Monthly Archives: March 2014

The dynamics of a "simple switch&quot

In order to describe the physics of a switch we need to introduce a dynamical model capable of capturing the main features of a switch, regardless if it is realized with a purely mechanical, electro-mechanical or electronic technology. According to the reasoning originally developed by Landauer7 we assume that the switch dynamics can be described by a single degree of

freedom (dof ) that is identified with x. Let’s suppose that x is a continuous variable (e. g. the

position of a cursor or the value of a magnetic field) that can assume two identifiable stable states: e. g. x<0 (logic state "0"), x>0 (logic state "1"). The two states, in order to be dynamically stable, are separated by some energy barrier that should be surpassed in order to perform the switch event. This situation can be mathematically described by a second order differential equation like:

mx= – dfaU (x) – myx + F (15)

Where F is an external force that can be applied when we want to change state, у is the frictional force that represent dissipative effects in the switch dynamics and

U (x) = – 2 x2 +1 x4 + c (16)

is the bistable potential shown in fig. 12. The additive constant, c, is an arbitrary constant that sets the zero level of the potential energy.

step 1 step 2 step 3

The dynamics of a &amp;quot;simple switch&amp;quot

Figure 13. Potential U (x) + F. First procedure: From left to right, step 1,2,3. Step 1 and step 3, F=0; step 2, F=-Fa.

Suppose that at a certain time t0, the system is x<0 (logic state 0) and F = 0. This is equivalent of picturing a material particle of mass m and position x, sitting at rest at the minimum of the left well in figurel5andis an equilibriumconditionforthe switch.

Accor dieg tothtt model ife swtrohevent irtpbs pronucedttisnecnssarytoapely an eeternal forcc F eaponieot bringing the particle from the left well (at rest at the bottom) into the right well (at rest at the bottom). Clearly this can be done in more than one way.

Atan exnmple – we start eisaissing what we call the first procedure: a three-step procedure based on the application of a large and constant force F=-F0, with F0 >0.

We start in step 1 (see fig. 13) with the particle on ^]^e left well and F=0. In step 2 we apply for a certain time F=-F0 in order to change the potential shape into U (x) – F0 x (see fig. 13, step 2).

Clearly after some time the particle will move toward the right until it reaches the bottom of eight спеП, ^с0^єі^^1 efterfewnseillalionc, rtsnttlesdpcto tiie presence of the dissipative force. Then, step 3, the force F is removed and the system returns to the unperturbed potential of Ff° . rc.°i thiewey0! ewitch eenntc anpnproducccL

Whct inthe minimum worp thstfhe forceF must perform to make the device switch from 0 to 1 (or equivalently from 1 to 0). The work is computed as:


p = J ,F(n)x. r (1 7)


where Xj and x2 are the starting and ending position of our particle.

In the above example the work is readily computed by considering that the total force acting on the particle is F0 + dU/dx and has caused a displacement from Xj=-1 to x2=1. The total work performed is easily computed to be L0 = 2 F0. Is this the minimum work? Clearly it is not.

In order to demonstrate that it is possible to switch with a less work, let’s consider the following 5-step procedure (second procedure, see Fig. 14): in step 1 and step 5 let F=0; in step 2 lower the potential barrier by applying a proper force F=-x. In step 3 apply an additional small

constant force – F1 that tilts the potential toward the left. Now F=-x-F1. At this point the material particle slowly moves toward the right. When the particle reaches the far right limit proceed to step 4 and remove the F=-x force. Finally in step 5, remove the additional force F=F1 and restore the original bistable potential.

step 1 step 2 step 3

The dynamics of a &amp;quot;simple switch&amp;quot

Figure 14. Potential U (x) + F. Second procedure: Step 1 and step 5, F=0; step 2 F=-x; step 3 F=-x-F1; step 4 F=-F1;

In order to compute the work performed on the particle observe that in step 1-2 and step 4-5 no work is performed because the applied force does not produce any displacement (or a negligible one). The only work performed happened to be during step 3 where it is readily computed as L1 = 2 F1. Now, by the moment that F1 << F0, as anticipated, we have L1 << L0. Based on this reasoning it can be concluded that, provided an arbitrarily small constant force is applied during the tilt, the resulting work will be arbitrarily small. Thus it can be concluded that in principle it is possible to perform the switching event by spending zero energy provided two conditions are satisfied: 1) The total work performed on the system by the external force has to be zero. 2) The switch event must proceed with a speed arbitrarily small in order to have arbitrarilysmalllossesdue tofriction.


Historically, pilots flew paths defined by VOR (VHF Omnidi­rectional Radiorange) radials or by nondirectional beacon sig­nals using basic display of sensor data. Such paths are re­stricted to be defined as a path directly to or from a navigation station. Modern aircraft use computer-based equipment, designated RNAV (Area Navigation) equipment, to navigate without such restrictions. The desired path can then be direct to any geographic location. The RNAV equip­ment calculates the aircraft position and synthesizes a dis­play of data as if the navigation station were located at the destination. However, much airspace is still made available to the minimally equipped pilot by defining the paths in terms of the basic navigation stations.

