Chapter 2 Basic Microwave Filter Design Theory
2.3 Filter Transformation
2.3.2 High-pass Bandpass and Bandstop Transformation
High-pass transformation: We can use the following substitution to convert a low-pass response to a high-pass response.
ωC
ω← − ω (2.24) It maps ω =0 to ω= ±∞ , and vice versa. The cutoff frequency occurs when
ω= ±ωC . The negative sign is necessary to convert inductors (capacitors) to realizable capacitors (inductors).
Bandpass transformation: Low-pass prototype is also transformed to bandpass response by replacing the frequency ω . Similarly, there is a mathematical substitution for this transformation as shown in Equation (2.25).
0 0 0
2 1 0 0
( ) 1( )
ω ω ω ω ω
ω←ω ω ω− − ω = ∆ ω − ω (2.25) , where ω0 is the center frequency; ω1 and ω2 denote the edges of the passband;
∆ is the fractional bandwidth (FBW) of the bandpass filter defined in Equation (2.26).
2 1
0
ω ω ω
∆ = − (2.26)
Bandstop transformation: The inverse transformation can be used to obtain a bandstop response. Substitution is shown below.
0 1 0
(ω ω )
ω ω ω
← −∆ − − (2.27) The related diagrams are shown in Figure 2.4.
PLR
PLR PLR
ω ω
ω
ω0 ω0
ω0
− −ω0
1
−1
( )a ( )b ( )c
ω2
− −ω1 ω1 ω2 −ω2 −ω1 ω1 ω2
Figure 2.4 Bandpass and bandstop frequency transformations. (a) Low-pass filter prototypeωC = . (b) Transformation to bandpass response. (c) Bandstop response. 1
Chapter 3
High-Power Low-pass Filter
3.1 Introduction
In modern science and technology, to pursue faster, wider system usage range, more efficient capability and more accurate performance, the high-power topic is deeply concerned [1]-[7]. Not only in academic researches, it becomes more popular in some business applications. More and more companies endeavor to invent new products in this area. For example, satellite output filters and multiplexers, wireless or radio base station transmitter filters and diplexers, etc.
Due to the high market demands for volume and mass production, there are more and more challenges in high-power operations. When designing a filter for these high-power requirements, there exist many effects which must be taken into account.
For examples, multipaction breakdown, ionization breakdown, passive inter-modulation (PIM) interferences and thermal-related high-power breakdown are all the possible phenomena in this field [7].
The presented high-power low-pass filter will be applied to the output stage of a communication system. The circuit prior to this filter is a power amplifier used to increase the intensity of output signals. However, it could increase the harmonic noises as well, so the low-pass filter is necessary used to suppress these harmonic noises which are produced from the power amplifier. Therefore, the power-resistant and high-temperature-resistant characteristics are stressed significantly in this product.
For practical low-pass filter implementation, the acceptable input power level is
about 100 to 150Watts. Compared to other higher power level applications, the thermal-related venting mechanism is the most important topic in this design. In following sections, complete design procedures and comparison between simulations and measured results are presented.
3.2 Filter Design Procedure
For the given high-power low-pass filter specifications, listed in Table 3.1.
1 Insertion Loss (DC~500MHz) <=0.8dB
2 Rejection (800MHz~2GHz) >=44dB
3 Input Connector SMA male
4 Output Connector N-type male
5 Pass Band VSWR <=1.45:1
6 Power Handling >=150W, CW
7 Box Size 4”x1.5”x1.5”
8 Temperature Range -10 to +60C
Table 3.1 Specifications of the high-power LPF.
The filter box mechanical drawing is shown in Figure 3.1.
Figure 3.1 The mechanical dimmensions of the high power LPF.
3.2.1 Prototype Establishment
First, we must determine the appropriate low-pass response model which can conform to the given specifications such as insertion loss, rejection and passband VSWR. For the practical implementation consideration, we use ladder network prototype which is beginning with a series inductor shown in Figure 2.2(a). In addition, an odd-order low-pass filter model is constructed for a symmetric structure.
In this chapter, we give a moderate design model by using Chebyshev polynomial approximation. It provides better design sense about response selectivity and less volume consumption of practical product.
