Mladen Vukomanović, mag. ing., rad za KDI [1,07 MiB]

Mladen Vukomanović, mag. ing., rad za KDI [1,07 MiB]
Analysis of Gap Waveguide Structures Using Mode
Matching Approach
Mladen Vukomanović,
University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia
[email protected]
Abstract – Gap waveguide structures combine the ideas of
parallel plate waveguides and electromagnetic band gaps to form
novel efficient high frequency waveguiding structures. Since the
concept is based on the use of periodic structures with bandgap
properties the analysis is very complex. Our proposed analysis
method is based on efficient use of Method of Moments and
Mode matching techniques to rigorously determine the scattering
of each element in the structure. To achieve this, the currents on
the elements are first determined by using the Green’s function
of a single element in a parallel-plate waveguide. The solution is
then expanded to all the elements from which dispersion
properties can be determined. Mode matching is the alternative
approach that is used to calculate scattering of each element. The
results of these approaches are verified using full-wave
commercial software. It is important to point out that the results
from the developed approach are obtained in a fraction of the
time needed by the commercial full-wave software to analyze the
same structure.
waves, corrugated surface supports or does not support
propagating waves along the structure, meaning that it is only a
one-dimensional soft surface. Better alternatives are uniform
2D periodic surfaces such as bed of nails surface or mushroom
surface (shown in Fig. 2(b)). These structures act as soft in any
direction when operating inside a certain frequency bandgap
and thus prohibit all surface wave propagation.
When such soft surface is paired with a PEC surface in a
parallel-plate configuration all lower parallel plate modes will
be suppressed if the dimension of the space between the plates
is less than a quarter of a wavelength. Essentially, due to the
stop band, the lowest order parallel plate modes are in cut-off
and the waves are forced to propagate only along the desired
directions imposed by ridges or strips as shown Fig. 1.
Keywords – Gap Waveguides; Mode Matching; Bed-of-nails
Gap waveguide is a parallel plate waveguiding structure
where one wall is replaced by a periodic metamaterial based
surface [1] (as shown in Fig. 1). The considered structures are
very promising for general microwave applications, but
especially for applications above 30 GHz. Applications could
include feeding networks for slot arrays, all-metallic
transmission lines for use in space, cavity mode suppression for
microstrip circuits (for solving packaging problems), etc. [1,2].
The major benefit of this guiding structure compared to the
microstrip line is the fact that no dielectric is needed and
therefore there are no dielectric related losses. Compared to
classical rectangular waveguides this structure can be
manufactured more easily and the dimensions can be smaller.
This is because instead of using metallic side walls this novel
waveguideing mechanism is based on electromagnetic bandgap
(EBG) properties of certain periodic surfaces. Essentially EBG
surfaces are here used as a practical realization of a soft
boundary condition [3]. By definition, soft surface stops waves
from propagating, whereas hard surface supports propagating
waves. Corrugated surface shown in Fig. 2(a) is a typical
example of a surface which can acts as a soft or a hard surface,
depending on the direction of propagation. It acts as soft
surface when corrugations are orthogonal to direction of
propagation, and as hard surface when corrugations are
longitudinal (oriented in the same direction as the wave
propagation). Depending on the direction of propagating
Figure 1. Gap waveguide structure with guiding strip/ridge
Figure 2. Other types of EBG surfaces suitable for building gap
waveguides. (a) corrugated surface, (b) mushroom surface
Since the crucial elements of these structures are periodic
surfaces with some inclusions, a general 3D code could be used
for the analysis of finite structures. However, this approach can
be very time consuming, and therefore it is necessary to
introduce an alternative which will keep all the electromagnetic
effects, but allow a faster first design.
In this paper a most common gap-waveguide problem is
analyzed, where the frequency stop band surface is realized
using a periodic array of cylindrical pins, so called “bed-ofnails”. The basis of the first presented approach is the Green’s
function of a single element from which by using Method of
Moments all the currents on all the elements and consequently
the scattered field can be determined. Second approach is based
on the Mode Matching technique and it gives better results
compared to the first approach. Finally, by reducing the
number of modes used and by applying acceleration techniques
a very efficient and also very accurate solution is obtained.
