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Combline filter design using AWR

RF for wireless design (North Carolina State University)

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ECE – 549 Design Project

Design of a Third‐Order Butterworth Combline Bandpass Filter

Submitted by
BAL GOVIND

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EXECUTIVE SUMMARY
This report details the architecture and performance of bandpass filter designed to operate with a
nominal center frequency of 8.5 GHz and a nominal bandwidth of 1.7 GHz. It has a characteristic
third-order Butterworth response and has no passband ripple. It is implemented in microstrip
technology, with a substrate of relative permittivity of 2.6 and uses gold metallization layers with
tantalum interconnects.
The filter is synthesized with dimensions selected for easy manufacturability, while minimizing
unnecessary multimoding from extraneous resonant modes, features usually prevalent in
microstrip structures. Further, the physical design is simple in that it comprises a system of three
parallel microstrip lines which, in operation, are electromagnetically coupled. This is an integrated
three-port network whose effective performance can be independently tuned for optimal
performance by using tuning capacitors interposed between the network and source and load ports
and/or by tuning shorted capacitors connected externally to the network. In effect, the advantage
of excluding lossy lumped components from the core of the filter is realized.
The filter is designed to work with optimal performance with a source and load reference of 50 Ω
with which no mismatch in phase velocities is expected. It has a high transmission response i.e., a
low insertion loss (𝑆21 ) within the passband, which by a conservative estimate is of the order of -
0.21 dB as predicted by an electromagnetic simulation. It also has an insertion loss of under -50
dB in the stopband. Further, the next passband is at 42 GHz and it, therefore, has a large margin
for useful operation (see Figure below). This has been validated by multiple approaches including
elementary 𝜋- arrangement transmission line lumped models as well as with commercially
available microwave simulator models incorporating effects of vias connecting the three-port
network into external control circuitry, with near identical results.

𝑆22 (dB)
𝑆21 (dB)

Wideband response for the designed third-order Butterworth filter with critical frequencies marked for
reference

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TABLE OF CONTENTS

I. Introduction ............................................................................................................................... 1
II. Circuit Transformations for lumped element models of Butterworth Filter ............................. 2

III. Microwave circuit model and simulation of third-order Butterworth filter ............................. 9

IV. Electromagnetic simulation of third-order Butterworth filter ............................................... 17

V. Conclusion .............................................................................................................................. 21

REFERENCES ............................................................................................................................ 22

APPENDIX .................................................................................................................................. 23

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I. Introduction
In this project, a third‐order Butterworth Combline Bandpass Filter is designed to operate with a
center frequency of 8.5 GHz, a bandwidth of 1.7 GHz and in a substrate of relative permittivity,
𝜀𝑟 , of 2.6. To this end, a systematic design procedure is undertaken to produce first, a lumped-
element model and second, a microwave coupled-line model. This is followed by a more realistic
prediction of performance by incorporating the three-port microwave model of the coupled lines
in a commercial electromagnetic simulation environment (National Instruments AWRDE). The
design’s nominal performance is validated by this approach as well. In this report, we will step
through the complete synthesis process of the bandpass filter.

Section 2 details the process of creating a lumped-element prototype of the filter, taking into
account tradeoffs and practical considerations in matching a 50 Ω source and load to our final
desired 𝜋-arrangement of short-circuited stubs. This involves a series of intuitive decisions leading
to selection of one pair among many combinations even and odd-mode line impedances to produce
a system impedance close to the desired 50 Ω system.

Section 3 then translates the 𝜋 -arrangement of stubs to a microstrip-model consisting of three

coupled short-circuited lines mounted on a dielectric substrate. Consideration is given to selection
of line dimensions which are in agreement with a robust design which eliminates effects of
multimoding and effects of mismatch in even and odd mode phase velocities. Due to the constraints
of the specification, the choice of a suitable substrate is limited to a 100 μm thick substrate.
Simulations are then done using the MCLIN model in AWR Design Environment to validate the
predicted performance of the lumped model in Section 2. We also see additional factors involved
in the practical design of a maximally flat filter relating to the degree of insertion and return loss
in passband and stopband and small artifacts produced from unwanted inter-line coupling.
Dimensions are selected for best performance while maintaining manufacturability of the
microstrip geometry.

Section 4 illustrates the implementation of the microstrip model in a 3D EM simulation

environment. Details are discussed in assigning boundary conditions and in convenient simulation
for better convergence of the solution. We investigate the performance using rectangular response
and polar response in passband and stopband and quantify the insertion loss and return loss of the
filter. Also discussed are features of the wideband response which shows periodicity in location of
passbands and inter-line extraneous coupling effects due to interaction of the three-port system in
a practical enclosure.

Section 5 presents a summary of the key findings during the synthesis process and reiterates the
importance of the design choices made in the selection of geometry of the filter. It also tabulates
performance metrics from ideal lumped-element model, MCLIN based model and 2.5D EM
simulation environment simulation results.

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II. Circuit transformations for lumped element models of Butterworth filter

The objective in this section is to first design a third order Butterworth filter in its lumped element
form and then realize it using parallel coupled microstrip transmission lines. It may be noted at
that the center frequency of the filter, each line would act as a λ/4 long resonator. Therefore, the
resultant system would behave as two pairs of coupled lines. We would then aim to reduce the
lumped element system to an equivalent pi-arrangement of resonant shorted stubs. Retrograde
analysis yields that that each shorted stub would be a resonant LC resonator. Let us now
systematically realize a design that can be implemented in microstrip technology.
STEP 1: Development of a lowpass prototype filter
The coefficients for a third order lowpass Butterworth filter with a corner frequency of 1 rad/sec
are 𝑔0 , 𝑔1 and 𝑔2 . These are tabulated in Table 2-4 of [1] for an input resistance of 1 Ω and an
output resistance of 1 Ω. There are two possible variants - a T-arrangement (Fig. 1a) and a 𝜋 –
arrangement (Fig. 1b). We prefer the latter analogue, wherein 𝑔0 = 1 F, 𝑔1 = 1 H and 𝑔2 = 1 F.

