Monthly Archives: March 2014

Overvoltage Transient Waveforms

Overvoltage Transient Waveforms

The definition of a transient waveform is critical for the de­sign of overvoltage protection circuitry. An unrealistic voltage waveform with long duration of the voltage or very low source impedance requires a high-energy protection device, resulting a cost penalty to the end-user. IEEE Std. 587 defines two ov­ervoltage current waveforms to represent the indoor environ­ment recommended for use in designing protection devices. Table 1 describes the waveforms, open circuit voltage, source impedance, and energy stored in the protection circuitry.

1. Category I. The waveform shown in Fig. 7 is defined as ‘‘0.5 jU. s-100 kHz ring wave.’’ This waveform is repre-

sentative of category I indoor low-voltage (ac lines less than 600 V) system transients. This 100 kHz ring wave has a rise time of 0.5 ^s (from 10% to 90% of its final amplitude), with oscillatory decay at 100 kHz, each peak being 60% of the previous one. The rapid rate of rise of the waveform can cause du/dt problems in the semiconductors. The oscillating portion of the waveform produces voltage polarity reversal effects. Some semi­conductors are sensitive to polarity changes or can be damaged when unintentionally turned on or off.

Overvoltage Transient Waveforms

Overvoltage Transient Waveforms

Overvoltage Transient Waveforms

Z

V

Z

Overvoltage Transient Waveforms

Figure 10. SCR crowbar overvoltage protection circuit for switching power supplier.

gation into the sensitive circuit; and (2) those that divert transients away from sensitive loads and so limit residual voltages. Attenuating a transient, that is, keeping it from propagating away from the source or keeping it from imping­ing on a sensitive load, is accomplished with series filters within a circuit. The filter, generally of low-pass type, attenu­ates the transients (high-frequency) and allows the signal or power flow (low-frequency) to continue undisturbed. Diverting a transient can be accomplished with a voltage-clamping de­vice or with a ‘‘crowbar’’ type device.

Filters. The frequency of a transient is several orders of magnitude above the power frequency (50/60 Hz) of an ac cir­cuit. Therefore, an obvious solution is to install a low-pass filter between the source of transients and the sensitive load. The simplest form of filter is a capacitor placed across the line. The impedance of the capacitor forms a voltage divider with the source impedance, resulting in attenuation of the transients at high frequencies. This simple approach may have undesirable effects, such as (1) unwanted resonance with inductive components located in the circuit resulting in high-peak voltages; (2) high capacitor in-rush current during switching, and (3) excessive reactive load on the power system voltage. These undesirable effects can be minimized by add­ing a series resistor (RC snubber circuit).

Voltage-Clamping Devices. A voltage-clamping device is a component having variable impedance depending on the cur­rent flowing through the device or on the voltage across its terminal. These devices exhibit nonlinear impedance charac­teristics. Under steady-state, the circuit is unaffected by the presence of the voltage-clamping device. The voltage-clamp­ing action results from increased current drawn through the device as the voltage tends to rise. The apparent ‘‘clamping’’ of the voltage results from the increased voltage drop in the source impedance due to the increased current. It must be clearly understood that the device depends on the source im­pedance to produce clamping. One is seeing a voltage divider action at work, where the ratio of the division is nonlinear (Fig. 9). The voltage-clamping device cannot be effective with zero source impedance. Table 2 lists various types of voltage – clamping devices and their features and characteristics.

Crowbar Devices. Crowbar-type devices involve a switch­ing action, either the breakdown of a gas between electrodes or turn-on of a thyristor. After switching on, the crow-bar de­vice offer a very low impedance path which diverts the tran­sient away from the parallel-connected load. These crowbar devices have two limitations. The first is their delay time, typ­ically microseconds, which leaves the load unprotected during initial voltage rise. The second limitation is that a power cur­rent from the steady-state voltage source will follow the tran­sient discharge current (called ‘‘follow current’’ or ‘‘power – follow’’).

