An Implementation of Compact Quarter-Wave-Like-Transformers Using Multi-Section Transmission Lines

This paper proposes a novel miniaturization technique of quarter-wave transformers (QWTs), implemented using multi-section transmission lines (MSTLs), based on the quarter-wave-like transformer (QWLT) theory. Multi-section QWLT characteristics are derived analytically and solved via appropriate optimization algorithms for associated transmission-line parameters. For an illustration purpose, twoand three-section QWLT prototypes with 50% physical size reduction from the corresponding QWT size operating at 2.4 GHz are fabricated using microstrips and tested. It is found that these prototypes yield acceptable return loss at 2.4 GHz without significant bandwidth reduction, comparing to the QWT result.


Introduction
Microwave devices such as amplifiers, directional couplers, power dividers, filters, and antennas are common integrated components of modern electronic circuits and communication systems.The physical dimension of microwave devices strongly depends on the operating frequency.As the world is moving into the digital era, sizes of electronic and communication components are significantly reduced to incorporate more features but yet maintain high performance.The challenges on size reduction, especially at low microwave frequency, thus have attracted many researchers in last few decades.Various techniques have been proposed to obtain high performance compact devices .
One of the miniaturization techniques is to load transmission lines (TLs) with appropriate lumped components or TL stubs to slow the wave propagation, which are known as slow-wave structures [1][2][3].This technique offers a significant size reduction and easy integration to integrated circuits (ICs).However, it still suffers from low quality factor and high power losses.Another approach is the low-temperature co-fired ceramic (LTCC) technology [4], [5].The multilayer sections of ceramic substrates with conductive layers provide a compact size, high conductivity, and lower surface roughness.However, these LTCC structures still suffer from loss dissipated into the composite metal materials.In addition, the method based on metamaterials is another interesting approach, where associated artificial TLs possess tiny unit cells.The electromagnetic properties of metamaterials can be controlled and designed by using left-or composite right/left-handed (CRLH) concept [6][7][8].However, associated structures are rather complex, thus not easy to design and also need to take care of the coupling effect between elements.
Based on slow-wave structures, another approach of miniaturizing microwave components is the technique of electromagnetic bandgap (EBG) structures.These structures can form passband and stopband characteristics to slow the wave propagation as well, where they can be fabricated on an underlying substrate, etched in a TL, or formed as a ground plane [9][10][11].However, their lattice design especially in three-dimensional structures is rather complicated since it requires an integration of many unit cells.Furthermore, slow-wave structures, formed by substrate integrated waveguides (SIWs), have also been proposed [12][13][14][15].The additional metal layer over the ground plane improves the signal carrying efficiency.However, the complexity and the cost of fabrication are still an issue.Recently, another slow-wave structure is based on compact microstrip resonant cells (CMRCs) [16][17][18].Their slowwave characteristic in passband regions is a result of suffi-cient discontinuity in structures.They thus offer more flexibility in geometry design, but it is limited only on symmetric cells.
The technique of multi-section TLs (MSTLs) stands out as another interesting approach for device miniaturization [19][20][21][22][23][24].It provides acceptable performance comparing with other techniques, yet with simple design and fabrication.Moreover, associated unit cells can be designed as symmetric or asymmetric structures, leading to more flexible dimension.One of good examples of MSTLs is in [21], where arbitrary lengths of MSTLs are proposed with good impedance matching and greater section lengths.Another example is the constant voltage-standing wave ratio (VSWR)-type TL impedance transformers (CVTs) and constant conductance-type TL impedance transformers (CCTs) [22][23][24].They are proved to have a wide bandwidth with a great reduction size designing based on the close form equations.Recently, a new type of MSTLs has been proposed [25].This work attempts to miniaturize quarter wave transformers (QWTs) using conjugately characteristic-impedance TLs (CCITLs) [26][27][28], known as quarter-wave-like transformers (QWLTs).The strengths of QWLTs over the standard QWT are their ability to support the slow-wave propagation associated with multi-section asymmetric unit cells and to reduce the total physical length of associated TLs based on the QWLT theory.
