Monthly Archives: April 2014

PNG Command Guidance Implementation

To implement PNG in a command guidance setting (i. e., no seeker), a differentiating filter must be used to estimate the LOS rate. As a result, command guidance is more sus­ceptible to noise than homing guidance. This issue is ex­acerbated as the engagement takes place further from the tracking station, noise increases, and guidance degrades. Within Reference 25, the authors address command-guided SAMs by spreading the acceleration requirements over tgo. The method requires estimates for target position, veloc­ity, acceleration, and tgo but takes into account nonlinear engagement geometry.

SAMPLE-AND-HOLD PERFORMANCE SPECIFICATIONS

A simplified model of a sample-and-hold circuit is a switch connected between the input and one terminal of a holding capacitor. The other terminal of the capacitor is tied to a ref­erence voltage (ground), and the output of the sample-and – hold is the voltage across the capacitor. The switch is con­trolled by a digital signal that determines the sampling time and the hold duration. Ideally, the output voltage tracks the input exactly when the switch is closed (track or sample mode) and stores a sample of the input voltage when the

Sampling moment

SAMPLE-AND-HOLD PERFORMANCE SPECIFICATIONS

Figure 3. Settling time and pedestal voltage during the sample-to – hold transition.

SAMPLE-AND-HOLD PERFORMANCE SPECIFICATIONS

switch opens (hold mode). In reality the behavior of the circuit deviates from this ideal model affecting the performance. The following subsections describe the specifications used to char­acterize sample-and-hold circuits.

Sample-Mode Specifications

The operation during the sample mode is similar to a voltage or current amplifier. Thus any specifications used to charac­terize an amplifier can be used to characterize the sample – and-hold circuit in sample mode. Some of the key specifica­tions in this mode of operation are offset, settling time, gain error, and nonlinearity. The offset is defined as the difference between the input and the output with no signal applied at the input. This is shown in Fig. 1. The time it takes for the output to settle within a certain error band around its final value with a step applied at the input is called the settling time. The size of the step is usually full scale as shown in Fig. 1 unless specified otherwise. Gain error and nonlinearity are both steady-state errors that describe the deviation of the magnitude transfer characteristic from a straight line with a slope of 1. As represented in Fig. 2, the gain error appears as the deviation of the slope of line P1P2 from 45° and can be expressed as Gerror = 1 — tan в. One definition of nonlinearity is the maximum deviation of the transfer characteristic from line P1P2. This is also shown in Fig. 2. Other sample-mode specifications include bandwidth, slew rate, distortion, and noise, which also characterize a general amplifier, are defined in a similar fashion.

REVIEW OF SUPERCONDUCTING COILS FOR FUSION Mirror Devices

In the last decade interest in the magnetic confinement in linear machine (mirror fusion) has strongly declined, but they were very popular in the 1970s. At the Kurchatov Institute in Moscow, a plasma trap named LIN-5 was built in 1970 by a split solenoid system with 0.2 m inner diameter and 5.8 T peak field at 1 kA. In 1975, for the LIN-5B machine, a 5 tonne, bath-cooled baseball type coil was wound with 6 km of square, monolithic NbTi conductor, 6.2 X 6.2 mm2 (2): at an operating current of 2 kA (75% of the short sample current), the peak field at the conductor is 5.6 T. The basic coil and conductor design is very similar to the US Baseball II-T wind­ing (see below).

In the United States, a superconducting magnetic mirror apparatus (SUMMA) consisting of four coils with 0.9 m outer diameter, 8.8 T peak field, and 18 MJ stored energy was built by NASA in the early 1970s. At Livermore, the first supercon­ducting baseball coil (Baseball II-T) was wound in 1971 with a square monolithic NbTi conductor, 6.35 X 6.35 mm2: the peak field in the pool boiling cooled winding is 7.5 T at 2.4 kA, with a stored energy of 12 MJ.