The Equation (2.21) introduced last chapter can be used to determine an appropriate degree for the proposed low-pass filter. The related specifications include passband insertion loss, stopband rejection, passband VSWR and the ratio of stopband to passband, denotes S. The Equation (3.1) gives the relationship between VSWR and reflection coefficient, S11.
max 11
min 11
1 1 V S
VSWR V S
= = +
− (3.1)
According to the given VSWR value, S11 is well determined and equal to -14.719dB. In other words, we choose the proper value of LR which is equal to 15dB. We obtain N ≥7.147 by means of the known parameters such as LA, ωs and ωp. So, the least filter degree is easily determined by (2.21) and it is chosen to a ninth-order model. The ladder network is plotted in Figure 3.2.
Second, for the sake of the given passband insertion loss limitation, we choose the proper ripple-level model to analyze the filter. There are two models with different ripple-levels as shown in Figure 3.3. The ripple values are ε =0.1 and ε =0.01, respectively.
Figure 3.2 The ninth-order low-pass filter prototype network.
Figure 3.3 The low-pass filter prototypes with different ripple-level, 0.1 (S11 & S21) and 0.01 (S33 & S43).
Comparing the responses of two different ripple-levels, it is observed that the larger ripple-level we choose, the better out-band selectivity but the worse passband return loss level. Therefore, we should consider a moderate and reasonable choice in our design seriously. Finally, the ninth-order Chebyshev low-pass filter prototype with ripple-levelε =0.01 is accomplished. The detailed prototype component values of Figure 3.2 are listed in Table 3.2.
L1/L5 (nH) L2/L4 (nH) L3(nH) C1/C4 (pF) C2/C3 (pF) 12.962 28.717 30.331 9.084 10.902 Table 3.2 Prototype values of ninth-order Chebyshev low-pass filter.
3.2.2 Material and Component Selection
Now, the reasonable model has been well determined for our design. Next, we will select suitable materials and components to implement it. In a lumped circuit filter, capacitors and inductors are the essential and fundamental components. The choice of superior lumped elements is one of the major issues about filter implementation. For high-power condition, the strong and good heat-venting materials and components are preferred to use. Later, some useful tips are introduced for achieving better filter performance and endurance.
According to these following steps, the summary estimations of capacitors are presented.
(1.) First, we should estimate the electric potential across each shunt capacitor.
We suppose the maximal input power of detection is about 200Watts and the network is set up on 50Ω-system. From simple electrical power equation, the r.m.s. value of electric potential is equal to 100Volts, which represents the reference withstanding voltage. Consequently, the suitable capacitors will be chosen based on this reference value.
(2.) Second, we calculate the 10pF capacitor impedance around the passband edge, about 500MHz. From the following equation, X is easily obtained. C
6 12
1 1
31.83
2 2 500 10 10 10
XC
πfC π −
= = = Ω
⋅ ⋅ ⋅ ⋅ (3.2) (3.) When V and X are determined, we can also estimate the r.m.s. value of C current across capacitors. From (1.) and (2.), we obtain the reference withstanding current Irms flowing through the capacitor which is equal to 3.1416 Amps.
According to this simple analysis, we choose ATC-100B-Porcelain-Superchip -Multilayer-Capacitors to implement the filter. This kind of capacitors has many product attributes such as high-Q, low noise, high self-resonance, low ESR/ESL and
ultra-stable performance characteristics. The basic specifications about the ATC-100B series are listed in Table 3.3. Especially, the WVDC parameter of capacitors conforms to the reference withstanding voltage we calculate before.
1 Capacitance Range 0.1~1000pF
2 High Q >=10000 @ 1MHz
3 DC Working Voltage (WVDC) 500V @ 0.1~100pF
4 Temperature Coefficient (TCC) +90 +/- 20ppm/oC @ -55oC to +125oC +90 +/- 30ppm/oC @ +125oC to +175oC Table 3.3 Specifications of ATC-100B series capacitors.
(4.) Finally, according to the estimated Irms and given ESR value plotted in Figure 3.4, we can calculate the power loss ratio of each shunt capacitor. The complete calculations are given by
2 3.14162 0.1 0.98( )
L rms
P =I ×ESR= × ≈ watt (3.3.a) 0.98 0.49%
200
L
T
P
P = = (3.3.b)
Figure 3.4 The ESR of ATC-100B series.