SHIMMING

The harmonic errors in the field of an as-built magnet divide into purely axial variations (axisymmetric zonal harmonics, which are accompanied by radial variations dependent on the elevation в from the z axis, but independent of ф) and radial variations (tesseral harmonics, which depend on ф, where ф is the angle of azimuth in the x-y plane).

In order to compensate for the presence of various un­wanted harmonic errors in the center field of the as-built coils, additional coils capable of generating the opposite har­monics are applied to the magnet. For each set of n and m in the associated Legendre functions, a current array can be

Figure 7. A set of axial shim coils for harmonic correction up to B(3,0). These coils generate small harmonics of 4th order and higher.

designed in the form of a set of arcs of varying azimuthal extent and symmetry and with various positions and extents along the z axis. The magnitude of the harmonic field that an array generates can be controlled by the current. This is the principle of variable harmonic correction for both MRI and NMR magnets. (Correction by means of ferromagnetic shims is not variable.)

The shimming of the unwanted harmonics is a process in two independent parts. First, there is the design of as many sets of coils as are needed to generate the compensating har­monics. Second is the measurement of the actual field errors to determine the magnitudes of the various harmonic compo­nents and the application of currents to the previously de­signed coils to provide the compensation. In fact, because su­perconducting shims must be built into the magnet prior to installation in the cryostat and cooldown, the range of har­monic errors in the field of the as-built magnet must be largely anticipated. Typically it might be assumed that the level of harmonic error decreases by a factor of three for each unit increase in n or m. Therefore, as a rough guide it has been found that compensation of up to B(3,0) for the zonal harmonics and up to B(2,2) for the tesseral harmonics is satis­factory in most cases for the superconducting shims of small bore NMR magnets. There will also be a set of room tempera­ture shims in a high resolution NMR system. Those will com­pensate for errors typically up to B(6,0) and B(3,3) in many cases. Typically there may be up to 28, but exceptionally up to 45 independent shims in all. They will be constructed ac­cording to a different principle from the superconducting shims. The shimming of MRI magnets is accomplished by cur­rent shims, typically up to n = 3 and m = 2, and by ferromag­netic shims.

Superconducting Axial Shims

These will be simple circular coils combined in groups so as to generate a single harmonic only (13). Thus, a coil to generate B(3,0) must generate no B(1,0) nor B(5,0). Because the super­conducting shim coils need to generate only a small fraction of the field due to the main coil, they generally need only com­prise one to three layers of conductor. For that reason the harmonic sensitivities can be calculated directly from Eqs. (4) and (5). A set of axial shims providing correction of B(n,0) harmonics for n = 1 through 3 are shown in Fig. 7. Note that,
for a fixed linear current density, only the angles defining the start and end of each coil are needed, together, of course, with the current polarities, either side of the center plane of the magnet, odd for n = 1, 3, 5, . . . and even for 2, 4, 6, . . ..

The set of coils illustrated in Fig. 7 generate negligible har­monics above the third order, B(3,0). The individual coils of each harmonic group are connected in series in sets, there being in each set enough coils to generate the required axial harmonic but excluding, as far as is practical, those harmon­ics that are unwanted. Thus, in the figure, coils labeled 2 gen­erate second-order B(2,0) but no fourth order. However, they do generate higher orders. The first unwanted order is B(6,0) but that is small enough that it may be neglected. So also with all higher orders because the denominator in the expres­sions of Eqs. (4) and (5) strongly controls the magnitude of the harmonic. Also illustrated in the figure is the effect on harmonic generation of the angular position of a circular cur­rent loop. Each of the dashed lines lies at the zero position of an axial harmonic. Thus, at an angle of 70.1° from the z axis, the B(4,0) harmonic of a single loop is zero. Two loops car­rying currents of the same polarity and suitable magnitude may be located on either side of the 70.1° line to generate no fourth-order harmonic yet generate a significant second order harmonic. Similarly, a coil for the generation of only a first order axial harmonic is located on the line for zero third or­der. The zero first-order harmonic line is at 90°, the plane of symmetry. In order therefore to generate a third order with no first, two coils must be used, with opposing polarities. The coils are all mirrored about the plane of symmetry, but the current symmetries are odd for the odd harmonics and even for the even harmonics. The loops may be extended axially as multiturn coils while retaining the property of generating no axial harmonic of a chosen order, if the start and end angles subtended by the coils at the origin are suitably chosen.