The Mirror Fusion Test Facility (MFTF) (see Fig. 1) was assembled at Lawrence Livermore National Laboratory in 1985 (3). It is the largest set of superconducting magnets for fusion, with a total mass >1200 t and about 75 t of strand. It consists of 8 C-type coils, 12 low-field, and 4 high-field sole­noids, including the A2 coils with a Nb3Sn insert (see Fig. 2). All magnets are pool cooled at 4.5 K by natural convection, with two-phase coolant outlet (<5% gas). The coils are all wound in the coil case, which acts as a cryostat, with ground insulation applied before winding. In the Yin-Yang coils, the winding form is fitted into the thick case by a copper bladder filled with urethane. The turn insulation is provided by G10 spacers, the layer insulation by perforated G-10 plates. The conductors for the solenoids (two types of NbTi and one Nb3Sn) consist of a thick multifilament composite soldered to the copper stabilizer (see Fig. 3). The joints are made by cold – welding of the composite. For the Yin-Yang coils, the conduc­tor is a square NbTi monolith with a perforated Cu strip wrapped around and soldered to increase the wetted surface. The conductor for the pancake-wound Nb3Sn insert, A21, is a flat multifilamentary composite soldered in the Cu housing after heat treatment (react and wind): the outer copper sur­face is oxidized to prevent solder wetting. All the conductors are designed to be cryostable. The fraction of /op//c is always smaller than 2/3. After cool-down and successful commission­ing of the magnet system, the project was discontinued in

1985. In 1990, the A2 coils were extracted and reassembled in the FENIX conductor test facility, which operated for three years at field levels as high as 13 T.

Tokamaks

The toroidal arrangement of plasma is the most promising confinement geometry, with the largest number of experimen­tal devices, including the pinch and reverse pinch, the stellar – ator group of machines, and the tokamaks, where a toroidal plasma current is initiated and sustained by a pulsed ohmic heating coil (central solenoid). Most of the superconducting magnets for fusion belong to the tokamak family, including four plasma experiments (T-7, T-15, TRIAM, Tore Supra) and a number of sizable technology demonstration devices (LCT, TESPE, DPC, Polo, ITER Model Coils).

12.4

І

I

1

ЖІІ

і*

‘vV

1

І

w

m

1.8

1

11

1

r446V////////

F

£

5.2

f

■<——- ►

6.9

t

2.1

Figure 3. Conductors for the MFTF mag­nets, from left to right: NbTi square conduc­tor with enhanced wet surface for the M, T, Al, and A20 coils, NbTi conductor for the S solenoids, react and wind Nb3Sn conductor for the A21 insert. (See Fig. 2 for coil identi­fication.)

The very first superconducting tokamak, named T-7, was built at the Kurchatov Institute, Moscow in 1975-1976 and first cooled down in 1977 (see Fig. 4). The toroidal winding system (4) consists of 48 circular double pancakes, in alumi­num case, with 60 turns in each coil. The average coil diame­ter is 1 m, the overall cold mass is 12 t and the stored energy 20 MJ at the nominal operation point. The torus was preas­
sembled into eight segments, individually tested before final assembly. The forced-flow conductor (see Fig. 5), is made from a strip of nine copper pipes, 2 mm inner diameter, with 16 multifilamentary and 32 single-core NbTi strands sitting in the grooves between the pipes and bonded to them by elec­troplating a 0.6 mm copper layer up to the final size of 28 X 4.5 mm. The cooling is by two-phase helium at 4.5 K, with all the pancakes connected in parallel. A 15-mm-thick copper shell sorrounds each coil and acts as an eddy currents shield for the poloidal field variations. The strands are not trans­posed and the conductor suffered from severe flux jumps, trig­gering quenches, during ramp up and ramp down. However, 80% of the design current (6 kA at 5 T peak field) was achieved. Operation of T-7 was discontinued at the Kurchatov Institute in 1987. Later, T-7 was transferred to the Chinese Institute of Plasma Physics, where it has been operating since 1996.

