TUTORIAL
A GENERAL DESIGN
PROCEDURE FOR BANDPASS
FILTERS DERIVED
FROM LOW PASS PROTOTYPE
ELEMENTS: PART I
B
Fig. 1 Circuit topologies
of low pass prototype
filters. ▼
g0 = 1.00
+
e1(s)
−
g0 = 1.00
andpass filters serve a variety of functions in communication, radar and instrumentation subsystems. Of the available techniques for the design of bandpass filters, those techniques based upon the low
pass elements of a prototype filter have yielded successful results in a wide range of applications. The low pass prototype elements are
the normalized values of the circuit components of a filter that have been synthesized for
a unique passband response, and in some cases, a unique out-of-band response. The low
pass prototype elements are available to the
designer in a number of tabulated sources 1,2,3
and are generally given in a normalized format, that is, mathematically related to a
parameter of the filter prototype.
This article presents a general design procedure for bandpass filters derived from low
pass prototype filters, which have been syn-
g2 = L1
gn−1
g1 = C1
g3 = C2
g1 = L1
g3 = L2
gn−2
gn+1 = 1.00
gn
gn−2
+
e2(s)
−
gn
n+1
+
e1(s)
−
g2 = C1
gn−1
+
e2
−
thesized for a unique filter parameter. A number of illustrated examples are offered to validate the design procedure.
LOW PASS PROTOTYPE FILTERS
Low pass prototype filters are lumped element networks that have been synthesized to
provide a desired filter transfer function. The
element values have been normalized with respect to one or more filter design parameters
(cutoff frequency, for example) to offer the
greatest flexibility, ease of use and tabulation.
The elements of the low pass prototype filter
are the capacitors and inductors of the ladder
networks of the synthesized filter networks as
shown in Figure 1. This diagram also depicts
Bandpass filters serve a
variety of functions in
communication, radar and
instrumentation subsystems.
= 1.00
K.V. PUGLIA
M/A-COM Inc.
Lowell, MA
Reprinted with permission of MICROWAVE JOURNAL® from the December 2000 issue.
©
2000 Horizon House Publications, Inc.
TUTORIAL
Γ (f')
L (f')
L (f') • 10
L1 = 9.78 nH
be adjusted (de-normalized) in accordance with
L2 = 9.78 nH
0
R'0 1
g1
R0 2π109
1
R
L1 = 0
g2
R'0 2π109
C1 =
−20
C2 =
C1
−30
C3 =
−40
−50
−60
S11
−70
−90
−100
R'0 1
g3
R0 2π109
1
R
L2 = 0
g4
R'0 2π109
C2 =
S21
0
−80
0
1
2
3
4
5
NORMALIZED FREQUENCY f' (Hz)
▲ Fig. 2
Low pass Chebyshev filter
prototype response.
−10
−20
−30
C3 =
−40
−50
0
Fig. 3 Low pass prototype filter
schematic. ▼
L = g2
S PARAMETERS (dB)
ATTENUATION (dB)
−10
0.5
1.0
1.5
FREQUENCY (GHz)
2.0
▲ Fig. 4 Chebyshev low pass filter
with a 1.0 GHz cutoff.
L2 = g4
and
C1 =
C2 = g3
1
C3
r 
ε = log –1  dB  – 1
 10 
g5
where
the two possible implementations of
the low pass prototype filter topologies. In both cases, the network transfer function is
()
Ts =
()
e1 (s )
e2 s
where
s = σ + jω, the Laplace complex
frequency variable
Clearly, the transfer function, T(s), is
a polynomial of order n, where n is
the number of elements of the low
pass filter prototype.
The illustrated circuit topologies
