Edge-Coupled Bandpass Filter Design

ELEC 4502 Fall 2018
Project 1:
Microstrip Edge-Coupled Band-pass Filter
Submitted by: Rashad Alsaﬀar - 101006781
Due Date: December 23, 2018
Section: A1O

ELEC 4502 Project 1 Report Rashad Alsaﬀar - 101006781
Contents
1 Introduction 3
1.1 Design Results Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Design Process 4
2.1 Theoretical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Equal-Ripple Low-Pass Filter Prototype . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Chebyshev Band-Pass Transformation . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3 Filter Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Simulated Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Edge-Couple Microstrip Design . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Ideal Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3 Non-Ideal Filter Design using MCLIN, MTAPER . . . . . . . . . . . . . . . . 8
2.2.4 Non-Ideal Filter Design using MCFIL, MSTEP . . . . . . . . . . . . . . . . . 10
2.3 Layout Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 MCLIN Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 MCLIN/MTAPER Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3 MCFIL/MSTEP Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 EM Co-Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Measured Results 14
3.1 Calibration Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Filter Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Comparison of Results 15
4.1 Measured vs. Ideal vs. Non-Ideal vs. EM . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Bandwidth/Center Frequency Comparison between Results . . . . . . . . . . . . . . . 16
5 Further Analysis 16
5.1 SMA Connectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 RO4350B Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3 Metal Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6 Conclusion 17
7 References 17
8 Appendix A: Sample Calculations 18
9 Appendix B: Matlab Code 20

1 Introduction
The creation of a band-pass filter can be established using edge-coupled half-wave resonators as is
the anticipated design. The use of short circuited quarter-wave resonators may also be approached,
however short circuit design is difficult within microwave microstrip design. The microstrip filter is
composed of three cascading edge-couple half-wave resonator components with each end connected
to a 50Ω source line. The circuit was generated and insulated onto a printed circuit board (PCB)
and was acquired for testing.
Filter design characteristics rely heavily on two factors: insertion loss and return loss. Insertion
loss is inversely proportional to filter bandwidth. Each quantity can be adjusted to manipulate fil-
ter performance. The addition of multiple resonator components would also increase insertion loss.
Return loss demonstrates signal return from the load to the source. A successful filter design will
demonstrate the lowest return loss at the center frequency of the filter, with generally low return loss
at low and high frequencies. This will assert the circuit experiences minimum power reflection from
load to source.
To attempt designing a microwave filter, a series of design steps were undertaken. First, a low-
pass prototype filter design was established, which was then converted to a Chebyshev band-pass
filter experiencing low-pass to band-pass transformation. Secondly, edge-coupled resonator charac-
teristics, i.e. even and odd mode impedances and coupling coefficients were calculated to determine
the appropriate microstrip dimensions for an ideal band-pass filter design. Finally, Agilent Design
Studio (ADS) was used to create a variation of circuit schematics ranging from ideal to realistic
components through several tuning segments and simulations.
The final ADS schematic was implemented into a layout where EM Co-Simulations were performed
to visualize realistic circuit behavior. A 3 inch × 2 inch shield made of vias was attached to the final
layout of the design.
1.1 Design Results Comparison
The tables below detail the specifications, theoretical, simulated, and achieved results for the filter
performance:
Specifications Theoretical Simulated Achieved
3.80GHz 3.800GHz 3.542GHz 3.4GHz
Table 1: Microstrip Edge-Coupled Band-pass Filter Center Frequency Comparison
900MHz 1010MHz 886.7MHz 821.9MHz
Table 2: Microstrip Edge-Coupled Band-pass Filter Bandwidth Comparison
0.5dB 0.481dB 3.052dB 2.813
Table 3: Microstrip Edge-Coupled Band-pass Filter Pass-band Ripple Comparison