The toroidal field coils of the T-15 tokamak, first operated in 1988 at the Kurchatov Institute in Moscow, are the largest worldwide application of Nb3Sn conductors (see Fig. 6) (5). The 24 circular coils, with average diameter 2.4 m, consist each of 12 single pancakes, cooled in parallel, with the He inlet at the inner radius joints. The turn insulation is ob­tained by wet winding to balance the uneven conductor con­tour. Two stacks of six pancakes are vacuum impregnated in two-halves steel cases, eventually bolted together. The react and wind method was applied. The forced-flow conductor (see

Figure 4. The 24 double pancakes of T-7 in the final assembly (cour­tesy of V. Keilin, Kurchatov Institute).

Fig. 5), is a flat cable of 11 nonstabilized Nb3Sn strands, bonded after heat treatment to two copper pipes by an electro­plated Cu layer, 1.2 mm thick. Over 100 km of conductor have been manufactured, in units of 200 m. Each coil was tested before assembly and achieved the specification, although a steady-state voltage was observed, by far larger than the joint voltage, in the range of 2.5 mV to 10 mV/coil. The design cur­rent in the tokamak is 5.6 kA at 9.3 T peak field. The limited size of the cryoplant (allowing a mass flow rate of only 0.38 g/s • conductor) and the large radiation loss limited the op­erating temperature to the range of 9 K to 10 K. The highest operation point was 3.9 kA at 6.5 T, 7 K to 8 K, in agreement with the single-coil test and in excess of the original strand specification. The field transients due to plasma disruption, up to 40 T/s, were withstood without quench, with an in­crease of the outlet temperature by 0.25 K. The design ground voltage is 1.5 kV. For fast discharge, 250 V was applied at the terminals, with a time constant of 104 s.

The TRIAM device at the University of Kyushu, Fukuoka (Japan), is a compact, high-field tokamak, first operated in

1986. The superconducting 16 D-shaped toroidal field coils, with ~3 m average perimeter, are cooled by a pressurized liquid helium bath at 4.5 K. The poloidal field coils, wound with normal conductor, are placed inside the TF coils. The magnet cold mass is 30 t. The toroidal field conductor (see Fig. 7), consists of a large Nb3Sn bronze composite (10.5 X 3.3 mm for high grade, over a half million filaments) soldered after heat treatment in a copper housing with roughened side surfaces to improve the heat exchange. Beside the copper housing (RRR = 90), a Cu-clad high-purity Al profile (RRR = 3000) is used as a stabilizer. Each coil is a stack of six double pancakes, housed in a steel case. Three conductor grades, with the same width and decreasing height, are used with soldered joints for the double pancake. The conductor is de­signed to be cryostable and can withstand an energy input up to 7.8 J/cm3 at the operating conditions: in case of plasma disruption, a normal zone may locally occur but is recovered within 0.3 s. At 6.2 kA, 11 T, the ratio /op//c is 0.6. No quench event has been reported after three years of operation, includ­ing plasma disruptions (6).

Figure 5. The forced flow conductors for T-7 (left) and T-15 tokamaks, bonded by copper electroplating.

The tokamak Tore Supra has been assembled at Cadara – che, France, in 1987 (7). The poloidal coils are wound from copper conductors. The 18 pool cooled circular TF coils, wound from NbTi superconductor operating at superfluid helium, are the largest magnet mass, 160 ton, cooled at 1.8 K. Each coil is made out of 26 double pancake, with an average diameter of 2.6 m (see Fig. 8). A cowound prepreg tape, 0.15 mm thick, is used for the turn insulation. The pancake spacers, 2.2 mm, are built by a perforated prepreg thin plate with glued glass-
epoxy bottoms. The ground insulation is obtained by over­lapped prepreg plates. A 2-mm-thick steel case is shrink-fit­ted to the winding and contains the atmospheric, 1.8 K He bath. A thick steel case, with thermal insulation, is shrink – fitted and cooled at 4.2 K. The conductor is a rectangular NbTi/Cu/CuNi multifilament composite, 2.8 X 5.6 mm, wound on the short edge to minimize the ac loss from the poloidal field variation. The temperature margin is ~2.5 K, with Tcs = 4.25 K at 1400 A, 9 T peak field. Little copper cross section is used in the conductor: for stability, the He bath enthalpy up to the A point is available, due to the very high thermal conductivity of He II. The heat exchanger is placed underneath the coil case, open only at the bottom. In case of quench, a He gas pressure builds on the top of the case/cryo­stat, and siphons the whole He volume within 3 s through the bottom opening, providing a very fast quench propagation and limiting the hot spot temperature below 80 K. In 1988, about six months after first operation, an interpancake short oc­curred at one coil during a fast discharge, with 1.5 kV across the coil and ~60 V across pancakes, well below the expected Paschen minimum for helium. The damaged coil was later replaced with a spare coil and the dump voltage was de­creased to 500 V in order to limit the pancake voltage to ~20 V, which was experimentally assessed as the safe threshold to avoid interpancake discharge. The poloidal field variations and the plasma disruption result in a temperature increase in the He II bath as small as 0.01 K.