Figure 8. Schematic of a set of radial shim coils for correction of a B(2,2) harmonic showing the positioning necessary to eliminate B(4,2) and B(4,4).

z

Superconducting Radial Shims

The radial shims are more complex than those for purely axial harmonics because the finite value of m requires a 2m – fold symmetry in the azimuthal distribution of current arcs, the polarity of current always reversing between juxtaposed arcs in one z plane (6,9). For instance, m = 2 requires four arcs, as shown in Fig. 8. However, as for m = 0, the set of current arcs shown in Fig. 8 will generate B(n, m), where n is 2, 4, 6, etc., or 1, 3, 5, etc., depending on even or odd current symmetry about the z = 0 plane. So, the positioning of the arcs along the z axis is again crucial to the elimination of at least one unwanted order, n. Fortunately, the azimuthal sym­metry generates unique values of the fundamental radial har­monic m. (Eight equal arcs cannot generate an m = 2 har­monic.) However, depending on the length of the arc, higher

The principles described earlier can be applied both in the design of shim coils and in the selection of main coil sets. A further observation from the zero harmonic lines of Fig. 7 is that the higher harmonics reverse sign at angles close to the plane of symmetry of the system. This implies that, to pro­duce single, high-order harmonics, coil positions close to the plane of symmetry must be chosen because the other coil loca­tions where the sign of the harmonic reverses are too far from the plane of symmetry to be usable; the coils lying a long way from the plane of symmetry generate weak high-order har­monics.

radial harmonics may be generated. For the shim coil config­uration of Fig. 8, the first unwanted radial harmonic is m =

6. The higher tesseral harmonics are much smaller than the fundamental because of the presence in the expression for the field of a term (r/r0)n. Generally, the arc length is chosen to eliminate the first higher-degree radial harmonic. As an ex­ample, if the arc length of each shim coil shown in Fig. 8 is 90° the B(6,6) harmonic disappears. The B(10,10) harmonic is negligible.

The superconducting shims are almost invariably placed around the outside of the main windings. Although the large radius reduces the effective strength of the harmonics they generate, the shim windings cannot usually be placed nearer to the center of the coil because of the value of winding space near the inner parts of the coil and because of the low critical current density of wires in that region due to the high field. A comprehensive treatment of shim coil design may be found in Refs. 6 and 9. Those references also include details of su­perconducting coil construction. It should be noted, however, that some expressions in Ref. 6 contain errors.

Ferromagnetic Shims

Ferromagnetic shimming is occasionally used in high field, small bore NMR magnets, but its principal use is in MRI magnets. It is in that application that it will be described. The principle invoked in this kind of shimming is different from that of shim coils. The shims now take the form of dis­crete pieces of ferromagnetic material placed in the bore of the magnet. Each piece of steel is subjected to an axial mag­netizing field at its position sufficient to saturate it. It then generates a field at a point in space that is a function of the mass of the shim and its saturation magnetization Bs with little dependence on its shape. For ease of example, a solid cylinder of steel will be assumed. The axis of the cylinder is in line with the field, as shown in Fig. 9. (In Fig. 9 the axis labeled z is that of the shim, not that of the MRI magnet itself. In fact, the shim will usually be placed at the inside surface of the bore of the MRI magnet.)

The field B, caused by the ferromagnetic shim, contains both axial and radial components. The axial component Bz is the correcting field required, and it adds arithmetically to the field of the magnet. The radial component adds vectorially to the field and produces negligible change in the magnitude of

Figure 9. Field vectors generated by a ferromagnetic shim in the bore of an MRI magnet. Bz adds arithmetically to the main field; Бг adds vectorially and so has negligible influence on the field.

the axial field. Therefore, only the axial component of the shim field must be calculated. If the saturation flux density of the shim is Bs, the axial shim field is given by

(14)

Bz = BsV[(2 — tan2 n)/(tan2 n + 1)5/2]/(4nz3)

where V is the volume of the shim and z and ^ are as shown in Fig. 9.

