Index of /rozprawy2/11308

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List of papers

This Thesis is based on and incorporates the following publications:

Chapter 3:

– I. Piekarz, J. Sorocki, S. Gruszczynski, K. Wincza, "Input match and output balance improvement of Marchand balun with connecting line," IEEE Microwave and Wireless Components Letters, vol. 24, no. 10, pp. 683–685, Oct. 2014.

– I. Piekarz, J. Sorocki, I. Slomian, K. Staszek, K. Wincza and S. Gruszczynski, "Marchand balun with connecting segment designed with the use of multi-technique compensation," in Proc. Mediterranean Microwave Symposium (MMS2014), Marrakech, Marocco, 2014, pp. 1–4.

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Compact single-layer microstrip Marchand type balun," in Proc. 16th Annual Wireless and Microwave Technology Conference (WAMICON) 2015, Cocoa Beach, FL, 2015, pp. 1–4.

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Miniaturized compensated quasi-lumped wideband marchand balun," in Proc. 21st International Conference on Microwave, Radar and Wireless Communications (MIKON) 2016, Krakow, Poland, 2016, pp. 1–3.

Chapter 4:

– I. Piekarz, J. Sorocki, K. Wincza and S. Gruszczynski, "Rat-race directional couplers operating in differential mode," in Proc. Radio and Wireless Symposium (RWS) 2017, Phoenix, AZ, 2017, pp. 184–18.

– J. Sorocki, I. Piekarz, K. Staszek, P. Kaminski, K. Wincza, and S. Gruszczynski, "Differential-mode branch-line directional couplers," International Journal of Information and Electronics Engineering, vol. 6, no. 4, pp. 226–229, Jul.2016.

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– I. Piekarz, J. Sorocki, K. Janisz, K. Wincza, S. Gruszczynski, "Wideband three-section symmetrical coupled-line directional coupler operating in differential mode," accepted for publication in IEEE Microwave and Wireless Components Letters.

– K. Janisz, I. Piekarz, K. Staszek, K. Wincza, S. Gruszczynski, "Differentially-fed coupled-line directional couplers with coupled-conductors of unequal width," in preparation.

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, J. Papapolymerou, "High-performance differentially-fed coupled-line directional couplers realized in inhomogeneous medium," in preparation.

Chapter 5:

– I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, "Coupled-line sensor with Marchand balun as RF system for dielectric sample detection," IEEE Sensors Journal, vol. 16, no. 1, pp. 88–96, Jan. 2016.

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Effective permittivity measurement with the use of coupled-line section sensor," in Proc. Radio and Wireless Symposium (RWS) 2016, Austin, TX, 2016, pp. 165–168.

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Coupled-line sensor setup for liquids and solids permittivity measurement developed with the use of 3D printing technology," accepted for 22st International Conference on Microwave, Radar and Wireless Communications (MIKON) 2018, Poznan, Poland.

– I. Piekarz, K. Janisz, J. Sorocki, K. Wincza, S. Gruszczynski, "Low-cost planar coupled-line sensor for permittivity measurement of low-loss dielectric materials in a wide frequency range," in Proc. 18th Conference on Microwave Techniques (COMITE) 2017, Brno, Czech Republic, 2017, pp. 1–5.

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Simplified three-strip coupled-line section as microwave sensor for dielectric materials measurement," in Proc. 9th International Kharkiv Symposium on Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves (MSMW) 2016, Kharkiv, Ukraine, 2016, pp. 1–3.

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List of Papers 15

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Multi-coupled-line microwave sensors for dielectric sample permittivity change detection," in Proc. International Conference on Electrical, Electronics and System Engineering (ICEESE) 2017, Kanazawa, Japan, 2017, pp. 1–5.

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Two-port measurement system with coupled-line sensor for detection of material permittivity change," in Proc. 18th Conference on Microwave Techniques (COMITE) 2017, Brno, Czech Republic, 2017, pp. 1–4.

– J. Sorocki, I. Piekarz, K. Wincza, S. Gruszczynski, "Differentialy excited coupled-line sensor for small dielectric samples detection," in Proc. 9th International Kharkiv Symposium on Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves MSMW’2016, Kharkiv, Ukraine, 2016, pp. 1–3.

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Calibration method of microwave measurement system for dielectric samples detection," in Proc. 9th International Kharkiv Symposium on Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves MSMW’2016, Kharkiv, Ukraine, 2016, pp. 1–3.

– K. Staszek, I. Piekarz, J. Sorocki, S. Koryciak, K. Wincza and S. Gruszczynski "Low-cost microwave vector system for liquid properties monitoring," IEEE Transactions on Industrial Electronics, vol. 65, no. 2 pp. 1665–1674, Feb. 2018.

Chapter 6:

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Microwave sensors for dielectric samples measurement based on coupled-line section," IEEE Transactions on Microwave Theory and Techniques, vol. 65, no. 5 pp. 1615–1631, May 2017.