2 Design Process
The creation of the filter was established through multiple domains; hand-written calculations were
performed to achieve theoretical results, which were then sketched into ADS. Simulations and tuning
were performed to gather appropriate design conditions for the final device response.
2.1 Theoretical Design
The edge-coupled half-wave microstrip resonator filter was required to have the following specifica-
tions:
− Center frequency fc = 3.80GHz
− Chebyshev response with pass-band ripple = 0.50dB
− Equal-ripple bandwidth = 900MHz
− Terminations = 50Ω
An equal-ripple low-pass filter prototype would have to be established first in order to convert to a
band-pass filter. This was achieved through applying filter design by the insertion loss method, refer-
enced in Pozar [1]. The insertion loss method allows for an improved filter design method with control
over pass-band, stop-band amplitude and phase characteristics to simulate desired characteristics [1].
A filter response is characterized by the power loss ratio:
2.1.1 Equal-Ripple Low-Pass Filter Prototype
The theoretical low-pass filter response was developed through Matlab code (displayed in Appendix
B). The given bandpass ripple Am was set to 0.5dB and center frequency ω was set to 3.8GHz.
A = 10 × log10[1 + (10Am/10
− 1)(cos2
(n cos−1
ω )], ω ≤ 1 (1)
= 10 × log10[1 + (10Am/10
− 1)(cosh2
(n cos−1
ω )], ω ≥ 1
where Am is defined as the band ripple and n is defined as the order of the filter.
ω =
ω0
ω2 − ω1
[
ω
ω0
−
ω0
ω
] (2)
where ω0 is the band center frequency and ω2 and ω1 are the upper and lower pass-band edges,
respectively. The low-pass prototype filter response is displayed below:
Figure 1: Low-Pass Filter Magnitude vs. Frequency Response

2.1.2 Chebyshev Band-Pass Transformation
The transformation from a low-pass prototype filter to a Chebyshev band-pass filter can be performed
using equation (2). Center frequency remained at 3.8GHz, with upper and lower band edges at
4.25GHz and 3.35GHz, respectively. Matlab code was used to plot the Chebyshev transformation
below:
Figure 2: Band-Pass Filter Magnitude vs. Frequency Response
2.1.3 Filter Transformation
An equal-ripple Chebyshev low-pass filter prototype was selected as part of the initial design process.
Table 8.4 given by Pozar [1] characterizes coupling coefficient element values g for a magnitude of
filter orders (N). See Appendix A for sample calculations of coupling coefficients.
Figure 3: Element Values for Equal-Ripple Low-Pass Filter Prototypes at 0.5dB ripple [1]
The list below displays the substrate properties for the Rogers 4350B material used for the filter
fabrication design [3]:
− H: substrate height = 62 mil
− r: dielectric permittivity = 3.66 F/m
− µr: dielectric permeability = 0.999994 H/m
− Cond: substrate conductivity = 5.96 × 107
S/m
− T: conductor thickness = 35 µm
− tan δ: loss tangent = 0.0037

2.2 Simulated Design
The edge-coupled filter utilizes parallel coupled microstrip transmission lines. A coupled transmis-
sion line experiences two different modes based on the wave propagation between the two coupled
lines as opposed to each individual line. The excitation of the two coupled lines may occur in phase
(even mode) or 180◦
out of phase (odd mode) [2]. The impedance seen by the propagating wave will
differ for even (Z0e) and (Z0o).
Coupled line even and odd impedances can be calculated using the coupling coefficients assigned
to a specific coupled line. The expressions below detail admittance inverter parameters given by
Pozar [1] for an Nth
order filter:
n = 1 ⇒ Z0J1 =
π∆
2g1
(3)
n = 2, 3, ..., N ⇒ Z0Jn =
π∆
2
√
gn−1 − gn
(4)
n = N + 1 ⇒ Z0JN+1 =
π∆
2gN × gN+1
(5)
where fractional bandwidth ∆ was previously calculated as 0.2368, appropriate coupling coefficients
were assigned, and Z0 is defined as the characteristic impedance of the transmission lines feeding
into and out of the filter, i.e. Z0 = 50Ω terminations.
2.2.1 Edge-Couple Microstrip Design
By substituting the previously selected coupling coefficients, even and odd impedances can be cal-
culated through the expressions below:
Z0e = Z0 × [1 + Z0Jn + (Z0Jn)2
] (6)
Z0o = Z0 × [1 − Z0Jn + (Z0Jn)2
] (7)
The table below details the calculated even and odd impedances for each coupled microstrip pair:
Coupling Pair Z0e (Ω) Z0o (Ω)
g1 85.785 37.515
g2 68.00586 39.896
g3 68.00586 39.896
g4 85.785 37.515
Table 4: Even/Odd Impedances for Coupling Microstrip Pairs

