Crossover coupler can also be designed using two branch-line hybrids in a cascade [3].
By applying standard hybrid analysis techniques, theoretically, the signal emerges only at the
diagonal port of the composite structure, with no insertion loss, and no power emerges from the remaining two ports such that high isolation between two crossing signal channels can be achieved. Nevertheless, the design is limited to that the two crossover traces have identical characteristic impedances. Chapter 2 extends the realization of planar crossover coupler of which the two crossover signal traces are allowed to have different characteristic impedances.
The circuit consists of two asymmetric branch-line hybrids [11-12] in cascade. Each hybrid is designed to have arbitrary power division and termination impedances. By properly selecting the structure parameters of each hybrid, a crossover coupler could be readily fulfilled.
Measured results of a fabricated circuit agree very well with the simulated counterparts.
In Chapter 3, the design of crossover junction with unequal port impedances in Chapter 2 is extended to have a dual-band characteristic based on the approach in [16]. According to the best of our knowledge, this is the first dual-band crossover coupler in open literature, so that its performance deserves investigation. An experimental microstrip dual-band crossover coupler is simulated and measured for demonstration.
Chapter 4 draws the conclusion of this work.
Chapter 2
Crossover Coupler Design using Asymmetric Branch-Line Hybrids
Crossover coupler is one of the solutions to the problem that two transmission lines cross each other. It enables the two signal paths to have as low as interference with each other. This chapter presents a crossover coupler design based on two asymmetric branch-line hybrids with arbitrary power divisions in a cascade. It is a new idea to design a four-port crossover junction with different termination impedances. Block ABCD matrix formulation is applied to the circuit to estimate the bandwidth. The relation between power division ration of asymmetric branch-line hybrids and performance of the crossover coupler is also investigated.
2.1 Crossover Coupler Design
A crossover coupler can be designed using two branch-line hybrids in a cascade [3]. By applying standard hybrid analysis techniques, theoretically, the signal emerges only at the diagonal port of the composite structure, with no insertion loss, and no power emerges from the remaining two ports so that high isolation between two crossing signal channels can be achieved. Nevertheless, the design has hitherto been limited to that the two crossover traces have identical characteristic impedances. This Chapter extends the realization of planar crossover coupler of which the two crossover signal traces are allowed to have different characteristic impedances. The circuit consists of two asymmetric branch-line hybrids [12-13]
in cascade. Each hybrid is designed to have arbitrary power division and termination impedances. By properly selecting the structure parameters of each hybrid, a crossover coupler can be readily fulfilled.
2.1.1 Asymmetric Branch-Line Hybrid with Arbitrary Termination Impedances
In [11-12], an asymmetric branch-line coupler with arbitrary termination impedance and arbitrary power division is proposed. Figure 2.1 shows this four port coupler terminated in arbitrary impedances Za, Zb, Zc and Zd. It consists of four -sections which form a ring. The impedances of four -sections are Z1, Z2, Z3 and Z4. When power is fed into port 1, phase difference of the two output waves at port 2 and port 3 is 90o. The ratio |S21|/|S31| is d1/d2 while port 4 is isolated. The design equations are as follows. For Za = Zb = Zc = Zd and d1 = d2, the result is the well-known conventional branch-line coupler [3].
2 Figure 2.1 Circuit schematic of an asymmetric branch-line hybrid.
2.1.2 Two Asymmetric Branch-Line Hybrids in a Cascade
Figure 2.2(a) shows the schematic of the proposed crossover coupler. It consists of seven quarter-wavelength sections with characteristic impedances Z1 ~ Z7. The termination impedances of the two pairs of the diagonal ports are Zo1 and Zo2. The circuit can be treated as two cascaded asymmetric branch-line hybrids as shown in Figure 2.2(b). Each hybrid possesses arbitrary power division and termination impedances. For impedance matching purpose, the two output terminations of the first hybrid, Zi1 and Zi2, are designed to be the same as the two input counterparts of the second circuit. Since the sections Za3 and Zb2 are in shunt connection, we have Z4 = Za3//Zb2. For coupler i (i = 1, 2) at the center frequency, let the outputs at their individual through and coupled ports be designated as –ji and –i, respectively. Here, i and i are positive real numbers satisfying i2
+ i2
= 1. Applying the standard hybrid analysis technique, we have
21 ( 1)( 2) ( 1)( 2) 1 2 1 2 design equations in [3] and [12], the characteristic impedances of the branch-line sections can be written as
1 2 1cos
Consequently, there are several degrees of freedom for the design since , Zi1 and Zi2 can be arbitrary. The only concern for choosing their values should be the realization of the line widths of the fabrication process.
