Polarization

Wave and antenna polarization. Polarization refers to the vector orientation of the radiated waves in space. As is known, the direction of oscillation of an electric field is always perpendicular to the direction of propagation. For an electromagnetic wave, if its electric field oscillation occurs only within a plane containing the direction of propagation, it is called linearly polarized or plane-polarized. This is because the locus of oscillation of the electric field vector within a plane perpendicular to the direction of propagation forms a straight line. On the other hand, when the locus of the tip of an electric field vector forms an ellipse or a circle, the electromagnetic wave is called an elliptically polarized or circularly polarized wave.

The decision to label polarization orientation according to the electric intensity is not as arbitrary as it seems; its causes the direction of polarization to be the same as the direction of the antenna. Thus, vertical antennas radiate vertically polarized waves, and horizontal antennas radiate horizontally polarized waves. There has been a tendency, over the years, to transfer the label to the antenna itself. Thus people often refer to antennas as vertically or horizontally polarized, whereas it is only their radiations that are so polarized.

It is a characteristic of antennas that the radiation they emit is polarized. These polarized waves are deterministic, which means that the field quantities are definite functions of time and position. On the other hand, other forms of radiation, for example light emitted by incoherent sources, such as the sun or light bulbs, has a random arrangement of field vectors and is said to be randomly polarized or unpolarized. In this case the field quantities are completely random and the components of the electric field are uncorrelated. In many situations the waves may be partially polarized. In fact, this case can be seen as the most general situation of wave polarization; a wave is partially polarized when it may be considered to be of two parts, one completely polarized and the other completely unpolarized. Since we are mainly interested in waves radiated from antennas, we consider only polarized waves.

Linear, Circular, and Elliptical Polarization. Consider a plane wave traveling in the positive z di­rection, with the electric field at all times in the x direction, as in Fig. 9(a). This wave is said to be linearly

 Fig. 9. Polarization of a wave: (a) linear, (b) circular, and (c) elliptical.

polarized (in the x direction), and its electric field as a function of time and position can be described by

 (32)

Ex — Exо sin(<yf — pz)

In general the electric field of a wave traveling in the z direction may have both an x and a y component, as shown in Fig. 9(b, c). If the two components Ex and Ey are of equal amplitude, the total electric field at a fixed value of z rotates as a function of time, with the tip of the vector forming a circular trace, and the wave is said to be circularly polarized [Fig. 9(b)].

Generally, the wave consists of two electric field components, Ex and Ey, of different amplitude ratios and relative phases. (Obviously, there are also magnetic fields, not shown in Fig. 9 to avoid confusion, with amplitudes proportional to and in phase with Ex and Ey, but orthogonal to the corresponding electric field vectors.) In this general situation, at a fixed value of z the resultant electric vector rotates as a function of time, the tip of the vector describing an ellipse, which is called the polarization ellipse, and the wave is said to be elliptically polarized [Fig. 9(c)]. The polarization ellipse may have any orientation, which is determined by its tilt angle, as shown in Fig. 10; the ratio of the major to the minor axis of the polarization ellipse is called the axial ratio (AR). Since the two cases of linear and circular polarization, can be seen as two particular cases of elliptical polarization, we will analyze the latter. Thus, for a wave traveling in the positive z direction, the electric field components in the x and y directions are

 (33)

Ex = EM sin {cot – fiz)

 (34)

Ey = Eyo sin(d»£ — f)z + S)

 Fig. 10. Polarization ellipse at z = 0 of an elliptically polarized electromagnetic wave.

where Ex0 and Ey0 are the amplitudes in the x and y directions, respectively, and 8 is the time-phase angle between them. The total instantaneous vector field E is

At z = 0, we have Ex = Ex0 sin at and Ey = Ey0 sin (at + 8). The expansion of Ey gives Ey = /^(sin cot cos S + cos cot sin S) (36)

/1 – (Ey/Eif

 Introduction of these into Eq.