The larger degree the filter is, the more the power loses. Another clever tip is the arrangement of capacitors. We can erect two or three shunt capacitors to achieve firmer mechanism, better heat-venting performance and reduce power loss ratio. Each
shunt stage includes three capacitors. In the Figure 3.2, C1 and C4 are composed of 2pF (2R0BW), 2.2pF (2R2BW) and 4.7pF (4R7CW) capacitors; C2 and C3 are composed of 1.5pF (1R5BW), 2pF (2R0BW) and 6.8pF (6R8CW) capacitors.
To obtain arbitrary inductances for our design, we use the self-made inductors.
For the sake of convenience on fabrication and the high-power-resistant features, the inductors are made of the semi-rigid coaxial cable with a 0,034” outside diameter, which is a 50-Ohm-Semi-Rigid Coaxial-Line from JYEBAO Corp. in Taipei Hsien, Taiwan. Then, we use the impedance analyzer, HP/Agilent 4191A RF Impedance Analyzer with frequency ranging from 1 to 1000 MHz, to measure the accurate capacitance and inductance around the design cutoff frequency. According to measuring experience, the half-turn of an inductor is about 10~15nH and a three-turn inductor is about 25~30nH. The inductances can be adjusted to fit the ideal prototype responses by tuning the space at intervals of spiral inductors.
The last considerations are connector choice and box design. From specifications, we buy the proper 50Ω SMA-male and N-type-male connectors from JYEBAO Corp.
The related connector configurations are shown in Figure 3.5.
Figure 3.5 The SMA male and N-type male connectors.
The box is an aluminum 6061-T6 material fabricated by machining which is plated with silver for possible soldering of capacitors and low loss. The metal housing, on the one part, provides better heat-venting efficiency and, on the other part, provides better isolation characteristic as well as to avoid the parasitic short-circuited effect. We can use basic concept of rectangle waveguide to estimate the cutoff frequency, which occurs in the aperture between cavities. The related rectangular waveguide is shown in Figure 3.6. The cutoff frequencies of the rectangular waveguide with width a and height b is
2 2
( ) 1
C mn 2
m n
f µε a b
= + (3.4)
, where ( )fC mn is the cutoff frequency of the rectangular waveguide at aperture region. That is, EM wave with a frequency higher than the lowest cutoff frequency (TE10 mode, ie. m=1, n=0) can propagate in the waveguide. Otherwise, it will be a evanescent decayed wave. In this case, the first two dominant modes are
10 18.69
TE ≈ GHz and TM11≈26.43GHz . They are all much higher than the frequency range we are interested.
Figure 3.6 The rectangle waveguide diagram.
3.3 Simulation and Measurement Comparison
The implemented product photos are shown in Figure 3.7, including both the vertical and lateral view. In these photos, we can clearly observe that the ninth-order low-pass filter arrangement and box structure. The mechanical design divides the whole cavity into five small cavities to push the cavity resonance to a much higher frequency. If the cavity resonant frequency occurs inside or near the filter passabnd, the filter performance may be degraded. To solve this problem, we separate the each cavity by metal wall and it can shift the resonance to higher frequency effectively.
(a) (b)
(c) (d)
Figure 3.7 Product photos. (a)Vertical view with cover. (b)Lateral view with cover.
(c)Vertical view without cover. (b)Lateral view without cover.
There are four samples of the high-power low-pass filters being fabricated. In this paper, we present only two samples (S/N:-001/-002). The measured frequency responses which include the model-001 and model-002 are shown in Figure 3.8 and Figure 3.9.
Figure 3.8 Frequency responses of Model-001.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Figure 3.9 Frequency responses of Model-002.
From the measured results, the passband (DC~500MHz) return loss and stopband (800MHz~2GHz) rejection all conform to the specifications. They are below 16.5dB and 49.5dB, respectively.
Further, there is broadband response of model-001 as shown in Figure 3.10 and
comparison to the filter without cover in Figure 3.11 and Figure 3.12.
Figure 3.10 Broadband responses of Model-001.
1 2 3 4 5 6 7 8 9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.0 1.0
-35 -30 -25 -20 -15 -10 -5
-40 0
freq, GHz
dB(S(1,1)) dB(S(3,3))
(b)
Figure 3.11 Comparison between with cover and without cover of Model-001. (a) Transmission coefficient. (b) Reflection coefficient.
