Chapter 1 INTRODUCTION
1.2 Thesis Organization
In chapter 2, by treating the wideband LC-filter as a matching network [13], the mechanism of the proposed band-pass filter can be easily understood graphically on the Smith chart. Besides, to further improve its stop-band rejection, two transmission zeros are then introduced, the component values of the LC filter can be obtained graphically through the Smith chart, and the relevant equations, which are functions of transmission zeros and poles, are also derived.
In chapter 3, the feasibility of the filter theorem is then supported by the agreements between the simulated results and its calculated counterparts from the developed formulas. A 3 – 10 GHz wideband filters fabricated in Rogers RO4003C (with εr = 3.5, tanδ = 0.0027, and thickness h = 0.508 mm) is demonstrated.
In chapter 4, we will conclude the characteristic of the proposed filter and compare with other type UWB filters.
6
It’s well-known that wide-band bandpass filters (BPF’s) have been a critical component for both scientific community and the communication industry. In chapter 1, we have listed different types of wideband bandpass filter and discuss the characteristic of each one. Then, in order to suppress out-of-band harmonic and reduce the circuit size in the same time, the proposed filter has shown in Fig.1.
In Fig. 1, the proposed LC filter only consists of five inductors and three capacitors, it has compact size. Besides, to avoid the mathematical entanglement, we resort to the Smith chart to reveal how simply by the three resonators this wideband LC-filter can achieve its superior wideband characteristic.
L1
Fig. 1. The circuit layout and the simulated S-parameters of the proposed wideband filter.
2.2 Matching criterion for a π-network
2.2.1 Matching mechanism of a π-network
As shown in Fig. 1, the proposed LC filter, which is a π-network, can achieve wideband characteristic, thus we will focus on what a π-network’s characteristic is. In Fig. 2(a), a π-network is shown, and the motion of impedance transformation on the Smith chart is shown in Fig. 2(b).
(a)
8
where Z0 is the ubiquitous microwave 50-ohm
In Fig. 2(b), we know that for an arbitrary value of susceptance b1 will have its corresponding value of reactance x2 to transform the impedance back to 50 ohm. In order to find out the relationship of b1 and x2, we have to solve the value of x1 first.
On Y-Smith-Chart, the constant g circle can be expressed
2 2 1 2
On the other hand, the constant b1 circle can be expressed
2 2 2
where u and v are the real part and imagine part of reflection coefficient respectively.
For g=1, u and v can be express as follow: Furthermore, the imagine part of normalized impedance can be expressed in terms of
u and v
In Fig. 2(b), it is clear that if the impedance transformation backs to 50 ohm, it has to satisfy the follow equation:
So far, the matching criterion (7) that can make the π-network match to 50 ohm is derived. On the other hand, we can also use ABCD matrix to solve a π-network in Fig. 2(a) to get the matching criterion (7), which will be described in Appendix.
Up to now we have figured out how the mechanism of matching network of the π-network is and the relationship between b1 and x1.Then, we will focus on how to choose the proper value of b1.
2.2.2 Choose of the value of susceptance b
1After the discussion above in section2.2.1, now we start to find out the value of susceptance b1. Initially, in fig. 3(a), we take a low-pass circuit, of which x2 = ωL/Z0
and b1 = ωCZ0, as an example to verify the feasibility of the derived matching criterion (8)
Fig. 3 A low-pass filter
Furthermore, to easily understand the mechanism of the π-network, we treat it as a matching network. Among the solutions that can meet the criterion (7), we just pick out three solutions as examples, b1 = 0.6, 1, and 1.67. As can be seen graphically on the Smith chart, there are three trajectories satisfying (7) can move
10
Fig.4 The Zin’s impedance transformation of a low-pass circuit
The solid trajectory is with b1 = 1, x2 = 1, the circled trajectory is with b1 = 0.6, x2 = 0.88, and the dashed trajectory is with b1 = 1.66, x2 = 0.88. When we set ω as 10 GHz, the values of L and C can be obtained in table.1 and its corresponding S-parameters are shown in Fig. 5. Notably, although the circled curve has the best matched condition (lowest S11), it has the poorest stop-band roll-off, on the contrary, the dashed curve has the poorest matched condition (lowest S11) while it has the best stop-band roll-off, thus the solid trajectory (b1 = x2 = 1) is a compromise between the dashed trajectory and circled trajectory and is preferred.
