Squint reduction in spiral antennas using
unbalanced feeds
Q-par/AJM/Squint-issues/1/2.0
Cover + 1 + 78 pages
June 2013
Andrew Mackay
This document has been prepared by Q-par Steatite Ltd., and may not be used or copied
without proper authorisation.
© Copyright 2013
Q-par Steatite Ltd., U.K.
Q-par/AJM/Squint-issues/1/2.0
ii
Author
Andrew Mackay
Date
June 2013
Issued by
Q-par Steatite Ltd
Barons Cross Laboratories
Leominster
Herefordshire HR6 HRS
UK.
Q-par/AJM/Squint-issues/1/2.0
iii
Document changes record
Issue
Date
Change summary
Issue 1.1
March 2013
Issue 1.2
April 2013
Issue 1.3
Issue 1.4
April 2013
May 2013
Issue 2.0
June 2013
Formulation for even/odd mode analysis and field
model.
Correction on dispersion characteristics. Missing αN
factor.
Some added material.
Corrected definition for µr0 and ǫr0 to include waveguide
geometric factor.
Inclusion of new software tool for determining characteristic impedance and propagation factors for multiply
layered coax with lossy materials. Theory of QTEM
modes and examples.
Q-par/AJM/Squint-issues/1/2.0
iv
Abstract
This document describes squint problems with spiral antennas using an unbalanced feed
line. It includes a dual mode analysis, a description of the physics of the problem based on
empirical evidence and an approximate field model. Examples are presented showing how
squint reduction can be achieved.
Q-par/AJM/Squint-issues/1/2.0
v
1
Squint and spiral antennas
1.1
Introduction
In cavity backed spiral and sinuous antennas “squint” refers to beam pattern asymmetry,
where the beam shape is not symmetric along a principal plane and where the beam maximum does not lie on bore-sight. It is generally considered undesirable but its definition is
rather loose. There are several problems; firstly there may be more than one local maximum
in the far field and secondly the angular position of the local maximum (or maxima) will
generally vary as a function of frequency and polarisation for a wide band antenna.
Assuming a single maximum for the gain, one common practise is to choose a polarisation
and a single principal plane that passes through the bore-sight of the antenna in angle space.
The squint may then be defined as the angle between bore-sight and the angle of maximum
gain. This may be referred to as the “zero-dB” definition referenced along a particular cut
and polarisation. Another common practise is to define a principal plane and polarisation
and define a reference level x dB below the maximum gain. The “x-dB” squint value is the
angle between bore sight and half the sum of the angles at the reference levels.
The problem with either definition is that the position of the maximum will rotate about
bore sight as a function of frequency so the value of the squint will depend on the chosen
principal plane. Furthermore, the polarisation state of the maximum may also change as a
function of frequency and may not coincide with the nominal polarisation of the antenna.
For example a spiral antenna will radiate a nominally circularly polarised wave, but unless
the axial ratio is zero for all angles and frequencies (which never occurs) the measured squint
will be different for a vertically or horizontally polarised wave. A better definition requires
measurements or predictions over a full range of solid angles and polarisation states, but is
often too time consuming to measure or predict. However, if simulation time is not an issue
it is convenient to define the radial squint. This is the angle θ = θr at which the total gain,
taken as a function of both θ and φ, is maximum (as would be measured by a polarisation
matched receive antenna).
Squint occurs because the antenna is fed by a feed which excites more than the odd (differential) antenna mode. This can be prevented by the use of a balanced feed, but a properly
balanced feed is usually more expensive than an unbalanced feed and difficult if not impossible to construct over a very large bandwidth.
It is common practise to feed the antenna spiral at its centre point using a coaxial or twoline microstrip, parallel strip, slot-line or similar structure feed. These are characterised
by two conducting wires or strips with a real or virtual earth return (see below). The two
wires or strips are attached to each of the two spiral arms of the antenna. If one of these
conducting elements (e.g. the outer coax or one of the microstrip conductors) is joined to
the antenna cavity or side wall then the feed will be unbalanced (except possibly at certain
spot frequencies). An unbalanced feed will generally be characterised by the presence of
Q-par/AJM/Squint-issues/1/2.0
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more than one propagating feed mode within the cavity; in particular an undesirable even
mode in addition to the required odd mode. Generally such an even mode will be excited at
nearly all frequencies1 . In contrast a balanced feed is symmetric with no difference, as seen
by the antenna, between either of the two lines making up the microstrip or coax feed. This
suggests that a balanced feed cannot have one line directly connected to the antenna cavity.
Balanced feeds are generally required to ensure there is negligible squint over a broad band.
There is an important distinction between a balanced feed and an impedance matched feed;
not all impedance matched feeds are balanced and not all balanced feeds are impedance
matched. For example a tapered microstrip feed, with one line grounded, may present a
near perfect impedance match over a wide bandwidth but may be far from balanced and
give rise to significant squint. We will make no distinction between a balanced feed and any
‘balan’ used to feed the antenna so for our purposes the terms will be synonymous.
Unfortunately, at microwave frequencies it is not possible to construct a perfect wideband
balan. In equivalent circuit terms a perfect balan may be represented by a perfect transformer. At low frequencies, e.g. below 1 GHz, real transformers may be made good enough
to provide good performance. At higher frequencies this is prevented by the properties of
real materials and, in particular, the absence of sufficiently low loss conductors and low loss
magnetic materials. Balans must therefore be constructed using canonical structures such
as slot lines and cavities whose bandwidth is inherently limited. Furthermore, the cost of
manufacture can become a significant issue.
In this research note we consider some practical methods, supported by analysis, for squint
reduction in unbalanced feeds.
1.2
Real and virtual earth feeds
A feed with a real earth return features a common earth return line which is not connected to
either of the arms of the antenna. For example, two coax lines whose outer shielded sections
are bonded together. Alternatively, two microstrip lines featuring a common ground plane.
An example of the latter is given in the even-odd mode analysis below. The conductive
structure of the common earth should be symmetrically placed on the axis centre line of the
antenna; i.e. it should be rotationally symmetric with symmetry angle of 2π/N where N is
an even integer. This is because a spiral antenna has the same symmetry and it is important
that the common ground structure does not excite even order modes through symmetry
breaking.
A virtual earth return has no physical conductor for a common earth. The feed consists only
of two wires or conductors such as a parallel strip configuration. In this case the even order
modes are defined by an excitation with both feed conductors at the same potential relative
to the cavity walls or cavity base. If the feed axis lies on the axis of the antenna then this
zero reference level occurs on axis, i.e. mid way between the two feed conductors, and may
be defined as the virtual earth line for the structure.
1 i.e. all frequencies with the exception of possible spot frequencies within the band.
Q-par/AJM/Squint-issues/1/2.0
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In any real antenna, with or without a virtual earth line, earth return currents flow in
the side and base walls of the cavity or in any radar absorbent material or other structure
within the cavity. The odd mode does not require such current flows and the odd mode
antenna impedance is fairly insensitive to the effect of structures (e.g. cavity base and walls)
which supports them. On the other hand, the even mode cannot exist without such currents
and hence the even mode antenna impedance is strongly dependent on these supporting
structures. Since squint is strongly dependent on the even mode antenna impedance for an
unbalanced feed it will in general be strongly dependent on the nature and composition of
the cavity walls and structure within the cavity.
At the risk of stating the obvious, to generate squint it is necessary to excite the antenna
with an even mode. When so excited the magnitude of the squint depends on the even mode
antenna impedance. The excitation may be represented by an even mode voltage, whereas
the effect of the excitation depends on the even mode antenna impedance. So despite the
fact that the even mode antenna impedance depends strongly on the cavity structure, no
squint is generated with a perfectly balanced feed where the even mode feed voltage is zero.
However, we need to consider the effect of both effects with an unbalanced feed.
In a real earth return feed both the currents flowing in the feed and in the cavity can give
rise to squint and these currents may affect the even mode excitation voltage as well as the
even mode antenna impedance. The cavity currents can give rise to two distinct propagating
even order modes associated with the fields within the feed and between the feed and the
cavity walls. If the cavity walls are perfectly conducting both even mode types are TEM in
nature. If the cavity walls are dielectric or free space (i.e. do not exist) then the external
mode is not true TEM. Figures 1-1 and 1-2 show examples of real earth return feed lines
and associated TEM modes for a cavity with conducting walls. The first example shows
two joined single-core coax lines. In this case the characteristic impedance2 of the internal
even and odd modes are the same. In the second example, a single screen is placed around
the two inner conductors. If the screen diameter is small enough there will be only a single
even and a single odd mode internal to the cable with characteristic impedances which are
generally different.
In isolation, i.e. in free space, a real earth return feed is characterised by two propagating
modes, one odd and one even, up to a high frequency (the first over-mode frequency of the
transmission line). In a cavity with conducting walls there are at least three propagating
modes, one odd and two even where the internal even mode characteristic impedance is
generally significantly smaller than the external even mode characteristic impedance. The
latter is approximately that of the TEM mode of the cavity plus feed with the feed considered
as a single conductor with characteristic dimension approximately that of the earth return
conductor. The relative strength of the two even order modes depends on the boundary
conditions and, in particular, how the common ground conductor is connected to the base of
the cavity. Thus the even mode feed voltage (as well as the even mode antenna impedance)
is dependent on cavity structure.
2 Not to be confused with the antenna even and odd mode impedances.
Q-par/AJM/Squint-issues/1/2.0
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If the common feed earth return line is electrically connected to the base of the antenna
only the internal feed modes can excite the antenna. In this case the antenna is excited by
a single even and a single odd internal feed mode and the even mode incident feed voltage
is independent of cavity structure. This construction is common practise for a real earth
return feed but, unless connected to a 180o hybrid or similar ‘transformer’ balancing device,
there will be an even mode present in the antenna excitation.
In contrast a virtual earth return feed must always feature the possibility of an even order
mode external to the feed, i.e. with earth return currents within the cavity that affects the
even mode excitation voltage. However, in practise (see later) the difference between even
mode voltages in a virtual earth feed and even mode voltages in a screened real earth feed
as shown in figure 1-2 is often small if the cavity contains an effective radar absorber.
1.3
An even-odd mode analysis
General analysis of squint for non balanced feeds appears to be a difficult topic with little
reliable published material. We begin by supposing the feed has a real earth return and
may be represented by a single even and a single odd mode and that the modal amplitudes
remain constant throughout the structure. This is assured if the feed is uniform. In addition
we will assume that one end of the feed lines is grounded, so that the feed may be connected
to a single unbalanced source.
We begin by considering the coupled microstrip feed illustrated in figure 1-3 where the
termination at ports 2 and 3 is represented by an antenna “black box” which should be
represented as a 2-port device. In basic analysis (e.g. [1]) of the coupled line the ports 2
and 3 would each be terminated to ground by a load impedance Za , but this is insufficient
for our purposes. As we will presently show, when the characteristic even mode antenna
impedance Zae and the odd mode antenna impedance Zao are different we need a “black
box” representation featuring two different impedance elements Za and Zb .
There are many different possible representations of impedances within the “black box”,
though not all are representable in terms of realisable passive impedances. We will discuss
this later, but for the present just observe the four cases illustrated in the figure.
We employ a modified form of the even-odd mode analysis in section 7.5 (pp380-381) of
[1]. The feed may be decomposed as the sum of an even mode excited by an electric field
(voltage) of strength αe and an odd mode of electric field strength αo . This is shown in
figure 1-4. The modes are added in such a way that the total field strength is unity at port
1 and zero at port 4.
Referring to figure 1-3, the amplitudes of the incident and reflected electric fields are represented by the voltages B1 , B2 , B3 , B4 , A1 , A2 , A3 and A4 where the suffix refers to the port
number. The Ai coefficients represent waves entering the structure and the Bi coefficients
represent waves exiting the structure. We assume a unit incident wave amplitude at port
Q-par/AJM/Squint-issues/1/2.0
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Antenna spiral front face
Pattern rotationally symmetric by 180 degrees
Outer wall may be conductor, free space or dielectric
Conductor base
Real earth return example (two joined sections of coax)
coax internal mode
First even−order modes
First odd−order mode
coax external mode
(Note that the relative strength of the fields within coax and
between coax and cavity walls depends on wall material
and boundary conditions)
Figure 1-1: Real earth return and lowest order propagating modes for a double coax
cable
Q-par/AJM/Squint-issues/1/2.0
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Antenna spiral front face
Pattern rotationally symmetric by 180 degrees
Outer wall, assumed here to be a perfect conductor
Conductor base
Earth return is a single coaxual screen
coax internal mode
First even−order modes
First odd−order mode
coax external mode
(Note that the relative strength of the fields within coax and
between coax and cavity walls depends on how the coax is
electrically attached to the cavity base)
Figure 1-2: Real earth return and lowest order propagating modes for a screened
twin wire conductor
Q-par/AJM/Squint-issues/1/2.0
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Representation of the feed plus antenna
port 4 connected to antenna cavity ground
Z0
A =1
1
A4
B4
B
3
A3
4
3
b
B2
A2
a
2−port antenna
‘‘black box’’
2
1
B1
source connected to port 1
with line characteristic impedance Z0
Representations of the antenna
b
b
a
a
b
a
Za
Za
Zb
Case 0. Valid represention only if
either Za or Zb are possibly
non−realiseable
a
Case 1. Valid representation only
if Z b is possibly non−realiseable
1
1
Za
Za
Zb
2:1
b
Zb
Zb
Case 2. Valid representation only
if Z a is possibly non−realiseable
1
Zb
Za
Case 3. Always realiseable
assuming a perfect transformer
Shown here with a
centre−tapped 2:1 turns ratio
Figure 1-3: A coupled microstrip line feed
Q-par/AJM/Squint-issues/1/2.0
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Reflection and transmission coefficients
for even and odd mode coupled lines
excited by port 1 voltages
α e and α o
Reflection and transmission coefficients
for one symmetry half, unit incident
voltage (canonical transmission line
terminated by antenna load)
α e Re
α e Γe
αe
Z0
α e Te
3
4
1
αe
Γe
Z ae
2
Transmission line
Z 0e
Z0
α e Γe
even mode
α e Te
Z ae
1
1
Z0
4
3
Z 1e
Γo
Zao
α o To
odd mode
αo R o
To
Z0
2
Transmission line
Z0o
α o Γo
Zae
Re
−αo To
1
αo
input impedance
2
−αo R o
−αo Γo
Z0
Z 0e
even mode
α e Re
−αo
Te
Z0
Z ao
1
1
Z 0o
odd mode
input impedance
2
Ro
Z
1o
Figure 1-4: Even odd mode decomposition of the coupled line with even mode fed
by a voltage αe and the odd mode by a voltage αo
Q-par/AJM/Squint-issues/1/2.0
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Zao
1 incident from a semi-infinite medium with characteristic impedance Z0 (we will assume
Z0 = 50Ω) so,
A1 = 1
(1-1)
We will also assume that there is a perfect reflection at port 4, so that
A4 + B4 = 0
(1-2)
We now represent this structure and its voltage coefficients as the sum of an even and odd
mode superposition. The even mode structure assumes an incident wave amplitude of αe
Volts at port 1, the odd mode structure assumes an incident wave amplitude of αo Volts at
port 1. The odd mode voltages at ports 4 and 3 are the negative of the voltages at ports 1
and 2 whereas the even mode voltages at ports 4 and 3 are the same as the voltages at ports
1 and 2. These are illustrated in the left half of figure 1-4. The coefficients Γe and Γo are
the even and odd mode reflection coefficients of a unit amplitude fed structure as illustrated
in the right half of figure 1-4. Similarly, Te and To are the transmission amplitudes at ports
2 and 3 and Re and Ro the reflection amplitudes into/from the even and odd mode antenna
loads Zae and Zao . The even and odd mode input impedances, looking into port 1, are
defined as Z1e and Z1o .
The modal decomposition implies,
A1
A2
A3
A4
=
=
=
=
αe + αo = 1
α e Re + α o Ro
α e Re − α o Ro
αe − α0
B1
B2
B3
B4
=
=
=
=
αe Γ e + αo Γ o
αe Te + αo To
αe Te − αo To
αe Γ e − αo Γ o
(1-3)
and
(1-4)
The reflection coefficients Γe and Γo are easiest to determine using characteristic impedances.
