Download The impedance transformation circle along a transmission line

NETWORK AND TRANSMISSION LINES
Group A
1) Coaxial cable
Coaxial lines confine virtually all of the electromagnetic wave to the area inside the
cable. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects,
and they can be strapped to conductive supports without inducing unwanted currents in them.
In radio-frequency applications up to a few gigahertz, the wave propagates in the
transverse electric and magnetic mode (TEM) only, which means that the electric and magnetic
fields are both perpendicular to the direction of propagation (the electric field is radial, and the
magnetic field is circumferential). However, at frequencies for which the wavelength (in the
dielectric) is significantly shorter than the circumference of the cable, transverse electric (TE)
and transverse magnetic (TM) waveguide modes can also propagate. When more than one mode
can exist, bends and other irregularities in the cable geometry can cause power to be transferred
from one mode to another.
The most common use for coaxial cables is for television and other signals with bandwidth of
multiple megahertz. In the middle 20th century they carried long distance telephone connections.
A type of transmission line called a cage line, used for high power, low frequency
applications. It functions similarly to a large coaxial cable. This example is the antenna feedline
for a longwave radio transmitter in Poland, which operates at a frequency of 225 kHz and a
power of 1200 kW.
Microstrip
A microstrip circuit uses a thin flat conductor which is parallel to a ground plane.
Microstrip can be made by having a strip of copper on one side of a printed circuit board (PCB)
or ceramic substrate while the other side is a continuous ground plane. The width of the strip, the
thickness of the insulating layer (PCB or ceramic) and the dielectric constant of the insulating
layer determine the characteristic impedance. Microstrip is an open structure whereas coaxial
cable is a closed structure.
2) Signal transfer
Electrical transmission lines are very widely used to transmit high frequency signals over
long or short distances with minimum power loss. One familiar example is the down lead from a
TV or radio aerial to the receiver.
Pulse generation
Transmission lines are also used as pulse generators. By charging the transmission line
and then discharging it into a resistive load, a rectangular pulse equal in length to twice the
electrical length of the line can be obtained, although with half the voltage. A Blumlein
transmission line is a related pulse forming device that overcomes this limitation. These are
sometimes used as the pulsed energy sources for radar transmitters and other devices.
Stub filters
If a short-circuited or open-circuited transmission line is wired in parallel with a line used
to transfer signals from point A to point B, then it will function as a filter. The method for
making stubs is similar to the method for using Lecher lines for crude frequency measurement,
but it is 'working backwards'. One method recommended in the RSGB's radiocommunication
handbook is to take an open-circuited length of transmission line wired in parallel with the feeder
delivering signals from an aerial. By cutting the free end of the transmission line, a minimum in
the strength of the signal observed at a receiver can be found. At this stage the stub filter will
reject this frequency and the odd harmonics, but if the free end of the stub is shorted then the
stub will become a filter rejecting the even harmonics.
Acoustic transmission lines
An acoustic transmission line is the acoustic analog of the electrical transmission line,
typically thought of as a rigid-walled tube that is long and thin relative to the wavelength of
sound present in it
3) Distributed element model
Fig.1 Transmission line. The distributed element model applied to a transmission line.
This article is an example from the domain of electrical systems, which is a special case of the
more general distributed parameter systems.
In electrical engineering, the distributed element model or transmission line model of
electrical circuits assumes that the attributes of the circuit (resistance, capacitance, and
inductance) are distributed continuously throughout the material of the circuit. This is in contrast
to the more common lumped element model, which assumes that these values are lumped into
electrical components that are joined by perfectly conducting wires. In the distributed element
model, each circuit element is infinitesimally small, and the wires connecting elements are not
assumed to be perfect conductors; that is, they have impedance. Unlike the lumped element
model, it assumes non-uniform current along each branch and non-uniform voltage along each
node. The distributed model is used at high frequencies where the wavelength approaches the
physical dimensions of the circuit, making the lumped model inaccurate.
Group B
1) Telegrapher’s equations
The Telegrapher's Equations (or just Telegraph Equations) are a pair of linear
differential equations which describe the voltage and current on an electrical transmission line
with distance and time. They were developed by Oliver Heaviside who created the transmission
line model, and are based on Maxwell's Equations.