(a) (b)
Figure 1. Variations of a third-order Butterworth lowpass filter, realized as (a) T-arrangement and (b) 𝜋 –
arrangement.

STEP 2: Elimination of the series inductor in the prototype lowpass filter

This step comprises of the replacement of the series inductor (1 H) of the chosen 𝜋- arrangement
with ideal inverters. The inverter is a two-port subcircuit. For an impedance inverter, of
characteristic impedance K = 1 Ω, the input impedance for a representative transmission line
2
terminated in a load is 𝑍𝑖𝑛 = 𝐾 ⁄𝑍 . The inductor in Fig. 1b is replaced by a parallel combination of
𝐿
two ideal inverters and an intermediate capacitor, 𝐶21 , given by
𝐿21 = 𝐶21 𝐾 2 (1)
The resultant transformation is shown in Figure 2. The corner frequency is still 𝜔𝑐 = 1 rad/sec.

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Figure 2. Replacement of series inductor in the prototype lowpass filter, with ideal inverter elements.

STEP 3: Transformation from a lowpass filter prototype to a bandpass filter prototype

We now aim to translate the lowpass filter’s center frequency of 0 rad/sec to the required
specification’s center frequency of 8.5 GHz, with a bandwidth of 1.7 GHz. The inductors in a
lumped-element circuit are first replaced by a series LC combination and the shunt capacitors are
replaced by parallel LC tank circuits. For the circuit in Fig. 2, capacitors 𝐶11 , 𝐶21 and 𝐶31 are
replaced with resonant LC combinations, while retaining the inverters. The following relations [1]
are used in deriving the new element values.
For 𝜔0 = 8.5 GHz, 𝜔1 = 7.65 GHz and 𝜔2 = 9.35 GHz,
𝛼 = 𝜔0 /(𝜔2 − 𝜔1 ) (2)
Here, 𝛼 = 5.
𝛼𝐶11
The new resonant capacitance, 𝐶1 ′ = (3)
𝜔0

𝐶1 ′ equals 93.62 pF and

1
the resonant inductance, 𝐿1 ′ = (4)
𝛼𝐶11 𝜔0

Here, 𝐿1 ′ = 0.00374 nH.

Similarly, 𝐶2 ′ = 187.24 pF, 𝐿2 ′ = 0.0018724 nH , 𝐶3 ′ =93.62 pF and 𝐿3 ′ = 0.00374 nH.
The bandpass filter now has three resonators and this is shown schematically in Fig. 3.
This is fine progress in the process of realizing a three stub model of the coupled line filter.

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Figure 3. Bandpass transformation consisting of hybrid lump-element and inverters, for 𝑓𝑐 = 8.5 GHz and
bandwidth of 1.7 GHz.

STEP 4: Transformation from a 1 Ω impedance system to 50 Ω impedance system

It now becomes necessary to scale every impedance in the network by the same amount. The
inductors, 𝐿1 ′, 𝐿2 ′ and 𝐿3 ′ are scaled up by a factor 50, the capacitors 𝐶1 ′, 𝐶2 ′ and 𝐶3 ′ are scaled
down by a factor of 50 and the inverters’ impedance (K) is scaled up by a factor of 50. The
transformation from Fig. 3 to the new prototype filter is shown in Fig. 5a. Also shown in Fig. 5
(b) is the rectangular response of two port parameters, 𝑆11 and 𝑆21 for the filter. We see that it has
an excellent passband characteristic, indicated by the precipitous fall in 𝑆11 in between the required
limits of 7.69 GHz and 9.39 GHz. We also see that it has a center frequency of 8.53 GHz, which
is very close to the required center frequency. This far, it would appear that our synthesis process
is accurate.

(a)

− 𝑆21
− 𝑆22
𝑆21 (dB)

𝑆22 (dB)

(b)
Figure 4. Impedance scaling to a 50 Ω system for the hybrid lump-element/ inverters bandpass filter for 𝑓𝑐 =
8.5 GHz and bandwidth of 1.7 GHz. (a) Current iteration of the circuit used in synthesis of a 𝜋 – arrangement
of stubs (b) rectangular response demonstrating maximally flat response in the required passband.

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STEP 5: Realization of LC resonators with stub elements

The poor performance of inductors at microwave frequencies necessitates their substitution with
transmission lines. We will denote input impedance of shorted stubs by 𝑍1 and the characteristic
impedance of the stubs by 𝑍01 . The aim is to develop a broadband realization of the LC resonant
𝜕𝑌𝑖𝑛
circuit. This is achieved by matching 𝑌𝑖𝑛 and of the lumped circuit and the transmission line
𝜕𝜔
circuit at 𝜔0 . Derivation of these conditions are shown in [1]. However, for brevity, the conditions
that the resonant capacitive element, C and the 𝑍01 of the stubs must satisfy are given as

4 2
𝐶0 = 𝐶 − + (5)
𝜔𝑟 2 𝐿 𝜔𝑟 𝑍01

𝜔𝑟 𝐿 𝜋
and 𝑍01 = (1 + ) (6)
4 2

In our design, we will also use the same length for all stubs. We will set the commensurate
(resonant) frequency of the lines, 𝜔𝑟 , as twice the resonant frequency, 𝜔0 , of the circuit. This is a
typical design choice. The resultant transformation from our previous design in Fig. 6 is shown in
Fig. 6.

