TUTORIAL
A GENERAL D ESIGN
PROCEDURE FOR BANDPASS
FILTERS D ERIVED FROM
LOW PASS PROTOTYPE
ELEMENTS: PART II
P
art II of this bandpass filter tutorial features several examples that illustrate the
general design procedure for bandpass
filters using resonators and coupling methods
of different types.
BANDPASS FILTER DESIGN
The general bandpass filter design procedure utilizes the normalized, low pass elements of the prototype
filter to determine the
r e q u ir e d r e son at or
…several examples that
coupling and coupling
to the external circuit,
illustrate the general design
that is, the source and
procedure for bandpass filters load. In each case, the
e xt e r n al sou r ce an d
using resonators and coupling
load are assumed to be
methods of different types.
50 Ω, although the filter could be matched
to other real impedances.
The general design procedure is listed in
Table 1. To illustrate the general bandpass filter design procedure, several examples are offered. It should be mentioned that the coupling parameters, that is, the capacitor matrix,
or even and odd mode impedances, for several
types of distributed resonators, may be determined with the aid of an electromagnetic
(EM) simulator. This is an extremely valuable
design tool because it eliminates the fabrication of models in order to determine the coupling of symmetrical, distributed resonators.
The illustrated examples include a lumped
element, Chebishev, bandpass filter using πtype resonators, a coaxial cavity bandpass filter
using λ/8 resonators with capacitive tuning
(comb-line), a coaxial cavity bandpass filter
using λ/4 resonators (interdigital) and a half
wavelength via-hole direct-coupled filter. Also
described are a dielectric resonator, a bandpass filter using disk-shaped resonators, an inductively coupled filter in coplanar waveguide
and a stripline or microstrip, Chebishev, bandpass filter using λ/2, side-coupled resonators.
Lumped Element Bandpass Filter
The lumped element, π-type resonator is
used in a five-section, 0.01 dB ripple, Chebishev filter. The low pass prototype elements
(g-values) may be calculated using the equa-
K.V. P UGLIA
M/A-COM Inc.
Lowell, MA
Reprinted with permission of MICROWAVE JOURNAL® from the January 2001 issue.
©
2001 Horizon House Publications, Inc.
TUTORIAL
TABLE I
BANDPASS FILTER DESIGN PROCEDURE
TABLE II
Determine the Filter Parameters:
Center frequency, f0
Bandwidth, ∆f
In-band ripple, r dB
Number of resonators, n
Rejection requirements, R(f')
Insertion loss, L(f)
Resonator type: lumped element, comb-line, interdigital, coaxial cavity, dielectric …
Special conditions: MFTD, Bessel, Gaussian …
Determine the Low Pass Prototy pe Elements:
Calculate elements
Acquire from tabular data
Calculate Coupling Coefficients:
k i,i +1 =
∆f
fo
1
for i = 1 to n –1.
g i g i +1
Determine Resonator Reactance or Susceptance Slope Parameter:
π-TYPE RESONATOR BANDPASS
FILTER DATA
i
gi
Cπ,I,I +1
Cia
Cib
0
1.0000
16.4
1
0.7563
5.1
37.7
45.6
2
3
1.3049
3.5
45.6
47.1
1.5773
3.5
47.1
47.1
4
1.3049
5.1
47.1
47.1
5
0.7563
16.4
45.6
37.7
6
1.0000
α or β.
Determine Coupling Reactance or Susceptance:
Ki,j+1 or Ji,i+1 for i = 1 to n–1; Ki,i+1 = αki,i+1 and Ji,i+1 = βki,i+1.
and the susceptance slope parameter
is calculated as
β = 1(2πf0)C = 0.063662 mhos
The resonator coupling elements are
determined using
Determine Input and Output Coupling Parameters.
Determine Filter Elements (for lumped element resonators).
Determine Filter Dimensions (for distributed resonators).
Optional:
2πf0C π
Computer simulation
Estimate filter insertion loss
Estimate filter group delay
▼ Fig. 1
= βk i, i + 1
for i = 1 to i = n – 1
Schematic of a π-type resonator.
L
C
i, i + 1
C
BANDPASS FILTER DESIGN
PROCEDURE
The bandpass filter design procedure is listed in Table 1. The values of