2.2.2 Ideal Filter Design
Calculated even and odd mode impedances can be substituted into the CLIN component, an ideal
edge-coupling resonator. The circuit below was designed within ADS, acting as an ideal band-pass
filter response which will be referenced and compared by future modifications of the filter.
Each CLIN component was designed using the calculated even and odd mode impedances displayed
in Table 4. The electrical length of each component was set to 90◦
, setting each half of the resonator
components to a quarter-wavelength long, at 3.80GHz.
Figure 4: CLIN Edge-Coupled Filter Circuit Schematic
The plot below details the ideal insertion loss of the ideal circuit above using S-Parameters S(2,1) and
50Ω impedance terminals. Results were plotted in decibels of magnitude as a function of frequency.
Figure 5: CLIN Ideal Insertion Loss Response
Insertion loss detailed in the plot above was evaluated through an ideal design. The plot experiences
three ripples, each corresponding to the center frequency, upper and lower frequency bandwidth lim-
its, similar to the characterization of a Chebyshev band-pass filter. Roll-off occurs close to the upper
and lower frequency bands as predicted.
The plot below details the ideal return loss of the ideal circuit from Figure 4 using S-Parameters
S(2,2) and 50Ω impedance terminals. Results were plotted in decibels of magnitude as a function of
frequency.

Figure 6: CLIN Ideal Return Loss Response
Return loss detailed in the plot above was evaluated through an ideal design. Design tuning will
involve the manipulation of existing values to adjust realistic responses towards ideal results.
2.2.3 Non-Ideal Filter Design using MCLIN, MTAPER
Dependant on mode-impedance, center frequency and electrical length, edge-coupled microstrips can
be developed using LineCalc within ADS. Inserting the mentioned parameters and synthesizing gen-
erates appropriate width, length and spacing of the edge-coupled resonator components.
The non-ideal circuit was generated using MCLIN and MTAPER components, utilizing defined
microstrip dimensions from LineCalc and established as microstrip coupled lines.
Due to the symmetrical conditions of the filter response, only half of the filter’s design characteristics
were to be calculated. The other half of the circuit would act as the reverse characteristics of the
first half. The table below details the calculated length (L), width (W) and spacing (S) dimensions
for each resonant component.
Design Parameter Value (mil)
W1 79.047638
W2 111.159449
W3 134.928346
L1 480.161417
L2 467.897638
L3 47.805118
L4 456.330709
S1 8.251339
S2 21.183425
Table 5: Non-Ideal MCLIN Design Characteristics
Two MLINs were placed on both ends of the filter, acting as a 50Ω transmission line to match with
a 50Ω source of the spectrum analyzer equipment used to measure the filter. A taper component
MTAPER was placed in between coupled resonator MCLIN components generated with different
widths, adjusting their lengths to fit the required board size. The second half of the filter is a
reversed setup of the first half, reversing the orientation of tapers and transmission line components.

Figure 7: MCLIN/MTAPER Edge-Coupled Filter Circuit Schematic
Note that MCLIN components contain their own parasitic capacitance, i.e. unwanted capacitance
due to connected conductor components within a circuit. The effects of parasitic capacitance causes
the S21 response to deviate from the CLIN S21 response.
The addition of the MTAPER components caused deviation in the results as opposed to the in-
dividual MCLIN components. Most noticeably, the response only generated two ripples as opposed
to three, as well as the shift in bandwidth and center frequency.
The plot below displays the insertion and return loss responses for the initial and modified MCLIN
schematic regarding the inclusion of MTAPER components:
Figure 8: Non-Ideal MCLIN Filter Response w/ & w/o MTAPER Components
Three equations detailed in Figure 12 above were given through ADS Help to determine center fre-
quency fc, bandwidth BW, and pass-band ripple PBR.
The results were derived from the MCLIN circuit response and recorded to the table at the right.
There are noticeable difference between the design specifications and the schematic response. Center
frequency and bandwidth have deviated by 0.03GHz and 0.756GHz, respectively. The addition of the
MTAPER components provided even further deviation from design specifications; center frequency
and bandwidth deviated by 0.128GHz and 0.553GHz, respectively.

