Index of /rozprawy2/11309

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List of papers

This Thesis is based on and incorporates the following publications:

Chapter3:

- J. Sorocki, I. Piekarz, K. Wincza, and S. Gruszczynski, “Right/left-handed transmission lines based on coupled transmission line sections and their application towards bandpass filters,” IEEE Transactions on Microwave Theory and Techniques, vol. 63, no. 2, pp. 384–396, Feb. 2015.

- J. Sorocki, I. Piekarz, K. Wincza, and S. Gruszczynski, “Semi-distributed approach to dual-composite right/left-handed transmission lines and their application to bandstop filters,” IEEE Microwave and Wireless Components Letters, vol. 25, no. 12, pp. 784–786, Dec. 2015.

- J. Sorocki, I. Piekarz, K. Wincza, and S. Gruszczynski, “Pseudo-highpass filters based on semi-distributed balanced composite right/left-handed unit cells,” Journal of Electromagnetic Waves and Applications, vol. 29, no. 16, pp. 2171–2177, 2015.

- J. Sorocki, I. Piekarz, S. Gruszczynski, and K. Wincza, “Low-loss wideband bandpass filters using semi-distributed unit cells,” in Mediterranean Microwave Symposium (MMS 2017), Marseille, France, Nov. 2017, pp. 20–30.

- J. Sorocki, I. Piekarz, S. Gruszczynski, and K. Wincza, “Low-loss pseudo-highpass filters using distributed-element unit cells,” in International Conference on Microwave, Radar and Wireless Communications (MIKON 2018), Poznan, Poland, May 2018, pp. 1–4.

Chapter4:

- J. Sorocki, I. Piekarz, S. Gruszczynski, and K. Wincza, “Miniaturized directional filter multiplexer for band separation in UWB antenna systems,” in International Symposium on Antennas and Propagation (ISAP 2015), Nov. 2015, pp. 1–4.

- J. Sorocki, I. Piekarz, S. Gruszczynski, and K. Wincza, “Low-loss directional filters based on differential band-reject filters with improved isolation using phase inverter,” IEEE Microwave and Wireless Components Letters, vol. 28, no. 4, pp. 314–316, Apr. 2018.

- J. Sorocki, K. Staszek, I. Piekarz, S. Gruszczynski, and K. Wincza, “Traveling wave directional filters with transmission zeroes using cross coupling,” submitted to IEEE Transactions on Micowave Theory and Techniques, 2018.

- J. Sorocki, I. Piekarz, S. Gruszczynski, and K. Wincza, “Cascaded loops directional filter with transmission zeroes for multiplexing applications,” in International Conference on Microwave, Radar

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14 List of Papers

and Wireless Communications (MIKON 2016), Krakow, Poland, May 2016, pp. 1–4.

- J. Sorocki, K. Janisz, I. Piekarz, S. Gruszczynski, and K. Wincza, “Frequency multiplexers with improved selectivity using asymmetric response directional filters,” submitted to IEEE Microwave and Wireless Components Letters, 2017.

- J. Sorocki, I. Piekarz, K. Wincza, and S. Gruszczynski, “Impedance transforming directional couplers with increased achievable transformation ratio,” International Journal of Microwave and Wireless Technologies, vol. 9, no. 3, p. 509–513, Apr. 2017.

- K. Staszek, J. Sorocki, P. Kaminski, K. Wincza, and S. Gruszczynski, “A broadband 3-dB tandem coupler utilizing right/left-handed transmission line sections,” IEEE Microwave and Wireless Components Letters, vol. 24, no. 4, pp. 236–238, Apr. 2014.

Chapter5:

- J. Sorocki, S. Koryciak, I. Piekarz, S. Gruszczynski, and K. Wincza, “Investigation on additive manufacturing with conductive PLA filament for realisation of low-loss suspended microstrip microwave circuits,” in International Conference on Electrical, Electronics and System Engineering (ICEESE 2017), Kanazawa, Japan, Nov. 2017, pp. 48–51.