the inductance and capacitors may be
determined using
1
f0 =
2π
and the input and output coupling
are calculated as
2πf0C 0,1 =
and
2
= 100 MHz
LC
2πf0C n ,n + 1 =
where
tions within the appendix or may be
determined from available tabular
data.
The filter parameters are
f0 = 100 MHz
∆f = 5 MHz
n=5
r dB = 0.01 dB
g0 = g6 = 1.0000
g1 = 0.7563
g2 = 1.3049
g3 = 1.5773
g4 = 1.3049
g5 = 0.7563
The unloaded quality factor for
this type of resonator at this frequency is Q u = 250. A schematic of the
resonator is shown in Figure 1.
L≈
10
= 100.0 nH
f0
and
C=
2
(2πf0 )2 L
= 50.66 pF
The inductance formula provides a
good estimate of a convenient value
of the resonator self-inductance from
which the resonator capacitance may
be calculated.
Next, the coupling values are calculated using
k i, i + 1 =
∆f
f0
1
g i , g i+1
for i = 1 to i = n – 1
∆f
β
f0 g0 g1Z0
β
∆f
f0 gn gn + 1Z0
The input and output coupling methods need not be the same as the resonator coupling. However, the input
and output coupling must satisfy the
relationship
βZ0
g gf
Qe =
= 0 10
2
in
∆f
J
( 0, 1 )
and
Qe
out
=
βZ0
(Jn ,n +1 )
2
=
gn gn + 1f0
∆f
Tab le 2 lists the π-type resonator,
bandpass filter data.
A schematic for the five-section, πt yp e r e son at or b an d p ass filt e r is
TUTORIAL
n
∑ gk = 1.98
dB
k =1
The computer simulation of the
transmission group delay is also presented in the results. The midband
group delay may be estimated from
the formula
( )
τd f0 =
1
2π∆f
n
∑ gk = 181
ns
k =1
It should be mentioned that the
on ly filt e r e le m e n t s wh ich we r e
tuned in order to achieve the desired
passband were the inductors of the πresonators.
COAXIAL CAVITY BANDPASS
FILTER λ/ 8 RESONATORS
The second bandpass filter design
example is the coaxial cavity type filter using λ/8 resonators with capacitive tuning or comb-line filter, so
called because the resonators are
fixed to ground on a single surface in
much the same configuration of the
teeth of a comb. This is a very popular filter structure because a moderate value of unloaded resonator quality factor may be achieved (and therefore low loss), and because a wide
rejection band is indigenous to the
structure by virtue of the wide separation in resonant modes. The resonator structure employed for the example filter is that depicted in Part I.
Unlike the traditional comb-line filter, which has no metallic obstacle
between resonators, the filter example employs rectangular cavities with
coupling slots as shown in Figure 4.
This structure yields a somewhat
higher unloaded quality factor because more of the field is enclosed.
The coupling between resonators is
controlled via the width of the slot, w.
The coupling coefficient between
resonators may be determined by
measurements of symmetrical-cou-
5.1
100
3.5
45.6 47.1
45.6
100
3.5
47.1 47.1
100
47.1 45.6
100
5.1
16.4
45.6
37.7
INDUCTOR VALUES IN nH
CAPACITOR VALUES IN pF
▲ Fig. 2
Schematic of a five-section π-type resonator bandpass filter.
pled resonators as a function of the
slot width or via analysis of the self
and mutual capacitance using an EM
simulator. Both methods were employed and resulted in acceptable
agreement, that is, less than 5 percent over the slot widths used in the
filter, between the experimentally determined coefficients and the measured values as shown in Figure 5.
To illustrate the filter design procedure a PCS1900 transmit band filter (1930–1990 MH z) is required
with minimum insertion loss (< 1.5
dB) and volume as the principal design specifications. A filter rejection
of 60 dB, minimum, in the PCS1900
receive band (1850–1910 MH z) is
also required.
From the earlier data, an eight-resonator filter is required to achieve the
rejection requirements. This may be
verified from direct calculation using