The purpose of this paper is to illustrate the implementation of compact multi-section QWLTs and to study their characteristics.Analytical formulas of multi-section QWLTs are first derived based on the ABCD matrix of associated unit cells.Note that this approach is more general, flexible and also includes the design method of the CVT/CCT as a special case.For illustration, two-and three-section QWLTs with 25% and 50% physical size reduction are designed and simulated at 2.4 GHz.In addition, the implementation of 50% physical size reduction of two-and three-section QWLT prototypes is then performed.Their characteristics are compared with those of the standard QWT.It is found that asymmetric multi-sections provide more flexibility in the design.It should be pointed out that the unique advantage of this QWLT approach over other techniques based on slow-wave structures for compact impedance transformers is that the total electrical length of QWLTs can be reduced to /q, with q > 4.However, for other slow-wave structures, their size reduction capability always has the quarter-wave (/4) limitation, where  is the effective wavelength associated with slow-wave structures at the operating frequency of interest.
This paper is organized as follows.Section 2 demonstrates the theoretical formulation based on the theory of QWLTs, including multi-section QWLT characteristics.Examples of the QWLT design and implementation for two-and three-section QWLTs are presented in Sec. 3, followed by results and discussions in Sec. 4. Finally, conclusions are given in Sec. 5.

Theoretical Formulation
In this section, the QWLT theory is briefly explained.Then, multi-section QWLT characteristics are studied analytically.Only reciprocal and lossless QWLTs are considered in the theoretical development.
Recently, it has been first demonstrated in [25] that applying CCITLs to the standard QWT can significantly reduce its physical dimension, where the new impedance matching component is called QWLTs as shown in Fig. 1.In [25], it is found that the magnitude of the characteristic impedances Z c  and the electrical length θ c of QWLTs of physical length l can be expressed respectively as follows: and 0 tan cot when and 0 .and 0 tan cot when and 0 where the argument ϕ of Z -,c is in the range of -90° ϕ  90°, β is the effective propagation constant of QWLTs, Z 0 is the feedline impedance, Z L is the load impedance, and the reflection coefficient  f,L is defined as Note that Z 0 and Z L are real and given for a specific impedance-matching problem.
In Fig. 1, Z in is the input impedance of the terminated QWLT.To match the lossless feedline, Z in must be real and equal to Z 0 , which can be readily obtained using (1) and (2).It should be pointed out that the argument ϕ must be chosen properly to obtain less electrical and physical lengths of QWLTs, compared to those of corresponding QWTs, as discussed in detail in [25].
Consider the ABCD matrix of the reciprocal QWLT in Fig. 1.In [26], it is shown that Z c  and θ c can be determined from its ABCD parameters as follows: Thus, it is required to determine the ABCD parameters to satisfy (1) and ( 2) simultaneously to obtain desired QWLTs, which is quite complicated and usually requires an optimization algorithm.Once the ABCD parameters are found, desired QWLTs can be implemented using MSTLs by determining corresponding TL parameters of MSTLs.
To simplify the above optimization problem, let us reconsider the ABCD matrix of the QWLT in Fig. 1.It can be readily shown that Z in and Z L are related to the ABCD parameters via the following relationship: Using ( 6) with the fact that the QWLT is reciprocal (AD -BC = 1) and Z in is real, one obtains two simple relationships as follows: Note that ( 7) and ( 8) are identical to those derived using the image parameter method [29].It should be pointed out that the ABCD parameters satisfying ( 7) and ( 8) automatically satisfy (1), ( 2), ( 4) and ( 5) as well since the original impedance-matching problem is the same.However, ( 7) and ( 8) look much simpler to solve for associated unknown parameters.
In this study, QWLTs are implemented using MSTLs.For illustration purpose, only two-and three-section TLs of a unit cell of length l are considered to form QWLTs with characteristic impedances Z c  and electrical length θ c as shown in Fig. 2(a) and (b) respectively, where Z n is the characteristic impedance of the nth TL, θ n = k n l n is its electrical length, k n is its propagation constant and l n is its physical length.In addition, it is assumed in this analysis that the propagation constants of each TL are identical (k n = k).