Table.1 The value of b1, x1, L, and C for three solutions for low-pass filter
Fig. 5 Three solutions of the low-pass circuit that meet the matching criterion (7).
On the other hand, in the case of high-pass circuit in Fig. 6, it also has the same phenomenon of high-pass filter of which x2 =1 /Z0ωC and b1 = Z0/ωL, We set ω as 3 GHz, the values of L and C can be obtained and listed in table.2 and its corresponding S-parameters are shown in Fig. 7. As a result, b1 = x2 = -1 is also suggested in high-pass filter. Thus far, the matching mechanism of the high pass circuit and low pass circuit has been clarified, and the matching criterion (7) is derived. Next, the proposed wideband filter mechanism will be elaborated.
Fig. 6 A high-pass filter
12
Table.2 The value of b1, x2, L, and C for three solutions for high-pass filter
Fig. 7. Three solutions of the high-pass circuit that meet the matching criterion (7).
2.3 Wideband Filter Mechanism
2.3.1 In-band design
After discussing the matching network of a π-network, in this section we will apply the method mentioned in section 2.2.1 and 2.2.2 to explain the proposed filter and give some simple design equations.
(a)
Fig. 8. (a) The prototypical wideband filter.
0 5 10 15 20
of 9.5 GHz are three matched frequencies (transmission poles)
The prototypical wideband filter is shown in Fig. 8(a). In Fig. 8(b), where f1 of 3.1 GHz (lower in-band), f2 of 5.4 GHz (mid in-band) and f3 of 9.5 GHz (higher in-band) are three matched frequencies (or transmission poles). To avoid the mathematical entanglement, we resort to the Smith chart which has been discussed in section 2.2 to reveal how simply by the three resonators this wideband LC-filter can achieve its superior wideband characteristic.
First of all, at lower in-band, the parallel circuit L1C1 appears inductive and the series circuit L2C2 appears capacitive, so the approximated circuit in Fig. 9 applies, and its corresponding high-pass characteristic with its matched frequency f1 (= 3 GHz)
14
0 5 10 15 20
Freq GHz
-60
-40 -20 0
-50 -30 -10
dB
Fig. 9. The simulated S11 and S21 of the equivalent high-pass circuit with L1=2.65nH, L2 = 1.14 nH, and C2 = 0.74 pF, where f1 is its matched frequency (transmission pole).
Fig. 10. Zin’s impedance transformation from the normalized termination resistance z0 = 1 and back up to the z0 starting point again.
As shown in Fig. 10, owing to the “shunt” inductor L1, the motion of Zin’s impedance transformation from the normalized termination resistance z0 = 1 to point A on the Smith chart is along the unit constant-conductance circle (g = 1), and we
obtain that the shunt inductor’s normalized susceptance equals b1; then, by series circuit L2C2, of which the normalized reactance is x2, the trajectoryof impedance transformation moves from point A to point B; finally, it moves back to the z0 starting point by the other shunt inductor L1. As derived in the previous section, b1 = x2 = -1 is the suggested matching criterion which makes Zin of a high-pass circuit equal 50-ohm.
Thus, in Fig. 10, the normalized susceptance b1 of the inductor L1 must be -1: microwave 50-ohm, thus L1 can be determined by the given ω1 as follow:
0
Secondly, at higher in-band, the parallel circuit L1C1 appears capacitive and the series circuit L2C2 is inductive, so the approximated circuit in Fig. 11 applies, and its corresponding low-pass characteristic with its matched frequency f3 (= 10 GHz) is
16
0 5 10 15 20
Freq GHz
-60
-40 -20 0
-50 -30 -10
dB
Fig. 11. The simulated S11 and S21 of the equivalent low-pass circuit with C1 = 0.32 pF, L2 = 1.14 nH, and C2 = 0.74 pF, where f3 is its matched frequency.