The practical application of ferromagnetic shims involves the measurement of the error fields at a number of points, and the computation of an influence matrix of the shim fields at the same points. The required volumes (or masses) of the shims are then determined by the inversion of a U, W matrix, where U is the number of field points and W is the number of shims. In an MRI magnet, the shims are steel washers (or equivalent) bolted to rails on the inside of the room tempera­ture bore of the cryostat. In the occasional ferromagnetic shimming of an NMR magnet, the shims are coupons of a magnetic foil pasted over the surface of a nonmagnetic tube inserted into the room temperature bore or, if the cryogenic arrangements allow, onto the thermal shield or helium bore tube. As in the design of the magnet, linear programming can be used to optimize the mass and positions of the ferromag­netic shims (e. g., to minimize the mass of material).

Resistive Electrical Shims

The field of an NMR magnet for high resolution spectroscopy must be shimmed to at least 10-9 over volumes as large as a 10 mm diameter cylinder of 20 mm length. If, as is usually the case, substantial inhomogeneity arises from high-order harmonics (n and m greater than 3), superconducting shims are of barely sufficient strength. This arises because of the large radius at which they are located, at least in NMR mag­nets (e. g., in the regions x and y of Fig. 6). In general, the magnitude of a harmonic component of field generated by a current element is given by

Bn

(15)

r" + 1 /r^ + 1

where r is the radius vector of the field point, and p0 is the radius vector of the source. Thus, the effectiveness of a remote source is small for large n.

In order to generate useful harmonic corrections in NMR magnets for large n and m, electrical shims are located in the warm bore of the cryostat. Although in older systems those electrical shims took the form of coils tailored to specific har­monics, modern systems use matrix shims. Essentially, the matrix shim set consists of a large number of small saddle coils mounted on the surface of a cylinder. The fields gener­ated by unit current in each of these coils form an influence matrix, similar to that of a set of steel shims. The influence matrix may be either the fields produced at a set of points within the magnet bore, or it may be the set of spherical har­monics produced by appropriate sets of the coils.

THE THICK LINEAR ANTENNA

The thin linear antenna is frequency-sensitive. In practical communication scenarios, the transponders use wideband sig­nals to increase the channel capacity and therefore needs an­tennas that can handle a large band of frequencies. One way of increasing the bandwidth is to use electrically thick wires.

A thick cylindrical dipole (Fig. 16) is the inexpensive way to increase the bandwidth of linear antennas. The increase in thickness leads to circumferential component Ц of the other­wise linear current. This can be handled with the integral equation formulation. Figure 17(a) and Fig. 17(b), respec­tively, show the variation of input resistance and reactance of the dipole with l/2a ratios 25 (thick), 50, and 10,000 (thin), where 2a is the diameter of the wire.

10"

Near fields:

H<p component at z = 1^5 m Radiated power = 8J75 mW at f = 1 MHz

10"

E

<

33

о

ф

MININEC

. „ NEC

Antenna:

Height = h = 75 m Radius = a = 0^3 m 15-Segment model

100 200 300 400

Radial distance (m)

500

MININEC compared to NEC

4

10c

10-

І10-

>10-

10-4

Antenna:

Height = h = 75 m Radius = a = 0.3 m 15-Segment model

Antenna: z – Component at wire surface Radiated power =

8.775 mW at f = 1 MHz

THE THICK LINEAR ANTENNA

Near fields:

3

Vertical and radial components of E-fields z = 1.5 m

Power radiated = 8.775 mW f = 1 MHz

2

Antenna:

ф

CL

Height = h = 75 m Er Radius = a = 0.3 m

4 15-Segment model

Ez

0

50 100 150 200 250 300 350

Radial distance (m)

0

400 450 500

Figure 11. Percentage difference in electric fields Ez and Er versus radial distance.