Let Z0e and Z0o be the characteristic impedances of the even and odd modes. Let the
characteristic propagation constants be βe and βo and let the feed line be of length L. The
impedances Z1e and Z1o represent the input impedances looking into port 1 for the two
modes terminated by an odd or even mode antenna impedance. For a spiral antenna above
a cavity the even mode and odd mode antenna impedances are generally different. If these
are Zae and Zao then the port 1 input impedances are given by,
Z1e = Z0e
Q-par/AJM/Squint-issues/1/2.0
Zae + jZ0e tan(βe L)
Z0e + jZae tan(βe L)
(1-5)
Page 9 of 78
and
Z1o = Z0o
Zao + jZ0o tan(βo L)
Z0o + jZao tan(βo L)
with
(1-6)
Γe =
Z1e − Z0
Z1e + Z0
(1-7)
Γo =
Z1o − Z0
Z1o + Z0
(1-8)
and
Using (1-2) together with (1-3) and (1-4) we find,
αo =
1 + Γe
2 + Γe + Γo
αe =
1 + Γo
2 + Γe + Γo
(1-9)
In order to determine the port 2 and port 3 coefficients we also need to determine Te , Re ,
To and Ro . This is rather more involved. First we determine the scattering S matrix for
the section of even or odd mode transmission line with respect to a port 1 and port 2
characteristic reference impedance of Z0 . See figure 1-5, illustrating the wave amplitudes at
the interfaces. Because the line is reciprocal and symmetric for both modes, we know that
S11 = S22 and S12 = S21 . The relevant interface equations for the even mode are,
(e)
1 + S11
= t1 + w 1
(e)
(1 − S11 )/Z0 = (t1 − w1 )/Z0e
(e)
t1 exp(−jβe L) + w1 exp(jβe L) = S21
(e)
t1 exp(−jβe L) − w1 exp(jβe L) /Z0e = S21 /Z0
(1-10)
and similarly for the odd mode. These may be solved to obtain,
(e)
S11
2 (1 + Z0 /Z0e ) − e−2jβe L (1 − Z0 /Z0e )
=
−1
(1 + Z0 /Z0e )2 − e−2jβe L (1 − Z0 /Z0e )2
(e)
S21 =
for the even mode and
(o)
S11
4e−jβe L Z0 /Z0e
(1 + Z0 /Z0e )2 − e−2jβe L (1 − Z0 /Z0e )2
(1-11)
(1-12)
2 (1 + Z0 /Z0o ) − e−2jβo L (1 − Z0 /Z0o )
−1
=
(1 + Z0 /Z0o )2 − e−2jβo L (1 − Z0 /Z0o )2
(1-13)
4e−jβo L Z0 /Z0o
=
(1 + Z0 /Z0o )2 − e−2jβo L (1 − Z0 /Z0o )2
(1-14)
(o)
S21
Q-par/AJM/Squint-issues/1/2.0
Page 10 of 78
port 1
port 2
characteristic impedance
Z 0e or Z 0o
Z0
Z0
t 1exp(−j β L )
t1
1
S 21
S 11
w1 exp(j β L )
w1
L
Figure 1-5: Determination of the scattering matrix for the odd or even mode transmission line, β ≡ βe or βo .
for the odd mode. Referring to the right hand of figure 1-4, and under the definition of the
scattering matrix, we then obtain,
(e)
(e)
Γe = S11 + S12 Re
(1-15)
(e)
S22 Re
(1-16)
Te
(1-17)
Γo = S11 + S12 Ro
(1-18)
Te =
where
Re =
and similarly,
(e)
S21
+
Zae − Z0
Zae + Z0
(o)
To =
where
(o)
S21
(o)
+
(o)
S22 Ro
(1-19)
Zao − Z0
To
(1-20)
Ro =
Zao + Z0
These equations may be solved to obtain expressions for Te , Re , To and Ro together with an
alternative derivation for Γe and Γo (useful for numerical tests),
(e) Zae −Z0
(e)
S
21
Zae +Z0
S21
Re =
Te =
(e) Zae −Z0
(e) Zae −Z0
1 − S22 Zae +Z0
1 − S22 Zae +Z0
(1-21)
(o)
To =
Q-par/AJM/Squint-issues/1/2.0
1−
S21
(o)
S22
Zao −Z0
Zao +Z0
Ro =
(o)
S21
1−
(o)
S22
Zao −Z0
Zao +Z0
Zao −Z0
Zao +Z0
Page 11 of 78
We now have explicit expressions for the Ai and Bi coefficients given models for Z0e , Z0o ,
Zae and Zao . Realisable representation for these impedances will be given in sections 1.3.4.
See also section 3.2.
1.3.1 Lumped element representations of the antenna
Referring to figure 1-3 we have represented the antenna by four cases. In case 0, ports 2 and
3 are terminated by lumped elements Za and Zb . This is the simplest representation but we
have no guarantee that either Za or Zb is realisable. Let us first consider the antenna as a
two port “black box” with terminals a and b connected to ports 2 and 3 of the coupled line,
respectively. We need to consider both its impedance matrix, Z ant defined by,
Z ant =
Zaa Zab
Zba Zbb
(1-22)
Saa Sab
Sba Sbb
(1-23)
and its equivalent scattering matrix, S ant ,
S ant =
where Sab = Sba since the antenna is reciprocal. The connection between the two is canonical,
as given is table 4.2, p211 of [1].
The scattering matrix is related to the coefficients A2 , A3 , B2 and B3 through,
A2 = Saa B2 + Sba B3
A3 = Sba B2 + Sbb B3
(1-24)
where we observe that there are two equations but three free scattering matrix parameters;
i.e. the scattering matrix and therefore the impedance matrix is underdetermined.
For Case 0, Zaa = Za , Zbb = Zb and Zab = Zba = 0. If Zae 6= Zao it is straight forward to
show that Za 6= Zb . This is not physically appealing since it is difficult to directly relate the
electrical imbalance in the model to the physical symmetry of the antenna. The reason is
that the zero potential point of the antenna does not coincide with the physical centre of
the antenna when an even mode is excited. We will show later that power can flow from
port 2 into port 3 or vice-versa which will mean that either Za or Zb can be non realisable
as a stand-alone passive device.
Making use of the freedom inherent in the model for the impedance or scattering matrix of
the antenna, we will in our other case representations assume Zaa = Zbb or, equivalently,
Saa = Sbb . This uniquely defines the impedance matrix in terms of the A and B coefficients
and ensures a symmetry to the impedance structure that mirrors the geometric symmetry
of the antenna. Cases 1, 2 and 3 all have this structure.
Q-par/AJM/Squint-issues/1/2.0
Page 12 of 78
Given Zaa = Zbb we obtain,
Saa = Sbb =
and
A3 B3 − A2 B2
B32 − B22
A2 B3 − A3 B2
B32 − B22
The canonical transformations are then given by,
Sab = Sba =
(1-25)
(1-26)
Zaa = Z0
2
2
1 − Saa
+ Sba
2
(1 − Saa )2 − Sba
(1-27)
Zba = Z0
2
2Sba
2
(1 − Saa )2 − Sba
(1-28)
and
Next we need to determine the lumped elements Za and Zb in our representations 1, 2 and
3. To do this, note the definition of the impedance matrix. If Va is the voltage at port a, Ia
is the current flowing into port a and Ib is the current flowing into port b then Zaa is the
impedance Va /Ia with no current flowing into port b, i.e. port b open circuit. Similarly, Zba
is the impedance Va /Ib with port a open circuit. We therefore find;
For Case 1
Zaa = Zbb = Za + Zb
Zab = Zba = Zb
(1-29)
It is possible to relate Za and Zb to the even and odd mode antenna impedances using the
assumed symmetry. This is shown in figure 1-6 for which we obtain,
Za = Za0
Zb = (Zae − Zao )/2
(1-30)
Since Zae and Zao are arbitrary realisable impedances it follows that Zb may not be realisable;
in particular if ℜ(Zao ) > ℜ(Zae ).
For Case 2
Zaa = Zbb =
Zb (Za + Zb )
Za + 2Zb
Zab = Zba =
Zb2
Za + 2Zb
(1-31)
Again referring to figure 1-6 we obtain,
Za =
2Zae Zao
Zae − Zao
Zb = Zae
(1-32)
Since Zae and Zao are arbitrary realisable impedances it follows that Za may not be realisable.
Q-par/AJM/Squint-issues/1/2.0
Page 13 of 78
For Case 3
We use the rule that a transformer with turns ratio N : 1 primary-to-secondary transforms
a secondary load impedance Z to a primary load impedance N 2 Z, a secondary voltage V to
a primary voltage N V and a secondary current I to a primary current I/N . We may then
show that,
Zaa = Zbb = Za + Zb
Zab = Zba = Za − Zb
(1-33)
and referring to figure 1-6 we obtain,
Za = Za0 /2
Zb = Zae /2
(1-34)
This is the most natural representation since there is a direct correspondence of lumped
element to antenna mode impedances and the lumped element values are realisable.
CASE 1 antenna representation
+V
Za
Port b
Zb
Za
Zao
Za
Za
Zao
Za
CASE 2 antenna representation
Odd mode equivalence
Zb
Za
Zb
Zb
Zb
Even mode equivalence
+V
Zao
Zb
Zao
Zb
Zae
Zae
0
+V
+V
2:1
Even mode equivalence
+V
+V
Zae
Zao
2:1
Zb
Za
1
Za
Zae
Zao
Port b
−V
0
−V
Odd mode equivalence
CASE 3 antenna representation
Zb
0
Za
Port b
−V
1
Zae
0
+V
+V
+V
Port a
Zae
Zb
−V
−V
1
+V
+V
+V
Za
Port a
Port a
Even mode equivalence
Odd mode equivalence
−V
0
0
Figure 1-6: Case 1, 2 and 3 decompositions in terms of even and odd mode antenna
impedances
Q-par/AJM/Squint-issues/1/2.0
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1.3.2 Special cases of interest
A case of interest is when the total reflection from the feed line plus antenna is close to zero,
i.e. B1 = 0. Using (1-4) and (1-9) this implies,
Γe =
−Γo
1 + 2Γo
(1-35)
If the antenna is to be squintless we require B2 = −B3 . Using (1-4) this implies that Te = 0.
This can be achieved if there are sufficient even-mode losses and if the feed line is sufficiently
long to permit this (see later). Alternatively, if there are no losses then this implies that
Γe = ±1. Using (1-35), if Γe = −1 then Γ0 = −1 which implies that both Te and To are zero
which corresponds to the unuseful case B2 = B3 = 0. If Γe = +1, then Γo = −1/3. This
corresponds to the ratio αo /αe = 3 which also happens to be the maximum realisable value
of this ratio (assuming B1 = 0) subject to the realisability requirements |Γo |, |Γe | ≤ 1.
Examination of (1-5) to (1-8) shows that this condition can be met over a non zero frequency
interval if and only if,
Z0o = Z0 /2
Z0e → ∞
(1-36)
This result makes physical sense but is unachievable in practice since it is not possible to
obtain a characteristic impedance Z0e → ∞ without a perfect magnetic material requiring a
lossless relative permeability µr → ∞. In practise, materials with large values of real valued
permeability are only found at low frequencies (< 1 GHz).
In conclusion we see that methods to reduce squint are available at low frequencies, typically
less than 1 GHz. Here we can readily obtain balan transformers or we can use magnetic
materials such as ferrite loaded composites which can be used to increase the characteristic
even mode impedance to values greater than that of free space. At higher frequencies we
must rely on electric losses and (to some extent, and if the operational frequency range
is not too high) magnetic losses. Such losses must be applied such that the even mode
transmission coefficient Te can be reduced without significant impact on the odd mode
transmission coefficient To , which is difficult to achieve on coupled microstrip since substrate
losses impact on both modes.
1.3.3 Power distribution
In order to quantify performance in an imperfect balanced feed, it is necessary to determine
the total power radiated by the antenna, the powers radiated at ports 2 and 3 or the powers
radiated in the particular antenna modes as well as the reflected power from the combined
system. It is also useful to check power balance; always a useful diagnostic. From basic
principals, the time-averaged power dissipated into a complex load,
1
P = ℜ(V I ∗ )
2
Q-par/AJM/Squint-issues/1/2.0
(1-37)
Page 15 of 78
where V is the voltage across the load, I ∗ is the complex conjugate of the current through
the load and ℜ(Z) represents the real part of a complex quantity Z. For our purposes, we
will define the peak voltage normalised (PVN) power,
P̂ = 2Z0 P
(1-38)
This is a useful normalisation when analysis uses E-field coefficients; for an incident wave of
field strength 1 Volt in a medium of characteristic impedance Z0 , P̂ = 1 Volt squared. This
means that all PVN powers dissipated in loads within the structure define the dissipated
power fractions, as a fraction of the incident power, given a source of 1 Volt in a medium of
characteristic impedance Z0 .
The PVN power dumped by port 2 into the antenna is thus,
P̂2 = |B2 |2 − |A2 |2
(1-39)
and the PVN power dumped by port 3 into the antenna is,
P̂3 = |B3 |2 − |A3 |2
(1-40)
The total power radiated by the antenna is thus P̂2 + P̂3 and we have the inequality,
0 ≤ P̂2 + P̂3 ≤ 1
(1-41)
by energy conservation. However, it is possible that either P̂2 < 0 or P̂3 < 0 if power flows
between the two ports. This will be shown in some later simulations and proves that Za or
Zb will not be realisable in our case 1 representation of the antenna as a two port network.
The reflected PVN power at port 1 is given as,
P̂ref = |B1 |2
(1-42)
and hence the total PVN power dumped into the outside world is given by,
P̂tot = P̂ref + P̂2 + P̂3
(1-43)
In general 0 < P̂tot ≤ 1 with equality to 1 if the transmission line is lossless, i.e. if the
characteristic impedances Z0o and Z0e are both real valued. This provides a useful diagnostic.
Next we need to determine the power radiated by each of the antenna (odd and even)
modes. Using our case 3 antenna representation this amounts to determining the power
absorbed by the lumped impedances Za = Zae /2 and Zb = Zae /2. Using the impedance
matrix representation of the case 3 network we evaluate the currents and voltages through
and across the loads. Referring to figure 1-7, we have,
Ia + Ib = Iγ = Vγ /Zb
Va − Vb = Vα − Vβ
(Vα − Vβ )/2 = Vδ = Iδ Za
Q-par/AJM/Squint-issues/1/2.0
(1-44)
Page 16 of 78
Zb
Vα
Ia
Va
Port a
Iγ
Vγ
Vβ
1
Ib
1
1
Port b
Iδ
Za
V
Vb
δ
Case 3 antenna representation
Figure 1-7: Case 3 network analysis for evaluating powers dissipated in the loads
So the VPN power dissipated into Za , representing the odd antenna mode, is given by,
P̂o = Z0 ℜ(Vδ Iδ⋆ ) =
Z0 |Va − Vb |2 ℜ(Za )
4|Za |2
(1-45)
and the VPN power dissipated into Zb , representing the even antenna mode, is given by,
P̂e = Z0 ℜ(Vγ Iγ⋆ ) = Z0 |Ia + Ib |2 ℜ(Zb )
(1-46)
But,
Va = A2 + B2
Vb = A3 + B3
(1-47)
and the currents may be determined from the inverse of the impedance matrix,
1
Va
Zaa −Zba
Ia
= 2
2
Vb
−Zba
Zaa
Ib
Zaa − Zba
(1-48)
so that we may obtain,
P̂o = Z0 |A2 + B2 − A3 − B3 |2
ℜ(Za )
4|Za |2
(1-49)
P̂e = Z0 |A2 + B2 + A3 + B3 |2
ℜ(Zb )
4|Zb |2
(1-50)
and
where, as we showed above, Za = Zao /2 and Zb = Zae /2.
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1.3.4 Realisable dispersive models for the characteristic impedances Z0e and Z 0o
When designing wide band feeds using lossy materials the impedances must be frequency
dependent and of a correct form to ensure realisability. A convenient way to do this is to
represent using an equivalent circuit. If an equivalent circuit exists in terms of resistors,
inductors and capacitors then realisability is guaranteed. The converse is also true; i.e. an
equivalent circuit must exist if the impedance is physically realisable, with the proviso that
the equivalent circuit may feature an infinite number of components.