Besides the four above described tokamaks, a number of prototype superconducting coils have been built under na-

Figure 6. The 24 Nb3Sn coils of T-15 assembled with the 12 hori­zontal ports (courtesy of V. Keilin, Kurchatov Institute).

tional or international auspices. The demonstration coils pro­vide valuable opportunities to learn about magnet and con­ductor technology. In such projects, the pressure for a conservative design is less strong and the performance mar­gins can be better explored than in a plasma experimental device. In the IEA Large Coil Task at the Oak Ridge National Laboratory (8), six large D-shaped magnets, 3 X 3.5 m bore, have been built to the same common specification using sub­stantially different design approaches (see Table 2). The coils, assembled as a tokamak (see Fig. 9), operated (1984-1985) at the same design point (8 T peak field) with margins ranging from 120% to 140%. For the first time, a cable-in-conduit Nb3Sn conductor (react and wind coil manufacture) was used in a large-scale application and, despite the broad resistive transition observed in selected coil sections (similar to T-15 behavior), the coil reached 8 T with Tcs = 8 K. The cryostable conductors for the bath-cooled coils (GD, GE, JA) could be eas­ily graded (both layer and pancake windings) and, using sol­dered copper profiles as stabilizers, achieved impressive re­sults in terms of effective use of strand: from 1.4 t of strand in the GE conductor to 8.2 t in the forced-flow conductors of EU (NbTi) and WH (Nb3Sn), cooled by supercritical helium at 3.8 K, 10 bar to 15 bar. However, two out of three pool cooled coils could not be dumped to the design voltage of 1 kV. The nuclear heat load was simulated by heaters on the inner ra­dius. The poloidal field coil variations were reproduced by a pulsed coil traveling inside the torus: the pulsed field test, with ДВц = 0.1 T, AB± = 0.14 T, to = 1 s, could be completed only for three coils (JA, CH, EU). Over 90% of the stored en­ergy could be dumped in the external resistors, except for the WH coil, with short circuited radial plates.

At the same time of the LCT project, two small-size experi­ments with D-shaped coils were carried out at the Forsch- ungszentrum Karlsruhe (Germany) and at Toshiba (Japan). The six toroidal coils of TESPE at Karlsruhe (9) have a 0.5 X 0.6 m bore and are pool-cooled at 4.2 K (8 t total cold mass). The coils are wound as double pancakes, shrink-fitted in steel housings insulated by glass epoxy laminate. The steel case is electron-beam welded and serves both as liquid helium con­tainer and mechanical reinforcement. The conductor is a sol­dered flat cable of 24 multifilament NbTi strands ф = 1.45 mm, operating at 7 kA, with a peak field of 7 T. The TESPE torus was first operated in 1984 with a test program focused on mechanical load and high-voltage safety issues. The double pancake built by Toshiba (10) in 1983 had a Nb3Sn cable-in – conduit conductor, 18.3 X 15.7 mm, with 486 strands, encased into a 1-mm-thick 316L steel jacket. The D-shaped coil, 1.1 X

0. 9 m, was wound after heat teatment and tested at 10 kA,

Nb3Sn composite

Cu housing

10.5

EB welded, Cu housing

CuNi cladded Al stabilizer

12.5

NbTi strands

PbSn solder

Figure 7. Soldered monolithic conductors, stabi­lized with high-purity aluminum profiles, for TRIAM (left) and the helical coils of LHD (right).

in a peak background field of 10 T, provided by a small split coil. The test included hydraulic friction factor and stability.