represent a filter prototype containing an odd number of circuit elements. To represent an even number
of elements of the prototype filter,
simply remove the last capacitor or
inductor of the ladder network.
For purposes of illustration, an example representing a Chebyshev filter is offered. The power transfer
function of the Chebyshev filter may
be represented by
T(f') = 10log{1 + εcos2[ncos–1(f')]}
for f' ≤ 1.0
T(f') = 10log{1 + εcosh2[ncosh–1(f')]}
for f' ≥ 1.0
rdB = inband ripple factor in decibels
These equations represent the
power transfer function of the
Chebyshev low pass prototype filter
with normalized filter cutoff frequency f' of 1.0 Hz. A graphical representation of the power transfer function
of the Chebyshev low pass prototype
filter is shown in Figure 2.
The low pass prototype filter parameters for the low pass Chebyshev
filter example are
n=5
R'0 = 1.0 Ω
In addition to the tabulated data of
low pass prototype filter elements,
the values may be computed via execution of the equations found in
Matthaei, et al.1, and repeated in Appendix A for convenience.
The schematic of the filter, which
was derived from Chebyshev, the low
pass prototype elements and the associated frequency response are shown
in Figure 4. Note that the 0.5 dB inband ripple results in a return loss of
–10 dB as expected.
A low pass filter may be converted
to a bandpass filter by employing a
suitable mapping function. A mapping function is simply a mathematical change of variables such that a
transfer function may be shifted in
frequency. The mapping function
may be intuitively or mathematically
derived. A known low pass to bandpass mapping function may be illustrated mathematically as
f' =
and
rdB = 0.5
A schematic representation of the
prototype Chebyshev filter is shown
in Figure 3. The prototype elements
are from Matthaei, Young and Jones1
where the normalized cutoff frequency is given in the radian format ω'1 =
1.0 = 2πf'1.
If this five-section prototype filter
were constructed from available tables of elements and a circuit simulation performed, the transfer function
would be exactly as represented in
the schematic. To construct the filter
at another frequency (1.0 GHz, for
example) and circuit impedance level
(R0 = 50 Ω), the element values must
R'0 1
g5
R0 2π109
f0  f f0 
 – 
∆f  f0 f 
where
f1f2
f0 = Îããã
∆f = f2 – f1
and f0, f1 and f2 represent the center,
lower cutoff and higher cutoff frequencies of the corresponding bandpass filter, respectively. If the substitution of variables is made within the
Chebyshev power transfer function,
the power transfer function of the
corresponding bandpass filter may be
determined as shown in Figure 5.
The schematic diagram of the
bandpass filter, which was derived
from the low pass prototype filter via
the introduction of complementary elements and producing shunt and se-
TUTORIAL
L (fa)
Γ (fa)
tributed resonators are shown in Figure 7.
Resonators may be characterized
by their unloaded quality factor Qu,
which is the ratio of the energy stored
to the energy dissipated per cycle of
the resonant frequency. Resonators
are also characterized with respect to
their reactance α or susceptance β
slope parameters, which are defined,
