CCE 5220 – RF and Microwave System Design
Tutorial 3: Transmission Lines
Question 1:
(a) A coaxial transmission line has an outer diameter of 22 mm and an inner copper
conductor of diameter 5 mm. The insulating material has a relative permittivity of 4.5
and relative permeability that can be taken as µ0= 4πx10-7 H/m. The operating frequency
of the line is 1 GHz. At 1 GHz take the a.c resistance to be 0.79 Ω/m and the leakage
conductance to be 9 mS/m. Determine:
(i)
(ii)
(iii)
(iv)
(v)
(b)
The value of the inductance per unit length
The value of the capacitance per unit length
The characteristic impedance of the line
The wavelength on the line
The velocity of propagation of the signal.
[Ans: 2.9632 x 10-7 H/m]
[Ans: 1.689 x 10-10 F/m]
[Ans: 41.88 < 0.233°]
[Ans: 0.141 m]
[Ans: 141 x 106 m/s]
A load impedance is connected at the end of a 1metre length of the line such that it is
correctly terminated, determine the value of the attenuation and phase shift at the
load. [Ans: 0.201 + 44.44j]
Question 2:
(a)
A transmission line is 70 km long and has the primary constants of resistance
R = 10 Ω/loop km; inductance L=2.5 mH/loop km; capacitance C= 15 nF/loop km
and leakage conductance G= 0.5 µS/loop km. Determine for an operating frequency
of 10kHz, the values of:
(i)
(ii)
(iii)
(iv)
(v)
(b)
(c)
The characteristic impedance
[Ans: 408.66 < -1.805°]
The propagation coefficient
[Ans: 0.385 < 88.19°]
The attenuation and the phase shift coefficient [Ans: 0.0122 + j0.385]
The wavelength on the line
[Ans: 16.32 km]
The speed of transmission of a signal
[Ans: 163.2 km/s]
The source generator at the sending end has an EMF Es= 10 V and a source
impedance equal to the characteristic impedance when operating at 10 kHz. Given
that the transmission line is terminated in its characteristic impedance, determine the
values of:
(i)
The sending end current and voltage [12.25 mA<1.805° & Vs = 5 V]
(ii)
The receiving end current and voltage [5.11 mA<-1.797 2.1<-1.828]
At a distance of 20 km from the load, determine
(i)
(ii)
(d)
The value of the current,
The attenuation and phase shift
[6.61 mA<-19.25 rad]
[0.61 8epers 19.25 rad]
Represent a loss-less transmission line by a nominal T section and deduce its ABCD
transmission line parameters.
CCE 5220 – RF and Microwave System Design
Owen Casha 2010©
1
(e)
Assuming that a transmission line is negligible if the length ℓ/λ ≤ 0.01, use simulation
or otherwise to deduce the error introduced by representing the line as the nominal T
of (d) above.
Question 3:
A loss-less transmission line has a characteristic impedance of 600 Ω. The terminating
impedance of the line is 200-j150 Ω. The line is 34 cm long and its wavelength at a
particular frequency is 8 cm. Given that the incident voltage is 10 V, determine:
(a)
The voltage reflection coefficient at the load
[Ans: 0.525<-148.88°]
(b)
The standing-wave ratio of the line
[Ans: 3.211]
(c)
The amplitudes of the voltage and current at the receiving end
[Ans: 4.164 V<-126.66°
(d)
The amplitudes of the reflected voltage and current at the sending end
[Ans: 3.55 V<20.56°
(e)
16.67 mA<-90°]
5.92 mA<-159.51°]
The input impedance, i.e. the impedance seen at the sending end
[Ans: 1440<0.64 rad]
Question 4:
(a)
Explain how a quarter wavelength loss-free line is used to match two impedances
together.
(b)
A 40 m length of an open-wire loss free line (µr=1 and εr=1) has a characteristic
impedance of 600 Ω and is supplied by a voltage source operating at 3 MHz. The line
is terminated in an open circuit. A 600 Ω resistor RA is connected at a distance of
25 m. Determine the input impedance of the transmission line. [Ans: 600 + 825.82j]
(c)
A length of a 50 Ω loss-free short circuited line is to be used to simulate a capacitance
of 50 pF at a frequency of 50 MHz. Determine the length of line required. [2.135 m]
(d)
A 50 Ω open-wire loss-less line (µr=1 and εr=1) has a load impedance of 50-j150 Ω. It
operates at a frequency of 600 MHz. Use the Smith Chart to determine:
(i)
(ii)
(iii)
(iv)
The voltage reflection coefficient at the load
[Ans: 0.83<-146°]
The Voltage standing wave ratio, VSWR
[Ans: 11]
The input impedance of a 1.1m length of line
[Ans: 4.5 – j Ω]
The length of line that has a purely resistive input impedance. State the value
of these resistances. [Ans: 0.101 m and 0.2265 m 4.5Ω 550Ω]
CCE 5220 – RF and Microwave System Design
Owen Casha 2010©
2