Figure 5. Realization of LC tank circuits with parallel combinations of capacitors and shorted stubs

STEP 6: Equalization of stub impedances

While our previous design has broadband performance, we see that the stubs have markedly
different characteristic impedances. Smaller characteristic impedances would also need to be
realized with wider microstrip lnes. We would, therefore, like to bring them closer to a system
impedance of 50 Ω. We also aim to design the microstrip lines to have the same width, for
manufacturability reasons.
This can be done by (a) developing a nodal admittance matrix in Fig. 6 and (b) developing a nodal
matrix of the scaled network and (c) equating the two to find the parameters to calculate the new
equalized stub impedances and equalized capacitances. The final tranformation, from the circuit
in Fig. 6, is shown in Fig. 7.

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Figure 6. Equalization of stub impedances in the hybrid capacitor-stub model in Step 6

STEP 7: Realization of inverters with stubs

The prototype filter in Step 6 consisted of three hybrid resonators with pairs of resonators separated
by inverters of equal impedances. Our new prototype should only consist of 𝜋 arrangement of
short-circuited stubs, each 𝜆/8 in lengt, and capacitors which can then can be realized by a
combline section. Of course, this involves the subsequent absorption of parallel impedance of
open-circuited stubs of negative impedance. The current prototype, devoid of inverters, is shown
in Fig. 8. It may be noted that the central stub is shared by the two outer pairs. Here, the series
stubs represent the coupling between parallel microstrip lines.

Figure 7. Realization of inverters with 𝜋 arrangement of capacitors and shorted stubs in Step 7

STEP 8: Scaling the impedance of hybrid capacitor-stub pairs

The design in Step 7 has transmission lines with very low characteristic impedances. We should
preferably have these in the range of 30 Ω to 80 Ω. We will now scale the outer two stubs to have
𝑍0 exactly equal to 50 Ω. In the process, however, the source and load impedance increase by the
same scaling factor. The transformed stub impedances and capacitances are shown in Fig. 8.

Figure 8. Impedance scaling of shorted stubs and inverters increases the system impedance as wellin Step 9

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STEP 9: Reducing the source and load impedances to 50 Ω

In this step, we reuse the properties of inverters to transform the impedances looking into the load
and source so they match 50 Ω. The newly introduced inverters are shown in Fig. 9. These are,
however, too large to be implemented as microstrip lines.

Figure 9. Introduction of inverters again to realize an input and out output impedance of 50 Ω in Step 9

STEP 10: Replacement of inverters with series-shunt capacitor networks

The 225 Ω inverters introduced in the last step can now be replaced with a series-shunt capacitor
network as the two-port network behaves as resistive loads, looking into the source and load. We
then get a parallel capacitor of a negative value a positive capacitor in series with the source or
load impedance. Each negative capacitor is then absorbed into one of two parallel capacitors in the
𝜋 -arrangements in the core of the filter, which then yield only positive capacitances in the resultant
circuit. The intermediate circuit and final prototype filter realized by transmission lines, are shown
in Fig. 11.

Figure 10. Transformations leading to the development of the final prototype filter realized by shorted stubs

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We once again check the frequency response of the new prototype filter to see if is consistent with
the maximally flat response we’d like to replicate from the lumped-element model. From Fig. 11,
we see that this is indeed the case, albeit a 0.2 GHz drop in center frequency and a slightly excessive
bandwidth of 1.79 GHz. This may imply that the design of the series stubs may have to be adjusted
a bit to reduce excessive coupling in the arms of the 𝜋 – arrangement.

− 𝑆21
− 𝑆21 − 𝑆22
− 𝑆22
𝑆21 (dB)

𝑆22 (dB)

(a) (b)
Figure 11. (a) 𝑆11 and 𝑆21 rectangular response for the stub- based model of the third-order filter and (b) Polar response,
showing maximum flatness in the passband at 8.31 GHz, with a bandwidth of 1.79 GHz.

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III. Microwave circuit model and simulation of third-order Butterworth filter

With the conclusion of the circuit design of the third-order maximally flat filter, progress can now
be made in realizing this with microstrip lines. Stubs will be realized by parallel coupled lines and
capacitors will be realized by surface-mount components. As indicated in the previous section, we
will realize the coupled lines as pairs of outer microstrip lines with one line shared between them.
Our analysis, however, neglects the mutual coupling between the two outermost lines. The
translation from the arrangement of stubs to a pair of coupled lines can then be realized by the
following relations
𝑍
𝑛 = 1 + 𝑍012 = 5.501 (7)
011

𝐾 = 1⁄𝑛 = 0.18 (8)

𝑍02 = 𝑛𝑍012 = 1238 Ω (9)
𝑍011 𝑍022
𝑍01 = 𝑛 𝑍 = 52 Ω (10)
011 + 𝑍022 + 𝑍012

We then relate this model to the physical realization of coupled lines and derive from them the
system impedance, two estimates of which are

𝑍0𝑆1 = 𝑍01 √1 − 𝐾 2 = 51.26 Ω (11)

𝐾2
𝑍0𝑆2 = 𝑍02 = 41.602 Ω (12)
√1−𝐾2

The difference in values of system impedance is because the shunt stubs have different parameter
values. We will use the geometric mean of these impedances to get the system impedance as