Another useful property of this representation is that, when modelling the characteristic
impedances of a transmission line, the correct associated frequency dependence of the propagation factor (wave number in the material) is automatically generated. The theory is
described in sections 3 and 4 of [2].
The characteristic impedance, K, of a transmission line (for our purposes K may be either
Z0e or Z0o ) and propagation constant β (where β is either βe or βo ) may be represented
in terms of an effective relative permittivity ǫr and an effective relative permeability µr or
the lumped series impedance Zs per unit length and lumped parallel admittance Yp per unit
length,
q
p
js p
β(s) = k0 ǫr µr =
ǫr µr = −j Zs (s)Yp (s)
(1-51)
c0
and
s
r
µr
Zs (s)
=
(1-52)
K(s) = η0
ǫr
Yp (s)
where c0 is the speed of light in vacuum, η0 is the impedance of free space (not to be confused
with the input line impedance Z0 ), η0 ≈ 377Ω and the signs to the square roots are chosen
so that ℑ(β) < 0 for passive materials assuming an ejωt time convention. Here, s = jω and
the functional dependence on s is shown explicitly.
If the propagating TEM fields are planar then ǫr = ǫr and µr = µr , the standard definitions
of relative permittivity and permeability. More generally they are different. The relations
may be inverted to obtain expressions as a function of ω given by,
µr (ω) = +
c0 Zs (ω)
jωη0
ǫr (ω) = +
c 0 η0
jω Zp (ω)
(1-53)
where we define Zp (ω) = 1/Yp (ω). Note the positive sign is shown explicitly. A negative sign
(mathematically permitted through inversion of (1-51) and (1-52)) is not physically possible
for passive materials. Questions of realisability for ǫr and µr (as defined for plane waves) are
covered in more detail in [2], but note there is a typographic error there in equation (13).
The expression for µr in [2] should have η0 s in the denominator as given in (1-53).
Q-par/AJM/Squint-issues/1/2.0
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The key idea for realisability is that, irrespective of the definitions of effective relative permittivity and permeability (which are dependent on the nature of the TEM modal fields),
the transmission line representation remains valid and the per unit length series impedances
and parallel admittances must be representable in terms of passive equivalent circuits. We
may then choose a base model with no losses and then add extra inductor/capacitor/ resistor
components to represent losses under various frequency dependent dispersions. The use of
such passive components is a necessary condition for realisability and is therefore strongly
recommended as a modelling method. The equivalent circuit representations are subject to
certain additional constraints associated with their properties at zero and infinite frequencies and associated with assumptions on the transmission line boundary conditions. For
example, in a TEM waveguide comprising perfect conductors, the series impedance elements
must look perfectly inductive at zero frequency and the parallel admittance elements must
look perfectly capacitive at infinite frequency.
In more detail, let the lumped impedance Zs per unit length and lumped admittance Yp
per unit length each be represented by equivalent circuits as shown in figure 1-8. When
represented by discrete components it is necessary to introduce an arbitrary scale length
D, but this does not effect the equivalent circuit representation. The extra constraints due
to the boundary conditions at zero and infinite frequency require that Yp is capacitive at
very high frequency and Zs must look inductive at very low frequency. This means that Zs
can be represented by the parallel combination of an inductor Ls and another quite general
impedance element Zs′ such that Zs′ contains no parallel inductive element. Similarly that Yp
can be represented by the parallel combination of a capacitor Cp and another quite general
impedance element Zp′ such that Zp′ contains no parallel capacitive element.
The simplest lossy realisation is given when Zs′ and Zp′ are purely resistive, i.e. Zs′ = Rs and
Zp′ = Rp . It is useful to define a characteristic line impedance in the absence of loss ZN , and
a lossless propagation constant βN . Let us then define3 ,
βN
(1-54)
k0
Both ZN and βN are real valued with no dependence on frequency. They are realisable only
if ZN ≥ 0 and αN ≥ 1. The realisability conditions are set for different reasons. That for
ZN is required in order to ensure the material is passive. That for αN is to ensure that the
velocity of the wave (either phase or group) does not exceed that of the speed of light in
vacuum. In notation, a subscript ‘e’ or ‘o’ refers to the associated quantity for an even or
odd mode respectively.
αN =
We now consider a starting point for the model in terms of requirements in the absence of loss.
For this purpose we consider a lossless effective relative permittivity ǫr0 and permeability
µr0 and relate these to impedances ZN and wavenumber modifier αN ,
r
µr0
(1-55)
ZN = η 0
ǫr0
3 Not to be confused with the voltage coefficients
Q-par/AJM/Squint-issues/1/2.0
αo and αe defined in the modal analysis section of this report
Page 19 of 78
and
αN =
p
µr0 ǫr0
(1-56)
where η 0 is the geometry dependent characteristic impedance when there is no dielectric/magnetic material present. This may be defined by,
η 0 = f 0 η0
(1-57)
where η0 ≈ 377Ω is the characteristic impedance of free space and f0 is a TEM mode
geometry factor. In free space f0 = 1. For a more general pure TEM waveguide mode4
f0 6= 1. For example, in concentric coaxial circular guide with inner conductor radius a and
outer conductor radius b,
1
b
f0 =
loge
2π
a
ZN and αN may be used to define the principal terms of an equivalent circuit representation
ensuring realisability. Thus we may define,
Ls = αN ZN /c0 = η 0 µr0 /c0
Cp = αN /(ZN c0 ) = ǫr0 /(η 0 c0 )
(1-58)
and
Zs (ω) = jωLs ||Rs
1
1
= Zp (ω) =
||Rp
Yp (ω)
jωCp
(1-59)
where the symbol || is the parallel addition operator,
x || y ≡
1
1/x + 1/y
(1-60)
Again, these forms apply equally to even and odd mode impedances with designation of a
subscript ‘e’ to denote the even mode and ‘o’ to denote the odd mode. Thus Lse and Rpe
would refer to an even mode inductance and resistance and similarly Cpo and Rpo would
refer to the odd mode capacitance and resistance. Note that Zs and Rs have units of Ohms
per metre, Zp and Rp have units of Ohm metres. Rs represents the magnetic loss and Rp
represents the electric loss in the transmission line.
A special case of interest is a coupled transmission line with electric (non-magnetic) losses
only, characteristic of carbon loaded radar absorbent materials and for which αN = 1. Choice
of αN = 1 is realisable in principal but not in practice because it requires that µr0 = 1/ǫr0 .
This is highly unlikely because there are no known lossless diamagnetic materials for which
µr = 1/ǫr for values of ǫr that differ significantly from unity. However it is useful to choose
αN = 1 to show the effect of changes in characteristic line impedance without any consequent
changes in the electrical length of the line.
4 For which there is a single contiguous region of homogeneous dielectric which separates two disjoint conducting
regions.
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Page 20 of 78
Transmission line made from discrete elements
D Zs
DYp
D
Characteristic scale length
D Z s and DYp have units of Ohms and Ohms −1
Realisable representation
Realisable representation
Zs
Zs
Cp
Zp
Zp
Ls
Figure 1-8: Lumped impedance representation of characteristic impedance of a
transmission line
If the losses are small enough such that they do not effect the electrical match at the ports
of the device, then it is possible to ignore the imaginary part of the characteristic impedance
and the losses are due only to the propagation constant. The power dissipation in such a
line decays by a factor of F = |exp(−jβL)|2 over a line of length L. Using our equivalent
circuit representation in the absence of magnetic losses, i.e. Rs → ∞, we obtain,
p
β(ω) = ω Ls Cp 1 −
If
j
ωRp Cp
1/2
λ0 ZL
≪1
2πRp
(1-61)
(1-62)
where λ0 is the free space wavelength then,
F ≈ exp
−L ZL
Rp
(1-63)
Thus, if a non-resonant electric radar absorber is characterised by a transmission loss of χ
dB per metre, we obtain the relationship between χ and Rp ,
χ≈
ZL
10 Log10 (e) ≈ 4.34ZL /Rp dB per metre
Rp
(1-64)
As an example suppose we consider an even mode loss, where we assume ZL = 200Ω < Z0
at 10 GHz. If we assume λ0 ZL /(2πRp ) = 0.2 may be considered a reasonably small loss
then we find Rp ≈ 5 Ω m. This equates to a transmission loss of χ ≈ 0.17 dB per mm.
Q-par/AJM/Squint-issues/1/2.0
Page 21 of 78
1.3.5 Virtual earth feed lines
With a virtual earth return feed, the above even/odd mode analysis is not strictly valid
because the fields between the pair of conducting lines and the rest of the cavity structure
cannot be expressed by a single mode bounded by the cavity wall. However, numerical
evidence from CST with a range of geometries suggests that when the cavity is filled with low
density RAM (excluding a small volume near the antenna front face and in the immediate
neighbourhood of the feed lies) that the existence of an additional perfectly conducting
cylindrical screen makes little difference to the shape and squint of the radiated antenna
fields, provided the screen is not too close to the feed pair. This is illustrated in figure
1-9. We conclude from this that the electric field between the feed line pair and the cavity
boundaries is very similar to the first even mode field that exists when the feed line pair is
screened.5
This indicates that the voltage induced on the antenna using a virtual earth return feed
can be approximately modelled by a dual mode real earth return feed provided the RAM is
low density; i.e. a RAM with relative permittivity whose magnitude is not too large. CST
models using Emerson RAM types ECCOSORB LS-18, LS-22 and LS-24 have been used to
make this observation. This indicates that the even/odd mode analysis described above is
appropriate.
5 This may not be the case if a different kind of RAM (e.g. high density or magnetic RAM) is used within the screened
section replacement.
Q-par/AJM/Squint-issues/1/2.0
Page 22 of 78
grounded screen
RAM
RAM
grounded line
grounded line
Input
Input
antenna front face
antenna front face
cavity
cavity
transmission line
transmission line
RAM
RAM
grounded screen
Input
WITH SCREEN
Input
WITHOUT SCREEN
Figure 1-9: Generated squint is found to be nearly the same with and without an
internal screen for a low density RAM filled cavity
Q-par/AJM/Squint-issues/1/2.0
Page 23 of 78
1.3.6 Tapered lines
Usually, a virtual earth feed includes a smooth impedance transition between the characteristic impedance of the input line Z0 and the impedance measured between the feed points
of the antenna, Zf . If the antenna lies in free space with no even mode excitation, αN o = 1
and Zf = 2Zao . In our case 0 antenna representation Zf = 2Za = 2Zb . In our case 1
representation Zf = 2Za (Zb irrelevant). In our case 2 representation Zf = Za || (2Zb ). In
our case 3 representation (physically most useful), Zf = 4Za .
According to spiral antenna theory, and found to be the case in practise, Zf ≈ 85Ω. But for
a uniform line with no reflection loss we require Z0o = Z0 /2 = 25 Ω and Zao = Z0o = 25 Ω.
This would require Zf to be 50 Ω. In order to transform between the required 85 Ω and
twice the odd mode impedance of the line requires a non-uniform line which is not modelled.
However, provided the line is long enough to permit a sufficiently gradual characteristic
impedance transformation over the bandwidth of the antenna, the line can be represented
as a uniform line with a perfect transformer between the end of the line and the antenna
load Zao or Zae for both the even and odd modes. While the impedance transformer will
generally be different for the even and odd modes, the effect is that the uniform line model
can be used with suitably transformed values of Zae and Zao .
In a virtual earth system comprising a double strip line of track width w, Z0o is approximately
proportional to 1/w (provided w ≫ t, for substrate thickness t) whereas Z0e is approximately
proportional to log(1/w). This means that a strip line taper will have little effect on Z0e while
having a significant effect on Z0o . For a gradually tapered line it is thus appropriate to model
Z0o and Z0a as the characteristic impedances near the beginning of the line, nearest ports 1
and 4, leave Zae unaltered and scale Zao by a factor of 50/85 ≈ 0.59 for a matched antenna.
In other words we assume Z0o = Zao = 25 Ω for a matched antenna under the uniform line
plus transformer model, while Zae should take its true (though usually unknown) value.
Q-par/AJM/Squint-issues/1/2.0
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1.4
Numerical examples
In this section we demonstrate the even/odd mode model both with and without losses. We
plot the PVN power dumped by ports 2 and 3, P̂2 and P̂3 , the total radiated power by the
antenna P̂2 + P̂3 , the total reflected power P̂ref and the power balance P̂tot as a function of
frequency for some representative couple lines. We also plot the powers radiated in the even
and odd modes P̂e and P̂o .
Determination of P̂e and P̂o is a prerequisite for determination of squint. Clearly feasible
means to reduce P̂e , and how this can best be achieved, is a useful study. However, to relate
these quantities directly to squint requires a model for the antenna beam patterns for the
even and odd antenna modes. This is something we consider in the next section. Here, we
give examples showing how P̂e and P̂o are related to some control factors.
1.4.1 Example 1. Lossless characteristic impedances
In this example we consider a 50 mm line (L = 50 mm). We assume characteristic line
impedances Z0o = 25Ω, Z0e = 50Ω, αN o = αN e = 1, Za0 = 25Ω and Zae = 100Ω. In this
case we assume no loss and no dispersion. An odd line impedance Z0o = 25Ω equates to a
50Ω strip-to-strip double strip impedance measured between strips. Figure 1-10 shows the
reflected power P̂ref and the powers radiated by the even and odd antenna modes, P̂e and P̂o
expressed in dB as a function of frequency. Figure 1-11 shows the powers exiting the ports
2 and 3, P̂2 and P̂3 expressed as scalars as a function of frequency. Note that the power out
of port 3 is negative, signifying that power always flows from port 2 to port 3 in this case.
Because there are no line losses, power balance P̂tot should be exactly equal to one. This is
found to be the case numerically to within arithmetic precision. One might expect squint to
be a serious problem with the assumed impedances, since powers in the even and odd modes
become equal at periodic frequencies.
1.4.2 Example 2. Lossless characteristic impedances
In this example we consider a 50 mm line (L = 50 mm). We assume characteristic line
impedances Z0o = 25Ω, Z0e = 100Ω, αN o = αN e = 1, Za0 = 25Ω and Zae = 50Ω. Again
we assume no loss and no dispersion. An odd line impedance Z0o = 25Ω equates to a
50Ω strip-to-strip double strip impedance measured between strips. Figure 1-12 shows the
reflected power P̂ref and the powers radiated by the even and odd antenna modes, P̂e and
P̂o expressed in dB as a function of frequency. Figure 1-13 shows the powers exiting the
ports 2 and 3, P̂2 and P̂3 expressed as scalars as a function of frequency. Note that here the
power out of both ports is positive.
Because there are no line losses, power balance P̂tot should be exactly equal to one. Again
this is found to be the case numerically to within arithmetic precision.
Q-par/AJM/Squint-issues/1/2.0
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Example 1. No loss. 50 mm line. Z0o=25 Ohms, Z0e=50 Ohms, Za0=25 Ohms, Zae=100 Ohms
0
Powers in dB
-5
Reflection loss, |B1|2
Even mode antenna power
Odd mode antenna power
-10
-15
-20
0
5
10
Frequency in GHz
15
20
Figure 1-10: Example 1, reflection and radiated antenna mode powers in dB
Example 1. No loss. 50 mm line. Z0o=25 Ohms, Z0e=50 Ohms, Za0=25 Ohms, Zae=100 Ohms
1
Powers as scalar fractions
0.8
0.6
Power out of port 2
Power out of port 3
0.4
0.2
0
-0.2
0
5
10
Frequency in GHz
15
20
Figure 1-11: Example 1, power flow out of ports 2 and 3, expressed as scalars
Q-par/AJM/Squint-issues/1/2.0
Page 26 of 78
Example 2. No loss. 50 mm line. Z0o=25 Ohms, Z0e=100 Ohms, Za0=25 Ohms, Zae=50 Ohms
0
Powers in dB
-5
-10
2
Reflection loss, |B1|
Even mode antenna power
Odd mode antenna power
-15
-20
-25
0
5
10
Frequency in GHz
15
20
Figure 1-12: Example 2, reflection and radiated antenna mode powers in dB
Example 2. No loss. 50 mm line. Z0o=25 Ohms, Z0e=100 Ohms, Za0=25 Ohms, Zae=50 Ohms
1
Powers as scalar fractions
0.8
0.6
Power out of port 2
Power out of port 3
0.4
0.2
0
-0.2
0
5
10
Frequency in GHz
15
20
Figure 1-13: Example 2, power flow out of ports 2 and 3, expressed as scalars
Q-par/AJM/Squint-issues/1/2.0
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1.4.3 Example 3. Characteristic impedances with electrical loss
In this example we again consider a 50 mm line (L = 50 mm). We assume lossy characteristic
line impedances defined by the lossless parts ZN o = 25Ω and ZN e = 100Ω, αN o = αN e = 1
and electrical losses defined by Rpo = 50 Ω m and Rpe = 5.0 Ω m. Antenna impedances are
taken as Za0 = 25Ω and Zae = 50Ω. This example is thus the same as example 2 with the
addition of a very small odd mode loss and a more significant even mode loss.