The most recent development project for tokamak’s toroi­dal field coil is the ITER TF Model Coil, to be tested at the Forschungszentrum Karlsruhe by the end of 1999 in the back­ground field of the EU-LCT coil (11). The race-track-shaped coil, with 2.5 X 1.4 bore, has an overall weight of 31 t and a stored energy of 60 MJ at 70 kA (compared to «400 t and «4 GJ in an individual full size ITER coil). The peak field is about 9 T, compared to 12.5 T in ITER. The main goal of the ITER TF Model Coil is to demonstrate the winding technique, which uses precision-machined steel radial plates where the pancake wound Nb3Sn cable-in-conduit conductor is encased after the heat treatment (react and transfer method).

The Demonstration Poloidal Coil (DPC) project at JAERI, Naka (Japan), aimed at comparing conductor design options

Channels 4.2 K Thick v. case

Polyimide-alumina

chocks

Thin case, 1 8 K

Ground insulation

Glass-epoxy chocks

Perforated pancake spacers

Superconducting pancake

Figure 8. Winding pack layout for the pool-cooled, NbTi toroidal field coils of Tore Supra (courtesy of B. Turck, Tore Supra).

for pulsed-field coils. Three Nb3Sn react and wind winding models with фяу = 1.3 m, DPC-EX (12), US-DPC (13) and DPC-TJ (14) were tested in the DPC facility in pulsed mode up to about 7 T to 8 T, sandwiched between two pulsed NbTi solenoids (DPC-U), connected in series. The conductors are all forced-flow cable-in-conduit (see Fig. 10). The jacket material is Incoloy for the US-DPC (used for the first time). The DPC – TJ had a double jacket: the outer one is 3-D machined without bending and fitted by spot-welding to the conductor after the heat treatment. The strand surface is bare copper for the DPC-TJ, and the coupling loss is 1000 times larger compared to the US-DPC and DPC-EX conductors, with Cr plated strands (2 ms). Ramp rate limitation was observed in the US – DPC, probably due to transposition errors in the cable. The NbTi strands in the cable-in-conduit of the background coils were insulated to avoid interstrand coupling loss: the conduc­tor turned to be unstable due to the inability to redistribute effectively the current among the strands.

The Polo coil, a NbTi circular winding with ф = 3 m, has been tested in 1994 at the Forschungszentrum Karlsruhe (Germany) (15). The cable-in-conduit conductor (see Fig. 10), has two separate hydraulic circuits: stagnant, supercritical He at 4 bar in the annular cable region and forced flow, 2 g/ s, two-phase He at 4.5 K in the central pipe. This design allows a homogeneous temperature along the conductor with a small pressure drop. The strand is a NbTi/Cu/CuNi compos­ite and the subcables have CuNi or insulating barriers, re­sulting in very low coupling loss, t = 210 да. Four stainless – steel corner profiles are laser-welded around the cable. Phase resolved partial discharge was first used at 4 K to assess the integrity of the glass-epoxy insulation. A midpoint electrical connection in the winding enables to create very-high-field transients in a half coil by a fast discharge of the other half coil. The coil has been tested up to 15 kA, 3.6 T. A degrada­tion of Ic by 30% in dc operation has been observed compared to the strand performance. Polo does not have ramp rate limi­tation, the stability criterion being the only limiting criterion. Very-fast field transient, up to 1000 T/s are withstood without quench. High-voltage operation, up to 23 kV, has been demon­strated.