Fig. 12. Zin’s impedance transformation of the equivalent low-pass circuit from the normalized termination resistance z0 = 1 and back to the z0 starting point again
As shown in Fig. 12, owing to the “shunt” inductor C1, the motion of Zin’s impedance transformation from the normalized termination resistance z0 = 1 to point B on the Smith chart is along the unit constant-conductance circle (g = 1), and we
obtain that the shunt capacitor’s normalized susceptance equals b1; then, by series circuit L2C2, of which the normalized reactance is x2, the trajectoryof impedance transformation moves from point B to point A; finally, it moves back to the z0 starting point by the other shunt capacitor C1. As derived in the previous section, b1 = x2 = 1 is the suggested matching criterion which makes Zin of a low-pass circuit equal 50-ohm.
Thus, in Fig. 12, the normalized susceptance b1 of the capacitor C1 must be 1, and can be derived as follow
Similarly, by series L2C2 the trajectoryof Zin’s impedance transformation moves from point B to point A, thus we can obtain that the normalized reactance x2 of L2C2
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x2 = -1 at the lower matched frequency f1 (transmission pole); at high frequency, it can be regarded as a low-pass circuit and meets b1 = x2 = 1 at the higher matched frequency f3 (transmission pole). Therefore, with the given f1 (ω1) and f3 (ω3), L1, C1, L2, and C2 can be manipulated by (9), (12), (14), and (15), respectively. Finally, substitute f1 = 3 GHz, and f3 = 10 GHz in (9), (12), (14), and (15), L1 = 2.65 nH, C1 = 0.32 pF, L2 = 1.14 nH, C2 = 0.74 pF can be derived. Its corresponding simulated results, as shown in Fig. 8, exhibiting three matched frequency (transmission zeros) 3.1GHz, 5.4GHz, and 9.5 GHz thereupon confirms the accuracy of the previously derived equations. In addition, with the derived components values, both the resonance frequency of the series circuit L1C1 and that of the parallel circuit L2C2 are at 5.4 GHz, that is, Zin equals the termination resistance Z0 (50 ohm) at this resonance frequency, thus, there is a matched frequency ω2 of 5.4 GHz. In conclusion, we can use transmission zero to design a filter’s in-band characteristic.
2.3.2 Stop-band design
In addition to the required in-band response characterized in the previous section, the stop-band rejection is now concerned and can be improved by introducing transmission zeros at the stop-band
(a)
0 5 10 15 20
In addition to the required in-band response characterized in the previous section, the stop-band rejection is now concerned and can be improved by introducing transmission zeros at the stop-band. With the short circuit nature of a series circuit, replacing C1 with the series circuit L3C1, as depicted in Fig. 13(a), can construct a transmission zero at f4, as illustrated in Fig. 13(b). If we adopt the same design methodology described in the previous section: at low frequency, it looks like a high-pass circuit and meets b1 = x2 = -1 at the lower matched frequency f1
(transmission pole); at high frequency, it can be regarded as a low-pass circuit and meets b1 = x2 = 1 at the higher matched frequency f3 (transmission pole). Thus, the
20 obtained, and its corresponding simulated results in Fig. 13(b) exhibiting two matched frequencies (transmission poles) at 3.1 GHz and 10 GHz, and a transmission zero at 13 GHz confirms the accuracy of the formulas above.