L (f a )


10
–1
–1  10
cosh 

r dB

10 – 1 
10


n=
cosh –1 ∆fn
( )
where
L(fa) = the attenuation required at
the specific rejection
frequency fa
r db = the in-band ripple factor
∆fn = the normalized bandwidth at
the rejection frequency,
▼ Fig. 4
Coaxial cavities with slot coupling.
w
that is
∆fn =
2 fa – f0
∆f
Because the reject band is close to
the passband, the passband was deliberately narrowed to 55 MH z so
that coupling adjustment could be facilitated in order to increase the final
filter bandwidth to 60 MHz. The filter parameters are
f0 = 1960 MHz
∆f = 55 MHz
n=8
r dB = 0.05 dB
π
θ0 =
4
Q u = 2250
1
= v0C g mhos
Y0 =
60
▼ Fig. 3
(dB)
4.343f0
∆fQ u
37.7
100
21
( )
IL f0 =
16.4
S11 (dB),
shown in Figure 2. A computer simulation was conducted on the fiveresonator bandpass filter using only
the inductors as variable or tuning elements. The results of the simulation
are shown in Figure 3. Note that the
bandwidth is 5.12 MHz as compared
to the design bandwidth of 5.0 MHz,
and that the midband insertion loss is
approximately 2 dB. The midband insertion loss of a bandpass filter may
be estimated from the equation
Computer simulation results.
0
800
−10
700
−20
600
−30
500
−40
400
−50
300
−60
200
−70
100
−80
90
100
110
FREQUENCY (1 MHz/ div)
0
TUTORIAL
The LP prototype g-values are
g0 = 1.0000
g1 = 1.0437
g2 = 1.4514
g3 = 1.9899
g4 = 1.6502
g5 = 2.0457
g6 = 1.6053
g7 = 1.7992
g8 = 0.8419
The coupling values are calculated
using
K i, i + 1 =
∆f
1
•
f0
g ig i+1
The slot width may be determined
from the measured coupling data of
symmetrical coupled resonators or
from an EM simulator that provides
the capacitance matrix data for the
particular structure. F or this example, both techniques are utilized.
F irst, the measured coupling data
from a pair of symmetrical coupled
resonators was subjected to polynomial regression of degree three. Next,
the coefficients of the polynomial
were used to generate an equation
for the coupling coefficient as a function of the slot width. Finally, a plot
of the coupling coefficient was used
to determine the slot width for each
coupled section.
The measured coupling data is displayed in Table 3 for three values of
slot width for square cavities of 0.875
inches and center conductor diameters of 0.375 inches. These dimensions produce a resonator characteristic impedance of approximately 60 Ω.
Performing a polynomial regression produces an equation for the
coupling coefficient Kc as a function
of the slot width, w such that
[
( )]
( )

