Corrugated Filter with Quarter-Wave Coupled Resonators

Design procedure

R. Goulouev

Corrugated waveguide filters are widely used in powerful
microwave applications in order to reject high frequency
spurious spectrum. Here a corrugated harmonic filter
of new type [1] using
quarter-wave-coupled E-plane corrugated reflection-zero
resonators providing high loaded Q is presented. The
expressions for scattering parameters of filter structure
are represented in terms of known variational approximations
and grouped into completed two s
tep design procedure, which combines a first-order synthesis,
based on quarter-wave-coupled prototype, with gradient
optimization procedure. The design procedure is realized
as MathCAD spreadsheet allowing performing complete
filter design
including synthesis, optimization and detailed verification
within several minutes.

Forward your comments & questions to R. Goulouev

E-mail: gouloue@ieee.org

Web Page: http://www.goulouev.com/

Waveguide

The filter [1] is represented as a number of wave guiding
elements of scattering matrices Si
and connected to each other by straight waveguide lines
of length L
i
.

Figure 1: Cascade of N+1 scattering elements.

S-parameters of cascade of any two elements with S-matrices
S0
and S1
can be expressed:

(1)

Where b
is propagation constant taking into account finite
conductivity of waveguide surface [2]

silver

(2)

copper

aluminum

nickel

where

(3)

are wavelength and wave number corresponding to free
space, a is waveguide width and b
is waveguide height. The initial value of loss factor
is multiplied by 1.7 practical correction in order
to take into account roughness of surface.

Single E-plane Cavity

The filter structure is based on shorted E-plane symmetric
stubs (cavities) grouped in resonating pairs. Scattering
properties of such a cavity can be evaluated using
E-plane T-junction model [3].

Figure 2: Single E-plane corrugation and its equivalent
circuit.

Where reactive elements of T-junction's equivalent circuit
from [3] modified for non-ideal waveguide impedances
are given by sequence of expressions

(4)

Here the short circuited waveguide susceptance Bs
representing the shorted waveguide section has to be
connected in parallel to Bd
susceptance of the T-junction circuit reactance to
simulate a single corrugation as given

(5)

Thus the reflection and transmission coefficients of
a single corrugation can be expressed from the equivalent
circuit and represented by

(6)

Resonators

A single resonator can be represented as a pair of two
corrugations connected to each other by a waveguide
section as shown on Figure 3.

Figure 3: A single resonator and its equivalent circuit.

The length d of the waveguide is chosen to provide reflection
zero at resonance frequency br.
Using the expressions (6) and (1) S-parameters of such
a resonator can be derived

(7)

The length of connecting waveguide d
can be found from the condition of reflection zero
at resonant frequency kr
as

(8)

Let represent the differential unloaded Q factor of
the resonator at the vicinity of b
r
as

(9)

where R(b)
and T(b
) are reflection and transmission functions from (7).
After simple manipulations the Q value can be expressed
from reflection and transmission functions of single
corrugation (6)

(10)

Now we can enter a simple relationship of loaded Q
of resonator with its dimension h das follows:

(11)

Filter Structure

By analogy with iris quarter-wave-coupled filters from
[2], here the resonators are integrated into a filter
structure by impedance inverting waveguide sections
of length t~lg/4
as shown on Figure 4 below

Figure 4: Corrugated filter consisting of single pairs
of corrugations

Filter structure with pseudo-maximally-flat frequency
response characterized by central frequency br and bandwidth
Db
can be synthesized using expressions (11) and (8) matching
required Q-values [2]

As scattering parameters of each resonator is known
(7), the frequency response of filter structure can
be calculated using consecutive cascading S-elements
as follows:

Interface Transformers

Because the dimensions of external waveguide line A x B
can differ from the dimensions of internal cross section
of filter structure a
x b
, the transformer sections are necessary. Here a single
quarter-wave step is used to match filter structure
with interface. In order to simulate the frequency
response of such a waveguide transformer, Macfarlane's
formulas [4] for HE-plane waveguide step can be used.

Figure 5: HE-plane junction and its equivalent circuit

Here the impedance transformation factor, admittance
of smaller waveguide and equivalent shunt susceptance
are expressed in following expressions

CORRECTION
: The previous expression for EH-plane junctions from
[4] are replaced with expression (16) obtained using
variational method [3].

Thus we can derive expression for S-matrix of HE-plane
step

and S-matrix of double step transformer as a cascade
(1) of two junctions

The dimensions of cross section At xBt and length of
transforming step can be synthesized from condition
of reflection zero at kr, the wave number corresponding
to the central frequency of filter's pass-band and
expressed as

(22)

As the quarter-wave transformer can be considered as
a reflection-zero resonator of low loaded Q (9), it
can replace the end resonators for most practical cases
in order to reduce length of filter and get rid of
practically unreal shallow corrugations.