respectively, as
L (fa) • 10
0
−10
−30
▲ Fig. 6 Chebyshev bandpass
filter schematic.
−40
−50
−60
L
−70
−80
Zin
Z, E
/4
Zin
OPEN
E
C
−90
E = λ/2
Zin
Z, E
ATTENUATION (dB)
−20
α=
(a)
−100
−5 −4 −3 −2 −1 0 1 2 3 4 5
NORMALIZED FREQUENCY fa (Hz)
L
Yin
▲ Fig. 5
Chebyshev bandpass filter
transfer function.
C
L
Yin
C
Z, E
C
Yin
λ
E=
and
β=
(b)
▲ Fig. 7
Typical resonant circuit; (a) series
and (b) shunt resonators.
ries resonators, is shown in Figure 6.
This is a basic low pass to bandpass
transformation, and unfortunately
sometimes leads to component values,
which are not readily available or have
excessive loss. As described later, the
mapping function need not be considered as part of the bandpass filter design procedure. It is presented here as
a supplement to the filter theory.
It bears repeating that the low
pass prototype filter elements, that is,
the g-values, are the result of network
synthesis techniques to produce a desired characteristic of the prototype
filter transfer function. These desired
characteristics might include a flat
amplitude response, maximum outof-band rejection, linear phase response, Gaussian or other amplitude
response, minimum time sidelobes
and matched signal filters.
RF AND MICROWAVE
RESONATORS
RF and microwave resonators are
lumped element networks or distributed circuit structures that exhibit
minimum or maximum real impedance at a single freTABLE I
quency or at multiRESONATOR SLOPE PARAMETERS
ple frequencies.
Resonator Type Reactance Slope, α
Susceptance Slope, β The resonant frequency f0 is the fre1
ω 0L or
Series LC
quency at which the
ω 0C
input impedance or
admittance is real.
1
Shunt LC
ω 0C or
The resonant freω 0L
quency may be furShunt π
ther defined in
2 ω 0C
terms of a series or
shunt mode of resoπZ 0
λ/2 line (short)
nance; the series
2
mode is associated
πY0
λ/2 line (open)
with small values of
2
input resistance at
Y
the resonant fre2
0
λ/8 line (short+C)
cot θ 0 + θ 0 csc θ 0
quency, while the
2
shunt mode is assoπY0
λ/4 line (short)
ciated with large
4
values of resistance
 πZ 0   λ g 0 
at the resonant freλ/2 WG line (short)
 2   λ 0 
quency. Some typical lumped and dis-
(
)
( )
ω 0 dX ω
2 dω
ω =ω 0
( )
ω 0 dB ω
2 dω
ω =ω 0
These are important resonator parameters because they influence Q u
and the coupling factor between resonators in multiple resonator filters.
Table 1 provides the reactance and
susceptance slope parameters of
some common lumped element and
distributed resonators.
Qu may also be defined in terms of
the reactance or susceptance slope
parameter as
Qu =
α
= βR sh
R se
where
Rse = resonator series resistance
Rsh = resonator shunt resistance
Together these resistances represent
the resonator loss. The bandpass filter design examples will illustrate the
utility of the slope parameters.
In many bandpass filter applications, particularly those applications
where the filter is deployed at the
front end of a receiver, it is important
to know the Qu for the resonators in
order to accurately estimate the insertion loss of the filter. The insertion
loss of a single transmission resonator
can be mathematically represented as