π
 1 +  K i, i + 1
2

Filter Simulation
In order to conduct a simulation of
the filter characteristics, an equivalent circuit is required. A suitable
equivalent circuit may be constructed
wit h t h e aid of t h e t wo-wir e lin e
equivalent of the coupled line shown
in Figure 7.
Y0
cot θ0 + θ0 csc 2 θ0
2
and the mutual coupling capacitors
β=
β • k i, i + 1 C g
=
υ0
2
The filter configuration is shown
in Figure 6. Note that all resonators
are the same diameter and that coupling to the input and output is via direct “tap” or contact to the first and
eighth resonator at a low impedance
point on the resonators. Alternate
coupling mechanisms are via capacitive probe at a high electric field
point of the resonator or via transformer coupling. Generally, the tap is
preferred because it eliminates the
need for another machined part, provid e s a m or e com p act filt e r, an d
makes the filter’s input and output a
direct short to ground at DC. The design data for the comb-line filter is
listed in Table 4.
The correlation coefficient R for the
polynomial regression was 1.00.
Alternatively, an E M simulator
may be utilized to determine the selfcapacitance C g and the mutual capacitance C m for various slot widths.
Subsequent data may also be subject
to polynomial regression analysis to
determine the correct slot width for
each coupled resonator.
The susceptance slope parameter is
calculated using
Filter configuration.
MEASURED
4.250
0
0.030
COUPLING K(w)
C i, i + 1 =
Kc(w) = –0.02717 + 0.10865w
– 0.03150w2
▼ Fig. 6
SIMULATED
becomes
0.026
4 × 40 tap
0.25 Dp. mim
18 plcs
4.125
2
0.022
0.125
0.018
1.125
2.125
3.125
0
0.014
0.125
0.1875
0.125 R. typ
0.010
0.40 0.44 0.48 0.52 0.56 0.60
SLOT WIDTH w (" )
0.125 R. typ
0
0.405
0.547
7
0.626
0.452
0.452
0.547
1.0625
1.125
1.1875
▲ Fig. 5 Coupling coefficient for λ/8 lines
vs. slot width.
0.449
4
1.625
0.325
2.125
2.250
TABLE III
0.1875
Coupling Coefficient
kc
0.400
0.01125
0.500
0.01928
0.600
0.02668
1.625
2.625
3.625
0.375 Dia OD
0.228 Dia ID
0.750 Long
8 plcs. typ
MEASURED COUPLING DATA
Slot Width (" )
w
0.625
2.0625
1.000
0.750 Ref
0
0.200
0
Dimensions in inches
TUTORIAL
If the coupled line equivalence is
applied repeatedly, a complete filter
equivalent circuit may be realized, as
shown in Figure 8, where the tuning
capacitance C t, resonator quality factor Q u and external circuit coupling
has been included. The complete filter was subjected to simulation and
optimization using only the tuning capacitance and external coupling taps
as the variable. The simulation results
are shown in Figure 9.
Note that the actual bandwidth is
52.8 MH z versus the design bandwidth of 55.0 MHz, and that the return loss is consistent with the 0.05
dB Chebishev response. The simulated midband insertion loss and group
delay are 0.85 dB and 37.1 ns, re-
spectively. The midband insertion
loss and group delay estimates are
( )
IL f0 =
4.343f0
∆fQ u
n
∑ gk = 0.85
dB
k =1
and
( )
τd f0 =
1
2π∆f
n
∑ gk = 35.9
ns
k =1
INTERDIGITAL
BANDPASS FILTER (λ/ 4-LINES)
The interdigital bandpass filter is
another popular type of microwave
filter implementation. They typically
have lower loss than comb-line structures and are easier to tune. They require resonators that are fixed to
ground at opposite
sid e s of t h e su p TABLE IV
porting housing as
COMB-LINE FILTER DESIGN DATA
sh own in Fi g u r e
10. Therefore, the
G
Coupling
Mutual
Slot
construction is not
Values
Coefficient
Capacitance
Width
gi
ki,I+1
Ci,I+1
Wi,I+1
as suitable for manu fact u r e as t h e
1.0437
comb-line filter.
0.02280
0.18402
0.547
I n m ost case s,
1.4514
the resonator length
0.01651
0.13327
0.465
is sligh t ly sh or t e r
1.9899
than λ/4 (typically,
0.01549
0.12498
0.452
0.9λ/4), wh ich al1.6502
lows the filter to be
0.01527
0.12326
0.449
tuned to the center
2.0457
frequency with the
0.01549
0.12498
0.452
tuning elements just
breaking the cavity
1.6053
0.01651
0.13327
0.465
wall. This facilitates
both ease of tuning
1.7992
and maximum un0.02280
0.18402
0.547
loaded quality fac0.8419
tor Q u of each reson at or. Th e cou -
Index
i
1
2
3
4
5
6
7
8
Yse = VoCm
PORT PORT
2
1
θ0
Yse = VoCg
PORT
1
θ0
PORT
2
θ0
▲ Fig. 7
Coupled line equivalent circuit.
θ0
pling to the input and output is accomplished via contact at the low impedance point of the resonator.
The design procedure for interdigital bandpass filters is very similar to
the comb-line filter design procedure. In this case, the example illustrates the design of a 500 MHz bandwidth filter centered at 10 GHz.
The filter parameters are
f0 = 10 GHz
∆f = 500 MHz
n=5
r dB = 0.1 dB
θ0 = π/4
Q u = 2500
Zc = 70 Ω
g0 = 1.0000
g1 = 1.1468
g2 = 1.3712
g3 = 1.9750
g4 = 1.3712
g5 = 1.1468
The coupling values are calculated
using
k i, i + 1 =
∆f
1
•
f0
g i , g i+1
the susceptance slope parameter becomes
πY0
4
and the mutual coupling capacitors
are determined using
β=
C i, i + 1 =
πC gk i, i + 1
4
The calculation of the resonator
coupling dimensions must be precede d b y se le ct ion of t h e r e son at or
ground plane spacing. In order to
properly define a distributed resonator and reduce evanescent modes,
certain geometric and aspect ratios
must be maintained. I t was found
empirically that selection of ground
plane spacing in accordance with
Z0
πλ 0 138
10
h≤
32
produced acceptable results.
The formula results from a resonator aspect ratio (ratio of length to
diameter) of 2 × 1. The approximate
ground plane spacing for a resonator
characteristic impedance of 60 Ω is
listed in Table 5.
Ω
Ω
045
GROUND PLANE SPACING
Z7
,8
,7
=2
=2
827
3Ω
Z6
Z4
Z5
,5
,6
=3
=3
05
01
5Ω
3Ω
01
=3
,4
Z3
TABLE V
Filter Frequency
f0 (GHz)
Ground Plane
Spacing h (" )
1
3.000
2
1.500
4
0.750
8
0.375
10
0.300
12
0.250
15
0.215
18
0.175
=
6
7
5
Filter equivalent circuit.
0
150
−10
125
λ/ 4
100
30
75
−40
50
−50
25
NORMALIZED MUTUAL
CAPACITANCE
S11(dB) S21 (dB)
Ω
°
OUTPUT
5°
▲ Fig. 8
8
Z
θ0
45 Ω
°
°
60
4
Zc
INPUT
3
Z
θ0
Zc
2
45 Ω
°
=4
θ0
Zc = 60
θ0 = 45° 1
5°
0
=4
5°
Z1
Z2
,2
,3
=2
=2
04
02
5Ω
7Ω
TUTORIAL
−60
0
1885
2035
FREQUENCY (15MHz/ div)
▲ Fig. 9
Simulation results.
For the present example, a ground
plane spacing of 0.275" has been selected, which requires a resonator diameter of 0.128" to obtain a resonator
characteristic of 60 Ω. This value is
calculated from
Z0 =
 4h 
1
log  =
π
d
v