In (4a) and (4b), µ0 is vacuum permeability, Jz is current
density on the cylinder, J0 is Bessel function of a first kind of
order zero, H0 is Hankel function of a first kind of order zero,
and ρ is radial distance from the cylinder. Propagation
constant kv is defined as:
 v 
kv  k 2    .
 h 
Exact expression for vector potential is shown in (4a) and with
it scattered electric field can be calculated as:
A. Cylinder in parallel plate waveguide
The starting point of the analysis is metallic (PEC)
cylinder inside an infinite parallel plate waveguide shown in
Fig. 3.
Ez  
j   2 Az
 k 2 Az  .
k 2  z 2
In (4a) current distribution along cylinder is unknown and to
find the complete solution PEC boundary condition is applied
on the surface of the cylinder:
nˆ  E  0,
Figure 3. Single cylinder inside a parallel plate waveguide
First goal was to find current distribution along the pin for a
known incident plane wave. The considered incident plane
wave is directed along the positive x axis. Assuming that the
current distribution along the pin is cylindrically symmetric
only the variation along z axis exists:
J  az    a  J z  z  e jt , 0  z  L .
In (7) E is total electric field on the cylinder and it corresponds
to the sum of incident and scattered field. After applying the
boundary condition integral equation is obtained where the
current distribution Jz is unknown. To compute this current
distribution along the cylinder the Method of Moments
approach is used. Figure 4 shows the computed current
distribution at frequency of 1 GHz. Length of the cylinder L is
8.66 mm, and height of the waveguide h is 12.16 mm.
Consequently, vector potential also has only one component:
A  az Az (  , z ) .
To get the exact expression for vector potential Green’s
function, a scalar Helmholtz equation has to be solved:
2 Az   , z   k 2 Az  o     a  J z  z  .
Az   , z   
j a0
 v Iv cos
v 0
Iv 
 2
v z 
 J 0  kv a  H 0  k v   ,   a
h  J 0  kv   H 0 2  kv a  ,   a
 J  z  cos
z 0
v ' '
z dz ,
1, v  0
2, v  0
Figure 4. Current distribution of a single cylinder in the parallel-plate
The result is expected since at the point where the cylinder is
connected with the parallel plate waveguide the current has the
maximum value and at the top of the cylinder the current is
ideally equal to zero or close to zero. Obtained current
distribution allows us to calculate exact values of the scattered
electric field.
B. Set of cylinders in parallel plate waveguide
One cylinder can be replaced with set of cylinders like in
Fig. 5. The function of the cylinders is to act as an artificial
magnetic conductor (AMC) in a certain frequency band and
the whole structure with the PEC top plate is now the basis of
gap waveguide.
Figure 5. Gap waveguide based on bed-of-nails AMC
It is assumed that the amplitude of all currents on the cylinders
is the same, but with a phase difference between them. If only
one row of cylinders along the y axis is considered, scattered
electric field from that row can be obtained and this result will
take into account contributions from all cylinders. Complete
formulation for the scattered electric field from one row of
cylinders is found to be:
Ez 
 a0
2hk 2
p 
v 0
 jkpDy s in ( )
 v z 
cos 
 J 0  kv a  I v
 h 
H 0(2) kv x 2  ( y  pDy ) 2
Fig. 7 shows the dispersion diagram of the gap waveguide
structure obtained using two methods: Moment Method
approach and general EM solver (CST Microwave Studio).
Length of the cylinder L is 8.66 mm, height of the waveguide
h is 12.16 mm and the period of the pins p is 3.75 mm. From
the dispersion diagram it can be seen that the stop band is
between 8 GHz and 12 GHz. There is a good agreement
between Moment Method and CST results, but only in the first
part of the diagram for lower frequencies. For higher
frequencies there is a visible deviation between the results.
Possible reason for this deviation lies in the fact that the field
in the gap between the cylinders and the top PEC plate is
perturbed by the current on the cylinder top. For that reason,
an alternative approach was developed based on Mode
Matching which takes this problem also into account.
, (8)
where Dy is the distance between cylinders and φ is the angle
of each cylinder in cylindrical coordinate system. This
procedure can be repeated further for other rows. The total
field will be the sum of incident field and scattered field from
the structure. To take into account other rows of the periodical
structure Bloch theorem is used, as in [5]. It is represented in
Fig. 6 where p is the distance between two rows.
Figure 7. Dispersion diagram for the gap waveguide calculated using
MoM (ABCD) approach and compared to CST simulation results.