Using the relation for Ex, we obtain sin cot = Ex/E and cos cot = (36) eliminates at, giving after rearranging

 If we define

E? 2 EXEV cos S Ey. 2 „

Eq. (37) takes the form

 + cE?= 1
 (38)

aE, – bEtEv

which is the equation of an ellipse, the polarization ellipse shown in Fig. 10. The line segment OA is the semimajor axis, and the line segment OB is the semiminor axis. The tilt angle of the ellipse is т. The axial ratio is

From this general case, the cases of linear and circular polarization can be found. Thus, if there is only Ex (Ey0 = 0), the wave is linearly polarized in the x direction, and if there is only Ey (Ex0 = 0), the wave is linearly polarized in the y direction. When both Ex and Ey exist, for linear polarization they must be in phase or antiphase with each other. In general, the necessary condition for linear polarization is that the time-phase difference between the two components must be a multiple of n. If 8 = 0, n, 2n, … and Ex0 = Ey0, the wave is linearly polarized but in a plane at an angle of ±n/4 with respect to the x axis (т = ±n/4). If the ratio of the amplitudes Ex0 and Ey0 is different, then the tilt angle will also be different.

If Ex0 = Ey0 and 8 = ± ж/2, the wave is circularly polarized. Generally, circular polarization can be achieved only when the magnitudes of the two components are the same and the time-phase angle between them is an odd multiple of nil.

Consider the case that 5 = 7t/2. Taking г = 0, from Eq. (33), (34), and (35) at t = 0 one has E =УЕу0,

and one-quarter cycle later, at cot = nil, one has E =*Ex0. Thus, at a fixed position (z = 0) the electric field vector rotates with time, tracing a circle. The sense of rotation, also referred to as the sense of polarization, can be defined by the sense of rotation of the wave as it is observed along the direction of propagation. Thus the above wave rotates clockwise if it is observed looking towards the source (viewing the wave approaching) or counterclockwise if it is observed looking away from the source (viewing the wave moving away). Thus, unless the wave direction is specified, there is ambiguity. The most generally accepted notation is that of the IEEE, by which the sense of rotation is always taken as that with the wave it traveling away from the observer. If the rotation is clockwise, the wave is right-handed or clockwise circularly polarized (RH or CW). If the rotation is counterclockwise, the wave is left-handed or counterclockwise circularly polarized (LH or CCW). Yet another way to define the polarization is with the aid of helical-beam antennas. A right-handed helical-beam antenna radiates (or receives) right-handed waves regardless of the position from which it is viewed, while a left-handed one radiates right-handed waves.

Although linear and circular polarizations can be seen as special cases of elliptical, usually, in practice, “elliptical polarization” refers to other than linear or circular. A wave is characterized as elliptically polarized if the tip of its electric vector forms an ellipse. For a wave to be elliptically polarized, its electric field must have two orthogonal linearly polarized components, Ex0 and Ey0. If the two components are not of the same magnitude, the time-phase angle between them must not be 0 or a multiple of n, while in the case of equal magnitude, the angle must not be an odd multiple of n/2. Thus, a wave that is not linearly or circular polarized is elliptically polarized. The sense of its rotation is determined according to the same rule as for circular polarization. So a wave is right-handed or clockwise elliptically polarized (RH or CW) if the rotation of its electric field is clockwise, and it is left-handed or counterclockwise elliptically polarized (LH or CCW) if the electric field vector rotates counterclockwise.

In addition to the sense of rotation, elliptically polarized waves are characterized by their axial ratio AR and their tilt angle т. The tilt angle is used to identify the spatial orientation of the ellipse and can

 Fig. 11. Polarization states of an electromagnetic wave represented with the aid of the Poincare sphere: (a) one octant of the Poincare sphere with polarization states, (b) the full range of polarization states in rectangular projection.

be measured counterclockwise or clockwise from the reference direction (Fig. 10). If the electric field of an elliptically polarized wave has two components of different magnitude with a time-phase angle between them an odd multiple of n/2, the polarization ellipse will not be tilted. Its position will be aligned with the principal axes of the field components, so that the major axis of the ellipse will be aligned with the axis of the larger field component and the minor axis with the smaller one.

The Poincare Sphere and Antenna Polarization Characteristics. The polarization of a wave can be represented and visualized with the aid of a Poincare sphere. The polarization state is described by a point on this sphere, whose longitude and latitude are related to parameters of the polarization ellipse. Each point represents a unique polarization state. On the Poincare sphere the north pole represents left circular polarization, the south pole right circular polarization, and the points along the equator linear polarization of different tilt angles. All other points on the sphere represent elliptical polarization states. One octant of the Poincare sphere with polarization states is shown in Figure 11(a), while the full range of polarization states is shown in Figure 11(b), which presents a rectangular projection of the Poincare sphere.