- I. Piekarz, J. Sorocki, S. Gruszczynski, K. Wincza, and J. Papapolymerou, “Suspended microstrip low-pass filter realized using FDM type 3D printing with conductive copper-based filament,” in IEEE Electronic Components and Technology Conference (ECTC 2018), San Diego, CA, USA, May 2018, pp. 1–4.

- J. Sorocki, I. Piekarz, S. Gruszczynski, K. Wincza, and J. Papapolymerou, “Application of additive manufacturing technologies for realization of multilayer microstrip directional filter,” in IEEE Electronic Components and Technology Conference (ECTC 2018), San Diego, CA, USA, May 2018, pp. 1–4. - J. Sorocki, I. Piekarz, S. Gruszczynski, K. Wincza, and J. Papapolymerou, “Application of 3D printing technology for realization of high-performance directional couplers in suspended stripline technique,” in preparation, 2018.

Chapter6:

- J. Sorocki, I. Piekarz, S. Gruszczynski, and K. Wincza, “Low-cost impedance tuner utilizing quadrature coupled-line coupler for load and source pull transistor measurement applications,” in 2016 IEEE Radio and Wireless Symposium (RWS 2016), Austin, TX, USA, Jan. 2016, pp. 169–171.

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1. Introduction

1.1. Significance of the research

The currently observed dynamic development of wireless communication systems enforces requirements for more efficient utilization of the available resources to satisfy the demand for systems’ capacity in terms of throughput and range [18], [19], [20]. To meet these demands, it is necessary in many cases to increase the number of transceivers and introduce appropriate multiplexing circuits and/or to increase the transmitting power while reducing the power loss within the transmitter and keeping the same or comparable physical volume of the devices. Therefore, an extensive research efforts are directed towards the development of novel solutions and technologies allowing for the realization of compact and lightweight systems with high level of components’ integration and increased power efficiency.

One of the directions to follow is the application of strip transmission line technique. Circuits realized in such a technique are quasi-planar structures supporting Transverse Electromagnetic (TEM) wave propagation what allows for easy modeling using classic transmission line theory and relatively fast design where 2.5D electromagnetic analysis is sufficient. Importantly, easy integration of passive and active, low and high frequency circuits into one board is possible in this technique. Moreover, such an approach can reduce mechanical design complexity and allows for achieving compact components as well as reduction of the entire system’s cost. A large variety of basic building blocks such as filters [21], [22], frequency multiplexers [23], directional couplers [24], [25], phase shifters [26], power dividing/combining networks [27], [28], antennas and antenna array feeding networks [29], [30], [31] etc., have been proposed over the years and implemented in strip transmission line technique. However, despite many advantages, such circuits are in many cases suitable only for low power applications due to relatively high power loss being a result of electromagnetic wave propagating fully or partially within lossy dielectric substrate. For example insertion loss of ∼ 1 dB, which is a typical value for cm-wave range circuits, leads to roughly ∼ 10% of the input power being transformed into heat.

As a result, nowadays an industry standard for realization of high power telecommunication transmitting front-ends is all-metal waveguide technique [32]. As such, waveguide circuits feature very low power loss since TE or TM wave modes can propagate in lossless air filling. However, the design process is time and computational power consuming due to the necessity of full 3D EM analysis. Moreover, physical realization requires precise mechanical machining while the resulting components are usually heavy and bulky. Furthermore, integration with other passive or active circuits, realized mainly in strip transmission line technique, requires additional waveguide-to-stripline transitions which

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16 1. Introduction

may introduce additional power loss.