2.3.3 Further improvement
To further improve the mid in-band S11, the design methodology should be modified as follow: it appears as a high-pass circuit and meets b1 = x2 = -1 at the low matched frequency f1 (transmission pole). Thus, the value of L1 can be determined by (9) (i.e. b1 = -1 at f1), and the normalized reactance x2 of L2C2 equals -1 as indicated in (10) (i.e. x2 = -1 at f1); besides, to achieve matched impedance at mid-band (ω2), we should locate the resonance frequencies of the series circuit L2C2 and the parallel
circuit L1C1L3 at ω2, thus Zin can equal the termination resistance Z0 (50 ohm) at ω2:
finally, the transmission zero locates at ω4, which is set by (14); therefore, combine (9), (10), (17), (20) and (21), L1, L2, L3, C1, C2 can be derived as follow: corresponding simulated result (solid curve) shown in Fig. 14 (b), exhibits three matched frequencies: f1 = 3.1 GHz, f2 = 5.4 GHz, f3 = 9.4 GHz, and one transmission zero: f4 = 13 GHz. Thus the accuracy of (22) ~ (26) is confirmed. Specifically, at the high matched frequency f3 (transmission pole), though the normalized susceptance b1
of the parallel circuit L1C1L3 is 1.42 rather than 1, and the normalized reactance x2 is
22
measurement results are shown in chapter 3.
From the previous discussion, we use transmission poles to design a filter’s in-band characteristic and use transmission zeros to design its stop-band characteristic.
Thus, with the given transmission zeros and poles, the desired S-parameters of the
wideband filter can be obtained. To examine the feasibility of the design methodology and the capability of the proposed wideband filter structure, a wideband filter was designed and fabricated. The measured results are shown in the next section.
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Chapter 3
LAYOUT AND MEASUREMENT
3.1 Layout of the proposed filter
Proposed wideband filter mechanism has been fully analyzed in chapter 2. In this chapter, quasi-lumped elements are used to realize the proposed LC-filter. As depicted in Fig. 14(b), the inductors L2, L3 and L4 are implemented by high-impedance microstrip line sections, and the shunt inductors L1 are implemented by a short-circuited microstrip stub. The capacitors C1 and C3 are implemented by low-impedance microstrip line sections, and finally the series capacitor C2 is realized by the mircostrip-to-CPW transition.
In the realization of proposed filter, the most difficult thing is to realize a series capacitor C2 in our layout. The following are some technique to construct series capacitor. First we consider microstrip gap as our series capacitor implementation. In order to reach the desired value of capacitor C2, the gap between two microstrip line sections will become much narrow. Due to the difficult of fabrication, microstrip gap is not suitable for our implementation. Secondly, the common used structure is interdigital capacitor which is shown in Fig. 15(a).
(a) (b)
Fig. 15 (a) The structure of interdigital capacitor (b) Its equivalent circuit
As shown in Fig. 15(b), the equivalent circuit of interdigital capacitor structure not only has desired series capacitor C but also accompanies two shunt parasitic capacitors CS. From Microwave engineering, it can be derived as follow
)
For our proposed filter circuit in Fig. 14, the value of C is 0.75 pF and value of Cs calculated by (27) will be 0.28 pF. The value of Cs is too large for us to ignore and we find that the parasitic shunt capacitor will destroy the proposed filter performance.
Thus the interdigital capacitor won’t be our chose to fabricate single series capacitor.
Finally, we use plate parallel capacitor to realize our circuit as shown in Fig. 16(a). It can not only produce the desired capacitor value but also eliminate the parasitic shunt capacitors. In addition, to reduce circuit size, mircostrip-to-CPW transition is used.
According to the measurement result in Fig. 17, it reveals that the plate parallel capacitor is available for implementation of pure series capacitor.
The proposed wideband band-pass filter was manufactured using Rogers RO4003C (with εr = 3.38, tanδ = 0.0027, and thickness h = 0.508 mm). Because the proposed ultra-wideband filter in Fig. 14 only has five inductors and three capacitors, the total circuit size can be miniaturized. Fig. 16 shows the three-dimensional circuit layout, the top-/bottom-layer layout of the proposed filter, and photograph. This layout reveals how the space was efficiently utilized. Besides, for the convenience of measurement, we add two 50ohm microstrip feeding lines on the both sides of the
26
(a)
(b)
0.23λ 0.31λ
(c)
Fig. 16 (a) The three-dimensional layout. (b) The top-/bottom-layer circuit layout of the proposed filter. (c) The photograph of the filter.