A. Cylinder in parallel plate waveguide
In this method, instead of first calculating the current along
the pin and from that the scattered electric field, the electric
field will be from the beginning represented as a sum of
cylindrical modes. As it is shown in Fig. 8, there is defined
electric field in the waveguide and in the gap between the
cylinder and the top plate.
Figure 6. Bloch theorem scheme
Electric and magnetic fields before and after the considered
row are observed. Comparing them S-parameters are obtained
and consequently ABCD parameters/matrix:
 E1   A B   E 2  ikB p  E 2 
 H1  C D   H 2  e
 H 2
  
 
 
Here, Bloch propagation constant is denoted as kB. From the
eigenvalues of ABCD matrix Bloch propagation constant for
all frequencies of interest was calculated. Those propagation
constants will form the dispersion diagram for this structure.
Figure 8. Single pin between parallel plates
Electric, oriented in z direction, and magnetic fields
between parallel plates, can be expanded in terms of
cylindrical modes as:
 n z   2
Escat   n cos 
 H 0  kn   ,
n 0
 hw 
H scat  j
 n z   2
 H1  k n   .
 hw 
 k
n 0
 ncos 
 n 
kn  k  
 .
 hw 
Electric and magnetic fields in the gap above the cylinder are
defined as:
 n
Egap   n cos   z  L   J 0 kn'  ,
n 0
 hg
H gap  j
 k
n 0
 n
 n cos 
 hg
 z  L  J1  kn'   ,
Expressions for the incident field that impinges at single pin
expanded in terms of cylindrical modes are [6]:
Einc  Einc ,n cos(
n 0
H inc  j
inc , n
n 0
n z
) J 0  kn   ,
 A       Ei    Einc  .
 D2      H i   H inc 
For different values n and m matrix D1 is filled with:
 n z 
 m z  ,
D1  m, n   H 0 2  kn a  cos 
 cos 
 dz
 w 
 hw 
matrix A with:
 n
 m z  ,
A  m, n   J 0  kn' a  cos   z  L   cos 
 dz
 hw 
 g
matrix B with:
B  m, n  
 m
 n z 
k  2
H1  kn a  cos 
( z  L)  dz ,
 cos 
 hg
 hw 
and matrix D2 with:
D2  m, n  
 n
 m
J1  kn' a  cos   z  L   cos 
 z  L  dz . (18d)
 g
 g
K matrix contains unknown coefficients α and β. In addition,
matrix Ei is filled with:
 .
 D1 
  B
where n is mode number, βn are unknown coefficients for each
mode, hg is height of gap between cylinder and a top plate and
kn’ is equal to:
 n
kn'  k 2  
 g
 Z  K    I  ,
Here n is mode number, αn are unknown coefficients for
each mode, hw is height of waveguide, k is total propagation
constant, η is impedance of the free space, and H0(2) are
Hankel functions of second kind of order zero. In addition, N
is maximum number of modes and kn is equal to:
Mode matching approach implies that we have to match
modes in (15a) and (15b) with test functions. The resulting
matrix equation is:
 n z 
 m z 
Ei  m, n   J 0  kn a   cos 
 cos 
 dz ,
 hw 
 hw 
and matrix Hi with:
 m
 n z 
H i  m, n   J1  kn a   cos 
 z  L  dz .
 cos 
 hw 
 hg
In (17) Einc and Hinc are amplitudes of incident electric and
magnetic field, respectively.
B. Row of cylinders in parallel plate waveguide
To get closer to a full periodical structure we have to add
one row of cylinders. Top view of one row is presented in Fig.
n z
) J1  kn   . (14b)
where Einc,n are amplitudes of outside incident electric field
and J0 and J1 are Bessel functions of order 0 and order 1
respectively. After we have defined all fields, boundary
conditions can be applied at the edge of the cylinder (ρ=a),
but only in the region above the cylinder:
Escat  Einc  Egap ,
Figure 9. One row of cylinders between parallel plates
H scat  H inc  H gap .
When we have more than one cylinder we have to include
coupling between them and therefore we have to modify
expressions for scattered electric and magnetic fields between
parallel plates:
 N
 n z   2
 jqcos0  p
Escat    ncos 
 H 0 (    q )e
q  n  0
 w 
H scat   j
 q  n0
 n z   2
 jqcos 0  p . (20b)
 H 0 (    q )e
 hw 
 n cos 
Here ρq is radial position of each cylinder and φ0 is angle
illustrated in Fig. 9. These series converge very slowly so we
used [7] to accelerate them.