εr
0C g
138
Cristal2 has given the closed-form
expressions for the even and odd
mode capacitance per unit length for
this line configuration, from which
the self and mutual capacitance may
be calculated (see Appendix A). The
graph of the mutual capacitance versus resonator center-to-center spacing is shown in Figure 11.
The design data for the interdigital
filter is listed in Table 6.
Filter Simulation
The equivalent circuit shown in
Figure 12 is utilized to conduct a
computer simulation. As in the case
of the comb-line filter, a contact at
the low impedance end of the res-
0.50
Interdigital filter construction.
onator serves as the input and output
coupling to the filter.
The results of the computer simulation using only the end capacitance
and tap points as variables are shown
in Figure 13. Note that the actual
bandwidth is 467 MHz versus the design bandwidth of 500 MHz, and that
the return loss is consistent with the
0.1 dB Chebishev response. The sim-
0.388
0.30
0.20
0.1966
0.10
0.1498
0
0.25 0.30 0.35 0.40 0.45 0.50
CENTER-TO-CENTER SPACING (" )
▲ Fig. 11
▲ Fig. 10
0.364
0.40
Mutual capacitance vs. spacing.
ulated midband insertion loss and
group delay are 0.30 dB and 2.3 ns,
respectively. The midband insertion
loss and group delay estimates are
( )
IL f0 =
4.343f0
∆fQ u
n
∑ gk = 0.30
dB
k =1
and
( )
τd f0 =
1
2π∆f
n
∑ gk = 2.2
ns
k =1
TABLE VI
INTERDIGITAL FILTER DESIGN DATA
Index
i
G Values
gi
1
1.1468
2
3
4
5
Coupling Coefficient
ki,I+1
Mutual Capacitance
Ci,I+1
Center Spacing
Si,I+1
0.03987
0.1966
0.364
0.03038
0.1498
0.388
0.03038
0.1498
0.388
0.03987
0.1966
0.364
1.3712
1.9750
1.3712
1.1468
TUTORIAL
Z1,2
θ
Ω
°
Z2,3
θ
°
Ω
Ω
Z
The via-hole coupling reactance is
calculated using
1910 Ω
= 85°
85°
Xi, i + 1 =
1
Zc = 60 Ω
θ0 = °
Z =
Ω
°
2
3
= 60
θ0 85°
4
Z =
θ
Ω
°
=
5
Ω
°
K i, i + 1

K
1 –  i, i + 1 
 Z0 
2
for i = 1 to i = n – 1
INPUT
T
OUTPUT
5°
Interdigital filter equivalent circuit.
0
12
−10
10
−20
8
−30
6
−40
4
−50
2
−60
9
10
11
FREQUENCY (0.2 GHz/ div)
0
GROUP DELAY (ns)
S11 (dB), S21 (dB)
▲ Fig. 12
X0,1 =
5°
The filter is constructed on 25 mil
alumina (Al2O 3) substrate material (εr
= 9.9). The available resonator quality
factor for this type of filter is 750 if
pure alumina with high quality metalization is employed. A schematic of
the filter is shown in Figure 15.
The resonator coupling values are
calculated using
k i, i + 1 =
Interdigital filter
simulation results.
∆f
f0
Xn ,n + 1 =
n
OUTPUT
INPUT
K0,1
▲ Fig. 14
K1,2
Kn−1,n Kn,n+1
1
g i , g i+1
HALF WAVELENGTH
VIA-HOLE COUPLED FILTER
This bandpass filter design example illustrates the utilization of seriestype resonators coupled by impedance inverters. The general schematic
of a bandpass filter, which employs
series-type resonators coupled by impedance inverters, is shown in Figure 14.
The design procedure is similar to
those previously explored. The filter
parameters are
πZo
2
The impedance inverter values become
f0 = 10 GHz
∆f = 700 MHz
n=5
r dB = 0.1 dB
g0 = g6 = 1.0000
g1 = 1.1468
g2 = 1.3712
g3 = 1.9750
g4 = 1.3712
g5 = 0.1468
θn = π –
∆f 1
f0 g0 g1
∆f
1
f0 gn gn + 1
2
(
1
φ j + φ j+1
2
3
4
5
IN
OUT
λ/ 2
λ/ 2
λ/ 2
λ/ 2
λ/ 2
1
2
3
4
5
IN
▲ Fig. 15
)
The actual via-hole diameter may be
determined from the via-hole model
within Series IV.™ This model was
utilized to obtain data that was further refined using polynomial regression. The results are shown in Figure
16. The design procedure is very similar to the direct-coupled waveguide
filter using inductive iris coupling
found in Matthaei, Young and Jones.1
and
1
2
and
and the input and output inverter values are
K n ,n + 1 = R b α

