Conclusion and Future Work

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 Z_oe , generally 3 ~ 5Z_oe ≈ Z_oo is sufficient to obtain good performance. There is Marchand compensated balun diagram as shown in Figure 4.1.

Z

L

2

Z

S 1

Z

S

a b

Z

0

Z

1

Z

2

Z

B

Housing

(a)

Z

2

S2

Z

Z

B 1

Z

S

Z

1

Z

_L

Z

0

a 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 Z_oc 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 N_a =N_b =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.

The most important issue for establishing the final model of balun is choosing the suitable element values which conform to the specifications in terms of bandwidth, load impedance and return loss. The coupled-line parameters can be calculated from these equations. In addition, it can be derived to the other practical transmission-line configurations from the proposed Marchand prototype. Based on the Marchand balun theory, we will use microstrip line (MS) combined with slotline (SL) and coplanar stripline (CPS) structures to implement the presented fifth-order balun network in next chapter. As the fourth-order Chebyshev filters having 5:1 bandwidth shown in the Figure 4.4, the corresponding diagrams of the presented fifth-order balun between the design specifications and element values can be also established. The complete design procedures and analyses will be discussed in chapter five.

RETURN LOSS(dB)

(a) Equivalent circuit parameters. (b) Coupled-line parameters.

The balun element values are all determined rapidly by checking the Figure 4.4.

These two diagrams are established on 50Ω-system and the impedance transformation ratio is equal to two. The impedance transformation is one of the important properties of balun. In next chapter, the presented balun is also mentioned that the design procedures for different values of impedance transformation ratio.

4.2.2 Chebyshev Model Establishment

The mathematical model should be established before the network synthesis. In general, Butterworth and Chebyshev polynomials are the common choices when we define the mathematical model for filter or balun design. In this section, Chebyshev approximation is used to synthesize the following second-order and third-order baluns.

The exact broadband synthesis theory [29, 30] is a well design method to analyze the network responses using in the network consisting entirely of equal wavelength lines, stubs and coupled lines. These are called commensurate lines. In next chapter, the complete design procedures of the novel fifth-order balun are presented.

Now, we have a short discussion about the Chebyshev model establishment of the second-order and third-order baluns by these following steps:

(1) An ideal balun is a three-port network which can separate the incident power into equal portions and 180-degree phase difference responses. For the two-port condition of exact synthesis method, balun must be reduced to the equivalent two-port network. For example, the two 50Ω output impedances of balun can be viewed as a series connection as a 100Ω load system.

(2) The Z-, Y-, and S-parameter representations are often used to characterize a microwave network with arbitrary number of ports. However, in many practical microwave networks consist of several cascade connection forms. In this case, by

multiplying ABCD-matrix of the individual two-port sub-networks is a convenient way to easily derive the whole network representation. The basic definition for a two-port network in terms of the total voltages and currents is given by

1 2 2 There are some useful transmission (ABCD-) parameters as shown in Table 4.1.

Z

Table 4.1 The ABCD parameters of some useful two-port circuits.

(3) According to Richard’s transformation definition, the distributed network composed of commensurate length of transmission-lines and load resistance can be treated as a lumped R-L-C network by using the complex frequency variable S for network analysis and synthesis. The complete exact synthesis method and Richard’s

transformation will be presented in chapter five.

The second-order model which is the minimum order of Marchand prototype balun consists of only a series open-circuited and a shunt short-circuited stub. A series open-circuited stub can be regarded as a series capacitor in Richard’s domain and a shunt short-circuited one is known as a shunt inductor. In the third-order model, a cascade transmission-line section is added behind the shunt short-circuited one.

From the Richard’s transformation theory, the corresponding transmission-line impedance transformations are introduced as follows [27].

0 , where Z denotes the distributed transmission-line impedance in real frequency ₀ domain; L and C are the corresponding lumped elements in complex plane (Richard’s) domain; Z is the corresponding impedance of unit element. The equivalent _UE circuits of second-order and third-order balun models are shown in Figure 4.5.

2

(a) Second-order prototype. (b) Third-order prototype.

(4) Due to the unmatched impedance property of the equivalent two-port network, the conversions between S-parameter and ABCD-parameter should be revised to the following equations.