Schematic representation of the elementary component of a transmission line.
The transmission line model represents the transmission line as an infinite series of two-port
elementary components, each representing an infinitesimally short segment of the
transmission line:




The distributed resistance of the conductors is represented by a series resistor
(expressed in ohms per unit length).
The distributed inductance (due to the magnetic field around the wires, selfinductance, etc.) is represented by a series inductor (henries per unit length).
The capacitance between the two conductors is represented by a shunt capacitor C
(farads per unit length).
The conductance of the dielectric material separating the two conductors is
represented by a shunt resistor between the signal wire and the return wire (siemens
per unit length).
The model consists of an infinite series of the elements shown in the figure, and that the values
of the components are specified per unit length so the picture of the component can be
misleading. , , , and may also be functions of frequency. An alternative notation is to
use
, , and to emphasize that the values are derivatives with respect to length.
These quantities can also be known as the primary line constants to distinguish from the
secondary line constants derived from them, these being the propagation constant, attenuation
constant and phase constant.
The line voltage
and the current
can be expressed in the frequency domain as
When the elements and are negligibly small the transmission line is considered as a
lossless structure. In this hypothetical case, the model depends only on the and elements
which greatly simplifies the analysis. For a lossless transmission line, the second order steadystate Telegrapher's equations are:
These are wave equations which have plane waves with equal propagation speed in the
forward and reverse directions as solutions. The physical significance of this is that
electromagnetic waves propagate down transmission lines and in general, there is a reflected
component that interferes with the original signal. These equations are fundamental to
transmission line theory.
If
and
where
are not neglected, the Telegrapher's equations become:
and the characteristic impedance is:
The solutions for
The constants
and
and
, starting at
at position
are:
must be determined from boundary conditions. For a voltage pulse
and moving in the positive
-direction, then the transmitted pulse
can be obtained by computing the Fourier Transform,
, attenuating each frequency component by
, of
, advancing its phase by
, and taking the inverse Fourier Transform. The real and imaginary parts of
computed as
where atan2 is the two-parameter arctangent, and
For small losses and high frequencies, to first order in
and
one obtains
can be
Noting that an advance in phase by
simply computed as
is equivalent to a time delay by ,
can be
2) Input impedance of lossless transmission line
The characteristic impedance
of a transmission line is the ratio of the amplitude of a single
voltage wave to its current wave. Since most transmission lines also have a reflected wave, the
characteristic impedance is generally not the impedance that is measured on the line.
For a lossless transmission line, it can be shown that the impedance measured at a given
position from the load impedance
is
where
is the wavenumber.
In calculating , the wavelength is generally different inside the transmission line to what it
would be in free-space and the velocity constant of the material the transmission line is made
of needs to be taken into account when doing such a calculation.
Special cases
[edit] Half wave length
For the special case where
where n is an integer (meaning that the length of the line
is a multiple of half a wavelength), the expression reduces to the load impedance so that
for all . This includes the case when
, meaning that the length of the transmission line
is negligibly small compared to the wavelength. The physical significance of this is that the
transmission line can be ignored (i.e. treated as a wire) in either case.
Quarter wave length
Main article: quarter wave impedance transformer
For the case where the length of the line is one quarter wavelength long, or an odd multiple of
a quarter wavelength long, the input impedance becomes
Matched load
Another special case is when the load impedance is equal to the characteristic impedance of
the line (i.e. the line is matched), in which case the impedance reduces to the characteristic
impedance of the line so that
for all and all .
Short
Main article: stub
For the case of a shorted load (i.e.
), the input impedance is purely imaginary and a
periodic function of position and wavelength (frequency)
Open
Main article: stub
For the case of an open load (i.e.
periodic
), the input impedance is once again imaginary and
3) Stepped transmission line
A simple example of stepped transmission line consisting of three segments.
A stepped transmission line[2] is used for broad range impedance matching. It can be considered
as multiple transmission line segments connected in series, with the characteristic impedance
of each individual element to be Z0,i. The input impedance can be obtained from the successive
application of the chain relation
where is the wave number of the ith transmission line segment and li is the length of this
segment, and Zi is the front-end impedance that loads the ith segment.