The polarization state described by a point on Poincare sphere can be expressed in terms of:

(1) The longitude L and latitude г of the point, which are related to the parameters of the polarization ellipse by

where t is the tilt angle with values 0 < т < n and є = cot 1(^ AR) with values —п/4 < є < +n/4. The axial ratio AR is negative and positive for right – and left-handed polarization respectively.

 і Polarization state 2т Fig. 12. One octant of the Poincare sphere, showing the relations of the angles т, є, y, and 8 that can be used to describe a polarization state.

(2) The angle subtended by the great circle drawn from a reference point on the equator, and the angle between the great circle and the equator:

great-circle angle = 2у and equator-to-great-circle angle = S

where y = tan — 1(Ey0/Ex0) with 0 < y < n/2 and 8 the time-phase difference between the components of the electric field (—n < 8 < +n).

All the above quantities т, є, y, and 8, are interrelated by trigonometric formulae (4), and knowing т, є one can determine y, 8 and vice versa. As a result, the polarization state can be described by either of these two sets of angles. The geometric relation between these angles is shown in Fig. 12.

The polarization state of an antenna is defined as the polarization state of the wave radiated by the antenna when it is transmitting. It is characterized by the axial ratio AR, the sense of rotation, and the tilt angle, which identifies the spatial orientation of the ellipse. However, care is needed in the characterization of the polarization of a receiving antenna. If the receiving antenna has a polarization that is different from that of the incident wave, a polarization mismatch occurs. In this case the amount of power extracted by the receiving antenna from the incident wave will be lower than the expected value, because of the polarization loss. A figure of merit, which can be used as a measure of polarization mismatch, is the polarization loss factor (PLF). It is defined as the cosine squared of the angle between the polarization states of the antenna in its transmitting mode and the incoming wave. Another quantity that can be used to describe the relation between the polarization characteristics of an antenna and an incoming wave is the polarization efficiency, also known as loss factor or polarization mismatch. It is defined as the ratio of the power received by an antenna from a given plane wave of arbitrary polarization to the power that would be received by the same antenna from a plane wave of the same power flux density and direction of propagation, whose state of polarization has been adjusted for maximum received power.

In general an antenna is designed for a specific polarization. This is the desired polarization and is called the copolarization or normal polarization, while the undesired polarization, usually taken orthogonal to the desired one, is known as the cross polarization or opposite polarization. The latter can be due to a change
of polarization characteristics during the propagation of waves, which is known as polarization rotation. In general an actual antenna does not completely discriminate against a cross-polarized wave, due to engineering and structural restrictions. The directivity pattern obtained over the entire direction on a representative plane for cross polarization with respect to the maximum directivity for normal polarization is called antenna cross-polarization discrimination, and it is an important factor in determining the antenna performance.

The polarization pattern gives the polarization characteristics of an antenna and is the spatial distribution of the polarization of the electric field vector radiated by the antenna over its radiation sphere. The description of the polarizations is accomplished by specifying reference lines, which are used to measure the tilt angles of polarization ellipses, or the directions of polarization for the case of linear polarization.

ICT – Energy – Concepts Towards Zero Power Information and Communication Technology

Edited by Giorgos Fagas, Luca Gammaitoni Douglas Paul and Gabriel Abadal Berini

A sustainable future for our information society relies on bridging the gap between the energy required to operate portable ICT devices with the energy available from portable/mobile sources.

The only viable solution is attacking the gap from both sides, i. e. to reduce the amount of energy dissipated during computation and to improve the efficiency in energy harvesting technologies.

This requires deeper and broader knowledge of fundamental processes and thorough understanding of how they apply to materials and engineering at the nanoscale, all the way up to the design of energy-efficient electronics.

This textbook is a first attempt to discuss such concepts towards Zero-Power ICT. The content is accessible to advanced undergraduates and early year researchers fascinated by this topic.

FREE FLIGHT

In April 1995, the FAA asked RTCA, Inc., an independent aviation advisory group, to develop a plan for air traffic man­agement called Free Flight (6). Free Flight hopes to extend airspace capacity by providing traffic flow management to air­craft during their en route phase. By October 1995, RTCA had defined Free Flight and outlined a plan for its implemen­tation (7).