Therefore, it is necessary to pursue and develop novel low-loss passive and active components in strip transmission line technique being suitable for high power applications and allowing for easy integration with other components. This would enable the realization of uniform transceiver systems satisfying the near future capability requirements. The main challenges and scientific problems that need to be solved, originating from the fact that guided wave propagating along the transmission line is being attenuated at the rate of attenuation constant, are:

– reduction of overall propagating wave’s path length by e.g., topology optimization, the development of alternative topologies, overall circuits’ size reduction, more efficient utilization of space, increased level of functionality integration, etc.

– reduction of dielectric related losses contributing to the attenuation constant by e.g., the introduction of loss reduced dielectric stack-ups such as suspended microstrip or the utilization of emerging or existing very low-loss dielectric materials,

– reduction of conductor related losses contributing to the attenuation constant by e.g., the realization of low-roughness metallization layers featuring high effective conductivity,

– minimization of radiated losses contributing to the attenuation constant by e.g., the use of self-shielded structures such as stripline,

The results of theoretical end experimental investigations will contribute to the development of knowledge and understanding in the field of microwave theory and techniques as well as analog electronics in general with potential of future industrial implementation in wireless telecommunication components and systems.

1.2. Scientific goals and organization of the Thesis

This Thesis presents a comprehensive study of various methodologies and design schemes allowing for realization of low-loss microwave circuits in strip transmission line technique. The main goal of this dissertation is to experimentally verify strip transmission line technique as the one suitable for realization of low-loss circuits for high-power systems, and to show that such a technique can be a competitive alternative for modern day telecommunication applications.

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

1. To develop low-loss microwave filters in strip transmission line technique,

2. To develop low-loss microwave power division/summation circuits in strip transmission line technique,

3. To develop a low-loss integrated high-power microwave transceiver front-end.

Fulfilling the above stated goals will allow for effective design of highly-integrated transmitting and receiving systems as a result of utilization of the same PCB manufacturing technology for realization of high- and low-power as well as high- and low-frequency components. The additional advantage is that well established transmission line theory can by employed for initial circuits‘ analysis and optimization which in combination with 2.5D electromagnetic analysis can reduce the requirement for the design time and computational resources.

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1.2. Scientific goals and organization of the Thesis 17

This Thesis is divided into seven Chapters with the following content:

Thesecond Chapter introduces the strip transmission line medium for Transverse ElectroMagnetic (TEM) wave propagation and provides the theoretical background for further considerations. Lossy transmission line model is provided and propagation of TEM wave is considered. Moreover, planar realization of the strip transmission line medium in a form of microstrip and stripline structures is presented.

The third Chapter describes the use of novel and alternative realization schemes of circuits as a way of power loss minimization. The realization of filters exploiting the periodic structures approach in which the filter is composed of identical, electrically small unit cells was studied. Such a realization allows for obtaining very compact structures since the increase of the filter‘s order is done by adding another unit cell to the cascade. The Author has proposed and developed various novel, left-handed metamaterial inspired unit cells being constructed of mostly or solely coupled and uncoupled transmission line sections, exhibiting all kinds of frequency selective response. Additionally, appropriate theoretical models were developed to allow for the analysis of the unit cells and resulting filter’s properties. The advantage of the proposed models is not only accurate in respect to electromagnetically calculated results over a wide frequency range but also allows for direct translation of parameters into the physical geometry due to the use of classic transmission line theory to describe the unit cells’ components. The applicability towards filters‘ realization was experimentally investigated, i.e. various unit cells’ topologies and realization techniques were considered to provide bandpass, bandstop and pseudo-highpass frequency response. Moreover, the further power loss reduction by unit cell’s topology optimization and realization technique alteration was studied on an example of pseudo-highpass filters. The initial topology requiring relatively lossy lumped components was replaced with the one requiring only transmission line section while the initial microstrip realization were replaced with suspended stripline medium, where wave propagation occurs partially within lossless air layer.