𝑍0𝑆 = √𝑍0𝑆1 𝑍0𝑆2 = 46.169 Ω (13)

We can then derive the even and odd mode characteristic impedances as
√𝑛−1
𝑍0𝑜 = 𝑍0𝑆 = 38.41 Ω (14)
√𝑛+1

𝑍0𝑆 2
𝑍0𝑒 =
𝑍0𝑜
= 55.48 Ω (15)

To determine the physical dimensions of the parallel coupled lines, we will generate a custom
lookup table for the given substrate permittivity of 𝜀𝑟 = 2.6. This is done by iteratively solving the
even and odd mode parallel coupled line equations (5.77) through (5.102) of [2]. The MATLAB
implementation in given in Appendix 1. Using the separation between the lines, s, height of the
substrate, h, and width of the lines, w, in ratios given by

g=s/h (16 a)

u= w / h (16 b)

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Using these ratios, combinations of 𝑍0𝑒 , 𝑍0𝑜 , 𝑍0𝑆 , even mode effective permittivity 𝜀𝑒𝑒 and odd
mode effective permittivity 𝜀𝑒𝑜 are generated. We then search for the combination that produces
near-equal values as given in equations (13) through (15). Table 1 shows a segment of this lookup
table with the candidate (highlighted in yellow) that satisfies the aforementioned criteria.

Table 1. Segment of generated table for dimensions, system impedances and eff. permittivities (𝜀𝑟 = 2.6)

g u 𝒁𝟎𝒆 (Ω) 𝒁𝟎𝒐 (Ω) 𝜺𝒆𝒆 𝜺𝒆𝒐 𝒁𝟎𝑺 (Ω)

0.1 0.3 208.33975 62.552 2.011 1.80706 114.158
0.2 0.3 199.07 75.066 2.02 1.8099 122.2458
. . . . . . .
. . . . . . .
. . . . . . .
0.5 2.88 55.85 38.418 2.2757 1.9815 46.324
. . . . . . .
. . . . . . .
. . . . . . .
14.9 6 28.83 29.258 2.282 2.18 29.046
15 6 28.83 29.260 2.282 2.183 29.046

Therefore, we use a value of g = 0.5 and u =2.88. We also see that we have obtained a system
impedance of 46.324 Ω, which is close to an ideal system impedance of 50 Ω. To reduce the
coupling between lines, we’d like the width of the lines to be a fraction of the length of, say, a fifth
or less. We’d also like the separation of the lines to be of a similar fraction of the length. These
will ensure that the bandwidth and center frequency do not vary greatly from our calculated design.
We also know that the length, L, of each line is given by
𝐿 = 𝜆𝑔 / 8 = 3 mm (17)

where, 𝜆𝑔 = c/f = 3 mm

At this juncture, it becomes necessary to consider the dimensional stability and manufacturability
of this design and ensure that multimoding is avoided. While a thicker substrate would be
preferable as it is less susceptible to cracking, it is more likely to support higher order and slab-
mode multimoding. We’d also like the length of the striplines to be at least five to ten times the
thickness of the substrate and five to ten times the width of the lines to prevent excessive coupling.
This places constraints on ratios u and g. If we were to use a substrate of standard thickness of 100
μm, our geometry is restricted to the following dimensions

h = 100 μm (18 a)
s = 50 μm (18 b)
w = 288 μm (18 c)

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h = 100 μm (18 d)

We will now make a few calculations to validate if effects of multimoding in the substrate have
been eliminated with this design.

First, the frequency for a Microstrip Dielectric Slab Mode,

𝑐 tan−1 𝜀𝑟
𝑓𝑐1 = = 36.81 THz (19a)
√2𝜋ℎ√𝜀𝑟 −1

for no discontinuity and for matching phase velocities and

𝑐
𝑓𝑐2 = = 465.13 GHz (19b)
4ℎ√𝜀𝑟

to account for discontinuities.

The frequency for higher order microstrip mode is

𝑐
𝑓𝑐2 = = 592.92 GHz (20)
4ℎ√𝜀𝑟 −1

It is preferable to also keep the substrate under λ/2 = 1.09 cm, which our design satisfies.

We then check for occurrence of Transverse Microstrip Resonance, given by

𝑐
𝑓𝑇𝑅𝐴𝑁 = = 361.54 GHz (21)
√𝜀𝑟 (2𝑤+0.8ℎ)

As the minimum of any these modes are above the operating frequency of 8.5 GHz for which our
filter is designed, this geometry is a great choice for bandpass characteristics.

We may now proceed with the microwave simulation of the coupled lines. The loss tangent of the
substrate is specified as 0.0002 and for a realistic prediction of performance, we will include the
presence of tantalum vias, which have a normalized resistivity (with reference to gold), 𝜌𝑡 = 5.369.
We will use gold metallization (normalized 𝜌 = 1) for both the conductor and the ground plane.
The conductor thickness is 2 𝜇m. We will use the standard M3CLIN model in National Instruments
AWR Design Environment’s element catalogue. Figure 12 (a) shows the complete schematic of
our Butterworth filter, including the effects of tantalum vias. Figure 12 (b) shows the relative
position of the microstrip lines, substrate and ground plane, reiterating the critical dimensions of
our setup. It is unfortunate that the AWR design environment does not a provision for a square via
and instead provides a cylindrical element instead. We will use this in our design, noting possible
differences in results that may arise in the forthcoming electromagnetic simulation.