THE THICK LINEAR ANTENNA

0

144

The Monitored ECG

There are two primary applications where a patient’s ECG is continuously monitored. Intensive care units within a hospi­tal often monitor the ECG of critically ill patients to observe the patient’s rate and rhythm. When the patient is suspected of having a life-threatening arrhythmia, it is best to monitor the patient in an environment where a rapid response and therapeutic intervention can be lifesaving. The other applica­tion of the monitored ECG is in the ambulatory, outpatient setting. The patient’s ECG can be monitored by a belt-worn device. In the 1950s Norman J. Holter (13), an American physicist, demonstrated that the ECG could be monitored while the subject was physically active. However, the technol­ogy of the day resulted in a very heavy backpack device weighing 85 lb and was impractical for routine use. Recording devices and their associated electronics and batteries eventu­ally became small enough to allow for a belt-worn tape re­corder using originally reel-to-reel technology, but now rely­ing exclusively on cassettes. These tape devices still have many inherent limitation, such as poor noise figures, low dy­namic range, and limited frequency response. If one is merely recording the patient’s rate and rhythm, then the tape tech­nology is adequate. However, high-resolution ECGs might also be useful when obtained from the ambulatory patient, and the tape technology is definitely limited for this appli­cation.

Newer digital recorders are currently available whereby the ECG is digitized and stored in either high density mem­ory chips or on actual hard disk drives. Current versions of the latter have removable drives with capacities exceeding 500 Mbytes. Depending on the application, the ECG may be sampled between 250 Hz and 1000 Hz. Originally, only one ECG lead was recorded on tape-based systems, but the new systems are not limited by the poor frequency response of tape systems or the physical size constraints of magnetic re­cording heads. With digital systems the number of simultane­ous (or near simultaneous) recordings is not particularly lim­ited, but three or four ECG leads are a practical number. Electrode positions for these monitored leads do not follow the conventions of the 12-lead ECG and are often similar to the bipolar limb leads, where the electrodes are placed a few inches apart over the chest, creating several lead fields through the heart.

For hospital monitoring, where the patient is being evalu­ated for a critical cardiac condition, only a few leads are re­corded, but a full 12-lead ECG is periodically recorded. In some cases the patients, although not acutely ill, are given the freedom to walk about the hospital and their ECG is tele­metered via radio frequencies by an antenna/receiver net­work. In such cases the goal is to monitor the patient’s rhythm in ‘‘normal’’ activities. The Monitored ECG

The massive amount of ECG information obtained during continuous monitoring is overwhelming. In the hospital ap­proach, the ECG signals are usually fed to a large system of monitor screens where specially trained technicians view the actual recording of 10 to 50 patients. In conjunction with com­puter-based software the high risk situations are quickly identified with appropriate communication to the medical staff, for example, ‘‘code blue.’’ This is not the case with outpa­tient monitors where the patient returns to the hospital 1 or 2 days later. Then the entire record is scanned with an inter­active software analysis program. Often just a compressed printout of the continuous ECG can be quickly inspected for an abnormal rhythm. An example of this ‘‘full disclosure’’ mode is shown in Fig. 11. A 7.5 min recording is shown in this format. Note that the first several minutes have a normal rhythm (there are several other abnormalities in this tracing, but they are beyond the scope of this article). The abnormal beats begin to appear in groupings of two or three. A condi­tion known as nonsustained ventricular tachycardia appears in the fourth trace from the bottom.

The event recorder is an extension to ambulatory re­cordings. In this case the patient wears a recorder for many days or even weeks. When the patients experience a symp­tom, such as chest palpitations or dizziness, they push an event button on the recorder which causes the recorder to save 1 min to 2 min of data before and after the event. The patient can call the physician office and transmit these data via a modem for rapid interpretation. The most advanced ver­sion of the event recorder is an implantable device that moni­tors the ECG for months or years and uses special monitoring software to record suspicious events without patient activa­tion. This type of unit can be interrogated at regular intervals over the phone or during regular visits.