The fourth Chapter presents the approach of circuits’ topology optimization focused on the performance improvement and functionalities integration as a way of power loss minimization. More effective circuits’ topologies of directional filters and directional couplers were studied allowing to obtain a given required electrical performance while reducing the overall circuits’ size and/or complexity which as a result leads to the reduction of power loss. The Author has proposed and developed various novel design techniques and topologies of the above presented microwave components. As for directional filters, a miniaturization technique used to obtain more compact structure was proposed along with isolation improvement technique to obtain the reduction of signal cross-talk as well as topologies allowing for filter’s selectivity improvement by the introduction of transmission zeroes into bandpass path. Different mechanisms of transmission zeros generation were studied such as cascading identical directional filters or introduction of loose cross coupling between loop resonators. Additionally, a novel realization scheme was proposed for frequency multiplexers realized as a cascade connection of directional filters. It was shown that each channel selectivity can be increased while keeping practically unchanged circuits’ size by exploiting the cascaded topology and proper filters’ design. As for directional couplers, a novel impedance transforming directional couplers were proposed

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18 1. Introduction

with the exemplary circuit providing equal power split and 4:1 impedance transformation. Such circuits find their application, among others, in multichannel power amplifiers where power division/summation networks are required and in many cases additional impedance transformation is of need to pre-match or match active devices. Moreover, a bandwidth enhancement technique was proposed for directional couplers in tandem configuration where by the realization of appropriate right- and left-handed phase shifters in-between a loosely coupled single-section directional couplers frequency responses similar to these of a two-section directional coupler can be obtained.

The fifth Chapter discusses the application of emerging materials and manufacturing technologies for the realization of low loss circuits in strip transmission line technique. The main focus is on additive manufacturing technologies including 3D printing and materials dedicated for these technologies. The detailed study is provided for the realization of low-loss suspended microstrip circuits where conductive thermoplastic materials are used to serve as a ground plane and structural enclosure, including the influence of bulk conductivity and printing parameters on total loss. Also, a combination of 3D printing of dielectric material and magnetron metal sputtering was proposed for circuits’ realization and verified on an example of a multilayer directional filter. Moreover, it was experimentally shown that the introduction of the 3rddimension of structure control allows for the realization of high-performance and low-loss circuits in suspended stripline technique on an example of a single-section 3-dB directional coupler.

The sixth section illustrates that power loss reduction allows to extend the functional range of a circuit on an example of a low-cost impedance tuner dedicated for push-pull transistor measurement. The usable range of realizable impedances is increased by replacing a larger and lossier rat-race coupler with a 6 times smaller coupled-line coupler allowing for measurement of power transistors featuring very-low input/output impedance. Such measurements are in many cases crucial for the design of high-power microwave amplifiers as in the developed high-power transceiver front-end, where the proposed tuner was used to find the maximum power terminating impedances at the transistor’s input and output.

The seventh Chapter concludes the major achievements presented in the Thesis and indicates further possible directions of the research and development of low-loss circuits for modern day wireless telecommunication applications.

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2. Strip transmission line medium for TEM wave propagation

The development of waveguide and other transmission lines for the low-loss power transmission at high frequencies was one of the early milestones in microwave engineering. Transmission in early RF and microwave systems relied on waveguides, two-wire lines, and coaxial lines. Waveguides have the advantage of high power-handling capability and low loss, but are bulky and expensive, especially at low frequencies. Two-wire lines are inexpensive, but lack shielding. Coaxial lines are shielded but are a difficult medium to fabricate complex microwave components. Planar transmission lines provide an alternative, in the form of stripline, microstrip lines, slotlines, coplanar waveguides, and several other types of related geometries. Such transmission lines are compact, low in cost, and capable of being easily integrated with active circuit devices, such as diodes and transistors, to form microwave integrated circuits.

In this chapter a transmission line that consist of two or more conductors and support transverse electromagnetic (TEM) waves is considered. A transmission line can be characterized by the propagation constant, the attenuation constant, and the characteristic impedance [33] which are derived here to provide theoretical background for further analysis.