1 2 3 4 5 6 7 8 9
0 10
0.2 0.4 0.6 0.8
0.0 1.0
freq, GHz
sqr(mag(S(1,1)))+sqr(mag(S(2,1))) sqr(mag(S(3,3)))+sqr(mag(S(4,3)))
Figure 3.12 Power conservation check of Model-001.
The measured frequency range is from 50MHz to 10GHz. In Figure 3.10, the box resonances occur at the high frequencies such as 4.4GHz, 5.6GHz, 7.6GHz and 9.4GHz. Because of the radiation loss at these frequencies, the transmission coefficients are not satisfied the -40dB level. The differences between the filter with cover and without cover are shown clearly in Figure 3.11 and Figure 3.12.
The detailed discussions and concepts are mentioned below.
(1.) There exist some noises propagating in free space. When inductor is exposed outside, it may couple with other power like an antenna. Because cover can give good isolation between inductors and outside noises, the insertion loss of filter with cover is better than the without one above 1GHz.
(2) Due to the coupling and parasitic effect, it will change the inductances we measured. Therefore, it may lead to slightly frequency response distortion.
(3.) Figure 3.12 shows the power conservation result. The dot-line response represents the filter with cover is better than the solid-line response which has no cover.
Finally, the high-power detection report is expressed in Figure 3.13. For protection reason, it is necessary that the -40dBm attenuator device is added in front of power meter. There is a dBm calculation equation as shown below.
10log ( / 0.00110 )
dBm= P Watt (3.5) To achieve 100Watts power, the detected value from power meter is must equal to 10dBm. From the measured report shown in Figure 3.13, both two models have good high-power-resistant property and their insertion differences are all below 1dBm from 400MHz to 500MHz.
Freq.(MHz)
400 410 420 430 440 450 460 470 480 490 500
L Diff.(dBm) (Without LPF-With LPF)
0.2 0.4 0.6 0.8 1.0
MODEL-001 MODEL-002
Figure 3.13 High-power detection report of Model-001/-002.
3.4 Conclusion and Future Work
High-power studies of a filter are the latest and popular topic. Most of all, looking for proper materials and components is the key point to implement the better designs. The basic specifications and characteristics of specific components can be well defined by using full-wave EM simulation tools and it provides better and faster design sense to network implementation. Besides some high-power effects mentioned before, there are some practical issues such as the multicarrier operation, sharp edge condition, design margin and prevention techniques are also covered. In this paper, low-pass filter is the most interesting design we focus on. The steeper selectivity and better stopband rejection performance will be obtained by introducing the transmission zeros by general Chebyshev polynomial form [33, 34]. In addition, according to the transformation methods in chapter two, some applications such as high-pass filters, bandpass filters, diplexers and multiplexers can be achieved [35, 36].
Chapter 4
Basic Balun Introduction
4.1 Introduction
With the recent growth in the telecommunication area, the demands for smaller, faster and more reliable device design and fabrication of circuits have been increasing.
Passive components play just as an important role as active devices in satisfying these requirements. In addition, great interest has been aroused from both academic and industrial areas toward wide-band technology. As a key component in wide-band wireless communication system, balun with high performance, compact size and low cost are highly demanded [14]-[20].
A balun is a device intended to act as a transformer which is matching an unbalanced circuit to a balanced one, or vice versa. A large number of analog radio frequency (RF) and microwave circuits require balanced inputs and outputs in order to reduce noises and high-order harmonics as well as improve dynamic range of circuits.
It is a key component in many wireless communication systems for realizing critical building blocks such as balanced mixers, balanced doublers, balanced frequency multipliers, balanced modulators, phase shifters, push-pull amplifiers and dipole antenna feeding networks.