C. Full periodical structure
Main idea to get full periodical structure is to observe each
row of cylinders as an individual scatterer, as it is in [8]. Every
scatterer is fully described with the scattering matrix. Since all
rows are equal the scattering matrix will be equal for all of
them. The scattered field of each row can therefore be
obtained by multiplying the scattering matrix and the matrix of
the incident field on that row. Total incident field that
impinges at some row is the sum of outside incident field and
scattered fields from all other rows placed before that row:
Eiin  Tik SEkin  Ti 0 EI .
Figure 10. Dispersion diagram for the gap waveguide calculated using
Mode Matching approach and compared to CST simulation results.
In comparison with the MoM approach results shown in Fig. 7
it is clear that the Mode matching approach results are in much
better agreement with the referent CST MS results. For that
reason all further results dealing with the realistic gap
waveguide structures will be obtained using the analysis based
on Mode Matching approach.
k 1
Here, Tik is the translation matrix that translates scattered
electric field of one row to another row, S is scattering matrix
of certain row, Ti0 is translation matrix of incident field, and EI
is the outside incident field that impinges on the whole
structure. Finally, the same matrix equation is obtained:
 Z 2 K 2   I 2 ,
where K2 is a vector of unknown coefficients for the incident
field in each row. From these coefficients scattered field of
each row of pins at a desired location can be obtained. If we
observe total electric field at some points along the structure
we can find phase difference between them, from which we
can calculate propagation constants for each frequency of
interest and present them in the dispersion diagram. Fig. 10
showns the dispersion diagram of the same structure as in Fig.
In all our examples before, the observed structure was
infinite and we performed summations through all elements in
a row. When we have a realistic structure we have only a few
elements in each row and therefore the summation through all
elements will not converge. For this reason we have to observe
each element as individual scatterer. We will use expressions
from (10a) to (14b). In Fig. 11 is presented dispersion diagram
of the real structure where we used twenty rows of cylinders
wherein each row contains 9 elements (central element plus
four on both sides).
Figure 11. Dispersion diagram for the realistic gap waveguide calculated
using Mode Matching
From Fig. 11 we can see that there is slightly propagation in
stop band, i.e. propagation constant is not equal to zero in stop
band like it is shown in Fig. 10. This is mostly due to the fact
that we have here a finite structure.
A. Groove gap waveguide
As it is mentioned in the Introduction, gap waveguide
structure guides electromagnetic waves if we make some
structural modifications like ridges, strips or grooves. This
chapter is oriented on groove gap waveguide shown in Fig. 12.
Main idea is to guide electromagnetic wave along the groove
for frequencies in stop band, thus relaying on the stop band
properties to prevent energy leakage from the groove.
Figure 12. Groove gap waveguide. (a) top view , (b) side view
Propagation constant for rectangular waveguide is equal [10]:
f 
k prop  k0 1   c  ,
 f 
where fc is cut-off frequency of rectangular waveguide and is
equal to:
 fc mn 
 m   n 
 ,
 
 w   hw 
2 
m, n  0,1, 2
For TE10 mode, in (23b) m is equal to one, and n is equal to
zero. In Fig. 15 is presented dispersion diagram of the groove
gap waveguide in stop band and we can see a good agreement
between our results and results obtained using CST. However,
there is difference between groove waveguide results and
normal waveguide, which indicates that the fields in these two
structures are not the same, because cylinders are very thin
and distance between them is relatively large compared to
their dimension, so groove does not act as rectangular
waveguide (cylinders does not imitate side walls of r.w.).
From Fig. 12 it can be seen that the groove is surrounded by
cylinders from both sides and thus only propagation along the
groove is allowed in the form of propagating TE/TM modes
similar to the modes in a normal rectangular waveguide.
1) First design
In this first design we used the same dimensions of the
structure as before: length of the cylinder L is 8.66 mm, height
of the waveguide h is 12.16 mm and the period of the pins is
3.75 mm. Groove length has to be equal around λ/2 and in
order to have propagation above 8 GHz, dimension W from
Fig. 12(b) was calculated to be equal 18.66 mm. In Fig. 13 is
presented CST design of the structure.
Figure 15. Dispersion diagram in stop band of groove gap waveguide
Figure 13. First design of groove gap waveguide in CST
The results of our approach are also compared with analytical
calculation, which are calculated for rectangular waveguide
(r.w.) with dimensions as in Fig.14.