THE METHOD OF MOMENTS SOLUTION

For many practical antennas and scatterers including linear antennas, the desired current distribution is obtained by nu­merically solving the integral equations. The Method of Mo­ments (MOM) is a technique to convert an integral equation to a matrix equation and hence solve the linear system by standard matrix inversion techniques. The MOM is very well documented in the literature (15), and only the basic steps will be briefly discussed below.

The magnetic field integral equation (MFIE) for the un­known current density can be rewritten as an inhomogeneous equation in operator form as follows:

L(Js) = 2n x H(r)

where m = 1,2,. . ., N; (■) is the inner product, the product of the two functions integrated over the do­main.

5. Express the set of algebraic equations given by Eq. (40) in the matrix form:

[Imn][an] — [gm]

where the matrix is given by

W L( Л) Wx, L( f2)

W2, L( f2) W2, L( f2)

J2*n<Wm, L( fn)) = {Wm, g)

[Imn] —

(43a)

where the right-hand side is a known quantity and L(Js) is an integrodifferential linear operator defined as

_WN, L( fN ) WN, L( fN ) ■■■ WN, L( fN )_

where aN and gN are the column vectors given by

(37)

a

1

L(JS) =Jsn x f Js(r’) x V’Gds’ 2п Jc

2

a

[aN] =

(44a)

XN.

W1, g)

<W2, g)

(38)

(44b)

where Js is the electric current on the wire, and G is the free space Green’s function.

Let us discuss the solution of inhomogeneous scalar equa­tion given by

L( f) = g

(39)

(40)

where f = f(x) is the unknown function to be determined, g(x) is a known function and L is a linear operator. The seven steps (16) in obtaining the solution of Eq. (38) is the same as the steps for the solution of Eq. (37).

The steps are as follows:

1. Expand f as

N

f=Y2 anfn

where the an’s are the unknown coefficients, and the fn’s are known functions of x known as expansion, or basis, functions.

2. Using Eq. (38) in Eq. (37), we get

N

У ‘ anL( fn ) = і

,<Wn, g).

Moment Method Solution for Radiation from Thin Wire

Two types of volume integral equations are used for the linear antenna and wire scatterer problem. These are the integral equation of Hallen type and the integral equation of Pockling – ton type. Hallen’s integral equation usually necessitates the postulation of a delta-gap voltage at the feed point and also requires the inversion of an (N + 1)-order matrix. The advan­tages of Pocklington’s integral equation is that it is possible to incorporate different source configurations and it requires inversion of a matrix of order N.

For a current-carrying perfectly conducting wire, the Hal – len’s integral equation obtained by solving the scalar wave equation is given by (1)

Iz (z’)6X^ _ ^ dz’ = cos (kz) +B2 sin(fe)] (45)

4п R

(41)

3. Define a suitable inner product f, g) defined in the range L of x:

< f, g) = f f (x)g(x) dx Jd

where

Iz(z’) = the current flowing through an elementary length of the wire

R = distance of the observation point from the elemen­tary length

I = the total length of the center-fed wire e, u, a = 0 are the primary constants of the medium in

which the antenna is radiating and k is the derived secondary constant, namely, the wave vector of the medium

^+k2)G{z, Z)

B1 and B2 are constants to be determined The Pocklington’s integral equation is given by

Iz (z’)

dz’ = —joeE[2]z (atp = a) (46)

exp(—jkR) 4jxR

where for thin wire the radius a < A, the free-space Green’s function G(z, z’) is given by

G(z, z’) = G(R) =

with R = /a[3] + (z — z’)2

Elz = the incident field (this is the source field for antennas and scattered field for scatterers)

(47)

able in Richmond’s work (17) and also in Ref. 8 are re­produced in Table 1.

