Chapter 1 Introduction
1.2 The Novel Marchand Balun
The word balun is an acronym for balanced-to-unbalanced transformer, used to connect balanced transmission-line circuits to unbalanced transmission-line circuits.
Balun is a key component in numerous applications such as balanced mixers, balanced doublers, balanced frequency multipliers, balanced modulators, phase shifters, dipole feed networks and push-pull amplifiers, etc.
Balun is a three-port network, which includes an unbalanced input port and two balanced output ports. In an ideal case, it can separate the incident power into equal portions and 180-degree phase difference. Sometimes, balun can be taken as the differential (E) port of magic-T, which is a basic but important component in many applications [10]-[13]. Magic-T is also widely used as an element in correlation receivers, frequency discriminators, balanced mixers, four-port circulators and reflectometers. A technique for converting balun into 180-degree hybrid by adding an in-phase power splitter is a common method [14].
Magic-T is a four-port network, which includes a sum-port (H-port), a difference-port (E-port) and two output ports. In addition, it is a bi-directional network that allows two usage modes for different purposes. One is that when signal transmitting from sum-port or difference-port, it can split the incident power into equal portions and in-phase or out-of-phase responses, respectively. The other mode is that it allows signals from two output ports to be summed or subtracted with relative power and phase, then export from H-port or E-port, respectively.
Over the past half-century, several different kinds of balun structures have been developed [15]-[17]. Early coaxial baluns were used exclusively for feeding dipole antennas. Later, planar baluns using stripline were developed for balanced mixers.
Current interest in transmission-line-type structure is focused on making it planar, compact, and more suitable for mixers and push-pull power amplifiers. In this paper, for the purposes of flexibility, wide-band response, better performance and easy to implement, we focus on planar Marchand balun design with PCB fabrication process [18]-[20].
In chapter four, several kinds of baluns will be mentioned briefly. Such as coaxial baluns, planar baluns, unplanar baluns, coupled-line baluns and transmission-line baluns are all included. Most of all, Marchand balun structure
analysis is the most important and interesting topic. The basic synthesis method and mathematical model of Marchand balun will be discussed clearly in this chapter.
Besides, we also give some basic concepts and characteristics about slotline structure [21]-[26]. It is a common transmission-line structure for the hybrid type with microstrip line to implement some practical networks.
In chapter five, based on Marchand balun theory, we present a novel broadband balun structure, which can extend the conventional fourth-order Marchand balun to the fifth or higher order. For practical fabrication, some types of transmission-line structures can be used to implement the fifth-order Marchand balun such as microstrip line (MS), slotline (SL) and coplanar stripline (CPS). Due to the entire network is composed of transmission-lines and stubs with identical electrical length, the exact synthesis method by Richard’s transformation can be used to derive the complete mathematical model of the presented novel balun [27]-[32]. We can arbitrarily design the parameters of balun such as impedance transformation ratio, bandwidth and return loss by solving these mathematical equations.
Both of the high-power low-pass filter and novel Marchand balun network, the complete circuit analyses, design procedures, simulations and measured results will be discussed in this paper.
Chapter 2
Basic Microwave Filter Design Theory
2.1 Introduction
A microwave filter is a two-port network used to control the frequency response for many different designers. It provides transmission at frequencies within the passband and attenuation in the stopband of the filter. Typical frequency responses include low-pass, high-pass, basspass and bandstop characteristics. Nowadays, there are many applications of microwave communication, radar or other test and measurement system.
In this chapter, we will introduce the modern procedure called the insertion loss method [8], [9]. It uses network synthesis techniques to design filters with a completely specified frequency responses. In general, the design begins with low-pass filter prototype. The common mathematical approximations such as maximally flat (Butterworth) and equal-ripple (Chebyshev) polynomials are often used to describe the filter responses. In addition, we can convert the basic low-pass prototype to other particular frequency range and impedance level by using some transformation rules.
2.2 Filter Design Procedure
The perfect filter responses would have zero insertion loss in the passband, infinite attenuation in the stopband and a linear phase response in the passband. But such filter does not exist in practical, so we should make some compromises in our design. Both of lumped or distributed realization can be used to implement the specific filters.
The insertion loss method is a systematic way to synthesis desired responses of filters. It allows a high degree of control over the passband and stopband amplitude and phase characteristics. According to the design procedure, we can easily evaluate the design trade-offs to best meet the application requirements. For non-periodic implementation purpose, we will focus on the establishment of lumped element prototype.
2.2.1 Power Loss Ratio Method
First, we define the analysis method based on reciprocal and lossless network conditions. For two-port networks, it is known that impedance and admittance matrices are symmetric for reciprocal networks and purely imaginary for lossless networks. Similarly, the scattering matrix which is the common parameter we use to describe network characteristics has its special properties. It is symmetric for reciprocal networks and unitary for lossless networks. A matrix that satisfies the conditions of Equation (2.1) and (2.2) is called symmetric matrix or unitary matrix, respectively.
[ ] [ ]
S = S t (2.1)[ ] [ ]
S * ={ }
S t −1 (2.2) Assume that the low-pass prototype is a passive, reciprocal and lossless two-port network. Based on these assumptions in power loss ratio method, a filter can be well-defined by its insertion loss or power loss ratio, denotes PLR:2
1
1 ( )
inc LR
load
P Power available from source
P = Power delivered to load = P = − Γ ω (2.3)
, where ( )Γ ω represents the reflection coefficient of the network. In symmetric two-port network, we can use S11 or S22 denotation to describe it. The insertion loss
(IL) function in dB is defined as , where M and N are all real polynomials in ω2. Substituting this form in equation (2.3) gives the following form.
