Monthly Archives: April 2014

Tactical Missile Maneuverability

Tactical radar-guided missiles use a seeker with a radome. The radome causes a refraction or bending of the incom­ing radar wave, which in turn, gives a false indication of target location. This phenomenon can cause problems if the missile is highly maneuverable. One parameter that measures maneuverability is the so-called missile (pitch) turning rate frequency (or bandwidth) defined by (2)

y — a, y(tf) — 0

(tf — t)2

def У




PNG Miss Distance Performance: Impact of System Dynamics

For the two cases considered above, the associated relative displacement y & RX satisfies

where y denotes the time rate of change of flight path angle and a denotes angle-of-attack (AOA). o)a measures

* LD, Sref =

Splan ‘




1 + 0.75 Splan


def Y t gQSref Cl







|L( jo>)^NVcR –

юа Vm

constant e > 0. This result however, requires that the guidance-control-seeker bandwidth ю satisfies








From this, it follows that юа decreases with increasing mis­sile altitude and with decreasing missile speed Vm.

Radome Effects: Homing-Robustness Trade-offs

Let ю denote the guidance-control-seeker bandwidth.

• Homing Requirement. If ю is too small, homing is poor and large miss distances result. Typically, we de­sire

юа<ю (10)

that is, the guidance-control-seeker bandwidth should be sufficiently large so that the closed-loop system “accommodates” the maneuverability capabilities of the missile, which implies that the guidance-control – seeker bandwidth ю must be large when юа is large (low altitude and high missile speed Vm).

• Robustness Requirement. If ю were too large, how­ever, it is expected that problems can occur. This result in part, is because of radome-aerodynamic feedback of the missile acceleration am into X. Assuming n-pole dynamics, it can be shown that the missile accelera-


where G = NVc represents the guidance system, F =

ю n

(——- ) represents the flight control system, R is the

і + ю

radome slope (can be positive or negative), and A = -—— denotes the missile transfer function from am


to pitch rate в For stability robustness, we require the associated open-loop transfer function

the rate at which the missile rotates (changes flight path) by an equivalent AOA. Assuming that the missile is mod­eled as a “flying cylinder” (8) with length L and diameter

D, it has a lift coefficient CL = 2a[1 + 0.75 plan a], where


Noting that am = VmY is the missile

= FG[X — R0] = FG[X — RAam] =

tion am takes the form

L=f FGRA = NVc

юа = – = a

1 + FGRA

pg Vm Sref






acceleration, Q = ^ pVm the dynamic pressure, W = mg the missile weight, and p the density of air, it follows that

to satisfy an attenuation specification such as – <s for some sufficiently small

s + ю

The lower inequality should be satisfied for good homing. The upper inequality should be satisfied for good robust­ness with respect to radome effects.

• When юа is small (e. g., at high altitudes or low speeds), designers make the guidance-control-seeker band­width ю small but sufficiently large to accommodate missile maneuverability (i. e., satisfy the lower in­equality). In such a case, radome effects are small and the guidance loop remains stable yielding zero miss distance after a sufficiently long flight. One can, typically, improve homing performance by increasing ю and N. If they are increased too much, radome ef­fects become significant, miss distance can be high, and guidance loop instability can set in.

• When юа is large (e. g., at low altitudes or high speeds), designers would still like to make the guidance – control-seeker bandwidth ю sufficiently large to ac­commodate missile maneuverability (i. e., satisfy the lower inequality). This, result generally, can be accom­plished provided that radome effects are not too sig­nificant. Radome effects will be significant ifVm is too small, (R^ N, Vc) are too large, or юа is too small (i. e., too high an altitude and/or too low a missile speed Vm ).

Given the above, it therefore follows that designers are generally forced to trade off homing performance (band­width) for stability robustness properties. Missiles using thrust vectoring (e. g., exoatmospheric missiles) experience similar performance-stability robustness trade-offs.