– I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, "Liquids permittivity measurement using two-wire transmission line sensor," submitted to IEEE Sensors Journal

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1. Introduction

In microwave electronics passive circuits are commonly designed with the use of unbalanced transmission lines, in which the signal is propagating between the lines and the common ground plane. Therefore, the most fundamental networks such as directional couplers [1–9], filters [10–14], phase shifters [5, 6], [15], etc., are developed as single-ended structures due to the simplicity in design and realization. On the other hand, the observed in recent years rapid development of the integrated circuit technology and shifting their frequency of operation towards higher frequencies enforces the design of amplifier circuits, mixers and other active devices [16–22] to have differential inputs and outputs. Such a circuit realization features superior interference rejection. Therefore, in order to connect single-ended-fed transmission line circuits with the active devices having differential inputs and outputs, multiple balun circuits converting balanced signals to unbalanced ones and vice versa need to be utilized. Among many baluns described in the literature [23–31], Marchand type baluns [23, 24], [26–31] are one of the most common due to their simplicity and wideband performance. Therefore, many variations of Marchand baluns were investigated for different purposes, such as matching and isolation improvement [24], [29–31], bandwidth enhancement [26,27] as well as miniaturization of the circuits [27,28].

In spite of the fact, that baluns allow for connection of differential active circuits with the single-ended passive ones, it is crucial to design these passive circuits to operate in a differential mode to avoid multiple single-ended to differential and vice versa signal conversion. This design approach allows to reduce the complexity of the system as well as allows for reduction of attenuation losses and increasing the attenuation of common-mode distortion. Following the recent trend of differential circuits’ utilization, methods for the description of differentially-fed circuits have been proposed, in particular mixed-mode scattering parameters were introduced, where differential, common and mixed S-parameters were defined [32–35]. Moreover, in literature, some examples of filters [36–40] having differential inputs and outputs were described, in which the transmission line sections were replaced with the differential ones, however, the coupled lines were single-ended-fed with respect to the common ground plane. Also, first examples of differential directional couplers have been recently described in [41,42] together with the method allowing for the design of differential directional couplers [42]. Nevertheless, the approach to the design of a coupler operating in a differential mode presented in [41] is significantly simplified, since in the assumed four-wire structure the influence of a ground plane is neglected and in [42] only a single-section differential coupler designed in a symmetric dielectric structure is analyzed.

To respond to the today’s electronics trends and requirements the Author of this Thesis proposed novel methods of Marchand baluns design featuring improved frequency characteristics and miniaturized

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size as well as proposed and analyzed various differential directional couplers including narrowband branch-line and rat-race couplers, and wide-band coupled-line couplers designed in homogeneous and inhomogeneous dielectric media.

Regarding balun circuits, the Author proposed various novel methods allowing for performance improvement and miniaturization [43–48] of Marchand baluns. In the developed compensation technique [43] the transmission line connecting the coupled-line sections in Marchand balun, which deteriorates significantly the entire circuit’s performance was compensated by a transformation into virtual coupled-line sections which were further incorporated into the initial coupled-line sections composing the balun. The Author also has shown, that the proposed compensation technique can be easily integrated with other methods allowing for circuit’s performance improvement [44], [46, 47] as well as circuits miniaturization [47,48]. Moreover, the Author proposed a novel approach dedicated for the design of impedance transforming Marchand type balun in a single-layer microstrip structure [45]. In the proposed method the design formulas presented in [29] were utilized together with the enhanced directional couplers proposed in [49], where metamaterial left-handed (LH) transmission lines inserted in-between two coupled-line sections allow for achieving higher nominal coupling of the resulting directional coupler, than the coupling level of the utilized coupled lines. Such a technique allows for realization of Marchand type balun terminated with arbitrarily chosen impedances, even in the dielectric structures featuring lower coupling coefficients such as single-layer microstrip.

Regarding the design of differential directional couplers, the Author proposed a novel approach for realization of branch-line and rat-race directional couplers operating in a differential mode [50, 51] suitable for application when different circuits having differential inputs/outputs are interconnected. Moreover, Author of this Thesis presented for the first time the investigation results on the realization of differentially-fed wide-band multi-section symmetrical coupled-line directional couplers. It was shown, that in order to realize particular coupling coefficient while maintaining symmetry of the circuit, pairs of metal strips need to be placed directly above each other in a given distance ensured by the dielectric stack-up. Since in case of multi-section couplers’ design it is required to realize different couplings for each coupled-line section [52], the Author proposed circuit’s realization in a multilayer structure with appropriately selected stack-up to satisfy the geometry requirements for each coupled-line section. Moreover, the Author presented an approach to the design of differentially excited couplers, in which the limitation of equal-width conductors is lifted [53]. The proposed realization allows to easily overcome the technological challenge in the design of multisection couplers [52] as well as allows for the design of directional couplers having different terminating impedances. Additionally, apart from the investigations provided for the differential coupled-line directional couplers designed in symmetric dielectric structure, the analysis of differential couplers realized in an asymmetric structure [54], which in practice is more convenient for circuit’s realization, was also presented. The Author of this Thesis analyzed the conditions for ideal differential coupled-line section realization, as well as proposed the methods for quasi-ideal coupled-line section realization by proper equalization of differential coupling coefficients. The compensating elements are calculated based on the analysis of capacitive matrix of the coupled-conductor system after appropriate matrix order reduction. Such an approach allows to equalize capacitive and inductive coupling coefficients of differential coupled-line directional couplers designed

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19

in inhomogeneous dielectric structure resulting in improvement of isolation and impedance match of the circuit.