Further, electric and magnetic fields in the cross section of the
groove gap waveguide were calculated in order to compare
them with the fields in the normal rectangular waveguide.
Field expressions for a rectangular waveguide are as in [11]:
Ez  E0cos(
Hx 
Hy 
k x
 jk y
y )e jkx x ,
y )e jkx x ,
y )e jkx x ,
Figure 14. Front side view of rectangular waveguide
k x  k 2  k y2 .
Fig. 16 presents a comparison of absolute values of electric
field in groove gap waveguide and rectangular waveguide.
Electric field in rectangular waveguide is normalized with
maximum value of electric field in groove gap waveguide.
2) Second design
In the second design we used dimensions of gap waveguide
to have stop band between 10 GHz and 20 GHz. Changing the
length of the cylinder and height of the waveguide, width and
marginal frequencies of the stop band can be approximately
calculated. Obtained dimensions are: length of the cylinder L
is 5.00 mm, height of the waveguide h is 6.00 mm, radius of
the cylinder a is 1.63 mm, period of the cylinders p is 6.50
mm and width of the groove w is 15 mm. In Fig. 18 is
presented second design.
Figure 18: Second design of groove gap waveguide in CST
In Fig. 19 is presented dispersion diagram of that structure in
stop band.
Figure 16. Electric fields in groove gap waveguide and in rectangular
We can see from Fig. 16 that shapes of electric fields are
almost equal. Using electric field two components of magnetic
fields were calculated and also compared with analytic
solutions. Results are presented in Fig. 17.
Figure 19. Dispersion diagram in stop band of groove gap waveguide
From Fig. 19 we can see a good agreement between all the
results. This indicates that the groove waveguide in this design
behaves almost identically as the normal rectangular
waveguide. This is further verified with electric and magnetic
field plots shown in Figs. 20 and 21. Better agreement in this
case is a consequence of the fact that this design with a
smaller gap between the pins and the top plate confines the
energy much better than the previous design.
Figure 17. Magnetic fields in groove gap waveguide and in rectangular
Fig. 17 shows that there is a partial agreement between the
results. Significant difference occurs at the edges of the
groove because in that area the field is not completely
contained due to the fact that there is a small leakage of
energy through the pins (since this is not an ideal soft surface).
Figure 22. Single element slot in gap waveguide technology [10]
Figure 20. Electric fields in groove gap waveguide and in rectangular
Figure 21. Magnetic fields in groove gap waveguide and in rectangular
From Fig. 20 and Fig. 21 can be seen a good agreement
between all results. As it is shown in Fig. 17, also here
significant difference occurs at the edge of the groove in
groove gap waveguide, because in that area the influence of
cylinders occurs.
Rigorous and efficient method for the analysis of gapwaveguide structures based on the bed-of-nails as the artificial
magnetic conductor is presented in this paper. Initial attempt
was to use Method of Moments approach to determine the
currents and fields inside the structure for single and multiple
cylinders. To determine the solution for the periodic array of
pins Bloch theory is used to determine the ABCD parameters
matrix. Result for this approach was unsatisfactory and
therefore another approach based on Mode Matching was
pursued. The field in the gap was expanded in to a set of
cylindrical modes and so was the field in the waveguide.
Using Mode Matching, boundary conditions on the boundary
between the gap and the waveguide were applied. From this,
dispersion diagram and scattered field could be obtained.
Determined dispersion diagrams were compared to full-wave
simulation with commercial software and have shown
excellent agreement. Also comparison was made between
groove gap waveguide and rectangular waveguide. It showed
that field confinement inside groove gap waveguide largely
depends on the selection of bed of nails parameters. Mode
Matching method used to obtain these results was shown to be
a very efficient and accurate approach for initial studies of gap
waveguide structures.
This work was supported in part by the project ''Passive and
Active Metamaterial Structures for Guiding, Scattering and
Radiation of Electromagnetic Energy", Unity through
Knowledge Fund (UKF), Croatia, 2013.
Gap waveguides are commonly used to feed arrays of
antennas. The most common antenna placed on the top plate
of gap waveguide is a slot antenna. Antenna is fed by a ridge
in ridge gap waveguide as in [10], or with groove in groove
gap waveguide. For this reason, the ridge and the slot antenna
are two new objects that will be introduced into this analysis
approach in future work. Example of a slot antenna placed on
a top plate of the gap waveguide is presented in Fig. 22.
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