The Free Flight system requires changes in the current method of air traffic control. Today, controllers provide posi­tive control to aircraft in controlled airspace. Free Flight will allow air carrier crews and dispatchers to choose a route of flight that is optimum in terms of time and economy. Eco­nomic savings will be beneficial both to the air carriers and to the passengers. Collaboration between flight crews and air traffic managers will be encouraged to provide flight planning that is beneficial to the aircraft and to the NAS. User flexibil­ity may be reduced to avoid poor weather along the route, to avoid special-use airspace, or to ensure safety as aircraft en­ter a high-density traffic area such as airports. The new sys­tem will offer the user fewer delays from congestion and greater flexibility in route determination (3).

Flights transitioning the airspace in Free Flight will have two zones surrounding the aircraft. A protected and an alert zone are used to provide safety for the flight. The size and shape of the zones depend on the size and speed of the air­craft. The goal is that the protected (or inner) zones of two aircraft will never touch. The aircraft may maneuver freely as long as its alert zone does not come in contact with another aircraft’s alert zone. When a conflict occurs between two air­craft alert zones, changes in speed, direction, or altitude must be made to resolve the conflict. The conflict resolution may be made by the air traffic manager or from the airborne collision avoidance system, TCAS (Traffic Alert and Collision Avoid­ance System).

The FAA and airspace users must invest in new technology to implement Free Flight. New communication, navigation, and surveillance systems are required to maintain situational awareness for both the air traffic manager and the flight crew. The FAA and aviation community are working together to phase in Free Flight over the next 10 years (8).

DISCRETE-TIME SC AND DIGITAL IMPLEMENTATIONS OF LC LADDER FILTERS

Both the SC filter and digital filter are sampled data (dis­crete-time) systems. The frequency response of a discrete­time system is adequately described in the z domain with z = esT, where s is the complex frequency and T is the sam­pling period. The frequency-domain design methods for SC filters and for digital filters have many similarities and often have the same z-domain transfer functions, in spite of the fact that SC filters are implemented as analog circuits whereas digital filters employ the digital arithmetic operations of addi­tion, multiplication, and delay and are implemented as com­puter programs or by dedicated hardware.

The frequency-domain design of discrete-time SC and digi­tal filters can be performed directly in the z domain. However, high-performance discrete-time filters may be designed by simulating continuous-time LC ladder filters as discrete-time filters. This is achieved in a way such that all of the above­mentioned favorable stability and sensitivity properties of LC filters are preserved. The transfer functions of the contin­uous-time LC ladder filter and its discrete-time counterparts are related by the bilinear transformation

s — (z – 1)/(z + 1)

In the following, we briefly discuss methods for converting continuous-time LC ladder filters into their discrete-time counterparts. The design details for SC and digital filter cir­cuit components and the treatment of parasitic and other nonideal effects are not considered here. Reference material on these topics can be found in the related articles in this encyclopedia and in Refs. 11 and 28-32.

Switched Capacitor Filters

SC filters that are based on LC ladder filters can be derived from RC-active filters that are themselves derived from LC ladder filters, preferably using the SFG simulation technique. In fact, the resistors in RC-active ladder filters can be simu­lated by switched capacitors, leading directly to the SC filter. There is a variety of different SC circuits for simulating the resistors in RC-active ladder filters. Different SC resistor cir­cuits, which are used to replace resistors in RC-active filters, may result in different frequency-domain relationships be­tween the transfer functions of the continuous-time LC ladder filter and its discrete-time SC counterpart. An example of SC resistor circuits is given in Fig. 8(a), which leads to the de­sired frequency-domain relationship given by the bilinear transformation.

In many SC filters, the ideal circuit capacitances are not significantly larger than the parasitic capacitances. In fact, the latter is around 20% of the former. Therefore, it is ex­tremely critical to only use those SC circuits that are not sen­sitive to parasitic capacitances. An example of a so-called stray-insensitive SC resistor circuits is given in Fig. 8(b). This circuit yields a different frequency transformation than the bilinear transformation. In order to achieve the desired bilin­ear transformation using simple SC ladder circuits, a so – called predistortion technique may be used that adds a posi­tive capacitance to one circuit component and subtracts an equal-valued negative capacitance from another circuit com­ponent, along with an impedance-scaling technique (30). It is noted that a similar technique is also used for LDI/LDD digi­tal filters.

