Download Huang Slides 2 V08

Antennas: from Theory to Practice
2. Circuit Concepts and Transmission
Lines
Yi HUANG
Department of Electrical Engineering &
Electronics
The University of Liverpool
Liverpool L69 3GJ
Email: [email protected]
Antennas: from Theory to Practice
1
Objectives of This Chapter
• Review the very basics of circuit concepts;
• Distinguish the lumped element system
from the distributed element system;
• Introduce the fundamentals of transmission
lines;
• Compare various transmission lines and
connectors.
Antennas: from Theory to Practice
2
2.1
Circuit Concepts
• Electric current I is a measure of the charge flow/
movement.
• Voltage V is the difference of electrical potential
between two points of an electrical or electronic circuit.
• Impedance Z = R + jX is a measure of opposition to an
electric current.
Antennas: from Theory to Practice
3
Lumped and Distributed Element Systems
• The current and voltage along a transmission line may
be considered unchanged (which normally means the
frequency is very low). The system is called a lumped
element system.
• The current and voltage along a transmission line are
functions of the distance from the source (which
normally means the frequency is high), thus the system
is called a distributed element system.
Antennas: from Theory to Practice
4
2.2
Transmission Line Theory
• A transmission line is the structure that forms all or
part of a path from one place to another for directing the
transmission of energy, such as electrical power
transmission and microwaves.
• We are only interested in the transmission lines for RF
engineering and antenna applications. Thus dielectric
transmission lines such as optical fibres are not
considered.
Antennas: from Theory to Practice
5
Transmission Line Model
A distributed element system is converted to a lumped one
Antennas: from Theory to Practice
6
Antennas: from Theory to Practice
7
Transmission line equation
Where the propagation constant:
Attenuation
const:
Phase
const:
Antennas: from Theory to Practice
8
The solutions are:
This is the characteristic impedance of the transmission line.
For a lossless transmission line, R = G =0, thus
The industrial standard transmission line normally has a
characteristic impedance of 50 or 75 Ω
Antennas: from Theory to Practice
9
Forward and reverse travelling waves
Velocity:
, so it is also called the wave number
Antennas: from Theory to Practice
10
Lossless transmission lines
• For a lossless transmission line, R = G =0,
Antennas: from Theory to Practice
11
Terminated Transmission Line
• Input impedance and reflection coefficient
Antennas: from Theory to Practice
12
Note: the power reflection coefficient is:
The input impedance
For the lossless case
Antennas: from Theory to Practice
13
Input impedance for special cases
• Matched case (G = 0):
• Open circuit (G = 1):
• Short circuit (G = -1):
• Quarter-wavelength case:
Antennas: from Theory to Practice
14
Example 2.1
A lossless
transmission line
with a characteristic
impedance of 50 
is loaded by a 75 
resistor. Plot the
input impedance as
a function of the line
length (up to two
wavelengths).
Input impedance for ZL = 75  and Z0 = 50  - a period function!
Antennas: from Theory to Practice
15
Return loss
• When the voltage reflection coefficient and power
reflection coefficient are expressed in logarithmic forms,
they give the same result, which is called the return loss
Antennas: from Theory to Practice
16
Example 2.5
A 75  resistor is connected to a low loss transmission
line with characteristic impedance of 50 . The
attenuation constant is 0.2 Np/m at 1 GHz.
a). What is the voltage reflection coefficient for l = 0 and
l/4, respectively?
b). Plot the return loss as a function of the line length.
Assume that the effective relative permittivity is 1.5.
Antennas: from Theory to Practice
17
Antennas: from Theory to Practice
18
Voltage Standing Wave Ratio (VSWR)
• The VSWR (also known as the standing wave ratio,
SWR) is defined as the magnitude ratio of the maximum
voltage on the line to the minimum voltage on the line
Antennas: from Theory to Practice
19
Antennas: from Theory to Practice
20
2.3
The Smith Chart and Impedance Matching
Antennas: from Theory to Practice
21
The Smith Chart
Antennas: from Theory to Practice
22
Example 2.7
Using a Smith
Chart to redo
Example 2.1,
and also display
the reflection
coefficient on the
Chart.
Antennas: from Theory to Practice
23
Impedance Matching
• Impedance matching is the practice of making the output
impedance of a source equal to the input impedance of the
load in order to maximize the power transfer and minimize
reflections from the load. Mathematically, it means the load
impedance being the complex conjugate of the source
impedance.
Ideally:
Generally speaking, resistors are not employed for impedance
matching The lumped matching networks can be divided into
three basic networks: L network, T network and pi () network.
Antennas: from Theory to Practice
24
Antennas: from Theory to Practice
25
Lumped T and P networks
T network which may
be viewed as another
reactance (jX2) added
to the L network
P network can be seen as
an admittance (jB2)
added to the L network
Antennas: from Theory to Practice
26
Example 2.8
A load with an impedance of 10-j100  is to be matched
with a 50  transmission line. Design a matching
network and discuss if there are other solutions
available.