Z1
Figure 2.2 (a) Proposed crossover coupler with arbitrary diagonal termination impedances. (b) Analysis of (a) by the transmission line model.
2.2 Circuit Bandwidth
2.2.1 Block ABCD Matrix for Four-Port Network
In Figure 2.3 (a) and (b), port voltages and currents are defined at various terminals of a four-port network for Y-, Z- and ABCD-parameter analyses. Admittance and impedance matrices are used to describe the relation between the port voltages and currents in Figure 2.3 (a). The admittance matrix [Y] of the microwave network relates these voltages and currents as shown in (2.5):
The impedance matrix can be defined in a similar fashion. The matrices [Y] and [Z] can be used to characterize a microwave network, but in practice when a microwave network consists of a cascade of two or more four-port networks, it is convenient to use a 4-by-4 ABCD-parameter for analysis purpose. The ABCD matrix of the cascade of two or more four-port can be easily obtained by directly multiplying the ABCD matrices of the each four-port, as shown in Figure 2.4. Mathematically, the voltages and currents of the two ports on the left hand side of the network are related to those on the right hand side by (2.6):
1 2 1 2 2
I1
Figure 2.3 (a) Voltage and current definitions in Z- and Y-parameter formulations. (b) Voltage and current definitions in block ABCD parameter formulations.
For the voltages and currents of a four-port in the block ABCD-parameter formulation shown in Figure 2.3 (b), the directions of currents at port 2 and port 3 are opposite to those in the Y- and Z-parameter formulation. By using similar definition for the traditional two-port ABCD matrix, each of the Ai-entries in block ABCD can be determined by applying input voltages at port 1 and port 4, and measuring the open-circuit output voltages at port 2 and port 3.
Similarly, each entry in block B is the ratio of input voltages to short-circuit output currents, those in block C is the ratio of input currents to open-circuit output voltages, and those in block D is the ratio of input currents to short-circuit currents. Detailed derivations are given in the following subsections.
I1
Figure 2.4 Block ABCD matrices for analysis of a cascade of two four-ports.
2.2.2 Conversion between the Block ABCD Matrix and the 4×4 Y Matrix
From (2.6), we have
Multiplying (2.8a) by Y34 and subtracting (2.8b) multiplied by Y24 from it, we can eliminate V4
and obtain the relation between V1 and V2. Then the transformation of Y-parameters to A1 can be derived as shown in (2.9a). All other elements of block A can also be derived similarly and are given in (2.9b) ~ (2.9d).
32 24 22 34
According to the relation of voltages and currents in the Y-matrix, the following equations can be derived with V2 = 0, V3 = 0 and I3 = 0:
Note that the direction of I2 in the Y-matrix is opposite to that in the block ABCD matrices.
The relation of V1 and V4 in (2.11a) is used in (2.11b). Then, the ratio V1/I2 can be derived as shown in (2.12a). Similarly, all other elements block B can be obtained as follows:
34
21
According to the relation of voltages and currents in the Y-matrix and the conditions V3 = 0, I2
= 0 and I3 = 0, the following equations can be derived: V4 can be expressed as a function of V2. Applying these relations to (2.14c) can eliminate V1
and V2. Then, the ratio of I1/V2 can be obtained as shown in (2.15a). All other elements of block C can also be derived as shown in (2.15b) ~ (2.15d):
11 32 24 22 34 14 22 31 32 21
From (2.6), D1 of block D is defined as the following equations can be obtained.
3 31 1 34 4 0
From (2.17a) V4 can be written as a function of V1. Substituting this function into (2.17a) and (2.17b), the ratio I1/I2 in (2.16) can be obtained. D2, D3 and D4 can be derived in a similar
Based the procedure shown above, the four-port Y-parameters can be readily transformed into block ABCD parameters. Next, we will show the formulation for transforming ABCD-parameters into Y-parameters for a four port network.