For the two-section QWLT in Fig. 2(a), the considered parameters are Z 1 , Z 2 , l 1 and l 2 where l = l 1 + l 2 and l is given for a specific impedance-matching problem.So, there are only 3 unknown parameters (Z 1 , Z 2 and l 1 ) to be considered for a given length l.For convenience in manipulation, the following parameters are defined: Using ( 9) and ( 10) in ( 7) and ( 8) and the ABCD matrix of two-section TLs [28], the characteristics of the twosection QWLT can thus be expressed in (11) and ( 12) as where 12 Z in ( 11) and ( 12) is defined as Thus, it is required to solve for Z 1 , Z 2 and l 1 simultaneously using ( 11) and ( 12) with an optimization algorithm.Due to the fact that there are more numbers of unknowns than numbers of equations, two-section QWLTs are not unique for a given design problem as illustrated in Sec. 4.
For the three-section QWLT in Fig. 2(b), the considered parameters are Z 1 , Z 2, Z 3 , l 1 , l 2 and l 3 where l = l 1 + l 2 + l 3 Similar to the case of two sections, five unknown parameters are considered only for a given length l (Z 1 , Z 2, Z 3 , l 1 and l 2 ).Similar to (11) and (12), the characteristics of the three-section QWLT can be written as ( 14) and ( 15): where Z 13 and Z 23 are defined as Thus, it is mandatory to solve for Z 1 , Z 2, Z 3 , l 1 and l 2 simultaneously using ( 14) and ( 15) with an optimization algorithm.Similar to the previous case, three-section QWLTs are not unique for a given design problem as well as illustrated in Sec. 4.

Design and Implementation of QWLTs
In this section, examples of QWLT design and implementation are discussed.Two case studies are multisection QWLTs with 25% and 50% physical size reduction compared to the corresponding QWT size.The operating frequency is at 2.4 GHz.
For illustration, QWLTs are designed using microstrips with the low loss substrate, called Arlon DiClad 880 (dielectric constant of ε r = 2.17, loss tangent of tan  = 0.0009 and substrate height of h = 1.524 mm).In this example, it is required to match a 100 Ω load to a 50 Ω input impedance.Using the Agilent Genesys EDA software [30], multi-section QWLTs implemented using microstrips are first simulated to obtain initial design parameters based on the minimum TL width of 0.2 mm on the substrate due to the limitation of the milling machine.Then, mutual coupling effects between TLs of multi-section QWLTs are taken into account using the finite integration technique (FIT) via the CST Microwave Studio (MWS) software [31].After optimizing design parameters, optimized multi-section QWLTs are fabricated and then tested using a network analyzer.
The characteristic impedance Z n and associated length l n in Fig. 2(a) and (b) are determined via the Agilent Genesys EDA software using (11) and ( 12) or ( 14) and (15) for two or three sections, respectively.Since QWLTs are modified from the standard QWT, their length l is then set to be equal to a quarter of wavelength multiplied by the reduction ratio p, that is l = p/4 where 0 < p  1.The ratio p is equal to 1 for the standard QWT and equal to 0.75 and 0.5 for 25% and 50% physical size reduction, respectively.To observe the slow-wave characteristic of multi-section QWLTs, the slow-wave factor (SWF) is considered.In [11], the SWF is defined as the ratio of free space wavelength to guided wavelength or the ratio of the effective propagation constant β of TLs of interest to the free space propagation constant k where β can be determined from (5) using θ c = βl for QWLTs.Typically, higher SWF yields shorter associated TLs.In addition to the SWF, the modified SWF is also applied in this paper for fair comparisons, which is defined as the ratio of the effective propagation constant of QWLTs to the propagation constant of QWTs.Furthermore, the group delay is also considered as an important factor for evaluating TL quality [11].Note that the voltage and current distributions along associated TLs are also determined to understand associated signal propagation characteristics better.These results will be provided in Sec. 4.

QWLT-parameter Range Evaluation
The Agilent Genesys EDA software is employed to determine the design parameters in Fig. 2(a) and (b), where the iteration numbers in the optimization algorithms, the gradient search and the pattern search, were set to 300 and 700 for two-and three-section QWLTs, respectively.As pointed out in Sec. 2, multi-section QWLTs are not unique for a given design problem.Tables 1 and 2 present the range of parameters of interest for ideal multi-section QWLTs with 25% and 50% physical size reduction, respectively.