Entire Corrugated Filter

Figure 6: Sketch of entire filter structure

Now dimensions of internal filter structure with end
resonators removed are obtained from (12), (11), and
(8) and expressed as follows:

(23)

where w=
Db/b
is relative bandwidth of filter.
Since all dimensions of filter are known, the S-parameters
of entire filter can be obtained as cascade of two
transformers (21) and internal corrugated structure
(13)

(24)

Response Optimization

As usual filter synthesizing procedures are based on
representation of waveguide structure as a circuit
of lumped elements, it is a very complicated problem
to extract the L-C parameters from waveguide junctions
and compensate their frequency dependence or overmoding.
Besides if even the problem is solved and the filter
has an ideal Chevychev or Zolotorev function response
in the vicinity of its pass-band, it is quite not obvious
the filter show rejection of far frequency spectrum
will be also so well. Therefore most designers rather
prefer to design from scratch a draft filter matching
the specification in common sense, and optimize its
pass-band and roll-off points. A simple gradient approach
is used here to adjust filter bandwidth. First of all
we extract physical parameters to be optimized from
constant and dependant parameters and group them in
two vectors as shown

(25)

Let declare a function evaluating average return loss
of pass-band [k0,k1] and isolation at specified point
k2 in respect to corresponding spec values R and T
in dB as

(26)

Now we can express any next vector approximation v1
from previous vector approximation v0 by scalar increment
dv

(27)

The procedure can be repeated any Nsteps times until
the minimum of function f is found

(28)

After optimization is completed the new optimized dimensions
can be found as inverse functions from (25)

(29)

Derivative Parameters

Most technical parameters can be obtained from S-response
of filter as follows:

Insertion and return loss in dB

Group delay

Group delay slope

Insertion Loss slope

Harmonics Rejection

As corrugated filters are based on E-plane structures
the filter response over frequency is a function of
waveguide propagation number b.
Therefore if a pass-band exists for TE10-mode, it must
be for any TEno-mode having the same b-plan.
Practically there is no way to eliminate spurious pass-bands
corresponding to spurious waveguide modes of higher
order except keeping TE10-mode clearance. That is a
property of E-plane corrugated filters of any type
[4]. Nevertheless, corrugated filters are commonly
used as low-pass filters because of their low cost,
low loss and extremely high power handling. Practically
designers of corrugated filters try to move those spurious
pass-bands out of important frequency bands required
to be rejected. Prediction of spurious rejection is
easier problem as the expressions (13) and (24) can
be used to evaluate frequency response of any TEno-mode
by substituting wavenumber k by given

A simple procedure [5] shows how to select optimal a
-dimension in order to provide rejection of modes of
higher order in particular frequency bands.

Peak Power Handling

Electrical breakdown can occur in air or vacuum due
to ionization or emission. Practically the peak power
handling of straight waveguide section can be approximately
evaluated as

(31)

This value should be reduced by 1+Q times for each resonator
of filter (12) take into account the power stored by
each resonator. Then the maximum peak power handling
can be estimated as

(32)

for filters with pseudo-maximally-flat response with
frequency bandwidth [f0,f1].

Dimensions of filter
body

Finally we can summarize the internal dimensions of
filter W x H x L

(33)

which are maximum dimensions of internal surface of
filter body. The real dimensions of filter assembly
must include thickness of metal and flanges in addition
to those simulated dimensions.

Manufacturing

The filter can be made from two symmetric half bodies
using regular milling or EDM machining. Though central
E-plane cut is recommended because of absence transversal
currents, central H-plane cut is commonly used in corrugated
filters production because of milling preferences.
If so, contact grooves should be used in order to provide
good electrical contact on flanges between filter cuts.
Electroforming is used more seldom, but allows creating
entire filter. Practically +/-0.002a
random tolerance on dimensions and positions of corrugations
might be allowed as it should not degrade much pass-band
performance of filter of this type. Reasonable radii
(not larger than 0.07a
) of milling tool might be allowed, as it does usually
not impact electrical properties of filer of this type.

Comparison with experimental
data

An experimental Ku-band WR75 filter has been designed
using the following design procedure. The following
dimensions has been specified

During production the real dimensions of filters are
assumed to be deviated from the designed dimensions
as +/-0.001'' random manufacturing tolerance were applied
to all corrugations and corners of all cavities were
rounded by milling tool of 0.064'' radius. After being
manufactured the both filters were measured using a
vector network analyzer and measurement errors are
expected to be +/-0.03 dB for insertion loss of pass-band,
+/- 2 dB for in-band return loss, +/-2 dB on roll-off
rejection and 70-80 dB noise floor limits. During testing
the following data of return and insertion loss over
WR75 waveguide bandwidth has been measured

Here the data simulated by (24) and data measured by
a network analyzer are put on the same plot

It is shown that deviation from performance simulated
to performance measured is in reasonable range corresponding
to applied manufacturing and measurement inaccuracy.
Far-out-of-band rejection has been measured for three
experimental filters. Test results have shown the rejection
evaluated by expression (24) is in practical match
with test data corresponding to TE10-mode in frequency
ranges up to 2.5-3.0 times filter pass-band. Thus accuracy
of the design expressions written above is matching
common practical requirements.