2
 1 +  2Q f – f0  
 l


f0  



–
log
L f = 10
   Q  2 

 1 –  l  

   Q u  
()
At f = f0, this equation may further
reduce to
TUTORIAL
COUPLING
0
1942
1958
La (f) (dB)
Rsh
−2
−3
−3.470
−4
The single resonator’s measured
performance.


1
L f = –10 log 
2
   Ql   
 1 – 
 
   Q u   
()
Solving for Qu, yields


Ql
Qu 
L(f 0 ) 


 1 – 10 20 
Therefore, a measurement of the
single resonator insertion loss at the
resonant frequency L(f0) and the –3
dB bandwidth ∆f is sufficient to accurately determine Qu of any resonant
structure. The loaded quality factor
Ql may be determined from a measurement of the resonant frequency
and the –3 dB bandwidth from the
equation
Ql =
f0
∆f
where
∆f = –3 dB bandwidth
This measurement technique may
also be employed to compare the
quality factors (Qu) of different types
of resonant structures or as a method
of comparing various plating or manufacturing techniques for the filter if
insertion loss is a critical parameter.
Consider an example. A PCS1900
transmit filter was required with min-
▲ Fig. 10 The single resonator’s
equivalent circuit.
where
▲ Fig. 9
A single resonator at 1960 MHz.
C
0
/4
−5
1925 1935 1945 1955 1965 1975
FREQUENCY (MHz)
▲ Fig. 8
COUPLING
−0.470
−1
λ/8
RESONATOR
imum insertion loss and minimum
size as critical design parameters.
Rectangular, coaxial λ/8 resonators
were determined to offer the minimum volume in an eight-resonator
filter consistent with the maximum
insertion loss requirement of less
than 1.0 dB at the center frequency
of 1960 MHz. A single resonator was
constructed of the type anticipated to
be used within the filter. The single
resonator is shown in Figure 8.
A single resonator structure was
fabricated and plated with silver in
order to obtain the maximum Q u .
The resonant structure was tuned to
the desired center frequency and the
insertion loss L(f0) and –3 dB bandwidth were measured. The measurement data is shown in Figure 9
where
L(f0 = 1.950 GHz) = 0.533 dB
∆f = 14.50 MHz
Qu 2250.
In executing the measurement, three
notes of caution are required in the
interest of accuracy: The coupling
probes to the cavity should be equal
and minimized to avoid load and
source resistance across the resonator; the source and load SWRs
should be kept low; and the input
SWR at f 0 should be minimized to
avoid mismatch loss.
The equivalent circuit of the single
resonator is shown in Figure 10.
Note that the circuit element, which
represents the resonator loss Rsh, has
been included. The value of Rsh may
be determined with the aid of the
susceptance slope parameter β from
Qu = βRsh
β=
[
( )
( )]
Y0
cot θ0 + θ0 csc 2 θ0 = 0.01836
2
1
π
for θ0 = and Y0 =
Mhos
4
70
or
Rsh = 122.5 kΩ
This measurement technique for
estimating the value of Q u is
completely general and applies to
lumped element and distributed
resonators.
RESONATOR COUPLING
Resonator coupling represents one
of the most significant factors affecting filter performance. There are several methods to couple resonators.
For ease of manufacturing and tuning, a common resonator type and
coupling method is generally preferable. Matthaei1 proposes what have
been termed J (admittance) and K
(impedance) inverters both to permit
a common type of resonator and to
serve as coupling elements for the
resonators.
The J inverters may be represented as the admittance of the element
or the value of the characteristic admittance of a quarter-wavelength line
in the equivalent circuit that couples
the resonators. Similarly, the K inverters may be represented as the impedance of the element or the value
of the characteristic impedance of a
quarter-wavelength line that couples
the resonators. This permits the general expression of the coupling between resonators to be mathematically written as
J
k i, i + 1 = i, i + 1
β
TUTORIAL
λ/4
Cπ
L
L
C−Cπ
C
For the special case where θ0 = π/4,
the coupling may be written as
Cr−Cπ
kc =
C
2υCm
2Cm
=


π
π
υC g  1 +  C g  1 + 
2
2


(a)
(a)
kc
Yse = vCm
λ/8
λ/4
(b)
▲ Fig. 11
Coupled (a) π-type LC
resonators and (b) λ/8 distributed resonators.
λ/4
λ/4
Ysh = vCg
(b)
kc
▲ Fig. 13
The symmetrical λ/4 coupled
resonator’s (a) proximity coupled λ/4 line
and (b) equivalent circuit.
λ/8
(a)
Z se = 1/vCm
λ/8
λ/8
λ/8
Z sh = 1/vCg
(b)
▲ Fig. 12
The coupled λ/8 transmission
line resonator’s (a) proximity coupled line
and (b) equivalent circuit.
and
K i, i + 1
α
for shunt-type and series-type resonators, respectively, where the coupling between the i th and the i th+1
resonators is represented by ki,i+1.
A similar approach to the general
design of bandpass filters employing
common types of resonators proposes
specific coupling elements in the case
of lumped resonator bandpass filters
k i, i + 1 =
or specific proximity methods of coupling in the case of distributed resonator bandpass filters. To illustrate,
consider the coupled π-type of L-C
resonators and the coupled λ/8 transmission line distributed resonators
shown in Figure 11.
In the case of the coupled π-type
L-C resonators, a series capacitor is
inserted between the resonators to
perform the coupling function, in
which case the coupling coefficient is
C
ω C
ω C
kπ = 0 π = 0 π = π
2ω 0C 2C
β
In the case of the coupled λ/8 transmission line resonators, the equivalent
circuit shown in Figure 12 is useful in
determining the coupling coefficient.
The coupling coefficient may be determined from the capacitive matrix parameters associated with the coupled
lines, that is, the capacitance per unit
length to ground Cg and the mutual capacitance per unit length between the
conductors Cm, where ν is the velocity
in the dielectric medium. The coupling
coefficient may be written as
kc =
( )=
Ysh cot θ0
β
[
( )
Ysh cot θ0
( )
( )]
Yse
cot θ0 + θ0 csc 2 θ0
2
2Ysh
=