The availability of superconducting bearings that are so nearly friction-free naturally leads to their consideration for flywheel energy storage. Flywheels with conventional bear­ings typically experience high-speed idling (i. e., no power in­put or output) losses of the order of about 1% per hour. With superconducting bearings, it is believed that losses as little as 0.1% per hour are achievable. When coupled with efficient motors/generators and power electronics (capable of losses as low as 4% on input and output), the potential exists for con­structing flywheels with diurnal storage efficiencies of 90%. Probably only one other technology is capable of achieving such high diurnal storage efficiencies: large superconducting magnetic energy storage, which employs superconducting coils hundreds of meters in diameter.

Electric utilities have a great need for inexpensive energy storage, such as flywheels, because their inexpensive baseload capacity is typically underutilized at night and they must use expensive generating sources to meet their peak loads during the day. A distributed network of diurnal-storage devices could also make use of underutilized capacity in transmission lines at night and add robustness to the electric grid. These factors are expected to become more important in the coming deregulation of the electric utility industry. Efficient energy storage would also be beneficial to renewable energy technol­ogies, such as photovoltaics and wind turbines.

With modern graphite fiber/epoxy materials, the inertial section of a flywheel rotates with rim speeds well in excess of
1000 m/s and achieves energy densities greater than those of advanced batteries. The kinetic energy in a (large) Frisbee – sized flywheel with this rim speed is about 1 kWh, and a per­son-sized flywheel could store 20-40 kWh. Although design concepts for flywheels that employ superconducting bearings with up to 10 MWh have been proposed, the most advanced experimental versions at present are in the 100 Wh to 1 kWh class.

Because superconducting levitation is versatile over a wide range of stiffness and damping, it has been suggested for nu­merous applications. Superconducting bearings, like magnetic bearings, do not require a lubricant, which could be a major advantage in harsh chemical or thermal environments. Su­perconducting bearings are particularly interesting for cryo­genic turbopumps. The low friction of the superconducting bearing allows its use in high-precision gyroscopes and gra­vimeters. The hysteretic nature of superconducting levitation has suggested its use in docking vehicles in space. As one vehicle approaches another it would experience a decelerating repulsive force. After the relative velocities have disappeared at a small separation distance, the vehicles would experience an attractive force if their distances tend to separate. The sta­ble levitational force suggests application in magnetically lev­itated conveyor systems in clean-room environments where high purity requirements mandate no mechanical contact. Trapped-field HTSs have been suggested for constructing ve­hicle magnets to be used in electrodynamic levitation of high­speed trains (see Magnetic levitation).

A Tapped-Delay Line Channel Model

In addition to providing channel information such as the rms delay spread, the power-delay profile фе(т), also provides a means for modeling the channel using a tapped-delay line (FIR) model. From Eq. (5), an(t) is the amplitude/gain coeffi­cient for a path arriving with delay Tn(t). A typical power-de – lay profile is shown in Fig. 14, which in the second figure, is uniformly sampled into equal delay bins. In general, the dif­ferent bins contain a number of received signals (correspond­ing to different paths) whose times of arrival lie within the particular delay bin. These signals are represented by an im­pulse at the center of each delay bin that has an amplitude with the appropriate statistical distribution (Rayleigh, Ri – cean, etc.). In deriving this model, two assumptions are made:

Time flat Frequency selective

Time selective Frequency selective

Frequency flat

Time selective

Time flat

Frequency flat






T 1 c



Symbol duration (Ts)

Figure 13. A typical power delay profile and the method of sampling the power delay profile to generate a tapped-delay line model.

• there are sufficient number of rays clustered together in each delay bin;

• the statistical distribution of the envelope is known.

The rate of sampling the power-delay profile is affected by the time resolution desired and also the bandwidth of the trans­mitted signal. The next step after sampling the power-delay profile is to use a threshold (say X dB below the peak of the power-delay profile), and using the threshold to truncate

A Tapped-Delay Line Channel Model

those samples below the threshold. This model can be imple­mented by using a tapped-delay line or FIK model, thereby allowing us to model any arbitrary channel.