K
1 –  n ,n + 1 
 Z0 
for j = 0 to j = n
Ki,i+j = αki,i+1 for i = 1 to i = n–1
K 0, 1 = R a α
K n ,n + 1
 2Xj, j + 1 
φ j = tan –1 

 Z0 
α=
Bandpass filter using series
resonators.
2
The coupling reactance is the result of the via-hole self-inductance,
which is controlled by the via-hole diameter and/or number. The length of
the resonators θn must be reduced by
an amount commensurate with the
magnitude of the coupling reactance
using the equations
for i = 1 to i = n – 1
and the reactance slope parameter is
2
K 
1 –  0, 1 
 Z0 
and
▲ Fig. 13
1
K 0, 1
OUT
Half-wavelength via-hole coupled filter.
TUTORIAL
References
TABLE VII
VIA-HOLE COUPLED FILTER DESIGN DATA
G
Values
Index
i
Coupling
(k)
Impedance
Inversion
(K)
Via
Reactance
(Ω)
Via
Diameter
(mils)
Resonator
Length
(°)
0
1.0000
–
15.487
17.130
3.0
–
1
1.1468
0.05585
4.387
4.421
10.0
157.78
2
1.3712
0.04256
3.343
3.358
14.0
171.16
3
1.9750
0.04256
3.343
3.358
14.0
172.35
4
1.3712
0.05585
4.387
4.421
10.0
171.16
5
1.1468
–
15.487
17.130
3.0
157.78
6
1.0000
–
–
–
–
–
The filter data are listed in Table
7. The via-hole, half-wavelength, direct-coupled filter was simulated
using the previously displayed equiv-
alent circuit. The results are shown in
Figure 17.
This via-hole, direct-coupled filter
is particularly suited to wider bandwidth filters because the dimensions
of the filter are more realizable than
the side-coupled filters. There are
also several implementations in addition to the microstrip medium, including stripline, finline, coplanar
waveguide and slotline.
The implementation of the inductive reactance, direct-coupled filter in
coplanar waveguide is shown in Figure 18. This implementation results
in low insertion loss if an air-dielectric is utilized.
16
12
8
4
0
2
4
S11 (dB), S21 (dB)
▲ Fig. 16
6 8 10 12 14 16 18 20
VIA-HOLE DIAMETER (mils)
Via-hole reactance vs. diameter.
0
6
−10
5
−20
4
−30
3
−40
2
−50
1
−60
9
ACKNOWLEDGMENT
The principal reference for the
content of this article is the work of
Matthaei, Young and Jones.1 Many of
the concepts within this reference
have been investigated and interpreted in order to provide a greater intuitive understanding of the bandpass
filter design process. This text is
strongly recommended for those having little familiarity with this work. ■
GROUP DELAY (ns)
REACTANCE (Ω)
20
0
10
11
FREQUENCY (0.2 GHz/ div)
▲ Fig. 17
Via-hole coupled filter
simulation data.
w0,1
w1,2
w2,3
w3,4
w4,5
w5,6
IN
▲ Fig. 18
OUT
1
2
3
4
5
θ1
θ2
θ3
θ4
θ5
Inductive-coupled filter in coplanar waveguide.
1. G.L. Matthaei, L. Young and E.M.T. Jones,
Microwave Filters, Impedance-matching
Networks and Coupling Structures, McGraw-Hill, New York, 1964.
2. E.G. Cristal, “Coupled Circular Cylindrical
Rods Between Parallel Ground Planes,”
IEEE Transactions on Microwave Theory
and Techniques, Vol. MTT-12, July 1964,
pp. 428–439.
3. W.J. Getsinger, “Coupled Rectangular Bars
Between Parallel Plates,” IRE Transactions
on Microwave Theory and Techniques, Vol.
MTT-10, January 1962, pp. 65–72.
4. A.I. Zverev, Handbook of Filter Synthesis,
John Wiley and Sons, New York, 1967.
5. G.L. Matthaei, “Interdigital Band Pass Filters,” IEEE Transactions on Microwave
Theory and Techniques, Vol. MTT-10, No.
6, November 1962.
6. G.L. Matthaei, “Comb-line Band Pass Filters of Narrow or Moderate Bandwidth,”
Microwave Journal, August 1963.
7. S.B. Cohn, “Parallel-coupled Transmissionline-Resonator Filters,” IEEE Transactions
on Microwave Theory and Techniques,
Volume MTT-6, No. 2, April 1958.
8. R.M. Kurzrok, “Design of Comb-line Band