GD

GE

WH

CH

EU

JA

Winding type

14 layers

7 double pan-

4-in-hand, 12

22 pancakes

2-in-hand, 7 dou-

20 double pan-

3 grades

cake, 3 grades

double pan­cakes

ble pancakes

cake, 2 grades

Cooling method

Pool boiling

Pool boiling

Forced flow

Forced flow

Forced flow

Pool boiling

Conductor

Soldered flat

Divided flat cable

Nb3Sn cable-in-

Square, soldered

Divided, flat

Soldered flat

cable, on edge

conduit

cable

cable-in-

conduit

cable, on edge

Non-Cu Jop

586 A/mm2

525 A/mm2

265 A/mm2

302 A/mm2

393 A/mm2

327 A/mm2

Winding Jop

27.4 A/mm2

24.7 A/mm2

20.1 A/mm2

30.1 A/mm2

25.7 A/mm2

26.6 A/mm2

He inventory

1320 1

1735 1

440 1

110 1

663 1

1425 1

Total weight

43.9 t

38.6 t

33.7 t

41.7 t

39 t

39 t

SC strand

2t

1.4 t

8.2 t

3.5 t

8.2 t

2.6 t

Test voltage

2 kV

2.5 kV

9.2 kV

10 kV

12 kV

3 kV

The most recent pulsed-coil development for tokamak is the ITER CS Model Coil (16), a layer-wound, two-in-hand so­lenoid to be operated at 46 kA, 13 T with 0.4 T/s field rate, scheduled to be tested at Naka (Japan) in 1999. The stored energy is 641 MJ, compared with 13 GJ in the full size central solenoid. The conductor is a Nb3Sn cable-in-conduit with a
thick-walled Incoloy 908 jacket, manufactured as extruded and drawn-down bars, assembled by butt-welding into a over­size pipe, up to 250 m long: the strand bundle is pulled into the jacket and eventually rolled to the final size. The conduc­tor is insulated by prepreg glass fabric with interleaved kap – ton foil, applied after the heat treatment by controlled un – springing of the individual layers.

Stellarators

The plasma confinement can be achieved in stellarators by a number of winding configurations. The two large projects us­ing superconducting coils are the Large Helical Device (LHD) at Toki, Japan, and the Wendelstein VII-X (W7-X), to be as­sembled at Greifswald, Germany. The superconducting mag­net system of LHD operates in dc mode and consists of a two helical coils (17) wound around the toroidal vacuum vessel and three sets of circular poloidal coils (18). The helical coils are pool-cooled: initial operation will be at 4.2 K, 6.9 T, 13 kA and, at a later stage, at 1.8 K, 9.2 T, 17.3 kA. A precision tool with 13 numerically controlled driving axes has been used to wind in situ the two helical coils (see Fig. 11). The turn and

Figure 9. The six D-shaped LCT coils assembled as a torus in the vacuum tank (courtesy of M. Lubell, Oak Ridge National Laboratory).

layer insulation is provided by glass epoxy spacers graded across the winding pack to provide the best mechanical sup­port in the stressed area and the largest wet conductor sur­face at the peak field (the highest field and the highest me­chanical stress are not at the same winding location). The conductor (see Fig. 7), is a NbTi flat cable soldered to a CuNi – cladded Al stabilizer into a copper housing, eventually sealed by two electron beam welds. The conductor is designed to be cryostable, with /op//c = 0.55. The forced-flow conductor for the poloidal coils (see Fig. 10), is a NbTi cable-in-conduit with 486 strands (ф = 0.76 mm or 0.89 mm), 38% void fraction. To improve the current sharing among strands, the surface is not coated, with a coupling loss constant of 300 ms. The tem­perature margin is 1.2 K to 1.6 K, /op//c = 0.33. The joints between the conductor sections are realized by filament join­ing, resulting in 0.14 nH resistance at full current. The OV coil, ф = 11.5 m (see Fig. 12), has been wound on the site, with prepreg turn insulation. The first operation of the LHD was successfully started in March of 1998.

The magnet system of the W7-X torus consists of 50 non – planar and 20 planar coils assembled in five modular seg­ments (19). A first nonplanar model coil was completed in 1997 and tested in 1998. The completion of the machine is scheduled by the year 2002. The forced-flow conductor to be used for all the coils (see Fig. 10), is a NbTi cable-in-conduit with 243 strands, ф = 0.57 mm, 37% void fraction. The square jacket is made of a hardenable Al alloy, coextruded around the cable: it is soft after extrusion and during the winding process. After hardening at 170°C, it provides the required stiffness to the winding pack. The temperature margin is >1 K and /op//c = 0.5.