π
Yse  1 + 
2

Another very popular type of resonator, which is frequently used in
microwave bandpass filters, is the
quarter-wavelength resonator. Figure
13 illustrates the coupling of symmetrical λ/4 resonators and the
equivalent circuit. Note that λ/4 resonators must be grounded at opposite
ends to prevent the null coupling
condition caused by cancellation of
the electric and magnetic field
modes.
The coupling coefficient for this
configuration may be written as
kc =
υCm υCm 4 υCm 4Cm
=
=
=
β
Y0 π
πυC g
πC g
4
For a given coupled line geometry,
the λ/4 lines offer closer coupling
than the comb-line configuration.
The input, output and adjacent
resonator coupling in a multi-element
bandpass filter are the parameters
that determine the amplitude, phase
and SWR over the passband of the
filter. This statement understates the
importance of resonator coupling to
the bandpass filter parameters.
Recall that the elements of the
low pass prototype filter, from which
the bandpass filter is derived, determine completely the characteristics
of the resulting filter. This fact will
become evident when the coupling
between resonators is disclosed to be
a function only of the fractional
bandwidth and the low pass filter
prototype elements. Fortunately, a
measurement technique is available
to verify the coupling values of symmetrical resonators, and may also be
utilized in multi-resonator filters.
That measurement technique will
now be explored.
The amplitude response of any
pair of symmetrical resonators may
be represented by1
TUTORIAL
EXTERNAL
COUPLING
RESONATOR
COUPLING
EXTERNAL
COUPLING
Ce
Cπ
L
Cr−Cπ
C−Cπ
C−Ce
L
Ce
EXTERNAL
COUPLING
Ce
kc
λ/8
C
C−Ce
EXTERNAL
COUPLING
Ce
C
(a)
RESONATOR 1
▲ Fig. 14
RESONATOR 2
Lumped element symmetrical resonators.
Yse
vCm
λ/8
e
()
L f = –10 log
C
In the equation, k is the coupling coefficient between the symmetrical
resonators, Qu is the unloaded quality
factor of each resonator and Qe is the
external quality factor. The external
quality factor is defined to differentiate the source and load coupling and
loss from the losses associated with
the individual resonators (Qu).
If the overcoupled condition is satisfied such that
1
1
k>
+
Qe Q u
it is possible to determine the resonator coupling coefficient from
2
f –f 
 1
1 
kc =  b a  + 
+

 f0 
 Qe Q u 
2
where f0, fa and fb are subsequently
defined.
The utility of the equations will be
demonstrated by two examples. Consider the symmetrical, lumped element resonators in the schematic
shown in Figure 14 where two, πtype, L-C resonators are coupled by
the capacitor C π , and the external
source and load are coupled to the
respective resonators by capacitor Ce.
Note also the Qu of each resonator is
L (f) (dB)
97.55
0
−1
−
−3
−0
0.837
7
97
99
101 103
FREQUENCY (MHz)
105
▲ Fig. 16
λ/8 symmetrical coupled
resonator; (a) schematic
and (b) equivalent circuit.
Symmetrical resonator’s
response.
Qu =
R
ω 0L
The following variables have been assigned
Qu =
R
= 1000
2πf0L
Qe =
2
 