Pass F ilters,” IEEE Transactions on Microw ave Theory and Techniques, Vol.
MTT-14, July 1966, pp. 351–353.
9. M. D ishal, “A Simple D esign Procedure
for Small Percentage Round Rod Interdigital F ilters,” IEEE Transactions on Microw ave Theory and Techniques, Vol.
MTT-13, September 1965, pp. 696–698.
10. S.B. Cohn, “Dissipation Loss in Multiplecoupled-resonator Filters,” Proceedings of
t h e I R E , Vol. 47, Au gu st 1959,
pp. 1342–1348.
Kenneth V. Puglia
holds the title of
Distinguished Fellow of
Technology at the
M/A-COM division of
Tyco Electronics. He
received the degrees of
BSEE (1965) and
MSEE (1971) from the
University of
Massachusetts and
Northeastern
University, respectively. He has worked in the
field of microwave and millimeter-wave
technology for 35 years, and has authored or
co-authored over 30 technical papers in the
field of microwave and millimeter-wave
subsystems. Since joining M/A-COM in 1971,
he has designed several microwave components
and subsystems for a variety of signal
generation and processing applications in the
field of radar and communications systems. As
part of a European assignment, he developed a
high resolution radar sensor for a number of
industrial and commercial applications. This
sensor features the ability to determine object
range, bearing and normal velocity in a
multi-object, multi-sensor environment using
very low transmit power. Puglia has been a
member of the IEEE, Professional Group on
Microwave Theory and Techniques since 1965.
TUTORIAL
APPENDIX A
NORMALIZED SELF AND MUTUAL CAPACITANCE OF ROUND RODS BETWEEN PARALLEL GROUND PLANES
OF ROUND RODS BETWEEN PARALLEL GROUND PLANES
Upon completion of specific elements of the design procedure and
the calculation of the normalized self and mutual capacitance per unit
TABLE AI
length, rod diameter and center-to-center spacing may be calculated.
APPROXIMATE VALUES
First, the filter ground plane spacing h must be selected. The selection
FOR GROUND PLANE SPACING
of a suitable ground plane spacing is bounded by the out-of-band spurious response, passband frequency and insertion loss. Certainly, the
Filter Frequency
Ground Plane
ground plane dimension should be less than one quarter wavelength
f0 (GHz)
Spacing h (" )
and no smaller than that required to achieve an unloaded Qu consistent with the insertion loss requirements. The maximum ground plane
1
3.000
spacing may be estimated in accordance with
2
1.500
Z0
πλ
h ≤ 0 10 138
4
0.750
32
8
0.375
This formula provides the approximate values for ground plane spacing
listed in Table A1:
10
0.300
Cristal2 has given the closed-form expressions for the even and odd
12
0.250
mode capacitance per unit length from which the self and mutual capacitance may be calculated from
15
0.215
C 0 (c ) =
18
0.175

   4

d
 1    
∞
 – ln 1 –  b   + 2 ∑ –1
 2   2 c   m =1

  b  


( )
m

 m πc  
ln  tanh 

 2 b  

and
C e (c ) =
NORMALIZED MUTUAL
CAPACITANCE
ε


πd
1 
4
b
ln 
2π 
4
 1–  d 
 

 2b 

ε

   4

d
 1    
∞

 m πc  
 + ln 1 –  b   + 2 ∑ ln  tanh 

2
2
c
 2 b  

    m =1 

  b  


For reasonable accuracy, the parameter m may be limited to 100. The
self and mutual capacitance is calculated using1
C g(c) = C e(c)
and
C m(c) = 0.25[C 0(c)–C e(c)]
The equations have been programmed using Mathcad for a ground
plane spacing of 0.275 inches and a rod diameter of 0.128 inches, that
is, Zo = 60 Ω. The data are shown in Figures A1 and A2.
▲ Fig. A1 Mutual capacitanve of coupled lines for h = 0.275 and d
= 0.128".
NORMALIZED CAPACITANCE
TO GROUND