The overall dimensions of these devices are 4.6 mm × 7.4 mm, which is approximately 0.23 λ × 0.31 λ, where λ is the guided wavelength of the microstrip structure at the center frequency f0 = 7.1 GHz. The dimension confirms the very compact size of the developed device.
3.2 Measurement
The full-wave simulated result of Fig. 14 is shown in Fig. 19, which is calculated by the Ansoft HFSS simulator. Fig. 17 shows the simulated (dashed curves) and measured (solid curves) S-parameters of proposed wideband filter. The filter has a measured 3-dB fractional bandwidth of 128% from 2.8 GHz to 11.4 GHz.
28
response ranging from 0.32 ns to 0.46 ns over the whole passband as shown in Fig.
18. Fig. 17. Simulated (dashed curves) and measured (solid curves) insertion loss, and
return loss of the fabricated filter.
5 10
Fig. 18 Simulated (dashed curves) and measured (solid curves) group delay of the fabricated filter.
Compared with Fig. 14, the simulation results in Fig. 17 have one additional transmission zero at frequency 0.6 GHz. The reason of additional transmission zero is
the coupling of adjacent inductor L1. Fig. 19 shows the layout of three different value of x of proposed filter and its simulation insertion loss results were shown in Fig. 20.
It is clear that additional transmission zero shifts left with increasing value of x
x x
Fig. 19. The top-/bottom-layer circuit layout of x=2 (circled curve), x=1 mm (dashed curves), and x=0 (solid curves) of the fabricated filter.
-80 -60 -40 -20 0
-70 -50 -30 -10
dB
S21
x =2mm x =1mm x =0mm
30
When compared with other publications in Table 3, initially we find that 3-dB bandwidth of UWB filters which were proposed in [8] ~ [11] vary from 108% to 139%., and 128% of the UWB filter we proposed. Secondly, concern about the stopband rejection, it is clear that the filters shown in Table [3] all have good out-of-band rejection of least 20 dB. Thus, it doesn’t make much difference in the performance of 3-dB bandwidth and stopband rejection. But the most important thing is that the UWB filter we proposed has the most compact circuit size compared with others. Therefore, the proposed compact ultra-wideband filter is promising for communication application.
Ref. 3-dB bandwidth stopband rejection (20dB) Circuit size In [8] 108%
Table. 3 Comparision with other publication in 3-dB bandwidth, stopband rejection, and circuit size
Chapter 4
CONCLUSION
4.1 Conclusion
In this paper, a simple design methodology for a compact ultra-wideband filter with wide-stopband has been thoroughly analyzed. By treating filter as a π-network circuit and solve the matching mechanism by using Smith chart. With the given specification, the desired value of each LC component can be calculated by equation (22) ~ (26).
On the other hand, to examine the feasibility of the design methodology and the capability of the proposed wideband filter structure, a wideband filter was designed and manufactured. By using quasi-lumped element and mircostrip-to-CPW transition, the layout of the proposed filter is designed. The measured results show that the filter prototype has 3-dB fractional bandwidth of 128% from 2.8 GHz to 11.4 GHz.
Furthermore, the return loss is greater than 11 dB within the pass-band, minimum insertion loss of 0.3 dB over the pass-band, superior 20 dB stop-band rejection to above 24 GHz, flat group delay of 0.4 ns within 0.15 ns variation over the pass-band, and very compact circuit size of 0.23 λ × 0.31 λ, where λ is the guided wavelength of the microstrip structure at the center frequency f0 = 7.1 GHz.
32
where Z0 is the ubiquitous microwave 50-ohm and
2 1
where b1 is the normalized susceptance of the capacitor, x2 is the normalized reactance of the inductor. Let |S11|2 = 0 or |S21|2 = 1, we can get
The matching criterion (31) is identical to (7)
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