Antennas: from Theory to Practice
27
Distributed matching networks
They can be formed by a quarter-wavelength transmission line, an
open-circuit/short-circuit transmission line, or their combinations.
Antennas: from Theory to Practice
28
Example 2.9
• A load with an impedance of 10-j100  is to be matched
with a 50  transmission line. Design two distributed
matching networks and compare them in terms of the
bandwidth performance.
Antennas: from Theory to Practice
29
A). a short circuit with a stub length l2 = 0.0325l;
B). an open circuit with a stub length l2 = 0.2825l.
Both have achieved a perfect matching at 1GHz but of different bandwidth
Antennas: from Theory to Practice
30
Frequency bandwidth limitation
• There exists a general limit on the bandwidth over which
an arbitrarily good impedance match can be obtained in
the case of a complex load impedance. It is related to the
ratio of reactance to resistance, and to the bandwidth over
which we desire to match the load.
• Take the parallel RC load impedance as an example,
Bode and Fano derived, for lumped circuits, a
fundamental limitation for it and it can be expressed as
Antennas: from Theory to Practice
31
Quality Factor and Bandwidth
• Quality factor, Q, which is a measure of how much
lossless reactive energy is stored in a circuit compared
to the average power dissipated.
where WE is the energy stored in the electric field, WM is the
energy stored in the magnetic field and PL is the average
power delivered to the load.
• Antennas are designed to have a low Q, whereas circuit
components are designed for a high Q.
Antennas: from Theory to Practice
32
where f1 and f2 are the
frequencies at which
the power reduces to
half of its maximum
value at the resonant
frequency, f0 and where
BF is the fractional
bandwidth. This
relation only truly
applies to simple
(unloaded single
resonant) circuits.
Antennas: from Theory to Practice
33
2.4
Various Transmission Lines
Antennas: from Theory to Practice
34
Two-wire Transmission Line
• Characteristic impedance (for lossless line):
Typical value is 300 Ω
Antennas: from Theory to Practice
35
• Fundamental mode
– Both the electric field and magnetic field are within the
transverse (to the propagation direction) plane, thus
this mode is called the TEM (transverse electromagnetic) mode.
• Loss
– the principle loss is actually due to radiation, especially
at higher frequencies. The typical usable frequency is
less than 300 MHz
• Twisted-pair transmission line
– the twisted configuration has cancelled out the
radiation from both wires and resulted in a small and
symmetrical total field around the line; but it is not
suitable for high frequencies due to the high dielectric
losses that occur in the insulation.
Antennas: from Theory to Practice
36
Coaxial Cable
Velocity in
a medium
Antennas: from Theory to Practice
37
• Fundamental mode:
– TEM mode below the cut-off freq
• Characteristic impedance:
The typical value for industrial standard lines is 50 Ω or 75 Ω,
do you know why?
• Loss
Antennas: from Theory to Practice
38
Cable examples
Antennas: from Theory to Practice
39
Microstrip Line
Effective relative permittivity:
thus
- determined by the capacity
Antennas: from Theory to Practice
40
• Characteristic impedance:
, W/d <1
, W/d >1
• Basic mode: quasi-TEM mode if the wavelength
larger than the cut-off wavelength:
Antennas: from Theory to Practice
41
• Loss
• Surface waves and cut-off frequencies
Antennas: from Theory to Practice
42
Antennas: from Theory to Practice
43
Stripline
• Characteristic impedance
Antennas: from Theory to Practice
44
• Fundamental mode: TEM mode if
• Loss
– Similar to that of microstrip, but little radiation loss
and surface wave loss.
Antennas: from Theory to Practice
45
Co-planar Waveguide (CPW)
where
Antennas: from Theory to Practice
46
• Characteristic impedance
where
Antennas: from Theory to Practice
47
• Fundamental mode: quasi-TEM mode
• Loss
– Normally higher than microstrip
Antennas: from Theory to Practice
48
Waveguides
• There are circular and rectangular waveguides which
have just one piece of conductor, and good for high
frequencies (high pass, and low stop).
Antennas: from Theory to Practice
49
Standard waveguides
The frequency range is determined by the cut-off frequencies
of the fundamental mode and the 1st higher mode. The cut-off
wavelength for TEmn and TMmn modes is given by
Antennas: from Theory to Practice
50
• Fundamental mode: TE10 mode
Thus its cut-off wavelength is 2a, and the operational
wavelength should shorter than 2a.
Antennas: from Theory to Practice
51
• Waveguide wavelength: the period of the wave
inside the waveguide.
• Characteristic impedance
Antennas: from Theory to Practice
52
Comparison of transmission lines
Antennas: from Theory to Practice
53
2.5
Connectors
Male (left) and female (right) N-type connectors
Antennas: from Theory to Practice
54
Antennas: from Theory to Practice
55