2 loss ratio formula. Next, some common mathematical polynomials combined with the power loss ratio formula can be used to describe the practical low-pass filter responses. Moreover, it is easy to convert the typical low-pass prototype into a bandpass or other filter types by using various transformation rules.
2.2.2 The Maximally Flat Prototype
The maximally flat approximation, which is also called the Butterworth or binomial response, is the simplest meaningful approximation to describe a practical low-pass filter. It provides the flattest possible passband response, better group delay performance but worse selectivity. For low-pass filter prototype, it is specified as follow.
2 2 , where N represents the order and ωC is the cut-off frequency of the filter. In general, ωC is equal to unity in an original normalized prototype. The range of filter passband is defined from ω=0 to ω ω= C and the power loss ratio is equal to 1+ε2 at the band edge. In addation, we often choose the proper ripple-level value to make PLR in -3dB point. It is means that half of power reflects back to the input port when ε =1. Like the binomial response for multi-section quarter-wave matching transformer, the first (2N-1) derivatives of power loss ratio are zero at ω =0. The mathematical approximation implies a very flat response across the passband. The squared S21-magnitude is also defined in Equation (2.10).
2 specifications. First, we introduce two parameters for calculating the required return loss and insertion loss which denote LR and LA, respectively.
2
, then we check the following two inequalities.
R p
return loss≥L for ω ω≤ (2.13)
A s
insertion loss≥L for ω ω≥ (2.14) , where ωp and ωs represent the passband and stopband frequencies, respectively.
The ratio of stopband to passband denotes S, which is defined in Equation (2.15) as follow.
1
s
p
S ω
=ω ≥ (2.15) Finally, an inequality to decide the necessary degree N of the Butterworth prototype filter can be derived as
20log ( )10
A R
L L
N S
≥ + (2.16)
2.2.3 The Equal-Ripple Prototype
The equal-ripple approximation, which is also called Chebyshev response, is another common approximation to filter design. Although the flatness of in-band response and group delay performance are worse than the Butterworth response, it has a sharper out-band selectivity and provides fewer elements for necessary rejection response of filter. For low-pass filter prototype, it can be specified as following function. , where N represents the order and ωC is the cut-off frequency of the filter. Different from Butterworth approximation, it introduces a parameter ε2 which determines the passband ripple-level. There are Chebyshev polynomials denote TN(x) defined as follows.
Equation (2.18) gives the first four Chebyshev polynomials and the higher-order polynomials can be found by using the following recurrence formula.
( )
2 1( )
2( )
N N N
T x = xT − x −T − x (2.19) The first four Chebyshev polynomials are plotted in Figure 2.1 and there are
several significant properties of Chebyshev polynomial expressed.
(1.) For − ≤ ≤1 x 1, TN
( )
x ≤ . Chebyshev polynomials oscillate between ±1 in 1 this range. This is the most important property of the equal-ripple characteristic and the region will be mapped to the passband of the low-pass filter.(2.) For x >1, TN
( )
x > . This region will be mapped to the range outside the 1 passband of the low-pass filter.(3.) For x >1, TN
( )
x increases faster with x as N increases.Figure 2.1 The first four Chebyshev polynomials,TN
( )
x .Similar as the Butterworth approximation analysis procedures, the squared S21-magnitude of Chebyshev approximation is defined as
2
21 2 2
( ) 1
1 N( )
C
S
T
ω ε ω
ω
=
+ (2.20) The proper degree resolution of Chebyshev approximation is expressed as follow.
2 1 2
10
6 20log [ ( 1) ]
A R
L L
N S S
+ +
≥ + − (2.21)
In this chapter, ladder circuit model is used to construct low-pass filter prototype for both Butterworth and Chebyshev approximations. In a normalized low-pass prototype where the source impedance and the cutoff frequency are normalized to unity, we can arbitrarily determine the filter degree and ripple-level to achieve the complete normalized element values denote g . Figure 2.2 shows different kinds of S N-degree ladder networks and element values are numbered from g to 0 gN+1.
g0
g3
g2
g1
1
gN+
gN or gN+1
gN
( )n even ( )n odd
(a)
g0 g1 g3
g2
N 1
g + gN
or gN gN+1
( )n even ( )n odd
(b)
Figure 2.2 Ladder circuits for low-pass filter prototypes. (a) Beginning with a series element. (b) Beginning with a shunt element.
2.3 Filter Transformation
Low-pass filter is the most important component we are concerned about in filter design. Based on the basic low-pass prototype, the other types of filter design procedures are also summarized in this section. The high-pass, bandpass and bandstop characteristics can be achieved easily by scaling in terms of impedance and frequency.
2.3.1 Impedance and Frequency Scaling
Impedance scaling: In the previously low-pass prototype design, the source and load resistances are set to unity except the Chebyshev prototype network with even degree. Supposed that source resistance scale to R and the new filter component 0 values are given by
' , where L, C, and RL are the normalized element values for the original prototype.
Frequency scaling: The cutoff frequency of the normalized low-pass prototype is equal to unity. The new model is accomplished by replacing ω by ω ωC. When combining impedance scaling factor, the new L, C values can be rewrite as follows.
' 0 k The related diagrams are shown in Figure 2.3.
PLR
Figure 2.3 Frequency scaling transformation. (a) Low-pass filter prototypeωC = . (b) 1 Frequency scaling. (c) Transformation to high-pass response.
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
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