02 01 02 01 To simplify the analysis, the impedance transformation ratio ⁰²

01

Z

Z is supposed to be 2 in this procedure. Finally, the complete mathematical representation of squared magnitude of transmission coefficient parameter denoting as S₂₁² can be derived by several minute and complicated calculations in Richard’s domain. It is a polynomial of the complex frequency variable S. Table 4.2 shows some useful ABCD-parameters in Richard’s domain.

Distubuted Circuit Lumped Circuit

1 1

Table 4.2 The ABCD parameters in Richard’s domain.

The squared magnitude of transmission coefficient of the second-order balun network can be expressed as

2

, where L and C are lumped element values in Richard’s domain. They represent a quarter-wavelength shunt short-circuited and series open-circuited stub in real frequency domain, respectively.

The third-order balun network can be defined as

2

, where Z is the characteristic impedance of unit element in Richard’s domain. It represents a cascade quarter-wavelength transmission-line in real frequency domain.

∆ is the fractional bandwidth (FBW) of the balun; F , ₆^' F , ₄^' F and ₂^' F are the ₀^' coefficients of S to the power of six, four, square and zero, respectively. They are functions of Z, L and C.

(5) For the reason that the equivalent two-port network of balun can be viewed as a bandpass filter response, so we choose a high-pass Chebyshev prototype in complex domain to model the desired design. The following two Chebyshev prototypes presented for second-order and third-order Marchand balun are shown in Figure 4.6

and 4.7. Specially, due to the special denominator form of ABCD-matrix of unit element, the Chebyshev polynomial is in need of some reasonable revisions. The method of mathematical function establishment is dilated in chapter five.

This is a second-order high-pass Chebyshev polynomial form shown below.

2

21 4 2 2 4

2 2 2

4

1 1

4 4

1 N( ^C) 1 ( ^{C C})

S S S S S S

T S S

ε ε

= =

+ +

+ − +

(4.19)

, where ε is expressed as ripple-level of Chebyshev response and its related response diagram is shown in Figure 4.6.

(a)

(b)

Figure 4.6 Normalized Chebyshev second-order prototype.

(a) High-pass response. (b) Bandpass response.

In thick-line model, the ripple-level is ε =0.1922 and the normalized cutoff frequency in Richard’s domain is S_C = j0.5 . Another thin-line model has

0.0934

ε = and S_C = j1.

This is a third-order high-pass Chebyshev polynomial form [30]

2

21 2

1 1 _N( )

S = F S

+ (4.20)

2 2 2

2 2

2 2

1 1

( ) 1 1

C C

C C

N m n m n

S S

S S

F S T T U U

S S S S

ε ^{    −     − }

=     − −    − 

(4.21)

, where F_N²( )S is revision type of original Chebyshev polynomial. The complete theory will be introduced later and its related response diagram is shown below.

(a)

(b)

Figure 4.7 Normalized Chebyshev third-order prototype.

(a) High-pass response. (b) Bandpass response.

Similar to the values of Second-order prototype, the ripple-level is ε =0.1922 and the normalized cutoff frequency in Richard’s domain is S_C = j0.5 in thick-line model. Another thin-line model has ε =0.0934 and S_C = j1. In general, the wider the bandwidth is, the worse the return loss is. Based on this simple trade-off concept, it gives a good sense for specific balun design.

The parameters of baluns such as impedance transformation ratio, bandwidth and return loss can be arbitrarily determined. The one or more reasonable numerical solutions will be obtained by solving these mathematical equations and discriminants.

They are the solutions which can exactly describe the presented network responses at all frequencies. Finally, the theoretical values of each quarter-wavelength section are well-defined in real frequency domain by converting the lumped element values.

The basic balun design procedures have been summarized. We will establish the new novel fifth-order Marchand balun prototype as same as the methods presented in next chapter.

4.3 Slotline

To realize the presented fifth-order planar balun on two-layer PCB process, the slotline structure is used to implement the several elements of network. A slotline is a planar transmission structure proposed for use in MICs by Cohn in 1968 [21]. The basic slotline configuration is shown in Figure 4.8.