2.2.4 Non-Ideal Filter Design using MCFIL, MSTEP
The MCFIL component is defined as a microstrip coupled-line filter section. Its advantage over a
simple MCLIN is its ability to adjust its width at each of its ports. The addition of the MSTEP
component allows for ports to be centered according to their set widths. With this advantage, lining
up individual microstrip components would be easier while avoiding shorted components.
The schematic designed below details a more appropriate edge-coupled bandpass circuit design tai-
lored closer to design characteristics and filter response.
Figure 9: MCFIL/MSTEP Edge-Coupled Filter Circuit Schematic
The plot below details a comparison of the the filter response above between the MCLIN/MTAPER
schematic, as well as the newly created MCFIL/MSTEP schematic:
Figure 10: Non-Ideal MCLIN/MTAPER vs. MCFIL/MSTEP Insertion & Return Loss Filter Response
The MCFIL/MSTEP schematic experiences slightly deviated values from the design specifications,
however with adjusted component dimensions via tuning, the filter response could be tweaked towards
its ideal counterpart.

2.3 Layout Generation
Generating a circuit layout of the device will allow for extensive EM simulations via ADS. The
required board size was 3.0in × 2.0in. Extensions would be made to the circuit to ﬁt the board size
speciﬁcations. The arrows on the input and output of the circuit act as pins used for EM simulations.
The RO4350B substrate was created within an EM setup.
2.3.1 MCLIN Layout
The layout below was generated from the single-component MCLIN schematic:
Figure 11: MCLIN Circuit Layout Generation
2.3.2 MCLIN/MTAPER Layout
A noticeable issue regarding the generation of the layout was the shorted connections between the
microstrips. This caused irregular circuit behavior and thus simulations could not be acquired with-
out providing necessary distance between components.
The addition of the MTAPER component helped solve this issue, and generated feasible results.
The layout generation below details the MCLIN/MTAPER layout:
Figure 12: MCLIN/MTAPER Circuit Layout Generation
2.3.3 MCFIL/MSTEP Layout
As part of the design process for an improved circuit, a layout was generated for the MCFIL/MSTEP
schematic designed in Figure 9:
Figure 13: MCFIL/MSTEP Circuit Layout Generation

2.4 EM Co-Simulation
With the addition to basic ADS simulations, EM co-simulations were performed to replicate results
similar to real life. Manufactured results would be based off the most recent EM co-simulations,
therefore it was imperative that circuit responses were accurate enough to produce a functioning
microwave filter.
For EM simulations to be attempted a substrate type and specification was required. As previ-
ously mentioned, the substrate used was RO4350B. Its defining characteristics are identifiable within
the MSub component in all established circuit schematics.
Figure 14: Rogers 4350B Substrate EM Substrate via ADS
The following generated layout was established from the MCLIN/MTAPER generated layout. A via
shield was created by establishing adequate spacing between the microstrip components, as well as
the manually placed vias.
Figure 15: MCLIN/MTAPER Final Circuit Layout Generation w/ Via Shield
The substrate file generated using RO4350B was applied to the layout above. An EM co-simulation
was executed, simulating realistic circuit behavior.
Due to the extended simulation time within EM co-simulations, a limited number of simulations
were able to be run to adjust results towards design specifications.

The plot below details the EM co-simulation response from the generated layout above:
Figure 16: Non-Ideal MCLIN/MTAPER EM Co-Simulation Response
The circuit experienced a center frequency shift towards 3.542GHz, with band-edges occurring at
3.097GHz and 3.983GHz. The addition of the MTAPERS, although helped resolve shorts within the
circuit, caused the circuit to experience massive deviance from design specifications.
The circuit also experienced a change in return loss S22 response pattern prior to adding MTA-
PERs. This was experimented with the MCFIL/MSTEP circuit, which generated the following plot:
Figure 17: Non-Ideal MCFIL/MSTEP EM Co-Simulation Response
While center frequency was still shifted from its intentional 3.8GHz location, the inclusion of MCFIL
and MSTEP components provided a closer center frequency, as well as a more accurate representa-
tion of the ideal filter response. The proper amount of ripple-bands are now visible and the design
maintains adequate bandwidth of 820MHz. While slightly off from design values, this would have
been a more practical approach to the filter design.
Another noticable difference is the smoothness of the insertion loss and return loss responses be-
tween Figures 16 and 17. The addition of MTAPERS generated a large amount of distortion during
the EM co-simulation where the MCFIL strategy provided a nearly distortionless response. Minimum
return loss is provided within Figure 17 at center frequency, as opposed to Figure 16.