Another alternative approach for designing SC filters is to directly simulate each inductor and each terminating resistor in the LC ladder filter. In this method, the interconnections of capacitors and simulated inductors/resistors are achieved by the so-called voltage inverter switchs (VIS), which contain active op-amps. This component simulation method guaran­tees the bilinear transformation between transfer functions of the continuous-time LC ladder filter and its discrete-time SC counterpart, which can be designed insensitive to parasitic capacitances. Nevertheless, considerable design effort and/or complicated switching signals may be required to achieve the low-sensitivity property.

In a way that is similar to the SC simulation of LC ladder filters, there are two alternative approaches for the digital simulation of LC ladder filters, namely the simulation of each LCR component and the simulation of the SFG represen­tation.

The component simulation method is achieved using wave digital filters (WDF) (11), where the circuit components of the LC ladder filter, such as inductors, capacitors and resistors, are directly simulated by corresponding digital domain com­ponents, such as delay registers and inverters. The parallel and serial interconnections of these digital components are facilitated by so-called parallel and serial adapters that con­tain adders and multipliers.

The SFG simulation of LC ladder filters employs lossless digital integrators and/or lossless digital differentiators (LDIs/LDDs) to replace the corresponding integration/differ­entiation operations in the continuous-time ladder SFG (28,31,32), where the continuous-time SFG may first be pre­distorted and impedance scaled in such a way that delay-free loops are avoided and all inductors and capacitors are individ­ually and directly realized as discrete-time LDI/LDD ele­ments. Each LDI/LDD element contains a delay register, a multiplier, and an adder. This SFG simulation approach does not require any special interconnection components and re­tains the one-to-one correspondence between the LC ladder filter parameters and its LDI/LDD digital counterparts. A distinguishing advantage of LDI/LDD ladder filters is that all of the state variables of inductor-simulating or capacitor – simulating LDI/LDD elements are independent of each other, thereby allowing concurrent implementation in parallel arith­metic schemes and flexible scheduling in bit-serial arithmetic schemes. It is noted that the very useful Brune sections can be perfectly implemented using the LDI/LDD method.

 2
 (c)

THE CONVENTIONAL PROCESS Alloy Fabrication

A high-purity fine-grained Nb-Ti alloy with chemical ho­mogeneity over both a large and small microstructural scale is an essential starting point to the production of Nb-Ti strands. The large liquid-solid phase separation shown in the phase diagram (Fig. 3), along with the high melting point of the Nb, makes it particularly difficult and expensive to produce a high-quality Nb-Ti alloy suitable for superconductor application. The main driving force for high homogeneity is the key role that precipitate quantity and morphology play in determining critical current den­sity, both of which are highly sensitive to composition. The development of a high-homogeneity Nb-Ti alloy was a cru­cial step in the advance toward high critical current Nb-Ti (see Ref. 15). The desired properties of the initial alloy billet are as follows:

1. The correct overall alloy composition to optimize Hc2, Tc and precipitation for pinning. The acceptable range is Nb-46-wt. % Ti to Nb-48 wt. % Ti.

2. Uniform composition over the entire billet to ensure optimum physical and mechanical properties over the entire filament.

3. Chemical homogeneity on a microstructural level in order to ensure uniform precipitation of the correct morphology (typically ± 1.5 wt. % Ti).

4. Low and controlled levels of impurity elements in or­der to ensure predictable superconducting and me­chanical properties.

5. Elimination of hard particles (typically Nb-rich) be­cause any particle that does not co-reduce with the alloy can result in filament drawing instability and ultimately strand breakage. The exterior of the final Nb-Ti rod must also be free of hard particles and must be smooth enough that it does not easily pick up particles during subsequent handling.

6. A fine (typically ASTM grain size 6 or smaller) and uniform grain size as it controls the distribution of precipitate nucleation sites. A fine grain size also im­proves diffusion barrier uniformity. Where high crit­ical current is less important, a larger grain size has been used increase ductility.

7. Low hardness (typically a Vickers hardness number of 170 or less) to ease co-deformation with softer sta­bilizer material.