Referring to section 1.3.4 this is equivalent to the use of a TEM mode absorber6 with an even
mode transmission loss of approximately 0.17 dB per mm and an odd mode transmission
loss of 0.017 dB per mm.
Figure 1-14 shows the reflected power P̂ref and the powers radiated by the even and odd
antenna modes, P̂e and P̂o expressed in dB as a function of frequency. Figure 1-15 shows the
powers exiting the ports 2 and 3, P̂2 and P̂3 expressed as scalars as a function of frequency.
Observe that although the average power radiated by the odd mode has been reduced by
between 0.5 and 1 dB, there has been a significant reduction in the power flowing into the
even mode.
Suppose the feed comprises a double strip line of the sort previously discussed (either with
real earth return or with virtual earth return). This result suggests that if there is scope for a
small reduction in average antenna gain as a function of frequency, squint can be significantly
reduced if a microwave absorber with sufficient loss can be placed in the neighbourhood of
the double strip line.
6 Note that a TEM mode absorber is one designed to work within a coaxial or other TEM mode configuration.
This is not the same as specified by a transmission loss for a plane wave at normal incidence, which is usually how
manufacturers specify transmission loss in carbon loaded absorbers.
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Example 3. 50 mm line. ZNo=25 Ohms, Rpo=50 Ohm metres, ZNe=100 Ohms, Rpe=5 Ohm metres
Za0=25 Ohms, Zae=50 Ohms
0
-5
2
Powers in dB
Reflection loss, |B1|
Even mode antenna power
Odd mode antenna power
-10
-15
-20
-25
0
5
10
Frequency in GHz
15
20
Figure 1-14: Example 3, reflection and radiated antenna mode powers in dB
Example 3. 50 mm line. ZNo=25 Ohms, Rpo=50 Ohm metres, ZNe=100 Ohms, Rpe=5 Ohm metres
Za0=25 Ohms, Zae=50 Ohms
1
Powers as scalar fractions
0.8
0.6
Power out of port 2
Power out of port 3
0.4
0.2
0
-0.2
0
5
10
Frequency in GHz
15
20
Figure 1-15: Example 3, power flow out of ports 2 and 3, expressed as scalars
Q-par/AJM/Squint-issues/1/2.0
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2
Antenna fields
2.1
Introduction
In the previous section we provided an even/odd mode analysis to establish the powers
radiated into the even and odd antenna modes given assumptions on the even and odd
characteristic impedances and antenna mode impedances. However, we need to establish
the antenna patterns before direct calculation of squint is possible. This is difficult to do
with any accuracy unless we conduct a full modal analysis. Numerical methods, such as
CST, can establish the antenna patterns although such simulations can only provide checks
on the quality of analytic models. It is common to simulate the odd modes using a source
point excitation of the two arms of the spiral. Even modes require a feed line model, with
(for example) a paired strip line electrically connected at one end and excited with respect
to the cavity housing.
In what follows we will use a very simple model for both modes. The simplicity provides a
generic analysis which does not account for detailed differences in the geometry of the spiral
antenna, the cavity shape and the details of the feed and absorber within the cavity. There
is scope for further refinement, based on full wave models, should this be useful.
In any analytic model the first requirement is that the even and odd modes should be
orthogonal. Specifically if the electric fields, represented as complex vector-valued functions
of the direction angles θ and φ are given by E e (θ, φ) and E o (θ, φ), then
ZZ
E e (θ, φ).E ⋆o (θ, φ) dΩ = 0
(2-1)
Ω
where the ⋆ represents the complex conjugate and dΩ represents an element of solid angle
over the surface of the unit sphere Ω. This means that the integrated power in the modes,
over all directions, is independent of the relative phasing between modes. Thus if the total
field is given by,
(2-2)
E tot = αe E e (θ, φ) + αo E o (θ, φ)
for arbitrary complex scalars αe and αo , then the peak electric field normalised power7 ,
P (θ, φ) = |E tot (θ, φ)|2 , may be integrated such that,
ZZ
ZZ
ZZ
2
2
2
|E e (θ, φ)| dΩ + |αo |
|E o (θ, φ)|2 , dΩ
P (θ, φ) dΩ = |αe |
(2-3)
Ω
Ω
Ω
I.e. the sum of the powers in the even and odd modes becomes equal to the power of the
sum of the even and odd modes. This is necessary for consistency with our previous model
of independent impedances for the even and odd modes. Note that in spherical coordinates,
ZZ
Z 2πZ π
X(θ, φ) sin θ dθ dφ
(2-4)
X(θ, φ) dΩ ≡
Ω
0
0
7 Note that P (θ, φ) differs by a factor of 2/η , where η is the characteristic impedance of free space, from the mean
0
0
power density definition as used in (for example) equation (5.75) of [3].
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The even and odd fields may be written in the form,
E e (θ, φ) = ue Ẽ e (θ, φ)
E o (θ, φ) = uo Ẽ o (θ, φ)
(2-5)
where ue and uo are normalisation constants and Ẽ e (θ, φ) and Ẽ o (θ, φ) represent the modal
fields. Furthermore, it is convenient to represent each of the vectors in terms of a θ directed
and φ directed component,
Ẽ o = Ẽθ,o θ̂ + Ẽφ,o φ̂
Ẽ e = Ẽθ,e θ̂ + Ẽφ,e φ̂
(2-6)
If we assume that the radiated power from the antenna is divided between the even and
odd modes and that the modes are orthogonal, then each mode must be independently
normalised. If |Ẽ e (θ, φ)|2 and |Ẽ o (θ, φ)|2 represent the even and odd mode directivities,
ZZ
|Ẽ o |2 dΩ = 4π
(2-7)
Ω
and
ZZ
where
Ω
|Ẽ e |2 dΩ = 4π
|Ẽ o |2 = |Ẽθ,o |2 + |Ẽφ,o |2
|Ẽ e |2 = |Ẽθ,e |2 + |Ẽφ,e |2
(2-8)
(2-9)
This fixes the magnitudes of the coefficients |ue | and |uo | but not their phases.
2.2
A simple model for the modal fields
In our basic model we will assume that the modal fields for a spiral antenna are independent
of frequency over the operational band of the antenna. This is not true, but is a fair
approximation for the odd mode8 . It is far less true for the even mode (as can be seen from
full wave analysis using CST), but it still remains a useful approximation.
We will next assume that the modes for a spiral are of similar form to those radiated by
a circular flat plate of radius a whose effective radius varies with frequency in a specific
manner. This allows us to use a modal form that had been much studied in patch antenna
design. In contrast, although numerical representations and measurements of real spiral
modes are available in the literature, no analytic representations of these modes seem to be
available.
8 One of the desirable features of a good spiral antenna is just this property
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Referring to section 5.2 (pp 320-324) of [3], the nth mode of a linearly polarised circular flat
plate antenna may be approximated by,
E n = Enθ θ̂ + Enφ φ̂
where
V ak0
cos nφ Jn′ (k0 a sin θ) .F3 (θ)
2
Jn (k0 a sin θ)
V ak0
sin nφ
cos θ .F4 (θ)
= nj n
2
k0 a sin θ
Enθ = −j n
Enφ
(2-10)
(2-11)
(2-12)
where we omit the radial dependence factor exp(−jk0 r)/r because we are only concerned
with the far-field gain of the antennas. Here, V represents a mode dependent excitation
voltage, Jn and Jn′ represent the Bessel and first derivative Bessel functions of order n, k0
represents the free space wave number and F3 and F4 represent mode independent correction
factors to account for substrate thickness and the effect of a ground plane.
These modes can be extended to represent a circularly polarised antenna using a complex
superposition. To do this, we add a second pair of modes whose amplitude is a factor j
times the first set when φ is transformed as φ → φ − π/(2n) for the nth mode. This is
legitimate because there is no preferred origin for φ and we are free to excite any mode with
any amplitude. This leads us to the new representation for a circularly polarised flat plate
antenna with,
′
′
E n = Enθ
θ̂ + Enφ
φ̂
(2-13)
where
V ak0
(cos nφ + j sin nφ) Jn′ (k0 a sin θ) .F3 (θ)
2
Jn (k0 a sin θ)
V ak0
(sin nφ − j cos nφ)
cos θ .F4 (θ)
= nj n
2
k0 a sin θ
′
Enθ
= −j n
′
Enφ
(2-14)
(2-15)
The odd numbered values of n may in principal be added in arbitrary manner to generate
a single spiral antenna odd mode and the even numbered values of n added in arbitrary
manner to generate a single even mode. Based on observation we find that the odd mode of
a spiral antenna can be approximated by the n = 1 mode. Can the n = 0 mode represent
the even mode? Let us assume that it can, a-priori, and examine the consequences.
Note that in numerical and analytical investigation it is sufficient to evaluate only the J1 (z)
and J0 (z) Bessel functions. All their derivatives and higher orders may be evaluated using
the Bessel identities, e.g.
J1 (z)
J1′ (z) = J0 (z) −
(2-16)
z
for arbitrary complex argument z (see [4], section 9.1.27).
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Let us first consider a large spiral antenna in free space fed by an odd mode. In this case
we assume that the radiation pattern is independent of frequency, dipole-like with a central
forward lobe and with a field that tends to some small value when θ = π/2. Examination of
2-14 suggests that the k0 a term in the Bessel derivative must be replaced by a constant, νo .
If the field is identically zero for θ = π/2 and does not vanish for smaller values of θ then
′
νo must be the first zero of the derivative of J1 (z), νo = j11
≈ 1.84118. However, it may be
desirable to assume a small but non-zero value of the field when θ = π/2. For the present
we will merely assume that,
′
1 ≤ νo ≤ j11
(2-17)
The lower limit of 1 is arbitrary, but corresponds to the value at which the effective circumference of the disk is equal to one wavelength. The power ratio |J1′ (1)/J1′ (0)|2 (the ratio of
the directivity in the θ component of the field at θ = 0 and at θ = π/2) is approximately −3.7
dB. Observations for a real spiral antenna suggest a much smaller value for the directivity
at θ = π/2 suggesting a value of νo somewhat larger than unity.
If we were to assume that the spiral antenna resonates at the same frequency for even and
odd modes with the same effective radius, then for the even modes we would replace the k0 a
term in the Bessel function by the same value as the odd mode. More generally we might
assume a different effective radius and replace by another constant, νe , if there is cause to
do so. We will discuss these alternatives below.
In addition, we will assume that the above description is valid only for 0 ≤ θ ≤ π/2. In the
lower hemisphere observation shows that the fields radiated by a cavity backed spiral are
small. We will therefore assume that all fields are identically zero for π/2 < θ ≤ π. Clearly
the fields should be small when θ = π/2 to minimise the non-physical field discontinuity
there.
These arguments lead to an approximation for the circularly polarised odd mode,
Ẽθ,o = −uo (cos φ + j sin φ)J1′ (νo sin θ) F3 (θ)
Ẽφ,o = uo (sin φ − j cos φ)
J1 (νo sin θ)
cos θ F4 (θ)
νo sin θ
(2-18)
and, if we assume the n = 0 mode is suitable for the even mode,
Ẽθ,e = −ue J1 (νe sin θ) F3 (θ)
Ẽφ,e = 0
(2-19)
where uo and ue are real valued positive normalisation coefficients, whose magnitude is
defined by (2-8) and (2-7). Note that it is straight forward to establish the orthogonality
(2-1) of the even and odd modes since the cos φ, sin φ are present only in the odd mode.
In [3] the functions F3 (θ) and F4 (θ) are taken as mode independent factors which model the
effect of a dielectric substrate of finite thickness and the effect of a ground plane. In free
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space9 ,
F3 (θ) = F4 (θ) = 1
(2-20)
and in the presence of a substrate with relative permittivity ǫr and thickness h and ground
plane,
p
2 cos θ ǫr − sin2 θ
p
F3 (θ) = p
(2-21)
ǫr − sin2 θ − jǫr cos θ cot(k0 h ǫr − sin2 θ)
2 cos θ
p
p
F4 (θ) =
(2-22)
cos θ − j ǫr − sin2 θ cot(k0 h ǫr − sin2 θ)
If we assume that the antenna is situated above a radar absorber with real part of dielectric
constant close to unity with k0 h large enough to absorb all energy in the downward direction
then,
p
cot(k0 h ǫr − sin2 θ) → j
(2-23)
and we have,
F3 (θ) ≈ cos θ
F4 (θ) ≈ 1
(2-24)
However, there is a problem which soon becomes apparent. The n = 0 representation of
the even mode has to be linearly polarised in the θ direction. A consequence of this is that
there will always exist an angle φ for which a linear combination of even and odd modes
(assuming both modes are present) is linearly polarised. Moreover, the value θ at which
this occurs is less than π/2 and significantly less when the even and odd modes are excited
in similar amounts. By definition, the axial ratio at this angle is infinite. This property
of the fields conflicts with observations where the axial ratio is bounded and generally not
very large even when there is significant squint present (for which even and odd modes are
excited to similar levels). This rules out the use of the n = 0 mode as a representation of
the even mode.
It might be expected that the use of the next even mode, n = 2, would solve these difficulties.
Unfortunately, it turns out that this is not the case. Although both n = 1 and n = 2 modes
are circularly polarised, the ratio of the θ and φ components of the n = 2 mode is different
to the ratio in the n = 1 mode. This again leads to a combination of modes which, through
interference, generates non-observed linear polarisations for values of θ ≪ π/2.
Numerical investigation of a spiral antenna excited by an even mode source, using CST,
shows that the field phase varies by a factor of 4π for non-zero θ which is characteristic of
an n = 2 mode. However, it is also observed that the θ and φ directed field magnitudes vary
with φ like cos(2φ + c) for some constant c. This behaviour cannot be modeled using any
single mode of the form (2-14) or (2-15) and we are forced to move away from the circular
disk model for the even mode.
9 We scale the functions differently and define
F3 = F4 = 1 in free space, loosing the factor of 2 employed in [3].
For our purposes a change by a constant factor is unimportant since we normalise the modes.
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In an effort to resolve the linear polarisation problem, we suggest a model which assumes a
modified n = 2 mode whose ratio between θ and φ components is taken to be the same as
the n = 1 mode. This has the effect of keeping axial ratio low even when there are significant
levels of squint, in line with observations of real spirals. However, it still predicts that the
magnitudes of the θ and φ directed even mode components are independent of φ. While
not satisfactory, it at least does not generate strongly non-physical behaviour. In this “base
line” model, we require νe = νo = ν and take,
Ẽθ,o = −uo (cos φ + j sin φ)J1′ (ν sin θ) F3 (θ)
Ẽφ,o = uo (sin φ − j cos φ)
J1 (ν sin θ)
cos θ F4 (θ)
ν sin θ
Ẽθ,e = ue (cos 2φ + j sin 2φ)J2′ (ν sin θ) F3 (θ)
Ẽφ,e = −ue (sin 2φ − j cos 2φ)
J2′ (ν sin θ)J1 (ν sin θ)
cos θ F4 (θ)
J1′ (ν sin θ) ν sin θ
(2-25)
′
Choice of ν, subject to (2-17), is still arbitrary. However we cannot take a value of ν → j11
′
without a non-physically large value of fields as θ → π/2 due to the presence of J1 in the
denominator of the Ẽφ,e expression. For current modelling purposes we will assume,
ν = π/2
(2-26)
This corresponds to an odd mode resonance for a disk whose diameter is half a wavelength
and to a value of the odd mode directivity in free space (for which F3 (θ) = 1) approximately
−13 dB at θ = π/2 relative to the directivity at θ = 0. Using this model with F3 (θ) = cos θ
and F4 (θ) = 1 we find that the even mode directivity has a maximum at θ ≈ ±50.2o .