Aircraft navigation requires the definition of the intended flight path, the aircraft position estimation function, and the steering function. A commonly understood definition of the intended flight path is necessary to allow an orderly flow of traffic with proper separation. The position estimation func­tion and the steering function are necessary to keep the air­craft on the intended flight path.

Navigation accuracy is a measure of the ability of the pilot or equipment to maintain the true aircraft position near the intended flight path. Generally, navigation accuracy focuses mostly on crosstrack error, although in some cases the alongtrack error can be significant. Figure 1 shows three com­ponents of lateral navigation accuracy.

Standardized flight paths are provided by government agencies to control and separate aircraft in the airspace. Path definition error is the error in defining the intended path. This error may include the effects of data resolution, mag­netic variation, location survey, and so on.

Position estimation error is the difference between the po­sition estimate and the true position of the aircraft. This com­ponent is primarily dependent upon the quality of the naviga­tion sensors used to form the position estimate.

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright © 1999 John Wiley & Sons, Inc.

“ Г


Intended path

Path definition Error

Path defined by Pilot or RNAV equipment

A_____ і

t t

Flight technical error

. j I Estimated position “rv – of aircraft

Position estimation error

True position of aircraft

95% Accuracy limit


, Intended path




95% Accuracy limit

99.999% Integrity limit


99.999% Integrity limit Figure 2. RNP-X accuracy and integrity limits.

Figure 1. Aircraft navigation errors.


Figure 3. Example of an airway chart.


i sfaVsS

Flight technical error is the indicated lateral deviation of the aircraft position with respect to the defined path. RNAV systems in larger aircraft have provisions to couple a steering signal to a control system to automatically steer the aircraft to the intended path. In less equipped aircraft, the RNAV sys­tem simply provides a display indication of the crosstrack dis­tance to the intended path, and the pilot manually provides the steering correction.


Josephson junctions are the fundamental building blocks for a variety of superconducting electronics appli­cations, including high-speed, low-power digital logic, and sensitive magnetic field and high-frequency elec­tromagnetic detectors. A Josephson junction consists of a “weak” connection between two superconductors which exhibits the Josephson effects (described below). While low-temperature superconductor (LTS) Joseph­son junction technology is well-developed, high-temperature superconductor (HTS) Josephson junctions are still relatively immature. Nonetheless, extensive HTS junction fabrication efforts are in progress due to the possibility of applying Josephson effects at temperatures compatible with reliable, low-cost refrigerators. In this article we discuss the more common approaches to HTS junction fabrication and optimization, with a focus on potential digital circuit applications.


There are two main types of defibrillators used today, the au­tomatic internal defibrillator and the external defibrillator.




Figure 2. Diagram of an implantable cardiovertor defi­brillator. The pulse generator is implanted in the pectoral region. A transvenous catheter electrode is threaded from the subclavian vein to the superior vena cava and into the right ventricle of the heart. This catheter also contains a pace/sense electrode on the tip. Implantation of this sys­tem only requires sedation of the patient and a local anes­thetic.

The automatic internal cardiovertor-defibrillator (ICD) is a 40 mL to 100 mL box with electrodes attached to it that extend either onto the epicardial (outside) surface of the heart or into the chambers of the heart (Fig. 2). This device monitors the cardiac rhythm and if ventricular fibrillation is detected deliv­ers a strong electrical shock, usually 10 J to 30 J. Currently, implantation of an ICD is the treatment of choice for patients who have survived an initial episode of sudden cardiac death and do not have a treatable cause for their arrhythmia. (1,2).


Figure 4. Probability of survival as a function of time in minutes from collapse to the beginning of cardiopulmonary resuscitation and time of defibrillation. Each contour represents a different time inter­val from collapse to the beginning of cardiopulmonary resuscitation. Note that the probability of survival from cardiac arrest drops as the time from arrest to the beginning of cardiopulmonary resuscitation increases and as the time to defibrillation increases. Reproduced with permission from the American Heart Association (98).

Collapse to defibrillation interval

The external defibrillator is a device distributed through­out the prehospital and hospital setting (the paddles popular­ized in television hospital dramas). These devices deliver a large electric shock, usually 100 J to 360 J, to the chest wall of a patient via either hand-held paddle electrodes or self­adhesive patch electrodes (Fig. 3). The external defibrillator is used to stop both ventricular and atrial tachyarrhythmias. Traditionally, the external defibrillator operator has analyzed the patient’s heart rhythm and decided whether or not to de­liver the electrical shock. Newer devices, intended to be used by minimally trained lay persons, such as police, firefighters, flight attendants, or even passers-by, will analyze the pa-


Figure 3. Diagram of electrode patch placement for external defi­brillation. One electrode is placed over the right border of the ster­num. The second electrode is placed on the left axillary line overlying the apex of the heart.

tient’s heart rhythm and determine whether or not a shock should be delivered without intervention from the operator. Perhaps someday defibrillators will be as common as fire ex­tinguishers. Quick action is vital to the survival of ventricular fibrillation. The rate of survival following an episode of ven­tricular tachycardia/ventricular fibrillation is inversely re­lated to the time that the patient’s heart has been in that rhythm before a shock is delivered (Fig. 4). Therefore, it is hypothesized that more patients would be saved if defibrilla­tion occurred as soon as possible by individuals likely to be near the person when his/her heart fibrillates (3).