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(a) (b)
Figure 12. (a) Microstrip design equivalent of lumped element model (b) relative dimensions of a single line and substrate

Let us analyse the performance of the microstrip model and step through the iterations we need
to undertake to achieve the performance metrics detailed in the filter specification.

In Figures 12 (a) and (b), we first look at the frequency response of our MCLIN model in both
rectangular response and polar response. We retain the calculated line dimensions and capacitor
values from the lumped model in Fig. 11 in the previous section. We find that, by linear circuit
analysis, the passband extends from 𝜔1 (-3 dB) = 6.85 GHz to 𝜔2 (-3 dB) = 8.922 GHz, which is
greater than we’d like and that the center frequency, 𝜔0 , is 7.886 GHz. This falls slightly below
the specifications on both counts. We also see a couple of zeros in the 𝑆11 response within the
passband. However, since these are in the order of -30 dB, these are hardly discernible as poles in
𝑆21 response and we are less concerned about their present effects. Nevertheless, it would behove
us to rid the passband of these zeros. Further, we see the additional zero in the 𝑆11 response at
around 11.5 GHz. We will account for this artefact shortly.

We should first aim to bring 𝜔0 to the desired value of 8.5 GHz. We, therefore, reduce the length
of the lines from 3000 𝜇m to 2600 𝜇m and observe the change in response as shown in Figures 12
(c) and (d). We see that 𝜔0 has increased to 8.61 GHz – a considerable improvement. The
bandwidth is still, however, 2.21 GHz - a little greater than the desired value. This, we know, is
because of inter-line coupling. While we have some intuition of its effects by modelling this effect
as series stubs in the transmission line circuit model in STEP 10 of the previous section, the mutual
coupling at these frequencies can hardly be captured without a microwave simulation model like
MCLIN.
To help solve the problem of broadened bandwidth, we increase the separation between the lines
from 50 𝜇m to 75 𝜇m and see how this reflects in the frequency responses shown in Figures 12 (e)
and (f). We see a maximally flat 𝑆21 response of our third-order prototype with almost no
discernible zeros in the 𝑆11 response in the passband between 7.65 GHz and 9.35 GHz.

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Our frequency response is now satisfactory and could use a little tuning of the 𝐶1 capacitor value
to achieve the best possible response. This is implemented and the finalized dimensions and
element values and the frequency response are shown in Fig. 14. On the Smith chart in Figures 14
(b) and (c), we see that as is swept between 2 and 14 GHz, it loops once into the origin at 8.5 GHz
at a slow rate and loops out quicker for frequencies outside the passband. We notice that as the
locus of 𝑆11 passes through the origin, the reflection coefficient is zero, indicative of maximum
transmission. We must also factor in the inductive effect of the tantalum via and note the
inhomogeneity in resistivity of the gold metallization and our chosen via material.

We must also try to reason why, from the 𝑆21 response of the MCLIN simulation, the initial
bandwidth has grown from that predicted by the lumped element model. This is likely because the
even mode impedance (55.85 Ω) and odd mode impedance (38.418 Ω) are different. This can also
be explained on the basis of different permittivity of the even mode (𝜀𝑒𝑒 = 2.27) and odd mode
(𝜀𝑒𝑜 = 1.9815), travelling at different velocities, giving greater coupling and, therefore, larger
bandwidth. We have also consistently seen a notch in the 𝑆21 response beyond the passband which
does not feature in the response for the lumped element model. This can be explained on the paths
of inter-line coupling in the physical layout. The first path comprises coupling between the two
outer striplines. The second path comprises coupling of the first to the second element and then
the second to the third element. The zero in the 𝑆11 response is then due to the mutual cancellation
of the transmission from these two paths and occurs at 10.2 GHz.

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Dimensions and capacitances Frequency Response (rectangular) Frequency response (Polar)

• 𝐿= 3000 μm 14
• h = 100 μm
• s = 50 μm
− 𝑆21
• w = 288 μm
− 𝑆22

𝑆22 (dB)
•

𝑆21 (dB)
h = 100 μm − 𝑆21
• Cb = 0.2535 pF − 𝑆22
• C1 = 0.2838 pF
• C2 = 0.458 pF

(a) (b)

↓ Reduce length to increase 𝜔 0

• 𝐿= 2600 μm
• h = 100 μm
• s = 50 μm − 𝑆21
• w = 288 μm
− 𝑆21 − 𝑆22
• h = 100 μm

𝑆22 (dB)
𝑆21 (dB)

− 𝑆22
• Cb = 0.2535 pF
• C1 = 0.2838 pF
• C2 = 0.458 pF

(c) (d)

↓ Increase separation to reduce BW

• 𝐿= 2600 μm
• h = 100 μm
• s = 75 μm − 𝑆21
• w = 288 μm
− 𝑆22
𝑆21 (dB)

𝑆22 (dB)

• h = 100 μm
− 𝑆21
• Cb = 0.2535 pF
− 𝑆22
• C1 = 0.2838 pF
• C2 = 0.458 pF

(e) (f)
Figure 13. Two Iterations in tuning of dimensions to achieve filter specifications (a) and (b) frequency response for parameters
derived from the original lumped element model (c) and (d) a bettered center frequency is achieved by reducing microstrip line
length (e) and (f) bandwidth is reduced by reducing coupling by increasing separation between parallel coupled lines.