From (2.5),
After some algebraic operations, the result is
1 4 2 3
All other elements in the Y-matrix can be obtained in a similar fashion:
14 1 2 2 1
1 4 2 2 4 2 2 3 4 1
Figure 2.5 shows a four-port coupler with arms of arbitrary lengths, arbitrary characteristic impedances and arbitrary terminations. Analysis of the crossover coupler in Figure 2.2 can be formulated by the block ABCD matrices. To find the entries of the block ABCD matrix, the Y-parameters of a four-port coupler in Figure 2.5 are derived first. By definition, Y11 = I1/V1 when port 2, 3 and 4 are short-circuited. Based on the input admittance of a loaded transmission line, the result is shown in (2.25a). All other entries in the Y-matrix can be obtained in a similar fashion, and the results are in (2.25b) to (2.25i).
11 1cot 1 2cot 2
Y jY
jY
(2.25a)22 1cot 1 3cot 3
Y jY
jY
(2.25b)33 3cot 3 4cot 4
Y jY
jY
(2.25c)44 2cot 2 4cot 4
Y jY
jY
(2.25d) Y12 Y21 jY1csc
1 (2.25e)14 41 2csc 2
Y Y jY
(2.25f)23 32 3csc 3
Y Y jY
(2.25g)34 43 4csc 4
Y Y jY
(2.25h)13 31 24 42 0
Y Y Y Y (2.25i)
These Y-parameters are then transformed into four-port block ABCD parameters as shown in section 2.2.2. Multiplying the ABCD matrices of each stage, the total ABCD matrix of the crossover in Figure 2.2 can be obtained. The overall S-parameter matrix can be calculated when the total ABCD matrix is transformed into Y matrix which is then transformed into the S-matrix. If a 20-dB return loss, 0.5-dB insertion loss and 20-dB isolation are referred, the bandwidth of the crossover coupler can be obtained by computation using a computer program.
The proposed crossover coupler in Figure 2.2 has four design parameters Zo1, Zo2, Zi = Zi1 = Zi2 and When the above parameters are determined, the impedances of the arms Z1~Z7
can be obtained from (2.3) and (2.4). Z01, Z02 and Zi are fixed and S-parameters are measured when port 1 and port 3 are terminated in Z01 and port 2 and port 4 are terminated in Z02. The
values of Z01 and Z02 will not affect the S-parameters but the ratio of Z01 to Z02 does, because Z1~Z7 will maintain the same ratio when the ratio of Z01 to Z02 is fixed. Then power division ratio is swept, and the bandwidth is calculated. The relation between and the calculated bandwidth are discussed as follows.
Figure 2.6 shows the calculated bandwidth for Zi1 = Zi2 = Z01. Each S-parameter may have its own response and bandwidth. Figure 2.6 (a) shows the bandwidth of |S11| and |S33|, Figure 2.6 (b) shows those of |S12| and |S34|, Figure 2.6 (c) shows those of |S13| and |S24|, Figure 2.6 (d) shows those of |S14| and |S23|, and Figure 2.6 (e) shows those of |S22| and |S44|. The other S-parameters can be known by the reciprocity theorem. In each port, curve 1 uses Z01:Z02 = 1:1, curve 2 uses Z01:Z02 = 1:2, curve 3 uses Z01:Z02 = 1:4, curve 4 uses Z01:Z02 = 2:1 and curve 5 uses Z01:Z02 = 4:1. When = 45o, the circuit has the best matching condition for all termination impedance ratios, while the circuit has the best isolation when the power coefficient is 20o or 70o. In Figure 2.7 and Figure 2.8, Zi1 and Zi2 are replaced by Z02 and Z Z01 02 , respectively. The trend of the circuit bandwidths of the S-parameters is similar to those in Figure 2.7.
3 2
4 1
Y
02Y
03Y
04Y
01Y ,
4
Y ,
3 Y ,
2
Y ,
1
4
3 2
1
Figure 2.5 Circuit schematic of a four port coupler.
0
4
0 25 50 75 100
bandwidth of S and S (%)
20 30 40 50 60 70 20 30 40 50 60 70
2244
(e)
Figure 2.6 Bandwidths measured by S-parameter responses. Port 1 and port 3 are terminated in Z01 and port 2 and port 4 are terminated in Z02. Zi1 = Zi2 = Z01. Curves 1 ~ 5 have Z01:Z02 = 1:1, 1:2, 1:4, 2:1, and 4:1, respectively. (a) |S11| and |S33|. (b) |S21| and |S34|. (c) |S13| and |S24|. (d)
|S14| and |S23|. (e) |S22| and |S44|.