2. Subdomain Expansion Functions. Subdomain expan­sions are attractive, convenient, and less expensive in terms of computer time. This stems from the fact that the current is matched on part of the integration path, whereas for the entire domain the integration is per­formed over the whole path and for all N terms of the expansions and coefficients determined.

Miller and Deadrick (8) provide a table containing the many basis and weighting functions which have been tested in com­puter runs. This table also compares the suitability of the use of different functions in different problems. The table is too big to reproduce here, and it is left to the reader to look up. It tabulates the method, integral equation type, basis function, current conditions for interior and end segments, the basis function in terms of unknown for unknown and end segments, weighting function for interior and end segments, number of unknowns, and specific comments on the applicability of the expansion functions.

THE METHOD OF MOMENTS SOLUTION

(48a)

(48b)

(48c)

The availability of high-speed computers, graphics, and soft­ware packages, along with the enormous development of mi­crocomputers, has made electromagnetic numerical computa­tion extremely viable. An attractive feature of numerical methods is their ability to handle arbitrarily shaped and elec­trically large bodies and bodies with nonuniformity and an­isotropy where exact solutions can only at best provide some physical insight. For large problems, it is possible to get a linear system with a minimum set of equations to achieve a certain accuracy.

An account of the area of numerical computation of thin wire problem is well-documented in the literature (1,3,6-13). As described in Ref. 8, there are many important computa­tional issues involved in thin wire problems. These are (a) segmentation of the structure and the correct number of seg­ments, (b) choice of right current expansion functions, (c) the thin wire approximation (radius a < A), (d) Roundoff errror due to matrix factorization, (e) near-field numerical anomaly, (f) treatment of the junctions of the segments, (g) wire-grid modeling, and (h) computer time required. Also, the errors (7) involved are of concern. There are two types of errors encoun­tered: (a) the physical modeling error, because in the absence of an exact solution for a variety of semicomplex and complex stuctures, it is the natural departure of the assumed struc­tural details from the actual structure, and (b) the numerical modeling error, since all numerical methods are approximate but sufficiently accurate for the application.

Formulation

Before we discuss the formulation of the thin-wire integral equation, comments on the expansion functions used in this study are in order. There are broadly two types of expansion functions:

The Electric Field Integral Equation and Its Matrix Representation

Figure 5 gives the geometry of the arbitrarily oriented straight wire. The wire is broken into segments, or subsec­tions. The mini numerical electromagnetic code (MININEC) relates the current distribution on the wire due to the inci­dent field. The integral equation relating the incident field Ei, magnetic vector potential A, and electric scalar potential ф are given by

—Ei • t = —joA • t — t • Уф

where

A = — f I(s)S(s)k(r)ds 4n J

Ф = ^ j q(s)k(r)ds

Table 1. Entire Domain Current Expansions Using Different Polynomials

A. The Polynomials

Fourier: I(z) = Ij cos(^x/2) + I2 cos(3nx/2) + I3 cos(5nx/2) MacLaurin: I(z) = Ij + I2x2 + I3x[4] + •••

Chebyshev: I(z) = I1T0(x) + I2T2(x) + I3T4(x) + ••• Hermite: I(z) = IjH0(x) + I2H2(x) + I3H4(x) + ••• Legendre: I(z) = I1 P„(x) + I2P2(x) + I3P3M + ••• where -1/2 < x = 2z/L < 1/2

B. Typical Results for Functions

L = 0.5A; a

= 0.005A; ft =

9

О

0

In

Fourier

MacLaurin

Chebyshev

Hermite

Legendre

1

3.476

3.374

1.7589

8.2929

2.2763

2

0.170

4.037

1.5581

14.3644

2.1005

3

0.085

3.128

0.0319

4.4135

0.0655

4

0.055

4.101

0.0112

0.3453

0.0421

5

0.040

1.871

0.0146

0.0073

0.0322

When the pulse functions of Eq. (52) are inserted in parenthe­ses, we obtain

0

1

■■■■■■■■■ n

-1

n

n+1 ■■■■

■■■■ N

-1

N

(a)

pi………. pn-1

pn+1 pN-1

n—1

n+1 ■

■ N-1

E {Sm ) ■

N

n

0

s™ — s

— s

5m+1

m1

m

m

Sm-1/2 +

m+1/2

2

2

ss

s

m1

m+1

m

m

= joiA(sm ) ■

Sm-1/2 +

S

m+1/2

(55)

2

2

The exact kernel treatment developed above is for observation points on source segments. For observation points near but not on the source, a segment has been developed by Wilton and MININEC has incorporated it (16).