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Dimensions and capacitances

• 𝑳= 2610 𝛍m
• h = 100 μm
• s = 75 𝛍𝐦

𝑆21 (dB)

𝑆22 (dB)
• w = 288 μm − 𝑆21
• h = 100 μm − 𝑆22
• Cb = 0.2535 pF
• C1 = 0.2748 pF
• C2 = 0.458 pF

(a)
− 𝑆21 − 𝑆21
− 𝑆21 − 𝑆22 − 𝑆22
− 𝑆22

(b) (c)

Figure 14. Finalized microstrip MCLIN-based design of third order Butterworth filter, after tuning capacitor 𝐶1 . (a)
Rectangular frequency response for 𝑆11 and 𝑆21 (b) and (c) polar frequency response for 𝑆11 and 𝑆21.

It also proves useful to examine the wide-band response of the filter shown in Fig. 15. Here, we
see the comparative response for, first, the transmission line model we derived in Section 2 (Fig.
15 b), with its circuit redrawn here (Fig. 15 a). The response is as expected for a Butterworth filter.
The additional coupling leading to the zero at 10.2 GHz in the 𝑆11 response does not feature here.
We do, however, see repeated passbands at 36.2 GHz and 70.12 GHz. The skirt in the 𝑆21 response
has a slow decay characteristic for all these passbands.

The wideband response for the MCLIN model is more interesting. We again see pass bands around
𝜔3 = 38.7 GHz and 𝜔4 = 42 GHz, not entirely in keeping with the lumped-element and
transmission line models. We see a very low insertion loss in the passband and an insertion loss of
- 2.74 × 10−6 dB and a return loss of -47.9 dB. We’d expect that the periodicity in the spacing of

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these passbands should be at multiples of half-wavelengths. But we also note that the impedances
of the chosen capacitors do not follow this trend, and is a consequence of choosing the
commensurate frequency, 𝜔0 , as 2𝜔0 . In practical operation, however, this is of little consequence
as transmission at such higher frequencies is attenuated to a greater degree.

− 𝑆21
− 𝑆22

𝑆21 (dB)

𝑆22 (dB)
(a) (b)

− 𝑆21
− 𝑆22
𝑆21 (dB)

𝑆22 (dB)
(c) (d)

Figure 15. Wideband response of the third-order prototype filter shows spurious pass bands at 𝜔3 = 38.7 GHz and 𝜔4 = 42 GHz
for (a) and (b) the lumped element model and (c) and (d) microstrip line filter model

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IV. Electromagnetic simulation of third-order Butterworth filter

In this section, we will translate our microwave circuit simulation synthesized using the MCLIN
model in the previous section to a more realistic electromagnetic simulation environment. We will
use the EMSight package within the AWR Design Environment. Here, AWR uses a back-and-
forth transfer procedure between a representative three-port network comprising our coupled lines
in an enclosure, and the remainder of the circuit, consisting of the short-circuit capacitors and input
and output ports.

The simulator uses a 2.5D simulation strategy and iteratively calculates 𝑆11 and 𝑆21 parameters at
each frequency across our chosen 2-14 GHz sweep. We must be judicious with computational
resources at this point and take into account the exponential rise in computational time needed for
greater accuracy. A regular grid is needed in both x and y directions. Our design has conveniently
chosen strip dimensions, i.e. in multiples of 5 μm. Therefore, we could use a grid size of 5 μm for
all of the solver’s iterations. We are also constrained to implementing tantalum vias which have a
square cross-section and have a side-length of 125 μm for minimal footprint and, thereby, lessened
coupling. We will again use 2 μm gold metallization for conductor and ground plane. We will
also use a perfect conducting wall for the top surface as well as for the four walls. To eliminate
unnecessary coupling, we use a line-to-enclosure side-wall separation as twice or thrice the
separation between adjacent lines. Fig 16 (a) shows the geometry of the computational space and
Fig. 16 (b) shows the three-port network embedded in the microwave circuit used in the MCLIN
simulation.

Let us now analyse the performance predicted by the electromagnetic simulation. Fig. 17 (a) shows
the 𝑆11 and 𝑆21 response in rectangular response and Fig. 17 (b) shows the same in polar co-
ordinates on a Smith Chart. We see that the center frequency has shifted to 8.9 GHz while the
bandwidth is relatively the same- around 1.73 GHz. This likely indicates that the line could have
been longer to effect a slightly lower center frequency. There are other avenues for investigation
such as the coupling and parasitic capacitance between the side wall of the vias and the bottom
surface of the conductor. Another reason in slight inconsistencies in results is the difference in
geometries of the square vias used here and the cylindrical vias used in the MCLIN solution.
Nevertheless, we do see very similar trends in that we see the typical looping in toward the origin
in both the 𝑆11 and 𝑆21 responses at the center frequency.
Of course, the type of metal deposition and patterning determines the level of accuracy of the
finalized filter. We also cannot, for instance, accurately predict how substrate permittivity and
metal conductivity will change with process conditions. We do, in any case, have a number of
surface-mount capacitors which can be tuned after fabrication to achieve the required response.

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0

𝐿 = 2610 μm • Enclosure dimensions:

3000 μm
3000 μm × 3000 μm
• Grid size = 5 μm in x and y

𝑠 = 75 μm
𝑤 = 288 μm

(a)

(b)
Figure 16. (a) Dimensions of parallel coupled lines and enclosure geometry in the computational space (b) EM
model implementing the 2.5 D PCL model as a subcircuit.

Of course, the type of metal deposition and patterning determines the level of accuracy of the
finalized filter. We also cannot, for instance, accurately predict how substrate permittivity and
metal conductivity will change with process conditions. We do, in any case, have a number of
surface-mount capacitors which can be tuned after fabrication to achieve the required response.
We then look at the wideband response, shown in Fig. 18. We see that the transmission response,
i.e. the insertion loss (IL) is at half-power at the corner frequency.