It consists of a dielectric substrate with a narrow slot etched in the metallization on one side of the substrate. The other side of the substrate is without any metallization. Slotline can be included in many microstrip circuits by etching the slotline circuit in the ground plane of the microstrip circuit. This type of hybrid combination allows great flexibility in the design of microwave circuits. Some of the

circuit elements which cannot easily be achieved in microstrip configuration can be incorporated with the slotline part of the circuit. For example, slotline could be employed frequently in short circuit, high impedance line, series stub and balun. In this section, the basic characteristics and concepts of slotline are briefly introduced [21]-[26].

W

ε

r

h

Figure 4.8 Slotline configuration

4.3.1 Slotline Analysis and Design Consideration

In slotline, the wave propagates along the slot with the major electric field component oriented across the slot in the plane of metallization on the dielectric substrate. The mode of propagation is non-TEM and almost transverse electric in nature. However, there is no certain rule existing about cutoff frequency because the slotline is a two-conductor structure unlike conventional waveguides.

There are numbers of techniques about the slotline analyses such as approximate analysis, the transverse response approach, Galerkin’s method in Fourier transform domain and finite-difference time domain technique. These methods give some basic concepts about field distribution, polarization, impedance and wavelength calculation about slotline configuration.

However, the various analyses listed above do not lead to any closed-form expressions for slotline wavelength and impedance. It becomes a serious handicap for

circuit analysis and design. Some attempts to overcome this difficulty have been reported and we can draw support from these related computed results or curves to obtain the correct characteristic impedance and wavelength of slotline. Moreover, the computer-aided tools can be applied to estimate these parameters of slotline. In later design, we use the Sonnet and ADS simulation tools to solve this problem.

In general, we assume that the metal conductor constituting the slot has zero thickness; however, it does not reflect the situation of reality. Some researches are shown that slotline wavelength (λ_S) increases and characteristic impedance (Z_OS) decreases when metal thickness (t) increases. Similar as microstrip line, the losses in slotline are the results of the lossy dielectric and the conductors with finite conductivity σ . Moreover, slotline is a dispersive transmission-line structure.

Because of the non-TEM nature of the slotline mode, the associated parameters such as characteristic impedance and phase velocity are not constants but vary with frequency.

In some special conditions, slotline could act as a resonant antenna and it lead to a considerable amount of energy dissipated in radiation. This is the most critical problem we care about in balun design which includes slotline structure. To achieve better slotline characteristic, radiation must be minimized. This is accomplished through the use of dielectric substrate of sufficiently high permittivity; consequently, the fields are closely confined to the slot region and the slot-mode wavelength will substantially less than the free space wavelength with negligible radiation loss.

4.3.2 Slotline Discontinuity and Transition

As in the case of microstrip circuits, the characterization of slotline discontinuities is necessary for slotline circuit designs. There are many types of short-end and open-end discontinuities as shown in Figure 4.9.

W εr

h

W εr

h h ^W ε_r h ^W ε_r

( )a

( ) b

Figure 4.9 Slotline discontinuities. (a) Short-end. (b) Open-end.

A short-end shown in Figure 4.9(a) is created by filling the slot with a conducting surface lying in the plane of the slot. The current flows in the metal surface around the end of the slot and there is appreciable energy stored beyond the termination. There exists some stored magnetic energy which gives rise to an inductive reactance in the end of the slot.

In addition, like an open-end in a microstrip line, the short-end in a slotline is not purely reactive. There are losses associated with the short-end due to the propagation of power in surface waves and radiation in the form of space waves. So, the equivalent circuit of a short-end is a series combination of an inductor and a resistor.

Unlike the microstrip short-end, the slotline short-end which omits the via-hole process is frequently used due to ease of fabrication.

Various types of open-end discontinuity are shown in Figure 4.9(b) such as flared, circular disc and combination of flared and half-disc open-end. The flared type is impractical because it requires a lot of substrate area and produces a lot of radiation loss. In practice, an open circuit is frequently used by means of a circular disc at the end of the slotline. The larger the radius of the disc is, the better the open-circuited

behavior becomes. However, it has a disadvantage that the circular disc will behave like a resonator especially if the width of the connecting slotline is narrow compared

behavior becomes. However, it has a disadvantage that the circular disc will behave like a resonator especially if the width of the connecting slotline is narrow compared

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