0 25 50 75 100
bandwidth of S and S (%)
20 30 40 50 60 70 20 30 40 50 60 70
1133
(a)
12 15 18 21 24
bandwidth of S and S (%)
20 30 40 50 60 70 20 30 40 50 60 70
1234
(b)
4
0 3 6 9 12 15 18 21 24
bandwidth of S and S (%)
20 30 40 50 60 70 20 30 40 50 60 70
2244
(e)
Figure 2.7 Bandwidths measured by S-parameter responses. Port 1 and port 3 are terminated in Z01 and port 2 and port 4 are terminated in Z02. Zi1 = Zi2 = Z02. Curves 1 ~ 5 have Z01:Z02 = 1:1, 1:2, 1:4, 2:1, and 4:1, respectively. (a) |S11| and |S33|. (b) |S21| and |S34|. (c) |S13| and |S24|. (d)
|S14| and |S23|. (e) |S22| and |S44|.
0
5 8 11 14 17
bandwidth of S and S (%)
20 30 40 50 60 70 20 30 40 50 60 70
1324
(c)
5 5.3 5.6 5.9 6.2
bandwidth of S and S (%)
20 30 40 50 60 70 20 30 40 50 60 70
1423
(d)
0 5 10 15 20 25 30 35 40
bandwidth of S and S (%)
20 30 40 50 60 70 20 30 40 50 60 70
2244
(d)
Figure 2.8 Bandwidths measured by S-parameter responses. Port 1 and port 3 are terminated in Z01 and port 2 and port 4 are terminated in Z02. Zi1 = Zi2 = Z Z01 02 . Curves 1 ~ 5 have Z01:Z02 = 1:1, 1:2, 1:4, 2:1, and 4:1, respectively. (a) |S11| and |S33|. (b) |S21| and |S34|. (c) |S13| and |S24|. (d) |S14| and |S23|. (e) |S22| and |S44|.
2.3 Simulation and Measurement
Two circuits are fabricated and measured to validate the design. Figure 2.9 (a) ~ (c) plot the simulation and measurement of the first fabricated circuit with Zo1 = 25 , Zo2 = 100 , Zi
= 100 and 46. The center frequency is designed at 2.45 GHz. The circuit simulation is done by the software package IE3D [20]. The substrate has r = 10.2 and h = 1.27 mm. The S-parameters are measured with reference port impedance 50 . The measured data are converted to S-parameters with designated port impedances with the aids of Y-parameters. If a 20-dB return loss, 0.5dB insertion loss and 20-dB isolation are referred, the estimation bandwidths are shown as curve 3 of Figure 2.7. The estimated bandwidths of |S11|, |S21|, |S31|,
|S41|, |S22|, |S23|, |S24|, |S33|, |S34| and |S44| are 92.9%, 16.5%, 13.4%, 4.1%, 10.3%, 4.1%, 12.2%, 23.4%, 16.3% and 10.4%. Figure 2.9 (a)~(c) show the simulation IE3D bandwidths of |S11|,
|S21|, |S31|, |S41|, |S22|, |S23|, |S24|, |S33|, |S34| and |S44| are 92%, 21%, 13.3%, 4.3%, 12.1%, 4.4%, 10.2%, 32.5%, 20.7% and 12.1% and the measured responses have 85.7%, 20.5%, 10.4%, 4.9%, 14.4%, 4.3%, 3.7%, 62.4%, 20.5% and 13.1% respectively. The measured data has good agreement with the simulation and the calculation. Figure 2.9 (d) shows the photo of the experimental circuit.
Figure 2.10 (a) ~ (c) plot the simulation and measurement results of the second fabricated circuit with Zo1 = 100 , Zo2 = 50 , Zi = 50 and 40. Again, a 20-dB return loss, 0.5dB insertion loss and 20-dB isolation are referred, and the estimation bandwidths are shown as curve 4 in Figure 2.6. The calculated bandwidths of |S11|, |S21|, |S31|, |S41|, |S22|, |S23|,
|S24|, |S33|, |S34| and |S44| are 9.48%, 17.5%, 14.8%, 5.8%, 15.36%, 5.74%, 16.73%, 9.98%, 16.87% and 12.9%, respectively. Figure 2.10 (a)~(c) show the simulation bandwidths of |S11|,
|S21|, |S31|, |S41|, |S22|, |S23|, |S24|, |S33|, |S34| and |S44| are 10.36%, 19.12%, 12.86%, 5.79%, 18.06%, 5.73%, 15.25%, 11.09%, 18.72% and 14.53% and the measured results have 11.63%, 19.59%, 6.43%, 5.51%, 15.31%, 6.12%, 11.94%, 14.39%, 18.98% and 14.08%, respectively.