Even though, transmission lines are lossy structures due to finite conductivity and/or lossy dielectric, these losses are usually small and for many practical problems may be neglected. However, when e.g. high power circuits with low line attenuation are necessary, resonant circuits in which very high quality factor are of need or low-noise feeding networks for active circuits are required, the effect of loss must be taken into account. Therefore, a lossy transmission line is considered here to allow for inclusion of the loss effects on a transmission line behavior [34].

2.1. Wave propagation in a lossy transmission line

A one dimensional section of transmission line shown schematically in Fig.2.1aas a two-wire line representation can be considered in terms of propagating voltage and current waves along z-axis. At a distance z, there is current I(z) traveling through each wire, and there is voltage difference V (z) between the wires. To calculate their distance and time relations, a transmission line model shown in Fig.2.1bcan be used. The section of line of infinitesimal length ∆z is modeled as a lumped-element circuit, where L, R, C, G, are per-unit-length quantities defined as follows:

– L is a series inductance per unit length, for both conductors, in (H/m), representing the total self-inductance of two conductors;

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20 2. Strip transmission line medium for TEM wave propagation

– R is a series resistance per unit length, for both conductors, in (Ω/m), representing the resistance due to the finite conductivity of the individual conductors;

– C is a shunt capacitance per unit length, in (F/m), representing the total self-capacitance due to the close proximity of the two conductors;

– G is a shunt conductance per unit length, in (S/m), representing the dielectric loss in the material between the conductors.

A finite length of a transmission line can be seen as a cascade of sections of the form shown in Fig.2.1b.

(a)

(b)

Figure 2.1: Definitions of voltage and current (a) together with lumped-element equivalent circuit for an incremental length of transmission line (b)

From the circuit shown in Fig.2.1b, Kirchhoff’s voltage law can be applied to give:

v(z, t) − R∆zi(z, t) − L∆z∂i(z, t)

∂t − v(z + ∆z, t) = 0, (2.1a)

and Kirchhoff’s current law leads to:

i(z, t) − G∆zv(z + ∆z, t) − C∆z∂v(z + ∆z, t)

∂t − i(z + ∆z, t) = 0. (2.1b)

Dividing2.1aand2.1bby ∆z and taking the limit as ∆z → 0 gives the following differential equations: ∂v(z, t) ∂z = −Ri(z, t) − L ∂i(z, t) ∂t , (2.2a) ∂i(z, t) ∂z = −Gv(z, t) − C ∂v(z, t) ∂t . (2.2b)

These are the time domain form of the transmission line equations, also known as the telegrapher equations.

For the sinusoidal steady-state condition, with cosine-based phasors,2.2aand2.2bsimplify to: dV (z)

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2.1. Wave propagation in a lossy transmission line 21

dI(z)

dz = −(G + jωC)V (z). (2.3b)

Following, the two equations2.3aand2.3bcan be solved simultaneously to give wave equations for V (z) and I(z): d2V (z) dz2 − γ 2_{V (z) = 0,} _(2.4a) d2I(z) dz2 − γ 2_{I(z) = 0,} _(2.4b) γ = α + jβ =p(R + jωL)(G + jωC) (2.5)

where 2.5 is the complex propagation constant, which is a function of frequency. Traveling wave solutions to2.4can be found as:

V (z) = V_o+e−γz+ V_o−eγz, (2.6a)

I(z) = I_o+e−γz+ I_o−eγz, (2.6b)

where the e−γzterm represents wave propagation in the +z direction, and the eγzterm represents wave propagation in the −z direction. Applying2.3ato the voltage of2.6agives the current on the line:

I(z) = γ R + jωL(V + o e −γz + V_o−eγz). (2.7)

Comparison of2.7with2.6bshows that the characteristic impedance Z0, can be defined as:

Z0= R + jωL γ = s R + jωL G + jωC, (2.8)

to relate the voltage and current on the line as follows: V_o+ Io+ = Z0 = −V− o Io− . (2.9)