  50
1
1 + 
= 100
  2πf0Ce R s   2πf0L


2
= 100 MHz
LC
C
k = π = 0.050
2C
With the assigned variables, the
amplitude response is shown in Figure 15. The two peaks in the amplitude response correspond to the frequencies fa and fb. Using the assigned
variables, the calculated coupling coefficient is kc = 0.05033.
A second illustrative example using coupled λ/8 resonators is shown
in Figure 16. An equivalent circuit
representation of the coupled lines,
external coupling and resonator loss
is required in order to quantify the
coupling factor.
The variables have been assigned
as
f0 =
1
2π
λ/8
Ysh = vCg
(b)
5
▲ Fig. 15
λ/8
−0.83
−
−7
−8
−
−10
95
C
102.45
La (f) (dB)


2




Qe 


1+





Qu 
Qe 


+k




2kQe
2 











2


 

Q

 1 + e 
2
  f – f0  
Qu 
 
– Qe2 
•+2
 
2
k
  f0  
 


 


 
2
4


 Qe   f – f0 


+4 

 
 k   f0 


2
Ce
1931
1959
0
−0.915
−1
−2
−3
−4
−5
−6
−7
−8
−8.377
−9
−10
1925 1935 1945 1955 1965 1975
FREQUENCY (MHz)
▲ Fig. 17
Symmetrical resonator’s
response.
Q u = βR =
RY0

π
2 1 + 
2

= 2250
Qe =
2
 
 
50
1
1 + 
= 250
  2πf0Ce R s   Z0 tan θ0


1
= 1950 MHz
f0 =
2πZ0 tan θ0C
k=
2Cm
= 0.020

π
Cg  1 + 
2

With the assigned variables, the
amplitude response is shown in Figure 17. The two peaks in the amplitude response correspond to the frequencies fa and fb. Using the assigned
variables, the calculated coupling coefficient is kc = 0.01999. Part II ex-
TUTORIAL
plores how to use the preceding principles and data to design bandpass filters using a variety of lumped and
distributed element resonators.
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. ■
References
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. 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 Bandpass Filters of Narrow or Moderate Bandwidth,”
Microwave Journal, Vol. 6, No. 8, August
1963.
7. S.B. Cohn, “Parallel-coupled Transmissionline Resonator Filters,” IEEE Transactions
on Microwave Theory and Techniques, Vol.
MTT-6, No. 2, April 1958.
8. R.M. Kurzrok, “Design of Comb-line
Bandpass Filters,” IEEE Transactions on
Microwave Theory and Techniques, Vol.
MTT-14, July 1966, pp. 351–353.
9. M. Dishal, “A Simple Design Procedure
for Small Percentage Round Rod Interdigital Filters,” IEEE Transactions on Microwave Theory and Techniques, Vol.
MTT-13, September 1965, pp. 696–698.
10. S.B. Cohn, “Dissipation Loss in Multiplecoupled Resonator Filters,” Proceedings of
the IRE, Vol. 47, No. 8, August 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.
APPENDIX A
CALCULATION OF THE CHEBYSHEV LOW PASS
PROTOTYPE ELEMENTS
For Chebyshev filters with resistive terminations and pass band
ripple, rdB, go = 1.00, and ω'1 = 1.0, the low pass prototype elements
may be computed using the following procedure.
Define:

 r

β = ln coth dB  
 17.37  

 β
γ = sinh 
 2n 
(
)
 2 k –1 π 
, for k = 1,2,…n
a k = sin 
 2n 


 kπ 
b k = γ 2 + sin 2  , for k = 1,2,…n
n
Then calculate:
g1 =
2a 1
4 a k –1a k
, gk =
, for k = 2, 3, … n,
γ
b k –1g k –1
g n+1 =1.00, for n odd,
 β
g n+1 = coth 2   , for n even.
4