The measured responses have good agreement with the simulation and calculation. Figure 2.10 (d) shows the photo of the realized circuit.
0
0
1 1.5 2 2.5 3 3.5 4
Frequency (GHz) -5
-10 -15 -20 -25 -30 -35 -40 -45
2434|S |, |S |, |S | (dB) 44
|S |24
|S |44
|S |34
-20 dB
Simulation Measuremen
t
(c)
(d)
Figure 2.9 The S-parameters of the first fabricated circuit with Zo1 = 25 , Zo2 = 100 , Zi = 100 and 46. (a) |S11|, |S21|, |S31|, |S41|. (b) |S22|, |S23, |S33|. (c) |S24|, |S34|, |S44|. (d) The circuit photo. W1 ~ W7 are 2.06, 0.48, 1.01, 1.1, 1.18, 0.43 and 2.19 mm. L1 ~ L7 are 11.06, 11.69, 11.42, 11.39, 11.35, 11.72 and 11.02 mm.
0
0
1 1.5 2 2.5 3 3.5 4
Frequency (GHz) -5
-10 -15 -20 -25 -30 -35 -40 -45
2434|S |, |S |, |S | (dB) 44
-20 dB
Simulation Measuremen
t
|S |24
|S |44
|S |34
(c)
(d)
Figure 2.10 The S-parameters of the second fabricated circuit with Zo1 = 100 , Zo2 = 50 , Zi
= 50 and 40. (a) |S11|, |S21|, |S31|, |S41|. (b) |S22|, |S23, |S33|. (c) |S24|, |S34|, |S44|. (d) Circuit photo. W1 ~ W7 are 1.33, 1.84, 0.74, 3.74, 0.26, 2.49 and 0.92 mm. L1 ~ L7 are 11.29, 11.12, 11.55, 10.7, 11.85, 10.95 and 11.47 mm.
Chapter 3
Dual-Band Crossover Coupler Design
Recent rapid progress in wireless communications has created a need of dual-band operation for RF devices, such as the global systems for mobile communication systems (GSM) at 0.9/1.8 GHz and wireless local area network (WLAN) at 2.4/5.2 GHz. This Chapter studies the design of crossover coupler with dual-band operation.
3.1 Elementary Two-Port for Dual-Band Operation
Figure 3.1 shows the elementary two-port for substituting a /4-section to achieve dual-band operation. The elementary two-port consists of two high-Z sections on both sides with two stubs being attached to their ends and a low-Z section in the middle. Let the admittance of the two shunt stubs be jB/2. Since the circuit is symmetric about its center, by setting the reference plane in the middle open- and short-circuited, respectively, the input admittances for even- and odd-mode, ye = y11 + y21 and yo = y11 - y21, can be readily derived as follows:
2 1 crossover junction operate with a dual-passband characteristic, each /4-section of the two asymmetric branch-line hybrids is replaced by the elementary two-port network, which is capable of providing 90 and 270 degrees at f1 and f2, respectively. The following equations can be derived.
21( )1 ( ( s1, s2) ( s1, s2)) / 2 s1/ T the solution. Inserting these solutions into (3.2c) and (3.2d), values of B(f1) and B(nf1), and hence the electrical length and characteristic impedance of open stubs, can be obtained.
Z
s1
s1
s1Z
s1
s2Z
s2
s2
t1
t1Z
t1Z
t1Figure 3.1 Two-port network for subsituting a quarter-wave section.
0 5 10 15 20 25 30 0
(degree)
y 21() R=4
R=2
R=1
s1
y21
(nf ) (f )
1y21 1
1 2 3 4 5
(a)
0 5 10 15 20 25 30
(degree)s1 R=4
R=2
R=1 y21
(nf )
(f )
1y21 1
0
y 21()1 2 3 5
4
(b)
Figure 3.2 Design graphs for determining s1 and R. (a) s2 = 5o. (b) s2 = 10o. Both s1 and s2 values are referred to f1.