Then2.6bcan be rewritten in the following form:

I(z) = V + o Z0 e−γz−V − o Z0 eγz. (2.10)

Converting back to the time domain, we can express the voltage waveform as:

v(z, t) = |V_o+|cos(ωt − βz + φ+)e−αz+ |V_o−|cos(ωt + βz + φ−)eαz, (2.11)

where φ±is the phase angle of the complex voltage Vo±. Moreover, the wavelength on the line is

λ = 2π

β , (2.12)

while the phase and group velocities are

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22 2. Strip transmission line medium for TEM wave propagation vp = ω β = λf = 1 √ µ, (2.13a) vg = ∂ω ∂β. (2.13b)

The phase velocity vpcorresponds to the propagation of a perturbation, i.e. it is the velocity at which

a fixed phase point on the wave travels, while the group velocity vg corresponds to the propagation of

energy. In a conventional medium, where permittivity and permability µ are greater than 0, the Electric field–Magnetic field–propagation constant (E, H, β) builds the right-handed (RH) triad, therefore vpand

vg are positive values describing a forward-wave propagation, outward from the source. However, if a

medium would feature , µ < 0, a left-handed (LH) triad is build [35]. Thus, as frequency is always a positive quantity, the phase velocity in a LH medium would be opposite to the phase velocity in a RH medium. Moreover, because β is known to be positive in a RH medium, it would be negative in a LH medium, hence phase as related to the phase velocity, propagates backwards to the source in the opposite direction than that of power as being related to the group velocity.

2.2. Power loss in a terminated lossy transmission line

The voltage and current wave propagating along a length l of a lossy transmission line terminated by the impedance ZL, as shown in Fig.2.2, with propagation constant β is attenuated at a constant rate of

α. Thus, γ = α + jβ is complex, however, it is assumed that loss is small, so that Z0is approximately

real. The actual power delivered to the load can be determined using the analysis given below.

Figure 2.2: A lossy transmission line terminated by the impedance ZL.

The voltage and current wave can be expressed as:

V (z) = V_o+(e−γz+ Γeγz), (2.14a) I(z) = V + o Z0 (e−γz− Γeγz_), _(2.14b)

where Γ is the reflection coefficient of the load and V_o+is the incident voltage amplitude referenced at z = 0. The reflection coefficient at a distance l from the load is:

Γ(l) = Γe−2jβle−2αl = Γe−2γl, (2.15)

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2.2. Power loss in a terminated lossy transmission line 23 Zin= V (−l) I(−l) = Z0 ZL+ Z0tanhγl Z0+ ZLtanhγl . (2.16)

The power delivered to the input of the terminated line at z = −l can be calculated as:

Pin = 1 2Re {V (−l)I ∗_{(−l)} =} |Vo+|2 2Z0 (e2αl− |Γ|2e−2αl) = |V + o |2 2Z0 (1 − |Γ(l)|2)e2αl, (2.17)

where2.14has been used for V (−l) and I(−l). The power actually delivered to the load is

PL= 1 2Re {V (0)I ∗_{(0)} =} |Vo+|2 2Z0 (1 − |Γ|2). (2.18)

The difference in these powers corresponds to the power lost in the line:

Ploss= Pin− PL= |V+ o | 2Z0 h (e2αl− 1) + |Γ|2(1 − e2αl) i . (2.19)

The first term in2.19accounts for the power loss of the incident wave, while the second term accounts for the power loss of the reflected wave. It must be noted that both terms increase as α increases.

The attenuation constant of a low-loss line can be found with a common and useful technique called the perturbation method. The method relies on the fields of the lossless line, with the assumption that the fields of the lossy line are not greatly different. Such an approach allows to avoid the use of the transmission line parameters L, R, C, G.