In this model,
|uo |2 =
2
Io
and
|ue |2 =
2
Ie
(2-27)
where,
Io =
Z
π/2
0
(J1′ (ν
J12 (ν sin θ) cos2 θ
2
sin θ)) |F3 (θ)| +
|F4 (θ)| sin θ dθ
(ν sin θ)2
2
2
(2-28)
and
Ie =
Z
π/2
0
(J2′ (ν sin θ))2 |F3 (θ)|2 +
J2′ (ν sin θ)J1 (ν sin θ)
J1′ (ν sin θ) ν sin θ
2
!
|F4 (θ)|2 sin θ dθ
(2-29)
These integrals need to be evaluated numerically. In the “base line” model, Io ≈ 0.106 and
Ie ≈ 0.0497.
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Using our previously defined even and odd mode power coefficients, the radiated far field
coefficient is given by,
q
q
−jχ
(2-30)
F (θ, φ) = P̂e Qe Ẽ e + e
P̂o Qo Ẽ o
where Qe and Qo are efficiency factors, 0 < Qe , Qo ≤ 1, that describe the fraction of the
even or odd mode power absorbed by the RAM within the cavity and χ is a real phase that
describes the phase relationship between the even and odd modes. This may be written as
the sum of two phases,
χ = χf + χs
(2-31)
where χf is the phase difference due to the differences between even and odd mode feed
phases and χs is the phase difference due to the spiral itself. Since we have assumed that
the effective radius of the antenna at which the antenna radiates is inversely proportional
to the wavenumber or frequency, it follows that to first approximation the path length from
feed point to radiation launch is also inversely proportional to wavenumber. This implies
that χs is a constant, independent of frequency. In reality, second order effects are likely
to be important and are probably dependent on the spiral geometry. Since χs represents a
phase difference between modes, and the even mode is not very independent of frequency,
χs is likely to be quite geometry dependent.
On the other hand, χf may be approximated as the difference in the real part of the even
and odd mode feed propagation factors,
χf = L ℜ(βo − βe )
(2-32)
For our purposes we will assume χs is a constant and define the origin of φ such that χs = 0.
The efficiency factors will also be dependent on frequency and the nature and position of
the RAM within the cavity. However, for present purposes we will simply assume that
approximately half the radiated energy of the spiral is absorbed in the RAM and thus set,
Qe = Qo = 0.5
(2-33)
The realised gain of the antenna is then given by,
G(θ, φ) = |F (θ, φ)|2
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(2-34)
Page 36 of 78
2.2.1 Some observations using the “base line” model
At normal incidence (θ = 0) the axial ratio (defined in dB) is always zero. Furthermore,
the axial ratio is independent of φ. Its value as a function of θ is determined by the odd
mode which is in turn controlled by the value of ν assumed in the model. Smaller values of
ν result in a more constant value of axial ratio closer to zero. For present purposes we will
assume ν = π/2 is a fixed model parameter. In this model the axial ratio is independent of
the ratio of even and odd modes and so is independent of frequency and of φ. This is not
found to be true in studies of real antennas.
Furthermore the model predicts that the magnitudes of the θ and φ components of the even
mode are independent of φ. This is not observed in numerical situations. However, attempts
to change this for example by introducing a non unity constant factor q in the even mode,
cos 2φ + j sin 2φ → cos 2φ + j q sin 2φ
sin 2φ − j cos 2φ → sin 2φ − j q cos 2φ
(2-35)
again fall foul of the axial ratio problem previously discussed. No simple solution of this
problem has been found.
The existence of a frequency dependent phase factor, χ, is to rotate the principal plane for
which the squint is a maximum. No attempt is made to accurately model this since there is
too great an uncertainty in the even mode. For example, numerical simulations suggest that
the even mode of a real spiral antenna varies significantly with frequency. In the “base line”
model, the even mode has a maximum gain at θ ≈ ±50.2o . This is within the range observed
for a real antenna and consistent with several references (e.g. [5]), although towards the top
end of observations.
It must be emphasised that the field model has serious limitations and is no substitute
for detailed predictions or measurements. Predicted values for squint and predicted beam
shapes are unlikely to be more than representative. The main purpose of the model is as a
tool to establish methods for squint reduction and to establish trends in behaviour.
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2.3
Numerical simulations
In this section we use the “base line” model to predict typical antenna patterns and predict
the radial squint assuming feed excitations of the kind made previously. Radial squint is
predicted using a simple scan over a regular mesh of θ and φ points. This is why it does not
appear to be a very smooth function of frequency. However, given the uncertainties in the
model, further refinement is probably unnecessary.
Note that care is required in the evaluation of the Bessel functions for small argument
in numerical implementation. The Bessel functions J0 (x) and J1 (x) are evaluated using
standard libraries in double precision arithmetic. Next, use is made of the Bessel identities
[4],
J2′ (x) = J1 (x) −
J2 (x) =
2J2 (x)
x
2
J1 (x) − J0 (x)
x
and the power series expansions for small argument to obtain,
x2
+ O(x4 )
J0 (x) = 1 −
4 2
x
x
4
J1 (x) =
+ O(x )
1−
2
8
x2
J2 (x) =
+ O(x4 )
8
x
J2′ (x) =
+ O(x3 )
4
x
J2 (x)
=
+ O(x3 )
x
8
for x → 0
2.3.1 Example 1 - continued
Here we consider our first example with lossless impedances as defined in section 1.4.1.
Figures 2-1 and 2-2 show the realised gain in dBi, relative to a polarisation matched antenna,
at 0.5 GHz and 4.5 GHz, for three values of φ = 0, 45o and 90o . There is no rotation of the
position of the gain maximum with φ since for a lossless TEM mode, βe = βo and hence
χ = 0 under our assumptions. In reality χ will vary somewhat with frequency.
Figures 2-3 and 2-4 show the corresponding predictions of axial ratio where there is clearly
no dependence on φ or frequency. Figure 2-5 shows an estimate of the radial squint as a
function of frequency.
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2.3.2 Example 2 - continued
Here we consider our second example with lossless impedances as defined in section 1.4.2.
Gains and axial ratios are qualitatively similar to those shown above so will not be plotted
here. Figure 2-6 shows an estimate of the radial squint as a function of frequency.
2.3.3 Example 3 - continued
Here we consider our third example with lossy impedances as defined in section 1.4.3. Figures
2-7 and 2-8 show the realised gain in dBi, relative to a polarisation matched antenna, at 0.5
GHz and 5.0 GHz, for three values of φ = 0, 45o and 90o . Figures 2-9 and 2-10 again show
the lack of dependence of axial ratio on φ or frequency. Figure 2-11 shows an estimate of
the radial squint as a function of frequency.
2.3.4 Example 4
In this example we consider a 50 mm line (L = 50 mm). We assume lossy characteristic
line impedances defined by the lossless parts ZN o = 25Ω and ZN e = 100Ω, αo = αe = 1 and
electrical losses defined by Rpo = 50 Ω m and Rpe = 1.5 Ω m. Antenna impedances are taken
as Za0 = 25Ω and Zae = 50Ω. This example is the same as example 3 with an increase in
the even mode loss.
Figure 2-12 shows the powers in the even and odd modes and figure 2-13 shows an estimate
of the radial squint. A comparison with example 3 shows that the addition of further even
mode loss is effective at reducing squint.
2.3.5 Some conclusions
It would appear that absorption of the even mode in the feed is effective at reducing squint.
The degree of squint reduction is, however, dependent on our field model which is known to be
inaccurate. Using reasonable values for characteristic impedances and antenna impedances
would suggest that the even mode may need to be better than ∼ −20 dB down on the odd
mode power in order to keep squint below 5 degrees.
It would seem sensible to study the effect of realistic radar absorbers in proximity to the
feed line in order to achieve the necessary relative reduction of even to odd modes.
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Example 1. No loss. 50 mm line. Z0o=25 Ohms, Z0e=50 Ohms, Za0=25 Ohms, Zae=100 Ohms
10
Realised gain (matched) in dB
5
0
-5
0.5 GHz, phi=0
0.5 GHz, phi=45
0.5 GHz, phi=90
-10
-15
-20
-80
-60
-40
-20
0
20
Angle theta in degrees
40
60
80
Figure 2-1: Example 1, Realised gain at 0.5 GHz
Example 1. No loss. 50 mm line. Z0o=25 Ohms, Z0e=50 Ohms, Za0=25 Ohms, Zae=100 Ohms
10
Realised gain (matched) in dB
5
0
-5
4.5 GHz, phi=0
4.5 GHz, phi=45
4.5 GHz, phi=90
-10
-15
-20
-80
-60
-40
-20
0
20
Angle theta in degrees
40
60
80
Figure 2-2: Example 1, Realised gain at 4.5 GHz
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Example 1. No loss. 50 mm line. Z0o=25 Ohms, Z0e=50 Ohms, Za0=25 Ohms, Zae=100 Ohms
20
Axial ratio in dB
15
0.5 GHz, phi=0
0.5 GHz, phi=45
0.5 GHz, phi=90
10
5
0
-80
-60
-40
-20
0
20
Angle theta in degrees
40
60
80
Figure 2-3: Example 1, Axial ratio in dB at 0.5 GHz
Example 1. No loss. 50 mm line. Z0o=25 Ohms, Z0e=50 Ohms, Za0=25 Ohms, Zae=100 Ohms
20
Axial ratio in dB
15
4.5 GHz, phi=0
4.5 GHz, phi=45
4.5 GHz, phi=90
10
5
0
-80
-60
-40
-20
0
20
Angle theta in degrees
40
60
80
Figure 2-4: Example 1, Axial ratio in dB at 4.5 GHz
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Example 1. No loss. 50 mm line. Z0o=25 Ohms, Z0e=50 Ohms, Za0=25 Ohms, Zae=100 Ohms
45
40
Radial squint in degrees
35
30
25
20
15
10
5
0
0
2
4
6
8
10
Frequency in GHz
Figure 2-5: Example 1, Radial squint in degrees
Example 2. No loss. 50 mm line. Z0o=25 Ohms, Z0e=100 Ohms, Za0=25 Ohms, Zae=50 Ohms
45
40
Radial squint in degrees
35
30
25
20
15
10
5
0
0
2
4
6
8
10
Frequency in GHz
Figure 2-6: Example 2, Radial squint in degrees
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Example 3. 50 mm line. ZNo=25 Ohms, Rpo=50 Ohm metres, ZNe=100 Ohms, Rpe=5 Ohm metres
Za0=25 Ohms, Zae=50 Ohms
10
Realised gain (matched) in dB
5
0
-5
0.5 GHz, phi=0
0.5 GHz, phi=45
0.5 GHz, phi=90
-10
-15
-20
-80
-60
-40
-20
0
20
Angle theta in degrees
40
60
80
Figure 2-7: Example 3, Realised gain at 0.5 GHz
Example 3. 50 mm line. ZNo=25 Ohms, Rpo=50 Ohm metres, ZNe=100 Ohms, Rpe=5 Ohm metres
Za0=25 Ohms, Zae=50 Ohms
10
Realised gain (matched) in dB
5
0
-5
5.0 GHz, phi=0
5.0 GHz, phi=45
5.0 GHz, phi=90
-10
-15
-20
-80
-60
-40
-20
0
20
Angle theta in degrees
40
60
80
Figure 2-8: Example 3, Realised gain at 5.0 GHz
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Example 3. 50 mm line. ZNo=25 Ohms, Rpo=50 Ohm metres, ZNe=100 Ohms, Rpe=5 Ohm metres
Za0=25 Ohms, Zae=50 Ohms
20
Axial ratio in dB
15
0.5 GHz, phi=0
0.5 GHz, phi=45
0.5 GHz, phi=90
10
5
0
-80
-60
-40
-20
0
20
Angle theta in degrees
40
60
80
Figure 2-9: Example 3, Axial ratio in dB at 0.5 GHz
Example 3. 50 mm line. ZNo=25 Ohms, Rpo=50 Ohm metres, ZNe=100 Ohms, Rpe=5 Ohm metres
Za0=25 Ohms, Zae=50 Ohms
20
Axial ratio in dB
15
5.0 GHz, phi=0
5.0 GHz, phi=45
5.0 GHz, phi=90
10
5
0
-80
-60
-40
-20
0
20
Angle theta in degrees
40
60
80
Figure 2-10: Example 3, Axial ratio in dB at 5.0 GHz
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Example 3. 50 mm line. ZNo=25 Ohms, Rpo=50 Ohm metres, ZNe=100 Ohms, Rpe=5 Ohm metres
Za0=25 Ohms, Zae=50 Ohms
45
40
Radial squint in degrees
35
30
25
20
15
10
5
0
0
2
4
6
8
10
Frequency in GHz
Figure 2-11: Example 3, Radial squint in degrees
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Example 4. 50 mm line. ZNo=25 Ohms, Rpo=50 Ohm metres, ZNe=100 Ohms, Rpe=1.5 Ohm metres
Za0=25 Ohms, Zae=50 Ohms
0
-5
2
Powers in dB
Reflection loss, |B1|
Even mode antenna power
Odd mode antenna power
-10
-15
-20
-25
0
5
10
Frequency in GHz
15
20
Figure 2-12: Example 4, reflection and radiated antenna mode powers in dB
Example 4. 50 mm line. ZNo=25 Ohms, Rpo=50 Ohm metres, ZNe=100 Ohms, Rpe=1.5 Ohm metres
Za0=25 Ohms, Zae=50 Ohms
45
40
Radial squint in degrees
35
30
25
20
15
10
5
0
0
5
10
Frequency in GHz
15
20
Figure 2-13: Example 4, Radial squint in degrees
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3
Analysis of multiply layered coaxial transmission lines
3.1
Introduction
In the analysis of the grounded 4-port network it is necessary to provide input values for
the characteristic impedance and propagation factor for the even and odd modes. The odd
mode characteristics can be approximated using analysis for strip lines, assuming the use
of radar absorbers between the line pair and the outer conducting shield has only a small
effect on these characteristics. On the other hand the even mode characteristics must, by
necessity, feature a structure with more than one region of dielectric. We require a region
of air around the strip line pair and at least one layer of absorber. For the purpose of
even mode analysis the strip line pair may be approximated10 as a single thick conductor
of radius a with circumference 2πa = W assuming the track width, W , is much larger than
the track separation, h, and that the outer radius b >> W . This is illustrated in figure 3-1
below. The coaxial approximation permits a much simpler analysis and an exact result in
the zero-frequency limit.
free space
W
b>>W
circumference W
h<<W
absorbing material
Cross section of strip line within a
shielded RAMed cylinder
Approximate equivalence for even mode.
Coaxial system.
Figure 3-1: Approximate even-mode equivalence of a shielded strip line pair and a
coaxial line
10 This approximation follows from analysis of the Method of Moments software, NEC, where it is shown that a wire of
radius a is approximately equivalent to a zero thickness strip of width W = 2πa.
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3.2
The characteristic impedance and propagation constant
An analysis is presented here describing the characteristic impedance and the propagation
constant from two perspectives, one in terms of the equivalent circuit values of the telegraph
equation and the other in terms of equivalent relative permittivity and permeability.