Various balun configurations have been reported for many applications in microwave integrated circuits (MICs) and microwave monolithic integrated circuits (MMICs). Several kinds of balun structures have been developed over the pass half-century such as coaxial baluns, unplanar baluns and planar baluns. Among them, the planar version of the Marchand balun is perhaps one of the most attractive baluns
due to its planar structure and wide-band performance [18, 19]. Furthermore, the multi-section half-wave or quarter-wave baluns are frequently used in microwave circuits as they can easily be realized with various design procedures and fabrication processes. They also provide satisfactory performances, namely, small insertion loss, low voltage standing wave ratio (VSWR) and good amplitude/phase imbalance between two balanced ports. In this chapter, because of the practical consideration of implementation, we also discuss some basic concepts and characteristics about slotline transmission-line structure [21]-[26].
4.2 Marchand Balun
Current interest balun design is focused toward making it planar, compact and broadband performance. In an ideal case, it can separate the incident power into equal portions and 180-degree phase difference responses. Due to the broadband performance and simplicity of implementation, planar Marchand balun is the most popular design choice in recent years. In this chapter, the related theories and analyses of Marchand balun are presented [18]-[20].
The first transmission-line balun was described in the literature by Lindenbald in 1939 and variations based on his original scheme soon followed. Among these was Marchand, who introduced a series open-circuited line to compensate for the short-circuited line reactance. In coupled-line balun design, as compared with a short coupled-line one, Marchand structure has less stringent requirements for Zoe , generally 3 ~ 5Zoe ≈ Zoo is sufficient to obtain good performance. There is Marchand compensated balun diagram as shown in Figure 4.1.
Z
L2
Z
S 1Z
Sa b
Z
0Z
1Z
2Z
BHousing
(a)
Z
2S2
Z
Z
B 1Z
SZ
1Z
LZ
0a b
(b)
Figure 4.1 Marchand compensated balun.
(a) Coaxial cross section. (b) Equivalent transmission-line model.
The compensated coaxial balun [17] basically consists of an unbalanced (Z1), an open-circuited (Z2), two short-circuited (ZS1 and ZS2) and balanced (ZB) transmission-line sections. Each section is about a quarter-wavelength long at the center frequency of operation. As shown in Figure 4.1(b), the open-circuited stub and short-circuited stubs are in series and shunt connection to the balanced line, respectively. Most of all, the ratio of the characteristic impedances of short-circuited and open-circuited stubs determines the bandwidth. In general, the higher the ratio is, the wider the bandwidth is. This is the basic but important concept to evaluate bandwidth of Marchand balun. In later design, choosing the proper transmission-line types is also a major issue.
4.2.1 Analysis and Synthesis Method
A Marchand balun can easily be realized by using a pair of coupled lines as shown in Figure 4.2 [8]. We can design a planar balun to meet the desired responses by properly selecting its parameters. First, the coupled-line type of Marchand balun will be presented. According to this equivalent model, we will use Richard’s transformation method to establish the reasonable solutions of second-order and third-order Marchand balun. In this chapter, the Chebyshev polynomial is chosen as the mathematical model for design a definable Marchand balun. The complete design procedures and calculating processes will be introduced in detail in later chapter.
Synthesis techniques of such baluns have been recently described and summarized in this section. A four-port coupled-line and its equivalent circuit are shown in Figure 4.2.
, ( oe, oo)
Figure 4.2 (a) Four-port coupled-line. (b) Its equivalent circuit.
, where Z1 and Z2 are the characteristic impedances of distributed unit elements; N is the transformer turns ratio. The relationship between Zoc and the coupling coefficient k of the coupled-line is given by
oc oe oo The characteristic impedances Z1 and Z2 are defined as follows.
1 1 2
The coupled-line Marchand balun and its whole equivalent circuit are shown in Figure 4.3. The balun has four reactive elements and is known as a fourth-order
Marchand balun. Matching section Z which is a non-redundant element makes it with improved broadband performance. Figure 4.3(b) and 4.3(c) show the equivalent circuit representations and Figure 4.3(d) shows the simplification when Na =Nb =N. In this case, the middle transformers in Figure 4.3(c) cancel each other.
The final equivalent circuit element values can be expressed in terms of coupled-line parameters Z , ac Z and k as follows. bc characteristic impedances of the coupled-line a and b, respectively.
The equations above can be rearranged to solve for the coupled-line parameters in terms of the equivalent element values as follows.
'
, ( a, a)
Figure 4.3 Coupled-line Marchand balun and its equivalent circuits.
Figure 4.3 Coupled-line Marchand balun and its equivalent circuits.