The impedance transformation circle along a transmission line whose characteristic impedance
Z0,i is smaller than that of the input cable Z0. And as a result, the impedance curve is offcentered towards the -x axis. Conversely, if Z0,i > Z0, the impedance curve should be offcentered towards the +x axis.
Because the characteristic impedance of each transmission line segment Z0,i is often different
from that of the input cable Z0, the impedance transformation circle is off centered along the x
axis of the Smith Chart whose impedance representation is usually normalized against Z0.
4) Transmission lines
Transmission lines are a common example of the use of the distributed model. Its use is
dictated because the length of the line will usually be many wavelengths of the circuit's operating
frequency. Even for the low frequencies used on power transmission lines, one tenth of a
wavelength is still only about 500 kilometres at 60 Hz. Transmission lines are usually
represented in terms of the primary line constants as shown in figure 1. From this model the
behaviour of the circuit is described by the secondary line constants which can be calculated
from the primary ones.
The primary line constants are normally taken to be constant with position along the line
leading to a particularly simple analysis and model. However, this is not always the case,
variations in physical dimensions along the line will cause variations in the primary constants,
that is, they have now to be described as functions of distance. Most often, such a situation
represents an unwanted deviation from the ideal, such as a manufacturing error, however, there
are a number of components where such longitudinal variations are deliberately introduced as
part of the function of the component. A well known example of this is the horn antenna.
Where reflections are present on the line, quite short lengths of line can exhibit effects
that are simply not predicted by the lumped element model. A quarter wavelength line, for
instance, will transform the terminating impedance into its dual. This can be a wildly different
impedance.
High frequency transistors
Fig.2. The base region of a bipolar junction transistor can be modelled as a simplified transmission line.
Another example of the use of distributed elements is in the modelling of the base region
of a bipolar junction transistor at high frequencies. The analysis of charge carriers crossing the
base region is not accurate when the base region is simply treated as a lumped element. A more
successful model is a simplified transmission line model which includes distributed bulk
resistance of the base material and distributed capacitance to the substrate. This model is
represented in figure 2.
Resistivity measurements
Fig. 3. Simplified arrangement for measuring resistivity of a bulk material with surface probes.
In many situations it is desired to measure resistivity of a bulk material by applying an
electrode array at the surface. Amongst the fields that use this technique are geophysics (because
it avoids having to dig into the substrate) and the semiconductor industry also use it (for the
similar reason that it is non-intrusive) for testing bulk silicon wafers.[2] The basic arrangement is
shown in figure 3, although normally more electrodes would be used. To form a relationship
between the voltage and current measured on the one hand, and the resistivity of the material on
the other, it is necessary to apply the distributed element model by considering the material to be
an array of infinitesimal resistor elements. Unlike the transmission line example, the need to
apply the distributed element model arises from the geometry of the setup, and not from any
wave propagation considerations.[3]
The model used here needs to be truly 3-dimensional (transmission line models are
usually described by elements of a one-dimensional line). It is also possible that the resistances
of the elements will be functions of the co-ordinates, indeed, in the geophysical application it
may well be that regions of changed resistivity are the very things that it is desired to detect.[4]
Inductor windings
Fig. 4. A possible distributed element model of an inductor. A more accurate model will also require
series resistance elements with the inductance elements.
Another example where a simple one-dimensional model will not suffice is the windings
of an inductor. Coils of wire have capacitance between adjacent turns (and also more remote
turns as well, but the effect progressively diminishes). For a single layer solenoid, the distributed
capacitance will mostly lie between adjacent turns as shown in figure 4 between turns T1 and T2,
but for multiple layer windings and more accurate models distributed capacitance to other turns
must also be considered. This model is fairly difficult to deal with in simple calculations and for
the most part is avoided. The most common approach is to roll up all the distributed capacitance
into one lumped element in parallel with the inductance and resistance of the coil. This lumped
model works successfully at low frequencies but falls apart at high frequencies where the usual
practice is to simply measure (or specify) an overall for the inductor without associating a
specific equivalent circuit.