From Tab. 1 and 2, it can be seen that all QWLT parameters vary with the required physical length and the number of TL sections of ideal QWLTs.Note that θ c for a given ϕ agrees well with the QWLT theory in Sec. 2. It should be pointed out that θ c is always less than 90° (for the QWT) as expected.When the QWLT physical length decreases, θ c tends to decrease and ϕ tends to increase.In addition, these QWLT matching networks provide a larger group delay caused by their dispersion [11], where the group delay of the corresponding ideal QWT is equal to 110.5 ps.Furthermore, the miniaturization capability can be further realized due to the SWF of greater than unity, where the SWF tends to increase when the QWLT physical length decreases.It can be observed that the three-section QWLT parameters tend to cover wider range than those of two-section QWLTs due to more degrees of freedom in the QWLT design.

Simulation and Measurement Results
From the above simulation results, multi-section QWLTs with 50% physical size reduction are chosen as an example in the simulation of signal propagation characteristics.Both two-and three-section QWLTs are selected at the same SWF of 1.19 for a fair comparison as shown in Fig. 3.The ideal QWLTs in Fig. 3(a) and (b) are terminated in a 100 Ω load and analyzed to determine their voltage and current distributions for a given input voltage and input impedance of 50 Ω.The corresponding ideal QWT is also analyzed for comparison, where its characteristic impedance is 70.7 Ω and electrical length is 90°.The voltage and current distributions are determined using terminated lossless TL equations [28], where the input voltage V in is set at 1 V. Numerical results of the magnitudes of voltage and current distributions shown in Fig. 4(a) and (b), respectively.Note that V in is located at the left end of twoand three-section QWLT as shown in Fig. 4(a).It is found that the distributions of both QWT and QWLTs exhibit similar trends, but different support, with equal voltage and current magnitudes at the input and load terminals.However, the QWLTs are significantly shorter and their voltage and current distributions are not smooth, with the slope discontinuity at associated TL junctions.
In addition, both SWF and modified SWF of ideal two-and three-section QWLTs in Fig. 3(a) and (b) are also considered as illustrated in Fig. 5.It can be seen that the SWF of ideal QWT is constant for all frequencies while those of ideal QWLTs slightly increase with frequency.Note that the SWFs of both ideal two-and three-section QWLTs are almost identical and less than that of ideal QWT.Furthermore, the graph of modified SWF is plotted as shown in Fig. 5(b).It is found that the modified SWFs of ideal QWLTs slightly increase with frequency and are approximately equal to 0.8.Although the modified SWFs are less than one, QWLTs with 50% physical size reduction can still be achieved in this case.
Next, the two-and three-section QWLTs with characteristic impedances and electrical lengths in Fig. 3 are implemented using microstrips.The microstrip dimensions are further optimized by taking into account of mutual coupling effects using the CST MWS software.The final QWLT designs for prototype fabrication are illustrated in Fig. 6.Note that the total electrical lengths of both twoand three-section QWLTs are identical and reduced to λ QWLT /6.04, which are less than λ QWT /4 of the conventional QWT.λ QWLT and λ QWT are effective wavelengths associated with QWLTs and QWT respectively, and they are assumed to be the same for a fair comparison.For size comparison, the QWT is also implemented using the same microstrip with the width of 2.73 mm and length of 24.8 mm.well.Note that all prototypes are fabricated on the finite substrate of 40 mm width.It is clearly seen that both QWLT prototypes are twice smaller in physical length than that of the QWT prototype (excluding the 50 Ω input port).The two-section QWLT characteristics are compared with other methods as shown in Tab. 3.All these methods are compared using the same condition.Note that for QWLTs, (11) and ( 12) are used for calculation.It is clearly seen that all methods provide similar solutions, indicating the validity of the proposed method.Note that the QWLT approaches also offer general design equations of asymmetric MSTLs leading to more flexibility in design and implementation.The issue of high impedance line can be solved by adding more TL sections with proper impedance and electrical length.Moreover, this paper also presents additional details on multi-sections; i.e., voltage and current distribution and SWF.It is found that, for two-section QWLTs, if θ c values are equal, then the bandwidths will be approximately equal.However, some differences are observed for higher number of TL sections.