The power flow along a lossy transmission line, in the absence of reflections, is of the form:

Pz = Poe−2αz, (2.20)

where Po is the power at the z = 0 plane and α is the attenuation constant we wish to determine. The

power loss per unit length along the line can be defined as:

Pl = −

∂P

∂z = 2αPoe

−2αz

= 2αP (z), (2.21)

where the negative sign on the derivative was introduced so that Plwould be a positive quantity. From

2.21, the attenuation constant can be determined as:

α = Pl(z) 2P (z) =

Pl(z = 0)

2Po

. (2.22)

The equation2.22states that α can be determined from the power on the line Poand the power loss per

unit length of line Pl. It is important to realize that Plcan be computed from the fields of the lossless line

and can account for both conductor loss and dielectric loss.

The power loss in a good conductor can be accurately and simply calculated in terms of the surface resistance of the conductor RSand the surface current ¯JS, or tangential magnetic field ¯HS:

Plcond= Rs 2 Z S | ¯JS|2ds = Rs 2 Z S | ¯Ht|2ds (2.23)

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24 2. Strip transmission line medium for TEM wave propagation

where R_S denotes a surface integral over the conductor surface and RS is determined from material

conductivity σ and current skin depth δS:

RS =

1 σδS

=r ωµ

2σ. (2.24)

On the other hand, the time averaged power dissipated in the volume V of the isotropic, homogeneous, linear medium due to conductivity, dielectric, and magnetic losses can be determined as: Pldiel = σ 2 Z V | ¯E|2+ω 2 Z V (00| ¯E|2+ µ00| ¯H|2)dv (2.25)

where R_V denotes a volume integral and 00 and µ00 are imaginary parts of the material‘s complex permittivity and permability, respectively.

2.3. Planar realizations of a transmission line supporting TEM waves

Two most commonly used planar types of a transmission line supporting TEM wave propagation are microstrip and stripline. This is primarily because they can be fabricated by photolithographic processes and are easily miniaturized and integrated with both passive and active microwave devices.

(a)

(b)

Figure 2.3: Stripline transmission line: geometry (a), electric and magnetic field lines (b).

The geometry of stripline is shown in Fig. 2.3 together with a sketch of the field lines. A thin conducting strip of width W is centered between two wide conducting ground planes of separation b, and the region between the ground planes is filled with a dielectric material. In practice stripline is

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2.3. Planar realizations of a transmission line supporting TEM waves 25

usually constructed by etching the center conductor on a grounded dielectric substrate of thickness b/2 and then covering with another grounded substrate. Variations of the basic geometry presented in Fig. 2.3 include stripline with different dielectric substrate thicknesses (asymmetric stripline) or different dielectric constants (inhomogeneous stripline). Air dielectric is sometimes used when it is necessary to minimize loss.

Since stripline has two conductors and a homogeneous dielectric, it supports a TEM wave, and this is the usual mode of operation. However, stripline can also support higher order waveguide modes. These can usually be avoided in practice by restricting both the ground plane spacing and the sidewall width to less than λd/2. Shorting vias between the ground planes are often used to enforce this condition relative

to the sidewall width. Shorting vias should also be used to eliminate higher order modes that can be generated when an asymmetry is introduced between the ground planes (e.g., when a surface-mounted coaxial transition is used). A stripline can be considered in a way to be similar to a coaxial transmission line — both have a center conductor completely enclosed by an outer conductor and are uniformly filled with a dielectric medium.

Because TEM mode of stripline is of primary concern, an electrostatic analysis is sufficient to give the propagation constant and characteristic impedance. Phase velocity can be expressed as:

vp= 1 √ µ00r = c r , (2.26)

where c is the speed of light in free-space, and thus the propagation constant of stripline is:

β = ω vp

= ω√µ00r=

√

rk0. (2.27)

The characteristic impedance of a transmission line can be determined under the assumption of very low loss as: Z0 = r L C = √ LC C = 1 vpC , (2.28)

where L and C are the inductance and capacitance per unit length of the line. Thus, Z0 can be found

when knowing C.