Moreover, the Author of this Thesis investigated another direction of the research on differentially excited coupled-line sections i.e., the possibility of utilization of differential excitation for sensor applications. Differentially excited coupled-line sections were proposed to serve as sensors allowing for dielectric sample detection in a wide frequency range. The major advantage of such a technique is its high sensitivity to the sample-under-test, when out-of-phase excitation is provided to the coupled-line section, which allows that the sample can be as narrow as the spacing between coupled strips. The developed by the Author measurement techniques were widely described in [55–65]. In the literature, there are many methods allowing for dielectric sample characterization [66–74], which can generally be divided into two classes depending on the bandwidth of operation, namely narrowband sensors (resonators) [66], [70–72] and broadband sensors [54–61,75], [64], [67,68], [73,74]. Usually, resonant techniques feature higher accuracies than non-resonant methods, since their principle of operation is based on the shift of resonant frequency. On the other hand, non-resonant methods in which the measurement of permittivity is related to the change of characteristic wave impedance and wave velocity of the utilized sensor feature limited accuracy, among others, due to the required calibration process, which introduces measurement errors [76]. However, such techniques allow for determination of the sample’s complex permittivity in a wide frequency range and feature increased selectivity in comparison to the resonant methods, which is very important for characterization of biological materials or materials having frequency dependent permittivity. Among non-resonant methods allowing for characterization of dielectric materials, the techniques based on planar circuits [54–61], [64], [73, 74] are the most attractive ones, since planar technology offers size miniaturization, low-cost and simple fabrication. The method shown in [73] uses only a single microstrip transmission line to determine the dielectric material permittivity. However, the sample under test is the substrate of the microstrip line itself, which significantly limits the scope of application of such a technique. In [74], a nondestructive technique of dielectric sample characterization was proposed; however, the main disadvantage of the presented method is the sample size, which needs to be comparable with the coplanar waveguide transmission line (the sample needs to cover coplanar line, slots between the line and ground plane, and also the ground plane itself).

The Author of this Thesis proposed a novel class of non-resonant and nondestructive sensors and associated methods for dielectric sample characterization utilizing differentially excited coupled-line sections in various configurations. The introduced theoretical models of the sensors allow for determination of sensor-sample complex permittivity based on the measured differential S-parameters. The investigated measurement techniques together with the calibration procedures were compared to describe their advantages and limitations in practical applications depending on the measured material type (liquid, solid), characterization purpose (detection, exact determination of complex permittivity) and frequency range, in which the material needs to be characterized. The usability of the developed sensors was proven experimentally by characterization of nonorganic, organic and biological samples. Moreover, the proposed sensors were used to develop a novel low-cost microwave vector system dedicated for monitoring the properties of liquid samples. The system is composed of the differentially-fed coupled-line sensor and five-port correlator allowing for measurement of both magnitude and phase

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of a signal dependent on Material-Under-Test. An exemplary microwave vector system was designed to operate at the frequency 2.4 GHz, manufactured and experimentally verified. The measurement results prove the increased sensitivity of the proposed coupled-line sensor on the dielectric materials in comparison to the methods described in literature. Moreover, the proposed system was utilized for measurements of several milk samples with various fat content and the obtained results prove the industrial and in future biomedical application of the presented approach.

In this work the Author aims to prove the following goals:

– Development of wideband differentially-fed coupled-line directional couplers.

– Analysis of possible capacitive and inductive coupling coefficients equalization in differentially-fed coupled-line sections.

– Utilization of coupled-line sections in sensor applications.

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[43] I. Piekarz, J. Sorocki, S. Gruszczynski, and K. Wincza, “Input match and output balance improvement of Marchand balun with connecting line,” IEEE Microwave and Wireless Components Letters, vol. 24, no. 10, pp. 683–685, Oct. 2014.

[44] I. Piekarz, J. Sorocki, I. Slomian, K. Staszek, K. Wincza, and S. Gruszczynski, “Marchand balun with connecting segment designed with the use of multi-technique compensation,” in 14th Mediterranean Microwave Symposium MMS’14, 2014.

[45] I. Piekarz, J. Sorocki, I. Slomian, K. Wincza, and S. Gruszczynski, “Compact single-layer microstrip Marchand type balun,” in Wireless and Microwave Technology Conference WAMICON 2015, 2015.