It should be pointed out that QWLTs can also be designed using the characteristic impedance profiles.For example, Figure 10 shows the variation of Z 1 and Z 2 with θ 1 for two-section QWLTs with 50% size reduction, where r 1 is fixed at 0.5 and θ 1 + θ 2 is approximately equal to 45°.

Conclusions
In this paper, compact QWLTs are implemented using the MSTL microstrip technology.For illustration, two-and three-section QWLT characteristics are derived analytically based on the ABCD parameters of QWLTs.In addition, associated voltage and current distributions, SWF, modified SWF and group delay of QWLTs are investigated as well.Simulation results show that an increase in the number of TL sections of QWLT unit cells clearly yields more flexible design QWLT parameters.The prototypes of two-and three-section QWLTs with 50% physical size reduction are fabricated and tested.It is found that simulated and measured results of |S 11 | for the QWLT prototypes agree reasonably well, without significant bandwidth reduction compared to that of the QWT.In the future, other potential TL structures for the QWLT implementation will be investigated as well for the performance improvement.In addition, QWLTs will be applied to miniaturize other useful microwave devices such as broadband multi-section transformers, power dividers, filters and antennas.About the Authors ...

Figure 8 (
Figure 8(a) shows the simulated and measured frequency responses of the magnitude of S 11 of the QWT microstrip prototype.In addition, similar results of twoand three-section QWLT microstrip prototypes are also illustrated in Fig. 8(b) and (c), respectively.It can be seen that measured results of |S 11 | of the QWLT prototypes agree reasonably well with simulated results.
Nonchanutt CHUDPOOTI received the first-class honors B.Sc. degree from the King Mongkut's University of Technology North Bangkok in 2012, where he is currently pursuing the Ph.D. degree.His main research focuses on the application in microwave microfluidic sensors, millimeter-wave substrate integrated circuit applications and Substrate Integrated Waveguide (SIW) applications.Prayoot AKKARAEKTHALINreceived the B.Eng. and M.Eng.degrees in Electrical Engineering from King Mongkut's University of Technology North Bangkok, Thailand, in 1986 and 1990, respectively, and the Ph.D. degree from the University of Delaware, Newark, USA, in 1998.From 1986 to 1988, he worked in the Microtek Laboratory, Thailand.In 1988, he joined the Department of Electrical Engineering, KMUTNB.His current research interests include passive and active microwave circuits, wideband and multiband antennas, and telecommunication systems.He was the Chairman for the IEEE MTT/AP/ED Thailand Joint Chapter during 2007 and 2008, and the president for ECTI Association, Thailand from 2014 to 2015.He is now working on the Senior Research Scholar Project for Thailand Research Fund (2015-2017).SILAPUNT (corresponding author) received the B.E. degree in Electrical Engineering from Chulalongkorn University, Thailand, in 1996, and the M.S. and Ph.D. degrees in Electrical and Computer Engineering from the University of Wisconsin-Madison, USA, in 1998 and 2004, respectively.In 2004, she joined the Department of Electronic and Telecommunication Engineering, King Mongkut's University of Technology Thonburi, Thailand, as a lecturer.Her current research interests include transmission lines and applications, microwave devices, photonics, and plasmonics.Since the beginning of her career, she has received several research grants from government funding agencies and private sectors.Danai TORRUNGRUENG received his B.Eng. degree in Electrical Engineering from Chulalongkorn University, Bangkok, Thailand, in 1993.He obtained his M.S. and Ph.D. degrees in Electrical Engineering from The Ohio State University in 1996 and 2000, respectively.Prior to joining Asian University, he worked as a senior engineer in the USA, involved in research and development of the urban propagation modeling project.At present, he is a full professor in the Electrical and Electronic Engineering Department, Faculty of Engineering and Technology, Asian University, Thailand.His research interests are in the areas of fast computational electromagnetics, rough surface scattering, propagation modeling, electromagnetic wave theory, microwave theory and techniques, antennas and sensors.