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− 𝑆21
− 𝑆22
𝑆21 (dB), 𝑆11 (dB)

(a)

− 𝑆21
− 𝑆22

(b) (c)

Figure 17. Electromagnetic simulation (EMSight) based response for the third-order Butterworth filter
derived from the MCLIN-based design. (a) Rectangular response for 𝑆11 and 𝑆21 demonstrating
maximally flat response in the 8.06 GHz – 9.81 GHz passband, with a slightly offset center frequency
of 8.9 GHz (b) and (c) polar frequency response for 𝑆11 and 𝑆21, showing typical Butterworth
response with a slow loop into the origin at 8.89 GHz.

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− 𝑆21
− 𝑆22
𝑆21 (dB), 𝑆11 (dB)

Figure 18. Wideband response for the electromagnetic simulation of the third-order Butterworth filter
derived from the MCLIN-based design. Periodic passbands are seen here at 8.8 GHz, 43.25 GHz.

This wideband response is in excellent agreement with the MCLIN-based model’s predictions.
Again, we see a very low insertion loss (𝑆21 ) of -0.21 dB within the passband and a return loss
(𝑆22 ) of -45.51 dB in the stopband. We also see the typical effects of coupling of the two outer
lines which a produce a zero in the stopband, at 15.2 GHz. This has a minimal effect, with this
unwanted response giving a insertion loss of 49.06 dB. These trends repeat with lesser efficacy at
a passband of 41.16 GHz to 45.34 GHz as well.

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V. Conclusion

In this report, we have realized a third‐order Butterworth Combline Bandpass Filter designed to
operate with a center frequency of 8.5 GHz, a bandwidth of 1.7 GHz. This was inplemented using
microstrip technology with a substrate of relative permittivity, 𝜀𝑟 , of 2.6 and with great accuracy.
We have stepped through the design procedure to produce a lumped-element model, a microwave
coupled-line model (MCLIN) and a 2.5D EM model (EMSight electromagnetic simulation
environment). The design’s nominal performance was validated by this approach at several
junctures in the design process by evaluation of insertion loss and return loss and by considering
both rectangular responses and Smith charts.

We took into account tradeoffs required for matching a 50 Ω source and load to the desired 𝜋-
arrangement of shorted stubs and considered even and odd-mode line impedances to produce the
desired system impedance close to 50 Ω. We also performed calculations to confirm that effects
of multimoding were eliminated. Dimensions were selected, keeping in mind the manufacturability
of the microstrip geometry and tolerances. We made useful observations on small artifacts
produced from unwanted inter-line coupling and checked for agreement in predictions between
MCLIN-based and EM simulation based approaches. We also saw periodicity in location of
passbands and analyzed the small coupling effects between the outer two lines in the model, as
was shown by the wideband response. The results are summarized in Table 2, for nominal
performance, comparing the expected nominal performance metrics for each.

Table 2. Performance metrics for three models used in the synthesis of the 𝑓𝑐 = 8.5 GHz, BW = 1.7 GHz 3rd Order
Butterworth Filter

Lumped- Microstrip Model (MCLIN) 2.5D EM simulation

Element
Model

Center Frequency 8.49 GHz 8.5065 GHz 8.7 GHz

Bandwidth 1.77 GHz 1.687 GHz 1.745 GHz

Other comments Multimoding • Insertion loss of -2.74 × • Insertion loss (𝑆21 ) of -0.21 dB
not 10−6 dB in the passband in passband
considered • return loss of -47.9 dB in • Return loss (𝑆22) of -45.51 dB
passband in passband
• No observable stopband • Weak stopband response of -49
response from coupling of dB, due to coupling of outer
outer two lines two lines

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REFERENCES

[1] Michael Steer. Microwave and RF Design - Microwave and RF Design Modules. Volume 4
3rd Ed. (2019)

[2] Michael Steer. Microwave and RF Design - Transmission Lines Volume 2 3rd Ed.(2019)

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APPENDIX

MATLAB Code to generate lookup-table for odd and even mode system parameters, from
vol2 textbook and determine dimensions of microstrip coupled lines
Parameters and dimensions
eta_0 = 376.73;
er= 2.6;
u_ratio = 0.3:0.01:6;
g_ratio = 0.1: 0.1:15;
f0 = 8.5e9;
c = 3e08;
lambda_0 = c/f0;

Make lookup table meshgrid

[U_RATIO,G_RATIO] = meshgrid(u_ratio,g_ratio); % generate a grid of u
ratio and g ratio values

% Calculation of Z0 (free-space impedance of microstrip)

F1 = (6 +(2.*pi-6).*exp(-(30.666./U_RATIO).^0.7528)); % 3.22
Z01 = (60.*log(F1./U_RATIO + sqrt(1+(2./U_RATIO).^2))); % 3.21
Z0 = Z01; % 3.20

Even mode calculations

mu = (G_RATIO.*exp(-G_RATIO) + U_RATIO.*(20+G_RATIO.^2)./(10+G_RATIO.^2));
% 5.88
m= (0.2175+(4.113+(20.36./G_RATIO).^6).^-0.251
+(1./323).*log((G_RATIO.^10)./(1+(G_RATIO./13.8).^10))); % 5.87
alpha = (0.5.*exp(-G_RATIO));
% 5.86
psi = (1 + (G_RATIO./1.45) + (G_RATIO.^2.09)./3.95);
% 5.85
phi = (0.8645.*(U_RATIO.^0.172));
% 5.84
b = (0.564.*((er-0.9)./(er+3)).^0.053);
% 5.83
a= (1 + (1./49).*log((U_RATIO.^4+(U_RATIO./52).^2)./(U_RATIO.^4 +
0.432))+(1./18.7).*log(1+(U_RATIO./18.1).^3)); % 5.82
Fe = ((1 + 10./mu).^(-a.*b));
% 5.81

epsilon_ee = (Fe.*(er-1)./2 + (er+1)./2);