Let us first consider a coaxial structure where there is only free space between the inner and
outer conductors, of radius a and b. In this case, in the low frequency limit the characteristic
impedance of the coax is given [1] by,
η0
b
η0 =
(3-1)
loge
2π
a
where η0 is the characteristic impedance of free space, defined by
r
µ0
η0 =
≈ 377Ω
ǫ0
(3-2)
and the propagation factor, which determines z-dependence of both the electric and magnetic
fields according to the factor exp(−γ0 z) in the forward +z direction, is given by
γ0 = jk0
(3-3)
where k0 is the free space wave number. Now suppose a coaxial line bounded by the same
inner and outer conductors of radius a and b, filled with a material with relative permittivity
ǫr (r) and relative permeability µr (r) as a function of r. The characteristic impedance is now
defined as Z and the propagation factor as γ, where exp(−γz) defines the z-dependence of
the fields. The definition of Z requires some care. For a true TEM mode, for which ǫr (r)
and µr (r) are independent of r, Z may be defined by,
Z=
v0
i0
(3-4)
where i0 is the current flowing down the centre conductor and v0 is the potential difference
between inner and outer conductor in a transverse plane (e.g. at z = 0). For the coax
geometry,
Z
b
Er (r) dr
v0 =
(3-5)
a
and
i0 = 2πaHφ (a)
(3-6)
where Er (r) is the radial component of the transverse electric field and Hφ (r) is the azimuthal
component of the transverse magnetic field. More generally, i.e. for non-TEM modes, there
is more than one possible definition of the characteristic impedance.
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For general modes in a coaxial structure, we may still define [7] a characteristic impedance
Z = Z1 , where
v0
Z1 =
(3-7)
i0
where
Z
(3-8)
v0 =
et . dl
P
where et is the transverse electric field and P is some path in the transverse plane between
inner and outer conductors and,
I
i0 = ht . dl
(3-9)
C
where ht is the transverse magnetic field and C is the closed path perimeter of the inner
conductor in the transverse plane.
For the coaxial geometry with non-constant ǫr (r) and µr (r), the TEM mode becomes a
quasi TEM (QTEM) mode. However, it can readily be shown that the QTEM mode is a
TM mode (see section 3.4). For the QTEM mode, et = r̂Er and v0 is a uniquely defined
potential independent of the path P. By definition, i0 is always uniquely defined for a
two-conductor coaxial structure. Consequently, Z1 is well defined for a QTEM mode.
However, this is not the only definition of characteristic impedance. Two others, defined in
[7], are given by Z = Z2 and Z = Z3 where,
Z2 =
|v0 |2
p⋆0
(3-10)
Z3 =
p0
|i0 |2
(3-11)
and
where p0 is the complex power defined by,
Z
p0 =
et × h⋆t .ẑ ds
(3-12)
S
where S is the transverse cross section at z = 0 and et and ht are the transverse electric and
magnetic field vectors as defined above. The ⋆ represents the complex conjugate.
In order that all three forms should be equivalent we require that,
v0 i⋆0 = p0
(3-13)
This is true for a TEM mode. However, under definition (3-8), (3-9) and (3-12) this is
generally not true for other modes and is not true for the QTEM mode of a coaxial structure
except when the QTEM mode is TEM. See appendix A for a proof.
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This raises some questions;
What is the physical consequence of inequivalence of definitions?
What is the ‘best’ definition to use for our application?
Physically, the characteristic impedance is used to determine how well another structure,
defined by a lumped impedance, can ‘match’ to the coax so as to determine the reflection
and transmission coefficients. For example, if a coax guide with characteristic impedance Z
is terminated by a load with characteristic impedance ZL = Z then there is no reflection
at the termination and no impedance discontinuity. A non-unique characteristic impedance
informs us that it is in general not possible to perfectly match the coax by such a load.
In circuit terms this is saying that fringing fields become important and a general multiple
mode matching procedure is required.
Since v0 , i0 and p0 are all well defined unique quantities it is not obvious which of these is
‘best’. Each of the three definitions for Z requires two of these three forms and each has a
clear physical interpretation. Another consideration is numerical accuracy; it is likely that
not all of these can be computed to the same degree of numerical precision.
If there is no way to directly measure the characteristic impedance then there is no requirement that Z be a response function. However there is an argument that some form for
the characteristic impedance could, at least in principal, be measured directly. In [8] the
argument is made that a good definition of the characteristic impedance should be a response function and therefore that is satisfies causality. Since the phase of the characteristic
impedance is unique and the same under all three definitions it follows that causality implies a Kramers-Kronig relationship and thus that there exists an amplitude function which
is unique under the minimum phase assumption. In [8] numerical evidence is presented to
show that definition (3-10) is almost indistinguishable from such a response function; i.e.
that (3-10) satisfies causality to a very good approximation. This may not be true of the
other definitions.
There are other good reasons for the use of (3-10). Firstly, from a numerical standpoint,
both v0 and p0 are evaluated as a sum of terms over the eigenvector whereas i0 requires
the first term nearest the conductor which may not be that accurate. Viewed as a response
function there is an argument that both v0 and p0 are easier to measure directly than i0 ,
though the argument concerning what is directly measurable “in principle” is not clear.
We may now define an effective relative permittivity ǫr and an effective relative permeability
µr defined in terms of Z and γ, by
r
µr
Z = η0
(3-14)
ǫr
and
γ = jk0
p
ǫr µr
(3-15)
At zero frequency the electric and magnetic fields do not couple so that µr and ǫr are
independent of each other. One consequence of this is that for a non-magnetic material with
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µr (r) = 1, then µr = 1.11 This implies that if µr (r) = 1 then γ serves to define ǫr and hence
for a non magnetic material at zero frequency,
Z=
η 0 jk0
γ
(3-16)
While dispersion is small, (3-16) remains accurate but it is not generally valid towards the
onset of higher order modes where we find that µr 6= 1 for non-magnetic materials.
As a function of frequency both ǫr and µr must satisfy Kramers-Kronig relations for realisable
dispersion. This may be modelled in terms of networks of frequency independent capacitors,
resistors and inductors. Some care is required here. For example it is standard practise
to model a uniform transmission line in terms of the per-unit-length series impedances and
parallel admittances illustrated in figure 3-2. This form is commonly used in the telegraph
L
R
C
G
Figure 3-2: Standard model for a transmission line (LRCG model).
equation to provide equivalent definitions for γ and Z (see p58 of [1] or section 2.6 of [7]),
s
R + jωL
Z=
(3-17)
G + jωC
γ=
p
(jωL + R)(jωC + G)
(3-18)
where R ≡ Rs is a resistance, L ≡ Ls is an inductance, C ≡ Cp is a capacitance and
G ≡ 1/Rp is a conductance. However, in general this equivalent circuit, while formally valid
at any given frequency, may not be a valid dispersive model. The equivalent circuit values
are functions of frequency which, individually, may not be realisable. In [7] expressions
(equations 33-36) are given for R, L, C and G in terms of the real and imaginary parts
of ǫr (r) and µr (r) where the frequency dependence is evident. Moreover, each component
requires a mixture of electric and magnetic terms. Note that for a QTEM (or TM) mode,
Hz = 0 so that C and G are dependent only on the electric fields and on the relative
permittivity. However, L and R depend on both magnetic and electric fields (and hence on
ǫr (r) and µr (r)). For a pure TEM mode L and R depend only on the magnetic fields.
In order to guarantee realisability, an alternative network representation should be constructed from frequency independent components. For a pure TEM mode this is possible
11 Similarly if ǫ (r)
r
= 1 then ǫr = 1, though this has less practical significance.
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using a representation of the form shown in figure 3-3 where L′ , R′ , C ′ and G′ are frequency
independent, as described in section 1.3.4. This is probably the simplest realisable dispersive
model featuring both electric and magnetic loss for a perfectly conducting coaxial structure,
valid when the QTEM mode is approximately TEM. A more complicated realisable dispersion model is provided later, in section 3.2.
R
L
C
G
Figure 3-3: Realisable representation for a TEM mode with frequency independent
components.
3.3
A quasistatic approximation with realisable dispersion
In this section an analytic model is constructed for the multi-layered coaxial model. This is
a quasi-static model valid well below the onset of the next higher order modes. However, the
characteristic impedance may change as a function of frequency due to the dispersive models
for the relative permittivity and permeability. The model is a generalisation of published
results (e.g. [9]) valid for an arbitrarily layered structure composed of lossy magnetic and
electric materials.
Consider a single layered coaxial structure with inner radius ri and outer radius ri+1 , filled
with a material whose relative permittivity is ǫr,i and relative permeability µr,i . Let the
capacitance per unit length of this structure be Ci . Then (see e.g. [1], pp60-62),
2πǫr,i
Ci
=
ǫ0
loge (ri+1 /ri )
(3-19)
where ǫ0 is the permittivity of free space. Note that the capacitance is independent of the
relative permeability. Equation (3-19) is defined for real ǫr,i , but we extend the definition and
define a complex Capacitance, C i , valid for a complex (lossy) ǫr,i . This provides a convenient
way of defining loss within a single complex variable, so avoiding the need for a separate
G term in (3-17) and (3-18) and adopt a bold font to signify the G term is included in its
complex definition.
In a multi-layered coaxial structure of N layers the total complex capacitance per unit length,
C, is formed as the series addition of all the single layer capacitances so that,
!−1
N
X
C
ǫ0
(3-20)
=
ǫ0
C
i
i=1
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Similarly, let the inductance per unit length of a single layered coax structure be Li . Then,
µr,i
Li
=
loge (ri+1 /ri )
µ0
2π
(3-21)
where µ0 is the permeability of free space. Note that the inductance is independent of the
relative permittivity. Equation (3-21) is defined for real µr,i but we extend the definition to
define a complex Inductance, Li , valid for a complex (lossy) µr,i . This provides a convenient
way of defining loss within a single complex variable, so avoiding the need for a separate R
term in (3-17) and (3-18). Again we adopt a bold font to signify that the R term is included
in its complex definition.
In a multi-layered coaxial structure of N layers the total complex inductance per unit length,
L, is formed as the series addition of all the layer inductances so that,
N
X Li
L
=
µ0
µ0
i=1
(3-22)
The complex characteristic impedance may be defined in terms of the complex capacitance
and complex inductance (with no separate R and G terms),
s
(L/µ0 )
(3-23)
Z = η0
(C/ǫ0 )
and the complex propagation factor,
γ = jk0
p
(L/µ0 )(C/ǫ0 )
(3-24)
The complex effective relative permittivity, ǫr , and the complex effective relative permeability, µr , are defined through the relations,
C
2πǫr
=
ǫ0
loge (b/a)
(3-25)
and
µ
L
(3-26)
= r loge (b/a)
µ0
2π
where b ≡ rN +1 and a ≡ r1 are the radii of the inner and outer conductors of the multi-layer
coax. Rearranging these gives the results,
C
b
ǫr =
(3-27)
loge
2πǫ0
a
and
2πL
µr =
µ0
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−1
b
loge
a
(3-28)
Page 53 of 78
We present results later showing how these predictions compare with a full modal analysis.
3.4
Finite element analysis
The quasi-static analysis is only valid for low frequencies and the accuracy of its predictions
cannot be determined a-priori without reference to a more exact analysis. This generally
requires a numerical treatment of some kind. For our purpose we will employ a finite element
analysis based on [6]. In principal, commercial software such as CST can furnish us with the
same results but we have found that the modal analyser integrated into the CST T-solver
does not predict the required p0 v0 form for the characteristic impedance of the QTEM modes
at high frequencies. HFFS or the CST F-solver are expected to do this. Following recent
discussions, we have also been advised that a post-processor is available for the T-solver
which also computes the necessary form, though questions remain concerning accuracy.
We start with the frequency domain Maxwell’s equations with ejωt implied time dependence,
∇ × E = −jωµH
∇ × H = jωǫE
(3-29)
with ǫ = ǫ0 ǫr and µ = µ0 µr and assume these are piece-wise constant. In cylindrical
coordinates the curl of a vector field A with no φ variation is given by,
∂Aφ
1 ∂(rAφ )
∂Ar ∂Az
∇ × A = −r̂
+ φ̂
−
(3-30)
+ ẑ
∂z
∂z
∂r
r ∂r
with standard notation subscript field components. The no φ variation is a requirement of a
TEM mode or QTEM mode in a coaxial structure. Also note that the QTEM mode requires
that there is no φ component of the electric field, Eφ = 0. The ∇ × E Maxwell equation
therefore implies that the z component of the magnetic field, Hz = 0, i.e. the QTEM mode
is a TM mode.
Equating φ̂ and r̂ components of the Maxwell fields gives the equations,
∂Er ∂Ez
−
= −iωµHφ
∂z
∂r
(3-31)
and
∂Hφ
= jωǫEr
(3-32)
∂z
Now assume that all fields propagate in the z-direction as defined by a propagation factor
γ, such that
−
Er (r, z) = e−γz Er (r)
Ez (r, z) = e−γz Ez (r)
Hφ (r, z) = e−γz Hφ (r)
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(3-33)
Page 54 of 78
This implies,
−γEr (r) −
∂Ez (r)
= −jωµHφ (r)
∂r
and
Er (r) =
γ
Hφ (r)
jωǫ
(3-34)
(3-35)
Combining these two equations gives,
(ω 2 µǫ + γ 2 )Hφ (r) + jωǫ
∂Ez
=0
∂r
(3-36)
Combining (3-36) with the z-component of the Maxwell equations,
1 ∂(rHφ )
= jωǫEz
r ∂r
(3-37)
permits a solution using finite differences. Using the conventional procedure of evaluating
the magnetic fields and electric fields on lattice points that are offset by half a lattice cell [6]
we obtain the discretised form for (3-36),
!
i+1/2
i−1/2
Ez
− Ez
2
2
i
(ω µi ǫi + γ )Hφ + jωǫi
=0
(3-38)
∆r
and for (3-37),
i+1
jωǫi i+1/2
1 ri+1 Hφ − ri Hφi
ǫi
E
=
∆r z
ǫi+1/2 (∆r)2
ri+1/2
(3-39)
and with displaced i value,
i−1
ǫi
1 ri−1 Hφ − ri Hφi
jωǫi i−1/2
E
=
−
∆r z
ǫi−1/2 (∆r)2
ri−1/2
(3-40)
Here, the index i refers to a value on a one dimensional grid location point and the i ± 1/2
refers to an evaluation displaced by plus or minus half an interval. We assume a regular
mesh with interval ∆r = ri+1 − ri for all i. Combining and collecting terms we obtain,
α1 Hφi−1 + α2 Hφi + α3 Hφi+1 = γ 2 Hφi for 1 ≤ i ≤ N
(3-41)
where for 2 ≤ i ≤ N − 1
−ǫr,i ri−1
ǫr,i−1/2 ri−1/2 (∆r)2 ǫr,i ri
ǫr,i ri
1
2
+
= −k0 µr,i ǫr,i +
(∆r)2 ǫr,i+1/2 ri+1/2 ǫr,i−1/2 ri−1/2
−ǫr,i ri+1
=
ǫr,i+1/2 ri+1/2 (∆r)2
α1 ≡ ai,i−1 =
α2 ≡ ai,i
α3 ≡ ai,i+1
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(3-42)
Page 55 of 78
where the ǫ and µ terms with r subscript are the relative permittivities and permeabilities
sampled at the grid points and we define,
ǫr,i−1/2
ǫr,i+1/2
ri−1/2
ri+1/2
=
=
=
=
(ǫi−1 + ǫi )/2
(ǫi+1 + ǫi )/2
(ri−1 + ri )/2
(ri+1 + ri )/2
(3-43)
With uniform sampling, ri = a + (i − 1)∆r and ∆r = (b − a)/N assuming a grid of N points.
Definition of terms with i = 1 and i = N requires application of the boundary conditions.
We are assuming perfect inner and outer conductors at radii a and b respectively which
1/2
N +1/2
requires that Ez = Ez
= 0. This is equivalent to the requirement that ǫ1/2 → ∞ and
ǫN +1/2 → ∞. Consequently, for i = 1, α1 is undefined and
α2 ≡ a1,1 = −k02 µr,1 ǫr,1 +
α3 ≡ a1,2 =
ǫr,1 r1
ǫr,1+1/2 r1+1/2 (∆r)2
−ǫr,1 r2
ǫ1+1/2 r1+1/2 (∆r)2
(3-44)
and for i = N , α3 is undefined and
−ǫr,N rN −1
ǫN −1/2 rN −1/2 (∆r)2
ǫr,N rN
= −k02 µr,N ǫr,N +
ǫr,N −1/2 rN −1/2 (∆r)2
α1 ≡ aN,N −1 =
α2 ≡ aN,N
(3-45)
Equation (3-41) represents a non-Hermitian complex matrix Eigenvalue equation with banded
matrix A comprising terms ai,j as defined above. It has, as solutions, the TM modes of the
structure both propagating and non-propagating (evanescent). The QTEM mode is thus
included as a solution. We are not concerned with the TE modes but these will also required
if an expansion of fields is required.12
In the absence of losses the modes are non-propagating if and only if they are below their
respective cut-off frequencies. In the presence of high losses the distinction can be blurred
since all modes are to a degree evanescent. The equation will also contain spurious solutions
due to the effects of numerical discretisation.