3 Measured Results
The figure below displays the printed circuit board layout for the filter:
Figure 18: Printed Microstrip Filter Design
3.1 Calibration Process
Measurements were recorded using a Vector Network Analyzer (VNA), however the device was re-
quired to be calibrated. Known loads were used to measure power within the coaxial cables that
would be used to connect the filter to the VNA via the SMA connectors soldered on the input and
output of the circuit. This would ensure minimum reflection and loss during testing, prioritizing the
response from the board.
The procedure involved utilized the Short Open Load Through (SOLT) method, using a given cali-
bration kit containing a short circuit, open circuit, load, and through connectors, connected to the
ends of the cables to be used. This would allow the VNA to recognize power reflection due to the
attached cables in order to properly test the filters at the appropriate input/output slots.
The calibration procedure began with setting up a frequency range of 1.5GHz to 6GHz to mea-
sure the devices. The short circuit termination was connected to the VNA and was calibrated via
the ”calibrate” button. Next was the attachment of the open circuit termination, followed by the
load termination. Calibration was once again performed by pressing ”calibrate”. The process was
performed individually for the source and load side of the VNA.
3.2 Filter Measurements
The following plot displays the insertion loss and return loss of the filter circuit via the VNA:
Figure 19: Measured Printed Filter Insertion/Return Loss Response

Measured results were processed through the calibrated VNA. The circuit experienced a center
frequency of 3.4GHz with a return loss of -4.686dB. The upper and lower band-edges were experienced
at 3.81GHz with a return loss of -8.801dB and 2.99GHz with a return loss of-5.089dB, respectively.
The bandwidth was measured at 822MHz, slightly deviated from the design specifications. As like
the EM co-simulation from Figure 16, the circuit experienced a largely deviated response as opposed
to the ideal filter model.
4 Comparison of Results
4.1 Measured vs. Ideal vs. Non-Ideal vs. EM
The following plots detail the multiple filter responses for insertion/return loss across multiple filter
circuit designs:
Figure 20: Measured vs. Ideal vs. Non-Ideal vs. EM Insertion Loss Responses
Figure 21: Measured vs. Ideal vs. Non-Ideal vs. EM Return Loss Responses
Each marker represents the center frequency of insertion/return loss plots. Marker M1 repre-
sents the measured filter, M2 represents the EM co-simulation pre-fabricated filter, M3 represents
MCLIN/MTAPER non-ideal ADS simulation, M4 represents the ideal CLIN filter, and M5 represents
the EM co-simulation MCFIL/MSTEP filter.

4.2 Bandwidth/Center Frequency Comparison between Results
The table below details the % error for center frequency across all discussed filter configurations in
Section 4.1:
Configuration f0 (GHz) % Error
Ideal 3.8 0
Non-Ideal 3.67 3.42
EM Co-Sim 3.54 6.84
Measured 3.4 10.53
Table 6: % Error Measurements for Filter Design Center Frequency
It can be seen that % error has increased throughout the design process towards the final mea-
surements of the filter. Center frequency was aimed to be accomplished by tuning through every
iteration, however due to the selection of MTAPERs and a minimal design time, preferred goals
couldn’t be accomplished. The table below details the % error related to bandwidth:
Configuration BW (MHz) % Error
Ideal 900 0
Non-Ideal 955.3 6.14
EM Co-Sim 886.7 1.48
Measured 822 8.67
Table 7: % Error Measurements for Filter Design Bandwidth
Bandwidth seemed very unpredictable with each design iteration of the microwave filter. Several
tuning cycles were established to maintain it within range. While the EM co-simulation response
proved the closest bandwidth to the design specifications, the measured filter experienced a larger
deviance.
5 Further Analysis
5.1 SMA Connectors
SMA (SubMiniature versionA) connectors were attached to the input and output of the microwave
filter. They ideally provide an impedance of 50Ω to match with the 50Ω of the input transmission
lines into the filter. The addition of connectors however results in discontinuities, i.e. changes in
conductor/insulator diameter, space available between components, dimension shifting. The SMAs
were connected using solder which introduces its own discontinuity issues. Alternative impedance
applied to the filter board would result in slightly larger return loss.
5.2 RO4350B Permittivity
The small dielectric constant of the RO4350B substrate used in the final filter fabrication is respon-
sible for slightly left-shifted center frequency and band-edges. The RO4350B datasheet [3] provides
two particular permittivity values for the substrate: 3.66 and 3.48±0.5. Each dielectric constant is
provided for different applications and testing methods. Anticipation of the effects of permittivity
could allow us to deviate our center frequency forward to counter the normal effect of the constant.