Relaxation oscillations are characterized by two time scales, and exhibit qualitatively different behaviors than sinusoidal or harmonic oscillations. This distinction is par­ticularly prominent in synchronization and desynchroniza­tion in networks of relaxation oscillators. The unique prop­erties in relaxation oscillators have led to new and promis­ing applications to neural computation, including scene analysis. It should be noted that networks of relaxation os­cillations often lead to very complex behaviors other than synchronous and antiphase solutions. Even with identical oscillators and nearest neighbor coupling, traveling waves and other complex spatiotemporal patterns can occur [31].

Relaxation oscillations with a singular parameter lend themselves to analysis by singular perturbation theory [32]. Singular perturbation theory in turn yields a geomet­ric approach to analyzing relaxation oscillation systems, as illustrated in Figs. 4 and 6. Also based on singular so­lutions, Linsay and Wang [33] proposed a fast method to numerically integrate relaxation oscillator networks. Their technique, called the singular limit method, is derived in the singular limit є ^ 0. A numerical algorithm is given for the LEGION network, and it produces large speedup com­pared to commonly used integration methods such as the Runge-Kutta method. The singular limit method makes it possible to simulate large-scale networks of relaxation os­cillators.

Computation using relaxation oscillator networks is in­herently parallel, where each single oscillator operates in parallel with all the other oscillators. This feature, plus continuous-time dynamics makes oscillator networks at­tractive for direct hardware implementation. Using CMOS technology, for example, Cosp and Madrenas [34] fabri­cated a VLSI chip for a 16 x 16 LEGION network and used the chip for a number of segmentation tasks. With its dy­namical and biological foundations, oscillatory correlation promise to offer a general computational framework.



If the azimuthal homogeneity of the magnetic field of the per­manent magnet is high, for example, if the magnet is a cylin­der with uniform magnetization throughout, the levitated magnet rotates freely above the superconductor. As long as the distribution of magnetic flux in the superconductor does not change, rotation encounters no resistance. If the magnet is spinning, the hysteretic loss in the superconductor de­creases the rotational rate.

In an electromechanical system, such as a magnetic bear­ing, the parameters of interest are the levitational force, stiff­ness, damping, and rotational loss. The 280 kPa levitational pressure is lower than that achievable in a conventional elec­tromagnetic bearing (~1 MPa) and significantly lower than that typically achieved in mechanical roller bearings (>10 MPa). The amount of mass levitated directly depends on the number and size of permanent magnets and superconductors available. In the present early period of technological develop­ment for superconducting bearings, several laboratories have stably levitated masses greater than 100 kg.

In practical superconductor bearings, the low levitational pressure available in the interaction between the permanent magnet and the superconductor is often augmented by vari­ous hybrid schemes in which interactions between pairs of permanent magnets provide the bulk of the levitational force. These interactions are unstable, as Earnshaw’s theorem pre­dicts, but the inclusion of a properly designed superconduct­ing component in the bearing is sufficient to stabilize the complete bearing. Augmentation takes the form of an Ever – shed-type design, in which a pair of permanent magnets is in attractive levitation, employs permanent magnets in repul­sive levitation, or uses active magnetic bearings (5).

The hysteretic nature of a superconducting bearing also makes damping of translational motion amplitude-dependent. For low-amplitude vibrations, damping is small, but quickly increases as the vibrational amplitude increases. This hyster – etic nature of the HTS bearing thus contributes to the ro­bustness of the system. The hysteretic nature also results in a larger uncertainty of the equilibrium position of the rotor than is typical in most rotating machinery. This uncertainty requires larger running gaps between moving and stationary parts.

The ease with which a permanent magnet spins, when lev­itated over a superconductor, and the absence of contact be­tween the surfaces, produce the illusion that the rotation is frictionless. In reality, small magnetic losses gradually slow the rotation. The losses are primarily the result of azimuthal inhomogeneities in the magnetization of the permanent mag­net, which produce hysteretic loss in the superconductor. Typ­ically, in permanent magnets with the best homogeneities, at a fixed radius above the rotating surface, the amplitude of the

ac component of the magnetic field is of the order of 1% of the average field at that radius. Although small, this inhomoge­neity is sufficient to cause a detectable decay in rotational rate when the magnet spins in vacuum.