12 The TE mode equation may be found in an analogous treatment of the electric field, leading to a finite difference
equation in Eφ .
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In matrix form the equation appears as,

a1,1 a1,2
 a2,1 a2,2 a2,3


a3,2 a3,3 a3,4


... ... ...


.
.


.
.


aN −2,N −3 aN −2,N −2 aN −2,N −1


aN −1,N −2 aN −1,N −1 aN −1,N
aN,N −1
aN,N














Hφ1
Hφ2
Hφ3
.
.
.
HφN −2
HφN −1
HφN














 = λ












Hφ1
Hφ2
Hφ3
.
.
.
HφN −2
HφN −1
HφN














where λ = γ 2 has N solutions representing the eigenvalues of the equation. For our purposes
most of these solutions are spurious. A selection method that always correctly picks out the
required QTEM solution would appear to be a very difficult task and a degree of pragmatism
is required. This is discussed presently. Solution of the eigenvalue equation is also non-trivial
though many standard algorithms are available. Ideally we desire a robust and efficient
banded solver for a non-Hermitian complex matrix. Unfortunately such solvers are state-ofthe-art and not readily available for non-commercial purposes. We will instead use a dense
matrix solver for non Hermitian matrices available for the LaPack freely available linear
algebra library. The one currently employed is the ZGEEV routine. This is not efficient for
< 200.
our purposes but is quite suitable for matrices of size N ∼
3.4.1 Eigenvalue selection
Selection of the correct eigenvalue and associated eigenvector, representing the QTEM modal
Hφ field, proceeds according to a relatively simple heuristic. This appears to work effectively
for lossy and non-lossy materials and for frequencies beyond the onset of higher order propagating modes, but is not guaranteed.
Suppose the N eigenvalues of the equation are designated as λi and write,
λi = γi2 = Ri ejΦi
(3-46)
with magnitude Ri and phase Φi . Then,
If the real part, ℜ(λi ) ≥ 0 for all i, we select the mode with minimum propagation loss. It is
possible for this to occur if the dielectric or magnetic losses
p are sufficiently large. The mode
with minimum loss is defined as that mode for which | (Ri ) cos(Φi /2)| is smallest.
If the real part of at least one mode, ℜ(λi ) < 0, then each such mode propagates with an
oscillatory component. If there is only one such mode then this mode is selected. If there is
more than one such mode, then the associated eigenvectors need to be examined. Assume
that one term of each eigenvector is normalised to unity (i.e. with zero imaginary part).
The correct eigenvector, associated with the QTEM mode, should not have terms whose
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real part changes sign. I.e. if one term is unity then all terms should have positive real
values. This is the eigenvector and associated eigenvalue that is selected. It may be possible
that for multiple propagating modes more than one eigenvector satisfies this criterion, but
to date we have not discovered one. We have, however, found examples where no eigenvector
maintains the sign of its real part, as described in one of the later examples. In this case it
is completely unclear how to determine the correct eigenvector or indeed if there is one since
the failure transition might mark a frequency at which eigenvectors are nearly degenerate. In
our software, a warning is flagged and the eigenvector is chosen with the minimum number
of sign changes. However, this may not be correct.
3.4.2 Determination of characteristic impedance
Once the correct eigenvalue and eigenvector have been determined, given by the index i = iq ,
the propagation factor
γ = γ iq
(3-47)
and it remains to determine the characteristic impedance. There are at least three alternative
ways to evaluate this, defined by equations (3-7), (3-10) and (3-11) as decribed previously.
Both (3-7) and (3-11) require an evaluation based on the current flow i0 , where
i0 = 2πaHφ (a)
(3-48)
with Hφ (a) given numerically by Hφ (a) ≈ Hφ1 which is based on an evaluation of the first
term of the eigenvector. We also have,
v0 =
Z
b
a
Er dr ≈ ∆r
where using (3-35),
Eri =
N
X
Eri
(3-49)
i=1
γ
Hi
jk0 η0 ǫr,i φ
(3-50)
and for the QTEM mode the complex power (3-12) is given by,
p0 =
Z
S
Er Hφ⋆
dS = 2π
Z
b
a
Er (r) Hφ⋆ (r) r dr ≈ 2π∆r
N
X
ri Eri .(Hφi )⋆
(3-51)
i=1
We employ the definition of the characteristic impedance as,
Z=
|v0 |2
p⋆0
where both the numerator and denominator involve sums over eigenvector terms. As discussed previously, this is expected to satisfy causality requirements to high accuracy.
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3.5
Dispersion models for the relative permittivity and permeability of layers
For our application we require effective methods to absorb the QTEM mode, used to
model the unrequired even mode of the transmission line feed. This means that the dielectric/magnetic layers must include significant losses so it is important to apply a realisable
dispersion model. Unfortunately the suppliers of radar absorbent materials rarely provide
such a model and equally rarely provide ǫr or µr as a function of frequency. It is therefore
necessary to construct our own models.
First we need to establish the connection between transmission line equivalent circuit parameters and ǫr and µr . For a TEM mode13 the characteristic impedance can be written in
the form,
r
µr
Z=
η
(3-52)
ǫr 0
where µr is the effective relative permeability, ǫr is the effective relative permittivity and η 0
is the characteristic impedance of the structure when the effective relative permeability and
permittivity are both unity. For a coaxial structure with inner radius a and outer radius b,
b
η0
loge
(3-53)
η0 =
2π
a
for a TEM wave we also have,
γ = jk0
p
ǫr µr
(3-54)
Now for any transmission line representation (valid for all mode types) with series impedance
element Zs ohms per metre and parallel admittance element Yp Siemens per metre, as illustrated in figure 3-4,
s
Zs
(3-55)
Z=
Yp
and
γ=
p
Zs Y p
Zs
(3-56)
Yp
Figure 3-4: General transmission line representation.
13 This result is generally not true for a non TEM mode, e.g. a QTEM mode at sufficiently high frequencies.
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equating these two forms gives the expressions14 ,
jZs
k0 η 0
(3-57)
jYp η 0
k0
(3-58)
µr = −
and
ǫr = −
Any magnetic/electric material used in a multi-layer coaxial structure may be represented
in a similar manner, i.e
jZs,i
µr,i = −
(3-59)
k0 η 0
and
jYp,i η 0
(3-60)
k0
within layer i of the guide. Realisability of materials is independent of the definition of η and
(3-59) and (3-60) could equally well be defined in free space (or any structure supporting a
TEM mode) with η 0 → η0 . Use of η 0 as a normalisation factor is useful, however, since it
means that if Zs,i → Zs and Yp,i → Ys then µr,i → µr and ǫr,i → ǫr .
ǫr,i = −
Note that if a QTEM mode becomes appreciably non TEM, µr,i and ǫr,i may still be defined
by (3-59) and (3-60) where Zs,i and Yp,i represent the per-unit-length values of a TEM wave
in some arbitrary structure which can support such a wave. However, in this case if Zs,i → Zs
and Yp,i → Ys it is no longer true that µr → µr,i and ǫr → ǫr,i .
A basic model was described for a TEM transmission line in figure 3-3. This can be taken
as a simple model for a material with both lossy electric and magnetic elements. A material
so defined is realisable because L′ , C ′ , R′ and G′ are independent of frequency and possesses
the correct asymptotic behaviour (see [2]) as ω → 0. In this case,
Zs,i =
1
1
+
jωL′i Ri′
−1
(3-61)
and
Yp,i = jωCi′ + G′i
(3-62)
In our current formulation we explore a more sophisticated model for defining µr,i and ǫr,i
where Zs,i and Yp,i are represented by the elements as shown in figure 3-5 for material i. For
each material, Ls and Cp are defined in terms of a real valued (lossless) dielectric constant
ǫr0 and a real valued (lossless) permeability µr0 .
14 There is a sign ambiguity in obtaining these expressions that is resolved by the fact that the imaginary parts of
µr
and ǫr must be negative for passive materials under exp(jωt) harmonic time dependence.
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We define the real valued refractive index α,15
α=
√
ǫr0 µr0
(3-63)
and the real valued characteristic impedance Zn by,
r
µr0
Zn = η 0
ǫr0
so that,
(3-64)
Ls =
η µr0
αZn
= 0
c0
c0
(3-65)
Cp =
ǫr0
α
=
Zn c0
η 0 c0
(3-66)
and
where c0 is the speed of light in vacuum. This is a convenient way of setting the ‘base’
characteristics before application of loss descriptors. The other component values are defined
directly in terms of their inductance, capacitance or resistance.
C m1
R m2
Zs
L m1
R m1
Ls
C e1
C
L e1
Yp
p
R e1
R e2
Figure 3-5: A more sophisticated dispersion model.
15 More generally,
α is related to γ by α = −jγ/k0
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3.6
Some examples
In these examples the emphasis will be on the predictions of the characteristic impedance.
The propagation factor is generally found to be more accurate and, for the finite difference
method, must be found before the characteristic impedance can be calculated. The CST
T-solver cannot predict the propagation factor of a lossless material without performing a
full 3D simulation since losses are not accounted for in the embedded general port solver.
Similarly, the characteristic impedance is defined as real in the T-solver which is not generally
correct.
Examples include a two layer lossless dielectric structure, a two layer structure with one
lossless magneto-dielectric layer, a two layer structure over a frequency range which includes
a higher order propagating mode and some structures containing losses.
3.6.1 A lossless two layer dielectric structure
In this example we consider an example published on the Internet [9]. The structure comprises an inner conductor radius a = 1.02 mm, an outer conductor radius b = 3.56 mm and
a mid-layer radius r1 = 2.54 mm. The first layer, for radius between a and r1 , is a lossless
dielectric with ǫr = 2.0 and µr = 1.0. The second layer, for radius between r1 and b, is a
lossless dielectric with ǫr = 5.0. HFSS results for the characteristic impedance are given at
frequencies 1 GHz, 10 GHz and 15 GHz.
Using the quasi-static approximation of section 3.3 we obtain a value of Z ≈ 48.51 Ω. Figure
3-6 shows the predicted value of Z as a function of frequency assuming N = 20, 50, 100, 200, 400
samples (∆r = 0.127, 0.0508, 0.0254, 0.0127, 0.00635 mm, respectively). Tabulated results
at the given frequencies are:
Frequency
1 GHz
10 GHz
15 GHz
CST (T-solver)
48.5 Ω
46.1 Ω
43.1 Ω
HFSS result
48.6 Ω
48.8 Ω
49.0 Ω
FD result (N = 400)
48.6 Ω
48.8 Ω
49.0 Ω
We see there is agreement to three significant figures with the HFSS result, but the CST
T-solver MWS predictions are poor as the frequency is increased16 . Since two significant
figure accuracy is suitable for most purposes, figure 3-6 suggests that sufficient accuracy can
probably be obtained with N = 100.
16 Discussions with CST support are ongoing as of 3/June/2013. It appears that use of the general port solver,
embedded in the T-solver (referred to as the MWS method) is not accurate for hybrid modes. It was recommended
either to use the F-solver (we don’t have a license for this) or to use a post processing method of which the P/U
method appears most accurate and/or suitable. It appears that the P/U post processing method is similar to ours both
in accuracy and in definition.
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Finite difference impedance estimates
52
characteristic impedance in ohms
51
50
49
48
N=20 samples
N=50 samples
N=100 samples
N=200 samples
N=400 samples
quasi-static result
47
46
0
5
10
15
20
Frequency in GHz
Figure 3-6: Predicted characteristic impedance. A lossless two layer non-magnetic
material, example 1.
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3.6.2 A lossless two layer magnetic structure
No independent results are available for comparison when one or more layers is magnetic.
In this example we again compare our FD results with the quasistatic approximation and
the T-solver in CST. The geometry is as above with a = 1.02 mm, b = 3.56 mm and
r1 = 2.54 mm. Here we assume the first layer, for radius between a and r1 , is free space with
ǫr = 1.0 and µr = 1.0. The second layer, for radius between r1 and b, is a lossless material
with ǫr = 3.50 and µr = 2.00. Figure 3-7 shows the predicted value of Z as a function of
frequency assuming N = 20, 50, 100, 200, 400 samples. The CST T-solver (MWS) predicts
a value of Z = 75.8 Ω at 0.2 GHz, in good agreement with the quasi-static result (for which
Z = 75.9 Ω) and with our FD results. However at 20 GHz the T-solver (MWS) predicts
Z = 47.0 Ω which is considerably at variance with our predictions.
Finite difference impedance estimates
84
N=20 samples
N=50 samples
N=100 samples
N=200 samples
N=400 samples
quasi-static result
characteristic impedance in ohms
82
80
78
76
74
0
5
10
Frequency in GHz
Figure 3-7: Predicted characteristic impedance.
tric/magnetic material, example 2.
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20
A lossless two layer dielec-
Page 64 of 78
3.6.3 A lossless two layer dielectric structure with onset of higher order modes
In this example we consider a more demanding prediction where there is more than one
propagating TM mode at the higher frequencies of the prediction range. We take a = 1.00
mm, b = 10.00 mm and r1 = 3.00 mm. The first layer for radius between a and r1 , is
free space with ǫr = 1.0 and µr = 1.0. The second layer, for radius between r1 and b, is a
lossless dielectric with ǫr = 2.05 and µr = 1.0. A second propagating TM mode begins at a
frequency of about 11.5 GHz.
Figure 3-8 shows the predicted value of Z as a function of frequency assuming N = 20, 50, 100,
200, 400 samples. The CST T-solver (MWS) predicts a characteristic impedance Z ≈ 87.1Ω
at 10 GHz, contrasting with our prediction of Z ≈ 117Ω at 10 GHz and a quasi-static
prediction of Z ≈ 118Ω.
Finite difference impedance estimates
140
characteristic impedance in ohms
130
120
110
100
N=20 samples
N=50 samples
N=100 samples
N=200 samples
N=400 samples
quasi-static result
90
80
70
0
5
10
15
20
Frequency in GHz
Figure 3-8: Predicted characteristic impedance. A lossless two layer dielectric with
a higher order propagating mode for frequencies beyond 11.5 GHz, example 3.
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3.6.4 A lossy two layer dielectric structure
Here we take a = 1.00 mm, b = 10.00 mm and r1 = 3.00 mm. The first layer for radius
between a and r1 , is free space with ǫr = 1.0 and µr = 1.0. The second layer, for radius
between r1 and b, is a non-magnetic lossy dielectric with µr = 1. Its relative permittivity as
a function of frequency is specified by ǫr0 = 1.50 and equivalent circuit values as illustrated
in figure 3-5 with Ce1 → ∞, Re2 → ∞ and Re1 = 1.00 Ω m.
Figures 3-9 and 3-10 show the predicted real and imaginary parts of the characteristic
impedance as a function of frequency, for three discretisations (N = 100, 200, 400) and
the quasistatic approximation. Note that in this case, because the constituent materials are
dispersive so is the quasistatic result. Figures 3-11 and 3-12 show the predicted real and
imaginary parts of the effective refractive index, α = γ/(jk0 ). The real part of α is shown
to be accurately predicted by the quasistatic approximation, though not so the imaginary
part at the higher frequencies. Neither the real not the imaginary parts of the characteristic impedance are accurately predicted by the quasistatic approximation at the higher
frequencies.
Figures 3-13 and 3-14 show the predicted real and imaginary parts of the effective relative
permittivity and figures 3-15 and 3-16 show the predicted real and imaginary parts of the
effective relative permeability. Observe that away from zero frequency, the effective relative
permeability is non-unity except in the quasistatic approximation. This is a result of our
definition of effective relative permittivity and permeability which assumes the mode is TEM
when in fact it is not.