5.3 Metal Impurities
Metal impurities can act as a source of power loss due to surface roughness; skin depth of a substance
decreases as frequency increases, causing conductor loss to increase. Power reflection can be affected
greatly by surface roughness, causing measured and simulated values to differ. As previously men-
tioned, the effect of solder onto the board and attachment of the SMA connectors directly contributes
to these metal impurities.
6 Conclusion
The design of a microwave filter encountered multiple mediums to develop a printed circuit board.
A theoretical low-pass to band-pass transformation prototype filter was implemented through Mat-
lab. Coupling coefficients were selected according to the 0.5dB equal-ripple table from Pozar [1],
contributing to the determination of the coupled microstrip pair dimensions.
Multiple design iterations were performed to tune the filter to accomplish its design specifications.
Different ADS components were used and all simulations were compared to the ideal filter response.
A generated layout was then sent as a Gerber file and fabricated onto a PCB. SMA connectors were
soldered and the VNA was calibrated and used to measure the filter response.
The measured filter response generally captured the performance of its last recent simulation through
an EM co-sim, however values were still deviated from the design specifications. Multiple EM sim-
ulations had to be established to visualize effects of tuning within the microstrip components. We
discovered the advantage of using MCFIL/MSTEP configuration as opposed to MCLIN/MSTEP,
delivering a closer response to the ideal microwave filter model.
7 References
[1]: Pozar, D. M., ”Microwave engineering”, New Delhi: Wiley India, 2017
[2]: R. Amaya, ”Project 1 - Design of an Microstrip Edge-Coupled Band-pass Filter”, 2018, De-
partment of Electronics, Carleton University
[3]: Rogers Corporation, RO4350B Laminates. Retrieved from https://www.rogerscorp.com/documents/
726/acs/RO4000-Laminates-RO4003C-and-RO4350BData-Sheet.pdf

8 Appendix A: Sample Calculations
Equations 1.7 - 1.12 within Lab Manual [2]
Calculation β:
β = ln[coth(
Am(dB)
17.37
)]
β = ln[coth(
0.5dB
17.37
)]
β = 3.548
Calculation γ:
γ = sinh(
β
2n
) ⇒ n = 3
γ = sinh(
3.548
2 × 3
)
γ = 0.626
Calculation a1 - an:
ak = sin(
(2k − 1)π
2n
] ⇒ k = 1, 2, ..., n
a1 = sin(
((2 × 1) − 1)π
2 × 3
]
a1 = 0.5
Calculation b1 - bn:
bk = γ2
+ sin(
kπ
n
)2
⇒ k = 1, 2, ..., n
b1 = 0.6262
+ sin(
1 × π
3
)2
b1 = 1.142

Calculation g0 - gn+1
g0 = g4 =
π
ω1
[
f2 − f1
f2 + f1
]
g0 =
π
1
[
4.25 − 3.35
4.25 + 3.35
]
g0 = g4 = 0.372
g1 =
2a1
γ
g1 =
2 × 0.5
0.626
g1 = 1.596
gk = 4 × [
ak−1ak
bk−1gk−1
] ⇒ k = 2, 3, ..., n
g2 = 4 × [
0.5 × 1
1.142 × 1.596
]
g2 = 1.096
Equation 3-5 - Admittance Inverter Parameters:
Z0J1 =
π∆
2g1
Z0J1 =
π × 0.237
2 × 1.596
Z0J1 = 0.483
Equation 6 - Even-Mode Impedance:
Z0e = Z0 × [1 + Z0J1 + (Z0Jn)2
]
Z0e = 50Ω × [1 + 0.483 + 0.4832
]
Z0e = 85.785Ω
Equation 7 - Odd-Mode Impedance:
Z0o = Z0 × [1 − Z0J1 + (Z0Jn)2
]
Z0o = 50Ω × [1 − 0.483 + 0.4832
]
Z0o = 37.515Ω

9 Appendix B: Matlab Code

Edge-Coupled Bandpass Filter Design

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