Design Example

Let us design a WR-75 corrugated filter matching the
following specification

Frequency Range, GHz 12.50 - 13.75

Insertion Loss, dB 0.35

Return Loss, dB 25

Rejection, dB

14.55- 15.6 GHz 50

21.3 - 24.0 GHz 80

28.2 - 31.1 GHz 60

Gain slope, dB/MHz 0.002

Insertion Loss Ripple, dBpp 0.02

Group Delay Variation, ns 1.0

Group Delay Slope, ns/MHz 0.03

Peak Power Handling W 3000

Interface WR75

Dimensions WxHxL, in 1.35x1.15x8.00

including flanges

A spec function can be specified in order to design
margins later

Interface dimensions A x B are specified as WR75 standard

However the internal
a
dimension of corrugated structure must be selected
in order to remove pass-bands of spurious TEno-modes
(30) from frequency bands specified by specification

Trying different a
-values from larger to smaller we find a=0.70''
provides clearance of TE20-,TE30- and TE40-modes. Selecting
b
and s
is less sophisticated and depends on requirements of
power handling and rejection of high frequency spectrum.
Practically
b ~ 0.05a - 0.1a and
s ~ 2.0b - 2.6b for 3-rd harmonic requirements.
For example

Now we select design bandwidth and order of filter,
for example

The design bandwidth can differ from the bandwidth specified
because it is used only to synthesize initial dimensions
of filter

By changing filter design bandwidth f0-f1
, filter order N and distance between resonators t we
can adjust real bandwidth and roll-off of filter

and fit it within the spec mask as shown on plot

Now we specify optimization constrains

Because the initial design is already close to meet
requirements, several steps of optimization Nst
should be enough to achieve near-band requirements
for return loss and rejection.

WAIT:

OPTIMIZATION MAY

REQUIRE COUPLE OF MINUTES

For more complicated specs more optimization steps might
be required. Now we upgrade dimensions of filter

and compute its frequency response over the frequency
range

The pass-band and roll-off seem to mach the spec. Now
we see where the spurious modes are expected to pop
up

The a-dimension of filter is selected right, so the
TE20-, TE30 and TE40-modes passing in frequency bands
other than specified. Now we can see details of pass-band
performance.

Insertion loss over pass-band

Group delay variation over pass-band

Group delay slope

Insertion loss (gain) slope

Power handling in vacuum (multipaction breakdown)

Power handling in air

Internal dimensions of filter body

Now we can assume the design to be completed, as all
specified parameters are in match with specification.

Conclusion

A corrugated harmonic filter of new type [1] using quarter-wave-coupled
E-plane corrugated reflection-zero
resonators providing high loaded Q. The expressions
for scattering parameters of whole filter structure
have been derived in terms of known variational approximations
and expressed in elementary functions
and compiled into a simulation tool. It has been shown
on practical example that accuracy of simulation of
frequency response of such filter is with good match
with measured data. The design expressions have been
represented as two s
tep design procedure, which combines a first-order synthesis,
based on quarter-wave-coupled prototype, with gradient
optimization procedure, presented as MathCAD spreadsheet.
It is shown that the design procedure allows performing
complete design including synthesis, optimization and
verification within several minutes.

References

1. R. Goulouev "
Corrugated waveguide filter having coupled resonator
cavities", US patent 6,169,466, 2001

2. R.E. Collin "Foundations for Microwave Engineering",
McGraw-Hill, 1966.

3. N. Marcuvitz "Waveguide Handbook", Peter
Peregrinus Ltd., 1993.

4. G.F. Graven and C.K. Mok "The Design of Evanescent
Mode Waveguide Bandpass Filters for a Prescribed Insertion
Loss Characteristic", IEEE, MTT-19, pp. 295-308,
March 1971.

5
. R. Levy "Tapered Corrugated Waveguide Low-Pass
Filters", IEEE, MTT-21, pp. 526-532, Aug. 1973.

6. R. Goulouev "Corrugated Filter Spurious Pass-Bands",
MathCAD spreadsheet, http://mathcad.adeptscience.co.uk/mcadlib/apps/corrfilt
.mcd or http://www.circuitsage.com/filter/corrfilt.mcd,
1999.