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[46] I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Wideband Marchand balun and bow-tie antenna for sensor applications,” in International Symposium on Antennas and Propagation ISAP 2015, 2015.

[47] I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Miniaturized compensated quasi-lumped wideband Marchand balun,” in 21th International Conference on Microwave, Radar and Wireless Communications, 2016.

[48] J. Sorocki, I. Piekarz, K. Wincza, and S. Gruszczynski, “Miniaturized microstrip Marchand balun and coupled-line section as microwave sensor for dielectric material detection,” in 9th International Kharkiv Symposium on Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves MSMW’2016, 2016.

[49] J. Sorocki, K. Staszek, I. Piekarz, K. Wincza, and S. Gruszczynski, “Directional couplers with reduced coupling requirements as a connection of coupled-line sections and left-handed transmission lines,” IET Microwaves, Antennas and Propagation, vol. 8, no. 8, pp. 580–588, Jun. 2014.

[50] J. Sorocki, I. Piekarz, K. Staszek, P. Kamisinski, K. Wincza, and S. Gruszczynski, “Differential-mode branch-line directional couplers,” International Journal of Information and Electronics Engineering, vol. 6, no. 4, pp. 226–229, Jul. 2016.

[51] K. W. I. Piekarz, J. Sorocki and S. Gruszczynski, “Rat-race directional couplers operating in differential mode,” in Radio and Wireless Symposium 2017, 2017.

[52] I. Piekarz, J. Sorocki, K. Janisz, K. Wincza, and S. Gruszczynski, “Wideband three-section symmetrical coupled-line directional coupler operating in differential mode,” accepted for publication in IEEE Microwave and Wireless Components Letters.

[53] K. Janisz, I. Piekarz, K. Staszek, K. Wincza, and S. Gruszczynski, “Differentially-fed coupled-line directional couplers with coupled-conductors of unequal width,” in preparation.

[54] I. Piekarz, J. Sorocki, K. Wincza, S. Gruszczynski, and J. Papapolymerou, “High-performance differentially-fed coupled-line directional couplers realized in inhomogeneous medium,” in preparation.

[55] I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Coupled-line sensor with Marchand balun as RF system for dielectric sample detection,” IEEE Sensors Journal, vol. 16, no. 1, pp. 88–96, Jan. 2016.

[56] K. W. I. Piekarz, J. Sorocki and S. Gruszczynski, “Effective permittivity measurement with the use of coupled-line section sensor,” in Radio and Wireless Symposium 2016, 2016.

[57] I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Calibration method of microwave measurement system for dielectric samples detection,” in 9th International Kharkiv Symposium

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on Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves MSMW’2016, 2016.

[58] J. Sorocki, I. Piekarz, K. Wincza, and S. Gruszczynski, “Differentialy excited coupled-line sensor for small dielectric samples detection,” in 9th International Kharkiv Symposium on Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves MSMW’2016, 2016.

[59] I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Simplified three-strip coupled-line section as microwave sensor for dielectric materials measurement,” in 9th International Kharkiv Symposium on Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves MSMW’2016, 2016.

[60] I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Microwave sensors for dielectric samples measurement based on coupled-line section,” IEEE Transactions on Microwave Theory and Techniques, vol. 65, no. 5, pp. 1615–1631, May 2017.

[61] I. Piekarz, K. Janisz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Two-port measurement system with coupled-line sensor for detection of material permittivity change,” in 18th Conference on Microwave Techniques COMITE 2017, 2017.

[62] I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Low-cost planar coupled-line sensor for permittivity measurement of low-loss dielectric materials in a wide frequency range,” in 18th Conference on Microwave Techniques COMITE 2017, 2017.

[63] K. Staszek, I. Piekarz, J. Sorocki, S. Koryciak, K. Wincza, and S. Gruszczynski, “Low-cost microwave vector system for liquid properties monitoring,” IEEE Transactions on Industrial Electronics, vol. 65, no. 2, pp. 1655–1674, Feb. 2018.

[64] I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Multi-coupled-line microwave sensors for dielectric sample permittivity change detection,” in 3rd International Conference on Electrical, Electronics and System Engineering ICEESE 2017, 2017.

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[66] S. Roberts and A. von Hippel, “A new method for measuring dielectric constant and loss in the range of centimeter waves,” Journal of Applied Physics, vol. 17, no. 7, pp. 601–616, Jul. 1946. [67] W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave

frequencies,” Proceedings of the IEEE, vol. 62, no. 1, pp. 33–36, Jan. 1974.

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[73] P. M. Narayanan, “Microstrip transmission line method for broadband permittivity measurement of dielectric substrates,” IEEE Transactions on Microwave Theory and Techniques, vol. 62, no. 11, p. 2784–2790, Nov. 2014.