The Nb-Ti alloy is prepared from high-purity Nb and Ti by consumable electrode vacuum-arc melting (where the electrodes are composites of Nb and Ti) and by electron – beam or plasma-arc melting. It is usually necessary to remelt the ingot two or three times in order to achieve the necessary chemical homogeneity. Primarily produced for the aviation industry, the high-purity source Ti is reduced from TiCl4 by Mg (the Kroll process). High-purity Nb is re­fined from lower-purity Nb by two or three electron-beam remelts. The lower-purity Nb source itselfis extracted from niobite-tantalite (Nb2O5 and Ta2O5) or pyrochlore (0.25% to 3% Nb2O5) ores by way of an intermediate ferroniobium alloy which is used on a relatively large scale for steel pro­duction. Table 1 lists the typical allowable ranges for im­purities, typified by specifications for the superconducting supercollider. The small level of allowable Ta has a histor­ical origin, and it is unlikely that additions of less than 1.5 wt. % Ta will have a significant impact on superconducting or mechanical properties. Increasing the level of Fe from 200 x-L/L (the specification of the superconducting super­collider as found in Ref. 16) to 500 xL/L (x-L/L is equivalent to the more commonly used ppm) actually has a beneficial effect as shown in Ref. 17.

The chemical inhomogeneities that may be observed in the alloy at this stage in production can be divided into two types based on size: macroinhomogeneities (those visible to
the eye) and microinhomogeneities (those requiring iden­tification using microscopes). The most common macroin­homogeneities are Ti-rich “freckles” and hard Nb-rich par­ticles. The Ti-rich freckles are so called from their appear­ance in ingot cross sections and are a result of Lorentz and buoyancy-driven flow of Ti-rich material between den­drites (see Ref. 18). Control of radial heat transfer and fluid flow in the melt pool eliminates the occurrence of freckles. Because of their relatively small size (typically 1 mm to 2 mm in diameter), compositional deviation (Ti-rich by 8 wt. % Ti to 10 wt. % Ti), and ductility, freckles are not in themselves particularly deleterious to strand production. The importance of the presence or rather the absence of Ti – rich freckles is as an indicator of good melt control. Ti-rich freckles are readily identified from flash radiographs of in­got cross sections. If flash radiography indicates that an ingot cross section is freckle-free, it is likely that smaller – scale microinhomogeneities, which are more difficult and expensive to quantify, have been kept to a minimum. A more serious macroinhomogeneity is the presence of hard Nb-rich particles which result in strand breakage failures (see Ref. 19). Nb-rich regions are a result of the high freez­ing point of Nb and can be eliminated by good process con­trol and remelting. The Nb-rich particles were the cause of many early strand failures but are rarely seen in modern production.

Microchemical inhomogeneity in the cast ingots of Nb-Ti is inevitable because of the coring produced by the large liquid-solid phase separation. The microhomogene­ity level can be qualitatively revealed by metallography using a composition-sensitive etch as shown in Fig. 4. In this example of a high-homogeneity-grade alloy, the micro­chemical variation is ± 1 wt. % Ti and has a wavelength of 100 xm to 200 xm. Commercial Nb-Ti alloys have mi­crochemical variations of ± 1 wt. % Ti to ± 4 wt. % Ti, with higher-homogeneity alloys costing more. Where high critical current density is less important, reduced micro­homogeneity can be acceptable in order to reduce cost but not to an extent that will reduce strand yield by causing drawability problems during subsequent processing.

The diameter of the initial cast Nb-Ti ingot ranges from 200 mm to 600 mm, and this is typically reduced to 150 mm by hot forging before being fully annealed in the single­phase в region (approximately 2hat 870°C). Extended an­neals can be used to reduce microchemical inhomogeneity but will increase the grain size and consequently reduce the density of precipitate nucleation sites.

 Figure 4. Microchemical inhomogeneity in an Nb-Ti alloy can be revealed using a composition-sensitive etch, as in this example of an high-homogeneity Fe-doped Nb-46 wt. % Ti alloy produced by Teledyne Wah Chang.

Table 1. Typical Nb-Ti Impurity Limits Based on Superconducting Supercollider Specifications

 Impurity Element Upper Limit (/uLfL) Impurity Element Upper Limit (/iL/L) Ta 2600 Cu 100 О 1000 Ni 100 С 200 Si 100 Fe 200 Cr 60 N 150 H 35 A1 100