CI RF Week 1
RGC
11/06
THE COCKCROFT INSTITUTE
RF Engineering for Particle Accelerators
Week 1: Transmission Lines
Notes
1. The impedance at the input of a transmission line having characteristic impedance Z0 and
length l which is terminated by an impedance ZL is given by:
Z in Z L  jZ 0 tan kl

Z 0 Z 0  jZ L tan kl
where k = 2π/λ and λ is the wavelength on the line.
2. The Voltage Standing Wave Ratio (VSWR) is defined as
1 
V
S  max 
Vmin 1  
where ρ is the voltage reflection coefficient of the termination of the line.
3. The return loss (in dB) is defined as
R  20 log 10 
Exercises
Note: The phase velocity of waves on an air-spaced TEM line is equal to the velocity of light in
free space (0.2998 × 109 m.sec-1).
1. A transmission line is terminated by a resistive load such that the VSWR on the line is 2.0:1.
Calculate the voltage reflection coefficient of the load, the ratio of the load impedance to the
characteristic impedance of the line and the ratio (expressed in decibels) of the amplitude of
the reflected wave to that of the incident wave (known as the Return Loss). (0.33; 2.0; -9.54
dB)
2. A transmission line has a characteristic impedance of 600Ω. Calculate the VSWR on the line
when it is terminated by and impedance (400 + j300)Ω. What is the VSWR if the load is
changed to a purely resistive one of 500Ω? (2.05:1; 1.2:1)
3. A loss-less transmission line of characteristic impedance 70 Ω is terminated by an impedance
R + jX. The VSWR is 2.0:1 and the position of the first voltage maximum is 1/12 wavelength
from the termination. Calculate R and X. (80.6 +j52.0 Ω)
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CI RF Week 1
RGC
11/06
4. A loss-less transmission line has a characteristic impedance of 100 Ω and is terminated by a
load of 75 Ω. If the line is 0.7 wavelengths long find:
(a) The input impedance, and
(b) The reactance which should be connected across the input of the line to make the
input a pure resistance. (77.9 – j13.2 Ω; 470j Ω)
5. Find the position and length of a short-circuit shunt stub which will match a load of (25 –
j25) Ω to a 50 Ω line when the wavelength on the line is 40mm.(2.52 mm from the load; 3.6
mm)
6. It is desired to connect a 75Ω transmission line to a load of impedance 150Ω using a quarterwave transformer. Determine the characteristic impedance of the transformer for perfect
matching. (106 Ω)
7. A loss-less short-circuited air-spaced stub with a characteristic impedance of 300Ω is to have
an input reactance of 400Ω. Find the length of the stub at a signal frequency of 2 GHz if:
(a) The stub is inductive, and
(b) The stub is capacitive. (59.6 mm; 15.4 mm)
8. Design a short-circuit stub to match and impedance of (73 + j42.5) Ω to a transmission line
with a characteristic impedance of 300 Ω at a frequency of 530 MHz. Assume that all the
lines are air-spaced. (231 mm long; 27.2 mm from the load)
9. A loss-less transmission line has a characteristic impedance of 100 Ω and is terminated by
impedance ZT. The VSWR on the line is 8.0:1. The distance between successive minima of
the standing wave is 100 mm and the distance of the termination from the first voltage
minimum is 81 mm. Determine:
(a) The load impedance, and
(b) The position and length of a single stub which would match the load to the line. (17 +
j66 Ω; 12.2 mm long; 70.4mm from the load)
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CI RF Week 1
RGC
11/06
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CI RF Week 1
RGC
11/06
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