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[75] I. Piekarz, J. Sorocki, K. Wincza, and S. Gruszczynski, “Sensitivity investigation of coupled-line micrwave sensor on dielectric material-under-test,” in 9th International Kharkiv Symposium on Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves MSMW’2016, 2016. [76] G. Guarin, M. Hofmann, J. Nehring, R. Weigel, G. Fischer, and D. Kissinger, “Miniature microwave biosensors: noninvasive applications,” IEEE Microwave Magazine, vol. 16, no. 4, pp. 71–86, May 2015.

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2. Theoretical Analysis of Coupled-Line Section Networks

2.1. Two-Conductor Transmission Line

Transmission lines supporting TEM (transversal electromagnetic) or quasi-TEM waves are widely used in microwave engineering. A single TEM transmission line composed of two conductors being in a close proximity is widely described in many scientific papers i.e., [1, 2] and can be analyzed using electromagnetic theory or by means of lumped-element modeling approach [3]. Stripline and microstrip shown in Fig.2.1and in Fig. 2.2are two widely used planar realizations of a transmission line due to the possibility of circuit miniaturization and ease of fabrication. The geometry of stripline and microstrip transmission lines are presented in Fig.2.1aand Fig.2.2a, respectively. The structures are constructed of a conductive strip having width w, which is suspended between two ground planes in the case of stripline or over one ground plane in the case of microstrip. The region between the ground plane in stripline or area between the conductive strip and the ground plane in microstrip is filled with a dielectric material having thickness equal to h and relative permittivity equal to εr. A sketch of electric and magnetic field lines for stripline and microstrip transmission lines are presented in Fig.2.1band Fig.2.2b, respectively.

(a) (b)

Figure 2.1: Cross-sectional view of stripline transmission line (a) and lines of electric E and magnetic H fields (b).

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(a) (b)

Figure 2.2: Cross-sectional view of microstrip transmission line (a) and lines of electric E and magnetic H fields (b).

In a lossless case the transmission line properties i.e., characteristic impedance Z0and phase velocity vof the guided TEM wave are equal [3]:

Z0= r L C (2.1) v=√c ε (2.2)

where inductance L and capacitance C of the line are per unit length, c is the velocity of light in free space and ε is the permittivity, which in the case of stripline technique equals relative permittivity of the utilized medium εrand in the case of microstrip equals effective permittivity of the medium εe f f [3]. In the case of transmission line system being composed of n conductors, assuming that n > 2, n–1 propagation modes can be distinguished. Thus, the analysis for single transmission line needs to be extended to the multiline system, which can be described using n-1 dimensional vectors of currents and voltages as well as n-1 by n-1 capacitance C and inductance L matrices.

2.2. Three-Conductor Coupled Transmission Lines

The three-conductor system shown in Fig. 2.3 is constructed of two signal conductors placed over a common ground plane and can be described by means of per-unit-length capacitances C1, C2 (capacitances of the first #1 and second #2 line) and Cm(mutual capacitance between lines) [4]. Q1, Q2 denote charges gathered on conductors #1, #2 and V1, V2 denote voltages of conductors #1, #2 (their potentials with reference to the ground conductor). In such a case the charges Q1, Q2can be expressed in terms of voltages and capacitances:

Q₁= C1V1+Cm(V1−V2) = (C1+Cm)V1−CmV2 (2.3) Q2= C2V2+Cm(V2−V1) =−CmV1+ (C2+Cm)V2 (2.4) Based on the above performed analysis, the capacitance matrix of two-strip coupled-line system can be determined as: [C] = " C₁₁ C₁₂ C21 C22 # = " C₁+Cm −Cm −Cm C2+Cm # (2.5)

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2.2. Three-Conductor Coupled Transmission Lines 29

whereas the inductance matrix of coupled lines is defined as [4]:

[L] = µ0ε0[C0]−1 (2.6)

where µ0, ε0 are permittivity and permeability of free space, whereas [C0] is a capacitance matrix of two-strip coupled lines assuming homogeneous dielectric medium having relative permittivity equal to 1.

Figure 2.3: Capacitance system of a three-conductor couple-line network. In this case C1= C2due to the identical geometrical

dimensions of the strips.

Assuming that conductors #1 and #2 have identical geometrical dimensions, as in Fig. 2.3, self-capacitances C1 and C2 are equal and the capacitance matrix can be expressed in terms of even-and odd-mode excitations, as it is shown in Fig.2.4[4]. For the even-mode excitation both conductors are excited with equal amplitude and in-phase voltages Ve(charges gathered on both conductors are equal Qe). In such a case there is no voltage between lines #1, #2 and the system shown in Fig.2.4ais simplified into the circuit shown in Fig.2.4b. For the odd-mode excitation both conductors are excited with equal amplitude and out-of-phase voltages Vo (charges gathered on both conductors are equal Qo). In such a case there is a doubled voltage between lines and the system presented in Fig.2.4ais simplified into the circuit shown in Fig.2.4c. For the calculated equivalent circuits (see Fig.2.4b,2.4c) the capacitances of coupled transmission lines for even- and odd-mode excitations in the presence of uniform dielectric medium having relative dielectric constant equal to 1 are equal [4]:

Coe= C1 (2.7)

Coo= C1+ 2Cm (2.8)

Having found the line capacitances for the even- and odd-mode excitations, the even- and odd-mode impedances of the coupled-line section in the presence of inhomogeneous dielectric medium can be calculated as [4]: Zoe= 1 c√CeCoe (2.9) Zoo= 1 c√CoCoo (2.10) where c is the free-space velocity of light and Ce, Co denote the even- and odd-mode per-unit-length capacitances in the presence of inhomogeneous dielectric medium equal:

Ce= Coeεe f f e (2.11)

Co= Cooεe f f o (2.12)

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where εe f f eand εe f f oare the even and odd mode effective dielectric constants.

(a)

(b)

(c)

Figure 2.4: Cross-section of two symmetrical coupled transmission lines for the even- and odd-mode excitations (a), an equivalent capacitance network for even-mode excitation (b) and an equivalent capacitance network for odd-mode excitation (c) [4].

2.3. Single-Ended Directional Couplers

Directional couplers are well-known components often used in modern microwave electronics, which over the years have been a subject of extensive studies [5–11]. Such networks consist of directly connected transmission lines or sections of coupled lines. The directly connected transmission-line components suffer from large occupied area and narrow operational bandwidth, therefore, coupled-line couplers are often utilized in modern telecommunication systems. A schematic diagram of a coupled-line directional coupler is presented in Fig.2.5. A single-ended coupled-line directional coupler is a four-port circuit, composed of two transmission lines being in close proximity. In such a case the electromagnetic (EM) wave propagating in one line is partially transferred to the coupled line and the direction of this EM wave propagation is dependent on the coupler’s construction. In forward directional couplers (see Fig. 2.5a) the EM wave in the coupled line is propagating in the same direction as the EM wave propagating in the excited line. In backward directional couplers, the EM wave in the coupled line is propagating in the opposite direction to the EM wave propagating in the excited transmission line (see Fig.2.5b).

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2.3. Single-Ended Directional Couplers 31

(a) (b)

Figure 2.5: Schematic diagram of a forward-wave directional coupler (a) and backward-wave directional coupler (b). Port number #1 is denoted as input.

In order to characterize the directional coupler presented in Fig.2.5afollowing parameters are defined [3]: – Coupling C C= 10 logP1 P3 [dB] (2.13) – Directivity D D= 10 logP3 P4 [dB] (2.14) – Isolation I I= 10 logP1 P₄ [dB] (2.15)

where Pndenotes the power at n-th port and the above mentioned couplers’ parameters are related with the expression:

D[dB] = I [dB]−C [dB] (2.16)

Assuming a reciprocal network, matched at all ports, the scattering matrix of a four-port directional coupler shown in Fig.2.5acan be written as follows [3]:

       0 S₁₂ S₁₃ S₁₄ S12 0 S23 S24 S13 S23 0 S34 S14 S24 S34 0        (2.17)

Assuming lossless network, following equations result from the unitarity or energy conservation conditions [3]: S₁₃∗ S23+ S14∗ S24= 0 (2.18) S₁₄∗ S₁₃+ S₂₄∗ S₂₃= 0 (2.19) S₁₄∗ |S13|2− |S24|2 = 0 (2.20)

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S∗₁₂S23+ S∗14S34= 0 (2.21) S∗₁₄S₁₂+ S∗₃₄S₂₃= 0 (2.22) S23 |S12|2− |S34|2 = 0 (2.23)

One way for2.20and2.23to be satisfied is if S14= S23= 0, which results in a directional coupler. Then:

|S12|2+|S13|2= 1 (2.24)

|S12|2+|S24|2= 1 (2.25)

|S13|2+|S34|2= 1 (2.26)

|S24|2+|S34|2= 1 (2.27)

The above presented analysis leads to the fact, that|S13| = |S24| and |S12| = |S34|. Further simplification can be made by choosing the phase references on three of the four ports [3] i.e., S12 = S34 = α, S13 = β ejθ, and S24= β ejϕ (α, β – are real values, θ , ϕ – are phase constants, which need to be determined). Since the dot product of rows 2 and 3 of the coupler scattering parameters yields to:

S∗₁₂S13+ S∗24S34= 0 (2.28)

The relation between phase constants can be written as:

θ+ ϕ = π± 2nπ (2.29)

Ignoring the integer multiples of 2π and assuming a symmetric coupler θ = ϕ = π/2 and the scattering matrix takes the following form:

[S] =        0 α jβ 0 α 0 0 jβ jβ 0 0 α 0 jβ α 0        (2.30)

Moreover, the parameters α and β meet the condition:

α2+ β2= 1 (2.31)

