Antennas: from Theory to Practice 2. Circuit Concepts and Transmission Lines

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Antennas from Theory to Practice2. Circuit
Concepts and Transmission Lines

• Yi HUANG
• Department of Electrical Engineering
  Electronics
• The University of Liverpool
• Liverpool L69 3GJ
• Email Yi.Huang_at_liv.ac.uk

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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.

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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.

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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.

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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.

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Transmission Line Model
A distributed element system is converted to a
lumped one
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Transmission line equation
Where the propagation constant
Attenuation const
Phase const
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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 ?
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Forward and reverse travelling waves
Velocity
, so it is also called the wave number
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Lossless transmission lines

• For a lossless transmission line, R G 0,

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Terminated Transmission Line

• Input impedance and reflection coefficient

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Note the power reflection coefficient is
The input impedance
For the lossless case
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Input impedance for special cases

• Matched case (G 0)
• Open circuit (G 1)
• Short circuit (G -1)
• Quarter-wavelength case

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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!
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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

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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.

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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

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2.3 The Smith Chart and Impedance Matching
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The Smith Chart
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Example 2.7

• Using a Smith Chart to redo Example 2.1, and
  also display the reflection coefficient on the
  Chart.

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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.
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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
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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.

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Distributed matching networks

• They can be formed by a quarter-wavelength
  transmission line, an open-circuit/short-circuit
  transmission line, or their combinations.

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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.

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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
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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

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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.
• Antennas are designed to have a low Q, whereas
  circuit components are designed for a high Q.

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.
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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.
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2.4 Various Transmission Lines
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Two-wire Transmission Line

• Characteristic impedance (for lossless line)

Typical value is 300 ?
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• 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 electro-magnetic) 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.

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Coaxial Cable
Velocity in a medium
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• Fundamental mode
• TEM mode below the cut-off freq
• Characteristic impedance
• Loss

The typical value for industrial standard lines
is 50 ? or 75 ?, do you know why?
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Cable examples
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Microstrip Line
Effective relative permittivity
thus
- determined by the capacity
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• Characteristic impedance
• Basic mode quasi-TEM mode if the wavelength
  larger than the cut-off wavelength

, W/d lt1
, W/d gt1
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• Loss
• Surface waves and cut-off frequencies

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Stripline

• Characteristic impedance

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• Fundamental mode TEM mode if
• Loss
• Similar to that of microstrip, but little
  radiation loss and surface wave loss.

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Co-planar Waveguide (CPW)
where
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• Characteristic impedance

where
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• Fundamental mode quasi-TEM mode
• Loss
• Normally higher than microstrip

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Waveguides

• There are circular and rectangular waveguides
  which have just one piece of conductor, and good
  for high frequencies (high pass, and low stop).

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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
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• Fundamental mode TE10 mode

Thus its cut-off wavelength is 2a, and the
operational wavelength should shorter than 2a.
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• Waveguide wavelength the period of the wave
  inside the waveguide.
• Characteristic impedance

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Comparison of transmission lines
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2.5 Connectors
Male (left) and female (right) N-type connectors
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