Propagative Path Loss Models

Free-Space Propagative Loss. When there is a LOS path be­tween the transmitter and receiver, the free-space propaga­tion model can be used to predict the signal strength. Such conditions occur in some satellite and terrestrial microwave communication links. Suppose that the distance between the transmitter and receiver is d meters, where d is in the far field. Then the free-space model (based on the Friis formula) gives

ing the propagative path loss (because the criteria for free – space propagation are not met). This topic has been very ex­tensively studied. Detailed information can be found in Kefs. 7, 10, 12, and 16. Computation of path loss is of particular interest to communication systems designers. Because the ac­tual KF communication environments encountered in practice are so numerous, a unified theoretical/analytical framework for estimating path loss is not feasible. Most system designers resort to empirical approaches and semianalytical methods, which have been validated by experimental/measured data, to estimate the path loss. The work of Okamura and Hata (13) is very widely used for path loss estimation. Okamura’s work is based purely on measured data, and Hata provided the empirical model to fit that data. The advantage of using empirical models and curve fitting to measured data is that it accounts for both known and unknown sources of path loss. On the other hand, the disadvantage is that the validity of the empirical model, derived from a set of data, is not guaran­teed for a different environment.

Let d be the distance between the transmitter and re­ceiver. Both theoretical and measurement-based models show that the average path loss [Lp(d)] increases directly propor­tional to dn, where n is called the path loss exponent. Typi­cally n > 2, as summarized in the Table 1. By contrast, in free space, N = 2. The path loss Lp(d) is given by

Lp(d) a dn

(44)

(45)

Lp(d) = Lp(d0) + 10ralog10

The reference point d0 is chosen such that Lp(d0) can be com­puted using the free-space path loss model.

Hata and COST-231 Models. This is one of the most widely used models for estimating path loss in KF communication channels. Based on extensive measured data, Okamura gen­erated sets of curves that characterize the median attenua­tion (50th percentile) Lp50, for a wide range of environments (range of carrier frequency ( fc), effective height of transmit­ting antenna (ht, eff), and distance d from transmitter. Hata (12) provided an empirical formulation from Okamura’s data, which shows good agreement (between the model and the measured data) for fc < 1.5 GHz. An extension of Hata’s model for frequencies up to 2 GHz is provided in Kef. 14. The Hata model and COST-231 models are given below: Hata Model

PtGtGrk2 (4 ж)2Ь

1

d2

Pr (d) =

(41)

where Pt and Pr are the transmitting and receiving power, respectively, with transmitting antenna gain Gt, receiving an­tenna gain Gr, and A is the wavelength of the carrier. The term L represents the losses in the system (L > 1). The path loss Lp is the difference between the transmitting and receiv­ing power expressed in decibels.

(4tt)2L

PtGtGTX2

P

Lp(dB) = 101og10 у = 10log

+ 20 log d (42)

Lp,50 = 69.55 + 26.16 log!0 fc — 13.82log!0 ht, eff — a(hr, eff)

As the signal propagative distance d increases, the received power decreases at 20 dB/decade, as seen from Eq. (42). An­other commonly used method to compute the signal power re­ceived Pr(d) is by measuring it relative to the received power Pr(d0) at a reference point (distance d0 from the transmitter) as given by

+ (44.9 — 6.55 log10 ht, eff) log10 d

(46)

Environment

Path Loss Exponent n

Free space

2

Urban cellular

2.7-4.0

In-building (non-LOS)

3.0-6.0

Shadowed urban cellular

4.0-6.0

(43)

Table 1. Path Loss Exponents for Different Environments

, d 2

PT(d)=PT(d0)[-£j ,d>d,

Outdoor Propagative Loss. In dealing with non-LOS envi­ronments, which is typical of most KF communication links, such as cellular/PCS, we need appropriate models for comput-

Lp (d) =

(dB) (48)

Pt

Log-normal shadowing Rayleigh/ Ricean fading

р i

(49)

COST-231 Model

Lp,50 = 463 + 33 9log10 fc — 13*82 log10 ht, eff — a(hr, eff)

+ (44.9 — 6.55 lOg10 ht eff) log10 d + CM (47)

where hr, eff is the effective height of the receiving antenna, a(hreff), is a correction factor based on hreff and CM is a correc­tion factor based on the propagative environment. The details regarding a(hreff), and CM are provided in Ref. 10. The range of values for which the Hata and COST-231 models are valid are summarized in Table 2.