Since stripline is a TEM line, the attenuation due to dielectric loss can be determined as

αd=

βtanδ

2 [N p/m] (2.29)

The attenuation due to conductor loss can be found by the perturbation method or Wheeler’s incremental inductance rule [34]. An approximate result is:

αc=    2.7x10−3_R SrZ0 30π(b−t) A f or √ rZ0 < 120 Ω 0.16RS Z0b B f or √ rZ0 > 120 Ω [N p/m] (2.30) with: A = 1 + W 2 b − t+ 1 π b + t b − tln( 2b − t t ), (2.31a)

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26 2. Strip transmission line medium for TEM wave propagation B = 1 + b 0.5W + 0.7t(0.5 + 0.414t W + 1 2πln 4πW t ) (2.31b)

where t is the thickness of the strip.

(a)

(b)

Figure 2.4: Microstrip transmission line: geometry (a), electric and magnetic field lines (b).

The geometry of a microstrip line is shown in Fig. 2.4together with a sketch of the field lines. The conductor of width W is printed on a thin, grounded dielectric substrate of thickness d and relative permittivity r.

If the dielectric substrate were not present (r = 1), microstrip would be a two-wire line consisting

of a flat strip conductor over a ground plane, embedded in a homogeneous medium (air). This would constitute a simple TEM transmission line with phase velocity vp = c and propagation constant β = k0.

The presence of the dielectric, particularly the fact that the dielectric does not fill the region above the strip (y > d), complicates the behavior and analysis of a microstrip line. Unlike stripline, where all the fields are contained within a homogeneous dielectric region, microstrip has some (usually most) of its field lines in the dielectric region between the strip conductor and the ground plane and some fraction in the air region above the substrate. For this reason microstrip line cannot support a pure TEM wave since the phase velocity of TEM fields in the dielectric region would be c/√r, while the phase velocity

of TEM fields in the air region would be c, so a phase-matching condition at the dielectric–air interface would be impossible to enforce.

In actuality, the exact fields of a microstrip line constitute a hybrid TM-TE wave and require more advanced analysis techniques. In most practical applications, however, the dielectric substrate is electrically very thin (d λ), and so the fields are quasi-TEM. In other words, the fields are essentially the same as those of the static (DC) case. Thus, good approximations for the phase velocity, propagation

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2.3. Planar realizations of a transmission line supporting TEM waves 27

constant, and characteristic impedance can be obtained from static, or quasi-static, solutions. Then the phase velocity and propagation constant can be expressed as:

vp= c √ e , (2.32) β = k0 √ e, (2.33)

where eis the effective dielectric constant of the microstrip line. Because some of the field lines are in

the dielectric region and some are in the air, the effective dielectric constant satisfies the relation:

1 < e< r (2.34)

and depends on the substrate dielectric constant, the substrate thickness, the conductor width, and the frequency. The effective dielectric constant of a microstrip line is given approximately by:

e= r+ 1 2 + r− 1 2 1 p1 + 12d/W. (2.35)

The effective dielectric constant can be interpreted as the dielectric constant of a homogeneous medium that equivalently replaces the air and dielectric regions of the microstrip line. The phase velocity and propagation constant are then given by2.25and2.26.

Considering a microstrip line as a quasi-TEM line, the attenuation due to dielectric loss can be determined as:

αd=

k0r(e− 1)tanδ

2√e(r− 1)

[N p/m], (2.36)

where tanδ is the loss tangent of the dielectric. The above provided formula account for the fact that the fields around the microstrip line are partially in air (lossless) and partially in the dielectric (lossy). The attenuation due to conductor loss is given approximately by

αc=

RS

Z0W

[N p/m]. (2.37)

For most microstrip substrates, conductor loss is more significant than dielectric loss. Exceptions may occur, however, with some semiconductor substrates.

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