πd
1 
4
b
ln 
2π 
4
 1–  d 
 

 2b 

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
250 300 350 400 450 500
CENTER-TO-CENTER SPACING (mil)
7.0
6.5
6.0
5.5
5.0
4.5
4.0
250 300 350 400 450 500
CENTER-TO-CENTER SPACING (mil)
▲ Fig. A2 Normalized self-capacitance
to ground for h = 0.275” and d = 0.128”.
TUTORIAL
APPENDIX B
ALTERNATE FILTER IMPLEMENTATIONS
In addition to the examples within Part II
of this tutorial, there are several alternative filter implementations that lend themselves to
the general design procedure and should be
cited. These alternative filter implementations
utilize resonators not fully exploited in recent
literature and for which recent CAD tools
have permitted extensive analysis and understanding. Two specific types of resonators are
suggested for special applications: cylindrical
dielectric resonators and helical resonators.
DIELECTRIC
RESONATOR
DIELECTRIC RESONATOR FILTER
Figure B1 shows the electrical and mechanical schematic of a three-section bandpass filter using cylindrical dielectric resonators. Resonators of this type are not typical
of conventional lumped or distributed element structures due to the existence of multiple resonant frequencies or modes. This resonator type permits the design of narrowband,
low loss filters due to the high unloaded quality factor (Q u > 10,000) in the principal or
dominant resonant mode.
DIELECTRIC
RESONATOR
COUPLING
SLOT
OUTPUT
COUPLING
LOOP TUNER
COUPLING
SLOT
TUNER
TUNER
DR2
DR3
DR1
Dielectric resonator filter.
TUNER
TUNER
TUNER
TUNER
SLOT
Helical resonator filter.
TUNER
SLOT
SLOT
▲ Fig. B2
OUTPUT
COUPLING
LOOP
LOW DIELECTRIC SUPPORT
H-FIELD LINES OF TE01δMODE
▲ Fig. B1
DIELECTRIC
RESONATOR
SLOT
A note of caution is in order with respect
to the multiple resonator modes and the environment necessary to suppress excitation of
these modes and thereby eliminate loss spikes
and spurious responses in the filter transfer
function. There are two techniques suitable
for such a determination. The first technique
requires experimental characterization of the
resonator modes and mode coupling via the
methods described in Part I. The second, and
until recently more difficult analysis, involves
analytical determination of the modes and
mode coupling through the utilization of a
three-dimensional electromagnetic simulation
tool. Some sophisticated filter designers have
made use of auxiliary modes to design socalled dual-mode or multi-mode filters that
utilize two or more of the resonant modes to
obtain very compact, low loss filters with excellent rejection band characteristics. The
general design procedure is not suitable for
this task.
The input, output and inter-resonator coupling is accomplished via the magnetic field
which is orthogonal to the circular plane of
the cylindrical resonator for the dominant
TE 01δ mode. The equivalent circuit of the dielectric resonator filter represents the electrical behavior in the principal mode only. It has
acceptable accuracy if the resonator environment and mounting arrangement has demonstrated elimination of the auxiliary modes.
To obtain the highest unloaded quality factor available from the cylindrical dielectric
resonator, a low loss dielectric support structure is required to separate the resonator from
the electrical ground boundary.
HELICAL RESONATOR FILTER
The helical resonator filter is another filter
type for which the general design procedure is
applicable. Helical resonator filters are particularly useful within the VHF and UHF bands
for low loss, narrowband applications, and
where the greater volume of a distributed resonator filter is prohibitive. The helical resonator is at least partially distributed due to
the combination of self-inductance and the
electrical length of the coil structure. The helical resonator is tuned via the distributed capacitance of the tuning screw at the high electric field point of the resonator. A direct tap at
the low electric field point of the resonator accomplishes input and output coupling. The
resonator spacing and coupling slot at the high
magnetic field point of the helical structure
facilitates inter-resonator coupling.
Figur e B2 shows the mechanical and
electrical schematic of a typical five-section
helical resonator filter. Once again, resonator
parameters and coupling coefficients may be
determined experimentally or analytically by
the use of EM simulation tools. Usually, a low
loss dielectric form that facilitates uniformity,
placement and structure supports the helical
resonator.