2.4. Differential Directional Couplers

The majority of directional couplers described in literature are designed to operate with nodal excitation of their coupled conductors [5–11]. However, recently it is common to design analog electronic circuits to have differential inputs and outputs due to superior interference rejection. Many examples of differential amplifiers, mixers and filters can be found in literature [14–18]. Following the trend of differential circuits’ utilization, also directional couplers operating in differential mode have been recently described [19,20]. A schematic diagram of a couple-line system constructed of four equal-width coupled conductors placed over a ground plane is presented in Fig. 2.6. Having known, that the

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2.4. Differential Directional Couplers 33

considered dielectric structure is a symmetric one, a modal approach can be applied and four propagating modes can be distinguished, i.e., even-differential and odd-differential modes as well as even-common and odd-common modes [20]. Cross-section of a symmetrical four-strip coupled-line system is presented in Fig.2.7, where capacitive elements related to particular conductors are visible. Assuming symmetry of the structure, four identical capacitive elements are associated with each of the remaining three conductors and the reduced per-unit-length capacitances for the differential and common modes are shown in Fig.2.8.

Figure 2.6: A schematic view of a four-conductor over ground-plane coupled-line system with indicated ports’ numbers [20].

Figure 2.7: Capacitive elements related to a particular conductor of the considered four-strip symmetric coupled-line geometry [20].

(a) (b)

(c) (d)

Figure 2.8: The equivalent capacitive elements for the structure shown in Fig. 2.7 related to even-differential (a), odd-differential (b), even-common (c) and odd-common (d) modes [20].

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Based on Fig. 2.8the reduced per-unit-length capacitances for the differential and common modes are equal to [20]: Cedi f f = 0.5C1+|C12| + |C14| (2.32) Codi f f= 0.5C1+|C12| + |C13| (2.33) Cecomm= 2C1 (2.34) Cocomm= 2C1+ 4|C13| + 4|C14| (2.35) and C₁= C11− |C12| − |C13| − |C14| (2.36) where Cedi f f is the even-mode differential capacitance, Codi f f is the odd-mode differential capacitance, Cecomm,Cocommare the even- and odd-common mode capacitances, respectively. Assuming homogeneous dielectric filling of the coupled-line geometry the modal impedances can be expressed as [20]:

Z_{e/o di f f /comm}= (vCe/o di f f /comm)−1 (2.37) The properties of circuits in microwave frequency range are typically described using scattering parameters, while the differential-mode circuits can be described using mixed-mode S-parameters, first introduced for the four-port circuits in [21–24]. The description of mixed-mode scattering matrix presented in [21–24] was extended in [20] for the description of eight-ports, allowing for calculation of mixed-mode scattering parameters from nodal simulation and measurement results. The mixed mode scattering matrix can be expressed as [21]:

                 b1di f f b_{2di f f} b3di f f b_{4di f f} b1comm b2comm b_3comm b4comm                  = " Sdi f f Sdi f f/comm S_{comm/di f f} Scomm #                  a1di f f a_{2di f f} a3di f f a_{4di f f} a1comm a2comm a_3comm a4comm                  (2.38)

The normalized power waves, for the chosen port indexing and the choice of differential and common excitations, are defined as follows [20]:

a1di f f =√1₂(a1− a2), a1comm=√1₂(a1+ a2) a2di f f =√1₂(a3− a4), a2comm=√1₂(a3+ a4) a3di f f =√1₂(a5− a6), a3comm=√1₂(a5+ a6) a_{4di f f} =_√1 2(a7− a8), a4comm= 1 √ 2(a7+ a8) b_{1di f f} =_√1 2(b1− b2), b1comm= 1 √ 2(b1+ b2) b2di f f =√1₂(b3− b4), b2comm=√1₂(b3+ b4) b3di f f =√1₂(b5− b6), b3comm=√1₂(b5+ b6) b4di f f =√1₂(b7− b8), b4comm=√1₂(b7+ b8) (2.39)

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2.4. Differential Directional Couplers 35

The mixed-mode sub-matrices2.38can be expressed as:

Sdi f f= 1 2          S11− S21− S12+ S22 S13− S23− S14+ S24 S15− S25− S16+ S26 S17− S27− S18+ S28 S31− S41− S32+ S42 S33− S43− S34+ S44 S35− S45− S36+ S46 S37− S47− S38+ S48 S51− S61− S52+ S62 S53− S63− S54+ S64 S55− S65− S56+ S66 S57− S67− S58+ S68 S71− S81− S72+ S82 S73− S83− S74+ S84 S75− S85− S76+ S86 S77− S87− S78+ S88          (2.40) Scomm= 1 2          S11+ S21+ S12+ S22 S13+ S23+ S14+ S24 S15+ S25+ S16+ S26 S17+ S27+ S18+ S28 S31+ S41+ S32+ S42 S33+ S43+ S34+ S44 S35+ S45+ S36+ S46 S37+ S47+ S38+ S48 S51+ S61+ S52+ S62 S53+ S63+ S54+ S64 S55+ S65+ S56+ S66 S57+ S67+ S58+ S68 S71+ S81+ S72+ S82 S73+ S83+ S74+ S84 S75+ S85+ S76+ S86 S77+ S87+ S78+ S88          (2.41) Sdi f f/comm= 1 2          S11− S21+ S12− S22 S13− S23+ S14− S24 S15− S25+ S16− S26 S17− S27+ S18− S28 S31− S41+ S32− S42 S33− S43+ S34− S44 S35− S45+ S36− S46 S37− S47+ S38− S48 S51− S61+ S52− S62 S53− S63+ S54− S64 S55− S65+ S56− S66 S57− S67+ S58− S68 S71− S81+ S72− S82 S73− S83+ S74− S84 S75− S85+ S76− S86 S77− S87+ S78− S88          (2.42) S_{comm/di f f}= 1 2          S11+ S21− S12− S22 S13+ S23− S14− S24 S15+ S25− S16− S26 S17+ S27− S18− S28 S31+ S41− S32− S42 S33+ S43− S34− S44 S35+ S45− S36− S46 S37+ S47− S38− S48 S51+ S61− S52− S62 S53+ S63− S54− S64 S55+ S65− S56− S66 S57+ S67− S58− S68 S71+ S81− S72− S82 S73+ S83− S74− S84 S75+ S85− S76− S86 S77+ S87− S78− S88          (2.43)

and can be calculated using following equation [22]:

Smixed mode= MSnodalM−1 (2.44) where M=√1 2                  1 −1 0 0 0 0 0 0 0 0 1 −1 0 0 0 0 0 0 0 0 1 _{−1 0} 0 0 0 0 0 0 0 1 −1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1                  (2.45)

Having the mixed-mode scattering matrix explicitly given, it can be shown that for a double symmetrical passive network, all the elements of sub-matrices2.42and2.43equal zero. From the symmetry of the network one can write [20]:

S21= S43= S65= S87, S31= S42= S75= S86 S41= S32= S76= S85, S51= S62= S73= S84 S61= S52= S74= S83, S71= S53= S64= S82 S₈₁= S54= S63= S72 S11= S22= S33= S44= S55= S66= S77= S88 (2.46)

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Additionally from the reciprocity we have:

Sxy= Syx (2.47)

By applying2.46and2.47to2.42and2.43one can easily prove that:

S_{comm/di f f} = Sdi f f/comm= 0 (2.48)

2.5. Baluns

A balun is a transformer converting a balanced signal to an unbalanced one or vice versa and therefore is usually used in various RF circuits, such as antennas, mixers, or power amplifiers [25–28]. Over the years, several different types of balun circuits were developed [25,27,29–33]. Due to the current interest in transmission line structures, which are easy in integration, the research is focused on planar realization of baluns [27,30–33].

Generally, balun is a 3-port device, as shown in Fig. 2.9with a single-ended input and differential output. Port #1 represents the single-ended/unbalanced port, whereas ports #2 and #3 create the differential/balanced port. The planar realization of a balun consists of two section of transmission lines, in which the first section divides the signal into two signals having equal magnitude and phase over a wide frequency range and the second section provides -90 deg and +90 deg phase shift of these two signals, so that the balanced output signals feature 180-deg phase difference [4]. Among many baluns presented in literature, Marchand type balun (see Fig. 2.10) is commonly used due to the wideband performance. A planar realization of the balun is often constructed of two identical quarter-wave-long coupled-line sections described by even- and odd-mode impedances Zoe, Zoo, terminated at single-ended port with Zin and at differential ports with Zout. In order to obtain an ideal impedance match between ports #2 and #3 an additional isolating circuit is required [33]. The coupled-line sections’ even and odd mode impedances (Zoe, Zoo) can be calculated based on the following equations [33]:

Zoe= p 2ZinZout k 1− k (2.49) Zoe= p 2ZinZout k 1+ k (2.50) Z0= √ ZoeZoo (2.51)

where k and Z0 are the coupling coefficient and characteristic impedance of the coupled-line sections composing the Marchand balun. In such a case the infinite sets of Zoeand Zoocan be obtained by changing k, where k is not a function of the impedances Zinand Zoutand influences only the banwidth of the balun, as it is shown in Fig.2.11[33], where frequency characteristics for various k values are presented.

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2.5. Baluns 37

Figure 2.10: Circuit diagram of Marchand balum circuit composed of two identical coupled-line sections.

(a)

(b)

(c)

Figure 2.11: Marchand balun frequency characteristics obtained for various k values assuming Zin= 50 Ω and Zout= 100 Ω:

(a) magnitude of S11, (b) magnitude of S21and S31, (c) phase difference between S21and S31[33].

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Index of /rozprawy2/11308

Contents

List of papers

1. Introduction

Bibliography

2. Theoretical Analysis of Coupled-Line Section Networks

2.1. Two-Conductor Transmission Line

2.2. Three-Conductor Coupled Transmission Lines

2.3. Single-Ended Directional Couplers

2.4. Differential Directional Couplers

2.5. Baluns

Bibliography