The Hata model has a correction factor for rural environ­ments. In general, the Hata model and the COST-231 model provide an example of the path loss computation in a outdoor, non-LOS environment. A variation, called the COST231-Wal – fish-Ikegami model can be used for transmitting antennas above or below rooftops and is accurate for d > 20 m. A num­ber of models similar to these discussed in this section are also used in practice. So, the choice of path loss models must take into account all aspects of the propagative environment, including transmission frequency, distance of transmission, polarization, antenna heights, surface refractivity, terrain ir­regularity, foliage, climate, ground conductivity, and ground dielectric constant (10).

Indoor Propagative Loss. An increasing number of wireless communications applications are designed for indoor environ­ments. Hence, there is considerable interest in indoor propa­gation and in models for it. Although the characteristics of indoor propagation vary slowly [quasi-static behavior (15)] as compared with outdoor propagation, a key difference is that propagation within a building is strongly influenced by a number of factors, such as building type, layout, construction material (amount of metal used), types of partitions, and height and placement of antennas. As a result, the variability in signal propagation and hence the path loss is quite signifi­cant. The model best suited for characterizing path loss in indoor propagation is similar to that for log-normal shadow­ing. The path loss at a distance d from the transmitter is given by

Lp(d0) + 10ralog10 ( -^- ) + £2

where П is a normal RV with standard deviation a and n is the path loss exponent. It was reported in Ref. 15 that the typical range of n is 3 to 4. A comprehensive list of the typical values of n and a for a variety of indoor environments is pro­vided in Ref. 10.

Table 2. Range of Validity of Hata and COST-231 Models

Parameter

Range of Validity

Hata

COST-231

Carrier frequency fc

150-1500 MHz

1500-2000 MHz

Effective transmit height ht, efr

30-200 m

30-200 m

Effective receive height hr, eff

1-10 m

1-10 m

Distance d from transmitter

>1 km

1-20 km

Correction factors

a(hr, eff)

a(hr, eff), CM

Transmit power Path loss (Lp)

Shadow in margin (Ls)

Fast fading margin (Lf) Receiver sensitivity

Thermal noise floor

Figure 16. Different components of link budget:—propagative loss, shadowing margin, and fading margin.

Augmented Proportional Guidance (APNG)

Advanced guidance laws reduce acceleration requirements and miss distance but require more information (e. g., time – to-go and missile-target range) (19). In an attempt to take into account a constant target acceleration maneuver at, guidance engineers developed augmented proportional guidance (APNG). For APNG, the commanded acceleration is given by

acA PNG(t) = NVcX(t) + – Nat = acpng(0 + ^ Nat (15)

for stability robustness which implies that the guidance-control-seeker bandwidth bandwidth ю must be small when Vm is small, (R!, N, Vc) are large, or юа is small (high altitude and low missile speed V ).

m

From the above, it follows that we require the guidance – control-seeker bandwidth ю to satisfy

RNVc

юа<ю<Є

ZEM

V

m

or acAPNG(t) = N( 2 ), where ZEM = y + ytgo + – attgo is

^go. 2

the associated zero effort miss distance. Equation (15) shows that APNG is essentially PNG with an extra term to account for the maneuvering target. For this guidance law,

(16)

1—–

tf

at

acPNGmax — ["

N — 2

larger maximum acceleration but less acceleration than PNG for t > 0.2632tf. As a result, APNG is more fuel effi­cient for exoatmospheric applications than PNG. Finally, it should be noted that APNG minimizes Jjf a2c(T)dT subject to zero miss distance, linear dynamics, and constant target acceleration (8).