% 5.80
phi_e = (phi./(psi.*((alpha.*(U_RATIO.^m)) + (1-alpha).*(U_RATIO.^-m))));
% 5.79

Z01e = (Z0./(1-Z0.*phi_e./eta_0));
% 5.78

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Z0e = (Z01e./sqrt(epsilon_ee));
% 5.7 6

Odd mode calculations

q = (exp(-1.366-G_RATIO));
% 5.101
r = (1 + 0.15.*(1 - exp(1-((er-1).^2)./8.2)./(1 + G_RATIO.^(-6))));
% 5.100
fo1 = (1- exp(-0.179.*G_RATIO.^0.15 - 0.328.*(G_RATIO.^r)./log(exp(1) + (G_RATIO./7).^2.8)));
% 5.99
p = (exp(-0.745.*G_RATIO.^0.295)./cosh(G_RATIO.^0.68));
% 5.98
fo = (fo1.*exp(p.*log(U_RATIO) + q.*sin(pi.*log(U_RATIO)./log(10))));
% 5.97
n = ((1./17.7 + exp(-6.424 - 0.76.*log(G_RATIO) - (G_RATIO./0.23).^5)).*...
log((10+68.3.*(G_RATIO.^2))./(1+32.5.*G_RATIO.^3.093)));
% 5.96
beta = (0.2306 + (1./301.8).*log((G_RATIO.^10)./(1+(G_RATIO./3.73).^10)) + ...
(1./5.3).*log(1+0.646.*(G_RATIO.^1.175)));
% 5.95
theta = (1.729 + 1.175.*log(1 + 0.627./(G_RATIO+0.327.*G_RATIO.^2.17)));
% 5.94
phi_o = (phi_e - (theta./psi).*exp(beta.*(U_RATIO.^n).*log(U_RATIO)));
% 5.93
Fo = (fo.*(1+10./U_RATIO).^(-a.*b));
% 5.92

epsilon_eo = (Fo.*(er-1)./2 + (er+1)./2);

% 5.91
Z01o = (Z0./(1-Z0.*phi_o./eta_0));
% 5.78
Z0o = (Z01o./sqrt(epsilon_eo));
% 5.76
Z0S = (sqrt(Z0e.*Z0o));

lookup_table = ([G_RATIO(:) U_RATIO(:) Z0e(:) Z0o(:) epsilon_ee(:) epsilon_eo(:) Z0S(:)]);

% create x-y-THETA2 dataset

% Make lookup table

% Extract rows in table in which Zoe, Zoo and Z0S match the filter design values
ind_1 = (lookup_table(:,3)>55 & lookup_table(:,3)<56 &...
lookup_table(:,4)>38 & lookup_table(:,4)<39 &...
lookup_table(:,7)>46 & lookup_table(:,7)<47);
first_cut = lookup_table(ind_1,:);

Dimensions of line
height = 0.0001 % thickness = 1mm
separation = (first_cut(1,1).*height)
width = (first_cut(1,2).*height)
epsilon_eff = (sqrt(first_cut(1,5)*first_cut(1,6)));

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lambda_g = (lambda_0/sqrt(epsilon_eff));
L = lambda_g/8
system_impedance = first_cut(1,7);

% OTHER NOTES
% 1. The center frequency could be different than your electrical design. That is because you had
to make a compromise in the effective
% permittivity you used to set the line length. Adjust the line length to get the center
frequency to agree with your ideal.

% 2. if when you run your synthesized design using the MCLIN coupled line model you find that the
bandwidth is larger than your electrical design.
% That is because there is more line?to?line coupling than you expected. This is mainly because
of the difference of the even? and odd?mode phase velocities. So you want to get the line?to?line
% coupling back to what you wanted and you do this by increasing the line separation.

% 3. Allowable substrate thicknesses are 1 mm, 635 ?m, 500 ?m, 375 ?m, 250 ?m, 200 ?m, and 100
?m.
% Use the thickest substrate available as that increasing manufacturability and hence yield.
% 4. Substrate loss tangent = 0.0002.
% 5. Manually ensure that there is no undesired multimoding.
% 6. Length of the strips to be at least several times the thickness of the substrate.
% 7. Also, you will want the length of the strips to be several times (at least 5 times) the
width of the strips or else there will
% be too much undesired coupling from various parts of one strip to parts of another strip.
% 8. Standard via sizes are in mils (thousandths of an inch) beginning at 5 mils (125 microns)
(then
% 10, 15, 20 mils etc). Thus the standard via dimensions are 125 ?m, 250 ?m, 375 ?m, 500 ?m
% etc. (In EM simulation you will need to treat the vias as having a square cross section rather
% than rounded.) Use gold metallization with tantalum vias. It will be good to use vias in your
% MCLIN?based simulation as then agreement with the EM results will be closer.

% g = s./h;
% u = w./h;
% gold_resistivity = 2.2;

Example output
height =
1.0000e-04

separation =
5.0000e-05

width =
2.8800e-04

L =

0.0030
Published with MATLAB® R2017a

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