vin
-
l = λ/4
Figure 3: Diagram for problem 6
(a)
Convert the length l = λ/4 to meters for the following cases
(i) f = 50 Hz and εr = 1 (found in our wall socket)
(ii) f = 16 MHz and εr = 4 (typical microcontroller)
(iii) f = 3 GHz and εr = 4 (typical CPU)
(iv) f = 5 GHz and εr = 4 (WiFi 802.11ac)
(v) f = 60 GHz and εr = 12 (WiFi 802.11ad)
(b)
Find the reflection coefficent at the load and the standing wave ratio SWR.
(c)
Find the input impedance looking into the transmission line and express the
voltage at the input Vin .
(d)
Calculate the power delivered to the load.
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