A figure of note for the rotational decay of a bearing is the coefficient of friction (COF), defined as the rotational drag force divided by the levitational force (weight of the levitated rotor). The COF for a mechanical roller bearing is of the order of 103. The COF for an active magnetic bearing is about 10 4 when parasitic losses for the feedback circuits and power for the electromagnets are factored in. Measured COFs for simple superconductor bearings are as low as 107. The para­sitic losses of a superconducting bearing are the power re­quired to keep the superconductor cold. For refrigerators that operate at about 30% of Carnot efficiency (the theoretical maximum) about 14 W of electricity are required to remove 1 W of heat at liquid nitrogen temperatures. Thus, the equiva­lent COF for an HTS bearing is about 2 X 10~6, about two orders of magnitude lower than the best alternative bearing.

Two magnetomechanical resonances occur in a magneti­cally suspended rotor with a polar moment of inertia greater than the transverse moment (i. e., a disk geometry): one is ver­tical and one is radial. In practice, the vertical resonance has a minimal effect on the COF in most superconducting bearing systems. The radial resonance occurs when the rotational fre­quency is close to that of the natural radial frequency of the rotor’s vibration. If the rotor has a transverse moment of iner­tia greater than its polar moment, then there is an additional resonance, having the form of a conical vibration, that is, with the top of the rotor moving to one side while the bottom of the rotor moves to the opposite side. Because superconducting bearings have low stiffness, the resonances occur at low fre­quencies on the order of several hertz. This, together with the large clearances possible with superconducting bearings, leads to a robust bearing system.

Figure 4 shows the COF as a function of rotational fre­quency for a cylindrical permanent magnet levitated 10 mm above a single YBCO cylinder and spinning in a vacuum chamber. One may divide the behavior into three regions: be­low the resonance, the radial resonance, and above the reso­nance. Below the resonance (f < 3 Hz), the losses are caused

с 0.00010




г7 ш

20 40 60 80

Rotational frequency (Hz)


Figure 4. Coefficient of friction versus rotational frequency for a 25.4 mm dia., 9.6 mm high cylindrical permanent magnet levitated 10 mm above a YBCO cylinder.









0.0010 by the inhomogeneity of the permanent magnet’s field. The resonance region (3-20 Hz), shown in Fig. 4, is relatively broad. In some systems, especially those with well-balanced rotors, the resonance is very narrow. Above the resonance (>20 Hz), the losses are affected by an additional factor, which is caused by the rotation of the magnet about its center of mass rather than its center of geometry or center of magne­tism. As shown in Fig. 4, above and below the resonance, the losses are mostly velocity-independent. However, detailed studies of losses in superconducting bearings show that small velocity-dependent effects are present which are intrinsic to the superconductors (19-21).

Because of size limitations encountered when high-perfor­mance bulk superconductors are produced, a large bearing system needs an array of superconductors. In the case of a single magnet levitated over an array of superconductors, an additional loss arises from magnetization of the individual su­perconductors upon levitation. The magnetization of the array leads to an ac magnetic field seen by the rotating permanent magnet and eddy current losses that depend on the electrical conductivity of the permanent magnet.

Some alternative bearing concepts that involve supercon­ductors exhibit even lower COFs. The Evershed-type hybrid has a COF of just over 10~8 (22). The velocity-dependent losses associated with superconducting arrays are also greatly reduced with this bearing design. A mixed-^ (where ^ is the magnetic permeability) bearing (23) has a COF of just over 10~9. In this bearing, a soft ferromagnetic cylinder (^ > 1) is levitated in attractive levitation between two permanent magnets and stabilized by a superconductor (^ < 1) placed between the rotor and each of the permanent magnets.