Normal Mode Helix

Let’s consider a helix with its axis along the z axis, centered at the origin [Fig. 2(a)]. The geometry of the helix reduces to a loop when the pitch angle a approaches zero and to a straight wire when it approaches 90°. Since the limiting ge­ometries of the helix are a loop and a dipole, the far field radiated by a small helix can be described by the radiation fields of a small loop and a short dipole when dimensions are small compared to a wavelength. The analysis of a small short helix is facilitated by assuming that the helix consists of a number of small loops and short dipoles connected in series as in Fig. 2(b). The diameter of the loops is the same as the helix diameter (D) and the length of the dipoles is approxi­mately the same as the spacing (S) between turns of the helix. Because the helix is small and short, the current distribution is assumed to be uniform in magnitude and phase over the entire length of the helix. For the same reason, the far-field pattern will be independent of the number of turns and thus can be obtained by considering the pattern of a single-turn helix which consists of a single small loop of diameter D and one short dipole of length S.

Assume that the complex amplitude of current is I and the angular frequency is w. The radiation electric field of the small loop of diameter D has only an Еф component, given by



Edt = rjk2IA sin (9

ф 4n r

where A = nD2/4 is the area of the loop, k = the propa­

gation constant, and r = V^Je the intrinsic impedance. The far field of the short dipole, or the Hertzian dipole of length S, has only an Ee component, given by


Eg = jcOf^IS——— sin в

4n r

The total radiation field for one turn is then given by



E = agEg + афЕф = {agjco^S + афф2А}1′:х-^ sin в (5)

T – C

4 nr

The normalized radiation field pattern f(e) of the normal mode helix is


f(9) = sin 9

which is the same as that of the Hertzian dipole and the small loop and is shown in Fig. 2(c). The field is zero along the axis (in the end-fire direction) and is maximum in the xy plane (в = 90°), which is normal to the helix axis.

Because Ee and Eф are 90° out-of-phase, as shown in Eqs. (3) and (4), the radiated wave is elliptically polarized. The axial ratio (AR) of the polarization ellipse of the far field is obtained by dividing the magnitude of Eq. (4) by that of Eq. (3):

I Eg I




2 SX


AR =



Normal Mode Helix

Normal Mode Helix



Normal Mode Helix

Figure 2. The normal mode helix. (a) Coordinate system. (b) Loop and dipole model. (c) Beam pattern.



Under this condition the radiation field is circularly polarized in all directions except of course along the axis where the ra­diation is zero. The polarization ellipse of the radiation from a helix of constant turn-length (L) changes progressively as the pitch angle a is varied. When a = tan-1(S/C) = 0 (the helix reduces to a loop), AR = 0, Ee = 0, E = а. фEф; thus the wave is linearly polarized with horizontal (or perpendicular) polarization. As a increases, the polarization becomes ellip­tical with the major axis of the ellipse being horizontal. When a reaches a value such that condition (8) is satisfied, AR = 1, and the polarization is circular. With the help of Eq. (2), the condition (8) leads to the following value of a:

where we have used k = 2n/A, r/k = шр. Because Ee and Eф are 90° out-of-phase, the polarization ellipse becomes a circle when |Eф| = |Eф|, indicating circular polarization. Setting AR = 1 yields

C = nD = V2SX or C = V2S7

—1 + Vl +L2 и


7 2?Г 4

nk—A X






AF =




As a increases further, the polarization again becomes ellip­tical with the major axis being vertical. Finally, when a = 90° (the helix reduces to a dipole), AR = oo, Eф = 0, E = aE«; thus the polarization is linear with vertical (or parallel) polar­ization. For small pitch angles (a < 1), Eq. (9) is simplified to

, = Ck/2

where aCP is in radians. For small pitch angles, circular polar­ization can occur at frequencies such that the circumference is very small compared to a wavelength (CA < 1).

From Eqs. (3) and (4), we note that the loop field Eф and the dipole field Ee, respectively, are proportional to the second and first powers of frequency. Correspondingly, radiation re­sistance of loop and dipole are proportional to the fourth and second powers, respectively. Thus, as frequency decreases, the dipole radiation predominates and the beam pattern is linearly polarized. In this linearly polarized frequency range, the normal mode helix has some interesting properties. Its beam pattern is essentially that associated with the dipoles, that is, a monopole of length NS above a ground plane. Its impedance, however, is significantly affected by the loops.

The normal mode helix is limited by its size. It has the same restrictions and limitations that apply to any electri­cally small antenna. But within those restrictions, it has cer­tain advantages over a dipole antenna of the same height. These include a lower frequency for resonance and a larger radiation resistance, both because of the longer path of the helical structure. While the dipole may require additional im­pedance-matching circuits to achieve resonance, the helix is resonant without supplementary matching elements. Another advantage over the dipole is that the helix is flexible and more resilient. The higher radiation resistance, resonant characteristic, and flexibility make the normal mode helix suitable for small antennas used in mobile communications.