3.2 Simulation and Measurement
A microstrip crossover coupler designed at 0.9/1.8 GHz is fabricated and measured for demonstration. The circuit is built on a substrate of r = 10.2 and h = 0.508 mm. The characteristic impedances of the branches in Figure 3.3 are Za1 = 39.36 , Zb1 = 33.17 , Za2 = 29.67 , Za3//Zb2 = 36.93 , Zb3 = 42.13 , Za4 = 27.83 , and Zb4 = 46.91 .
Termination impedances of the port 1 and port 3 are Zo1 and these of the port 2 and port 4 are Zo2. Each branch is subsituted by the two-port in Fig. 3.1 by using the solutions in Fig. 3.2(b) with R = 2, s1 = 15o and s2 = 10o.
Figure 3.4 compares the measured results with the simulation data obtained by the IE3D [18]. Figure 3.4(a) and 3.4(b) plots the circuit responses when port 1 is excited, and 3.4(c) and 3.4(d) draws the results when signal is fed to port 2. In Fig. 3.4(a), it can
be observed that the measured return losses |S11| and the isolations |S41| at the two designated frequencies are better than -20 dB. If a 15-dB return loss is referred, measured data indicate that |S11| has bandwidths of 10.8% and 4.6% and |S11| has 3.5%
and 4.2% at f1 and f2, respectively. Fig. 3.4(b) shows the isolation between port 2 and port 1 |S21| and the crossover response |S31|. The measured |S31| = –0.89 dB and –1.1 dB at f1 and f2, respectively. When input is at port 2, the measured return loss |S22| in Fig.
3.4(c) is better than -20 dB and has bandwidths of 15% and 5.8% at f1 and f2, respectively, for a 15-dB reference. The experimental |S32| indicate that the isolations between ports 2 and 3 are better than 20 dB in both bands. In Fig. 3.4(d), the measured
|S42| = –0.69 dB and –1.24 dB at f1 and f2, respectively. In Fig. 3.4(a) through 3.4(d), reasonably good agreement between the simulation and measurement can be observed.
Fig. 3.4(e) shows the photograph of the experimental four-port dual-band crossover junction.
3 2
4 1
Z
i2Z
i1Z
b2Z
o2Z
o1Z
o2Z
o1Z
a4Z
b4Z
b3Z
a3Z
a2Z
b1Z
a1Figure 3.3 Two cascaded asymmetric branch-line hybrids to form a new crossover coupler with arbitrary diagonal port impedances Zo1 and Zo2.
0.5 0.75 1.0 1.25 1.5 1.75 2.0
0.5 0.75 1.0 1.25 1.5 1.75 2.0
(e)
Figure 3.4 Simulation and measured responses of the fabricated dual-band crossover junction. (a) |S11| and |S41|. (b) |S21| and |S31|. (c) |S22| and |S32|. (d) |S12| and |S42|. Zo1 = 50
and Zo2 = 25 . (e) Photograph of the experimental circuit.
Chapter 4 Conclusion
A new design for crossover coupler with different diagonal port impedances is devised by using a cascade of asymmetric branch-line couplers with arbitrary power division. It allows the port termination impedances to be arbitrary. Circuit analysis in term of block ABCD matrix and Y-matrix formulation is used to calculate the bandwidth associated with each S-parameter. In the second part of this thesis, the crossover is devised to have dual-band operation. Each quarter-wave section is replaced by an elementary two-port shown which is designed to achieve proper phase changes at the two designated frequencies. One microstrip circuit for operation at 0.9/1.8 GHz is realized. The measured responses not only validate the idea but also agree well with the simulation.
There is still some room for improving the performance of the proposed crossover couplers. First, is it possible to synthesize a passband of crossover couplers like a filter? The passband function can be maximally flat response, Butterworth response or elliptic function response. Furthermore, the size of a microstrip realization can be too large to be used in some applications. It needs miniaturizing circuit size and keeping the flexibility of the port termination impedances at the same time. Second, if a passband response can be synthesized for a crossover coupler, many existing design techniques for dual-band bandpass filter can be useful for microwave passive circuit.
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[14] K.-K. M. Cheng and F.-L. Wong, “A novel approach to the design and implementation of