This example is not completely arbitrary and is representative of the kind of geometry and
RAM parameters we might wish to employ in practise. It shows quite complicated behaviour
with a characteristic impedance Z and refractive index α that is far from constant. Since the
imaginary part of Z is significantly smaller than the real part it is probably fair to ignore the
imaginary part for matching purposes. The real part varies by about 30% about its mean
value and so might be assumed constant in a first order approximation. The refractive index
probably needs to be treated differently. At the higher frequencies its real and imaginary
parts can probably be assumed to be constant, but there is a marked peak in the loss near
3 GHz. There appears to be a frequency range directly above 3 GHz where increasing the
frequency decreases the loss and hence where the ordinarily expected increase in absorption
of this mode due to total path loss will not occur.
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Finite difference and quasistatic impedance estimates
140
Real part of characteristic impedance in ohms
130
120
110
100
90
N=100, Real part
N=200, Real part
N=400, Real part
Quasistatic, Real part
80
70
60
0
5
10
Frequency in GHz
15
20
Figure 3-9: Predicted real part of characteristic impedance with a lossy layer. Example 4.
Finite difference and quasistatic impedance estimates
Imaginary part of characteristic impedance in ohms
20
15
10
N=100, Imag part
N=200, Imag part
N=400, Imag part
Quasistatic, Imag part
5
0
0
5
10
Frequency in GHz
15
20
Figure 3-10: Predicted imaginary part of characteristic impedance with a lossy layer.
Example 4.
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Finite difference and quasistatic alpha
Real part of effective refractive index, alpha
1.5
1.4
N=100, Real part alpha
N=200, Real part alpha
N=400, Real part alpha
Quasistatic, Real part alpha
1.3
1.2
1.1
1
0
5
10
Frequency in GHz
15
20
Figure 3-11: Predicted real part of effective refractive index α. Example 4.
Finite difference and quasistatic alpha
Imaginary part of effective refractive index, alpha
0
-0.05
-0.1
-0.15
N=100, Imag part alpha
N=200, Imag part alpha
N=400, Imag part alpha
Quasistatic, Imag part alpha
-0.2
0
5
10
Frequency in GHz
15
20
Figure 3-12: Predicted imaginary part of effective refractive index α. Example 4.
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Finite difference and quasistatic effective permittivity
Real part of effective relative permittivity
2.5
2
1.5
1
N=100, Real part permittivity
N=200, Real part permittivity
N=400, Real part permittivity
Quasistatic, Real part permittivity
0.5
0
0
5
10
Frequency in GHz
15
20
Figure 3-13: Predicted real part of effective relative permittivity. Example 4.
Finite difference and quasistatic effective permittivity
imaginary part of effective relative permittivity
0
-0.2
-0.4
-0.6
N=100, Imag part permittivity
N=200, Imag part permittivity
N=400, Imag part permittivity
Quasistatic, Imag part permittivity
-0.8
-1
0
5
10
Frequency in GHz
15
20
Figure 3-14: Predicted imaginary part of effective relative permittivity. Example 4.
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Finite difference and quasistatic effective permeability
1.2
Real part of effective relative permeability
1.1
1
0.9
0.8
N=100, Real part permeability
N=200, Real part permeability
N=400, Real part permeability
Quasistatic, Real part permeability
0.7
0.6
0
5
10
Frequency in GHz
15
20
Figure 3-15: Predicted real part of effective relative permeability. Example 4.
Finite difference and quasistatic effective permeability
Imaginary part of effective relative permeability
0
-0.01
-0.02
-0.03
N=100, Imag part permeability
N=200, Imag part permeability
N=400, Imag part permeability
-0.04
-0.05
0
5
10
Frequency in GHz
15
20
Figure 3-16: Predicted imaginary part of effective relative permeability. Example 4.
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3.6.5 The effect of changing the loss parameter
If we assume we can change the carbon loading of a radar absorber and that a suitable model
of the carbon loading is given by the previous model then it is natural to ask how the total
absorption, described by the real part of γ, or the absorption per unit frequency, described
by the imaginary part of α change as a function of frequency.
In this example we assume a discretisation fixed by N = 200 and consider the structure of example 4 above, with changes to the value of Re1 . Values assumed are Re1 = 0.1, 0.2, 0.5, 1.0
and 5.0 Ω metres. All other parameters are as given above. These predictions are interesting.
Firstly, this is one example where our heuristic fails to determine a unique eigenvector for
the case Re1 = 0.5 Ω metres at the top end frequencies. We have simply omitted data above
19.5 GHz where it appears the wrong eigenvector is selected.17
Secondly, there appears to be a definite optimum value for Re1 somewhere near Re1 = 0.2 Ω
metres. Below optimum, the characteristic impedance looks capacitive over most of the
band. Above optimum, the characteristic impedance looks inductive over most of the band.
The optimum value gives the most loss to the refractive index and appears to correspond to
a nearly non-reactive characteristic impedance. This is clear evidence for the importance of
the RAM characteristic. Not “just any” RAM will do the job well.
17 Our software issues a warning when the heuristic fails, but still attempts to find the correct value by picking the
eigenvector with the minimum number of sign changes.
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Finite difference impedance estimates
Real part of characteristic impedance in ohms
140
120
100
80
R1e=0.1 Ohm metres, Real part
R1e=0.2 Ohm metres, Real part
R1e=0.5 Ohm metres, Real part
R1e=1.0 Ohm metres, Real part
R1e=5.0 Ohm metres, Real part
60
0
5
10
Frequency in GHz
15
20
Figure 3-17: Predicted real part of characteristic impedance with a lossy layer. Example 5.
Finite difference impedance estimates
Imaginary part of characteristic impedance in ohms
20
15
10
5
0
-5
-10
R1e=0.1 Ohm metres, Imag part
R1e=0.2 Ohm metres, Imag part
R1e=0.5 Ohm metres, Imag part
R1e=1.0 Ohm metres, Imag part
R1e=5.0 Ohm metres, Imag part
-15
-20
0
5
10
Frequency in GHz
15
20
Figure 3-18: Predicted imaginary part of characteristic impedance with a lossy layer.
Example 5.
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Finite difference, alpha
Real part of effective refractive index, alpha
1.5
1.4
R1e=0.1 Ohm metres, Real part
R1e=0.2 Ohm metres, Real part
R1e=0.5 Ohm metres, Real part
R1e=1.0 Ohm metres, Real part
R1e=5.0 Ohm metres, Real part
1.3
1.2
1.1
1
0
5
10
Frequency in GHz
15
20
Figure 3-19: Predicted real part of refractive index α. Example 5.
Finite difference, alpha
Imaginary part of effective refractive index, alpha
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
R1e=0.1 Ohm metres, Imag part
R1e=0.2 Ohm metres, Imag part
R1e=0.5 Ohm metres, Imag part
R1e=1.0 Ohm metres, Imag part
R1e=5.0 Ohm metres, Imag part
-0.35
-0.4
0
5
10
Frequency in GHz
15
20
Figure 3-20: Predicted imaginary part of refractive index α. Example 5.
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A
The inequivalence of definitions of impedance for the coaxial QTEM mode
When considering a mode which is not TEM, there is more than one definition of the characteristic impedance as described in section 3.2. For a QTEM mode of a coaxial structure,
the definitions for p0 , v0 and i0 are given by,
Z
Z b
⋆
rEr (r)Hφ (r)⋆ dr
(a-1)
p0 =
Er Hφ ds = 2π
a
S
v0 =
and
Z
b
Er (r) dr
(a-2)
a
i0 = 2πaHφ (a)
(a-3)
for inner conductor radius a and outer conductor radius b. There is equivalence of definitions
of the three characteristic impedance forms (3-7), (3-10) and (3-11) only if
v0 i⋆0 = p0
For a QTEM mode we have,
(a-4)
γ
Hφ (r)
jωǫ(r)
Er (r) =
(a-5)
where γ = γ(ω) is the propagation constant and ǫ(r) = ǫ0 ǫr (r) is the permittivity.
Thus we have,
v0 i⋆0
and
2πaHφ⋆ (a) γ
=
jω
2πγ
p0 =
jω
Z
b
a
Z
b
a
Hφ (r)
dr
ǫ(r)
r Hφ (r) Hφ⋆ (r)
dr
ǫ(r)
(a-6)
(a-7)
As a special case, for the TEM mode, Hφ (r) = Hφ0 /r for constant Hφ0 and ǫ(r) = ǫ0 ǫ0r where
ǫ0r is a constant. It is then straightforward to show that,
p0 = v0 i⋆0 =
Hφ0 Hφ0⋆ loge (b/a)
ǫ0 ǫ0r
(a-8)
proving equivalence of definition for the TEM mode. More generally, p0 = v0 i⋆0 requires
satisfaction of the integral equation,
a Hφ⋆ (a)
Z
b
a
Hφ (r)
dr =
ǫ(r)
Z
b
a
Hφ (r)Hφ⋆ (r)
dr
ǫ(r)
(a-9)
with the field satisfying the eigenvalue equation, using (3-36) and (3-37),
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d
−ω µ ǫ(r)Hφ (r) − ǫ(r)
dr
2
1 d
(rHφ (r))
rǫ(r) dr
for λ = γ 2 = λ0 where λ0 is the QTEM eigenvalue.
= λHφ (r)
(a-10)
To show that (a-9) and (a-10) cannot generally be simultaneously satisfied for a QTEM mode
can be difficult. Probably the easiest approach is to obtain a non-trivial solution example
for (a-10) and show it fails to satisfy (a-9).
As far as we are aware there is no exact non-TEM analytic solution of (a-10) available at
non-zero frequency. Our strategy will thus be to obtain a perturbation solution; one which
is perturbed from a pure TEM mode. We then show that in general such a solution fails to
satisfy (a-9), proving the inequivalence result.
We are at liberty to choose any realisable permittivity profile to demonstrate inequivalence.
For our purpose we will assume the permeability µ is a constant and the relative permittivity
is given by,
(r − a)
ǫr (r) = ξ + δ.
(a-11)
a
where 0 < δ ≪ 1 is the perturbation parameter and ξ > 1 is a constant, which we will
assume for present purposes is real.
Under this perturbation, the eigenvalue equation (a-10) will have a perturbed eigenvalue.
We will therefore represent,
γ 2 = −ω 2 µǫ0 (ξ + δ.f )
(a-12)
where f is some constant of order O(1) which needs to be determined. Since (a-12) represents a perturbation to the TEM mode it guarantees that the perturbed mode is QTEM18
Similarly, the solution to Hφ (r) must take the perturbed form,
Hφ (r) =
Hφ0
+ δ.g(r)
r
(a-13)
where Hφ0 is a constant and g(r) is some function yet to be determined. Substituting (a-11),
(a-12) and (a-13) into (a-10), we may equate coefficients of order 1 to confirm the unperturbed solution and in δ to obtain the perturbation. The unperturbed solution has a
vanishing derivative in (a-10) and so all zero order terms cancel as required. Equating terms
of order δ yields the equation,
d 1 d
1 (1 + f )
2
0
r g(r) = −ω µǫ0 Hφ
−
(a-14)
dr r dr
a
r
18 Unless there is mode splitting degeneracy which would be indicated by a subsequent failure to obtain a solution to
order O(δ).
Q-par/AJM/Squint-issues/1/2.0
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This may be integrated twice to obtain,
2
r
k1 r k2
r
2
0
g(r) = −ω µǫ0 Hφ
− (1 + f ) loge r − 1/2 +
+
3a
2
2
r
(a-15)
where k1 and k2 are constants of integration. The constants k1 and f are determined by
the boundary condition requirement that Ez (r) = 0 when r = a and r = b. In terms of the
magnetic field,
d
r Hφ (r) = 0
for r = a, b
(a-16)
dr
and thus,
d
r g(r) = 0
for r = a, b
(a-17)
dr
This gives,
k1 = ω 2 µǫ0 Hφ0 1 − (1 + f ) loge a
(a-18)
and
f=
so that we may rewrite g(r) as,
g(r) = −ω
2
µǫ0 Hφ0
(b/a) − 1
loge (b/a)
−1
r
r
k2
r2
+ (f − 1) − (f + 1) loge (r/a) +
3a 4
2
r
(a-19)
(a-20)
Note that f describes the perturbation to the eigenvalue and that if we define ρ ≡ a/b then
f (ρ) is a monotonic function f (ρ) > 0 for ρ > 1 and f (ρ) = 0 when ρ = 1. The function
g(r) describes the perturbation to the magnetic field with arbitrary constant k2 , which can
be fixed by requiring that g(a) = 0 when a = b. This auxiliary condition ensures there is no
perturbation to the field for arbitrary perturbation strength δ in the no material limit. If so,
ω 2 µǫ0 Hφ0 a2
k2 =
12
(a-21)
We may now substitute (a-13) using this expression for g(r) into (a-9) and equate coefficients
of order O(1) and O(δ). The zero order coefficient (to order O(1)) of the left hand side (LHS)
and right hand side (RHS) of (a-9) are given by,
Hφ0 Hφ0⋆
b
loge
LHS1 = RHS1 =
ǫ0 ξ
a
confirming that to zero order, the TEM mode satisfies the equivalence of definitions of
characteristic impedance. The first order coefficient (to order O(δ)) of the LHS and RHS
are given by,
Z Hφ0⋆ b
ag ⋆ (a)Hφ0
(r − a)Hφ0
b
LHSδ =
dr +
(a-22)
loge
g(r) −
ǫ0 ξ a
ξar
ǫ0 ξ
a
Z Z
(r − a)Hφ0
Hφ0 b ⋆
Hφ0⋆ b
g(r) −
g (r) dr
(a-23)
dr +
RHSδ =
ǫ0 ξ a
ξar
ǫ0 ξ a
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Consequently we can only obtain equality of terms if,
Z b
g ⋆ (r) dr = ag ⋆ (a) loge (b/a)
(a-24)
a
Using (a-20),
Z
b
⋆
a
g (r) dr = −ω
2
ǫ0 µHφ0⋆
(b3 − a3 ) (b2 − a2 )(f − 1) (f + 1) b2
(b2 − a2 )
+
−
loge (b/a) −
9a
8
2
2
4
+k2⋆ loge (b/a)
(a-25)
whereas,
⋆
ag (a) loge (b/a) = −ω
2
µǫ0 Hφ0⋆ a2
loge (b/a)
1
f
+
12 4
+ k2⋆ loge (b/a)
(a-26)
So unless ω = 0 (for which (a-24) is true for any constant k2 ) it is always possible to choose
a and b such that (a-24) is false. This completes our proof.
Q-par/AJM/Squint-issues/1/2.0
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B
References
1
D. M. Pozar “Microwave engineering”, Wiley and Sons 1998 (second edition).
2
A. J. Mackay “Dispersion requirements of low-loss negative refractive index materials
and their realisability”, IET Microwaves, Antennas and Propagation, Vol 3, Issue 5,
August 2009, pp 808-820.
3
R. Garg, P. Bhartia, I. Bahl, A. Ittipiboon “Microstrip antenna design handbook”,
Artech House, 2001.
4
M. Abramowitz, I. A. Stegun “Handbook of Mathematical Functions”, Dover New
York press 1970 (9th ed.).
5
R. G. Corzine, J. A. Mosko “Four-arm spiral antennas”, Artech House, 1990.
6
R. F. Huang, D. M. Zhang “Application of mode matching method to analysis of axisymmetric coaxial discontinuity structures used in permeability and/or permittivity
measurement” Progress in Electromagnetic Research, PIER 67, 2007, pp 205-230.
7
R. B. Marks, D. F. Williams “A general waveguide circuit theory”, Vol 97, No 5,
September-October 1992, Journal of Research of the National Institute of Standards
and Technology, pp533-561.
8
D. F. Williams, B. K. Alpert “Characteristic impedance, power and causality”, IEEE
Microwave and Guided wave letters, Vol. 9, No. 5, May 1999, pp181-182
9
Alex R. (full name not known) “Multi dielectric coax”
See http://www.microwaves101.com/encyclopedia/coaxdual.cfm
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