Expansion of Currents

The currents are expanded in terms of pulse functions as shown in Fig. 5, excluding the end points where the currents are chosen as zeros to satisfy boundary conditions at the ends. The current expansion is given by

n

(b)

Figure 5. Wire segmentation with pulses for current and charges. (a) Unweighted current pulses. (b) Unweighted charge representa­tion. The whole length is broken into several segments. Each segment is assigned a pulse, and the pulses represent the assumed current distribution.

t is a unit vector tangential to the wire at any distance along the path of integration which is the length of the wire and k(R) is given by

I(s) = Y InPn (s)

(56)

2п exp(-jkR)

(49)

Using this current expansion in Eq. (48b) and after mathe­matical manipulations which are available in Ref. 1 and are not detailed here, we get the linear system matrix equation in N unknowns:

R

ф =0

The continuity equation given below determines the relation­ship between the charge q(s) and the rate of change of current with distance:

[Vm ] = [Zmn][In ]

(57)

1 dl

<?(s) = —–Г-

jo ds

(50)

where m, n = 1, 2, . . ., N, [Zmn] is the square impedance matrix, and [Vm] and [In] are applied voltage and current col­umn vectors:

The MININEC solves the integral equation using the follow­ing steps:

1. The wires are divided into small equal segments such that the length of the segment is still large compared to the radius of the wire (a < A, typically 1/100th of a wavelength). The radius vectors m, n = 0, 1, 2, . . . are defined with reference to a global origin.

2. The unit vectors are defined as

1

k (rm+1/2 rm-1/2 ) ‘ (sn + 1/2tm, n,n + 1/2

Zm — —

4п jrne

m+1 /2,n, n + l Sn + 1 – Sn

m+1/2 ,n-1,n

n-1/2tm, n-1/2,n

) –

(58)

+ sn

+ ■

sn s

n-1

tm-1/2,n, n-1 tm-1/2,n-1,n

+

sn+1 – sn

ss

n1

n

(51)

n+1 /2 ‘

‘n+1 ~’n

K+l ~Гп

The testing and expansion functions are pulse functions which are defined by

This matrix has elliptical integrals involved in it. These ellip­tical integrals can be evaluated numerically.

The above equations are valid for any radius other than small, for which the expression for ф breaks down and Har­rington (18) provided an approximate formula for ф. This is given by

(59a)

(59b)

(60)

Pn ( s ) =

(52)

2n As L a.

exp( jkrm )

n ° n – 1

2

rn 1 + rn

(54)

r

r

n + 1/2 ■

2

where the points sn+1/2 and sn+1/2 are the segment midpoints and are given by

Sn+1 + Sn Sn +1/2 — 2 ’ Sn-l/2

In terms of global coordinates,

1 for sn-1/2 < s < sn+1/2

0 otherwise

_ Гп +Г„_! гг —1/2 2

ss

(53)

Inclusion of Nonradiating Structures

The Ground Plane. When the wire structure near the ground plane is assumed to be perfectly conducting, an image

4п rm

The integral is given by

tm, u,v = / k0 (sm – s’) ds’

* = ^_іпГд£1_Д

nAs L aJ 4п

for m = n

for m

is created. The structure and the ground plane is equivalent to the structure and the image.

The voltage and current relationship is given by

THE METHOD OF MOMENTS SOLUTION

(61)

Figure 7. Geometry of the Tee antenna. Typical dimensions are shown.

+ z,

m,2N-n+1

Wire Attached to Ground. When a wire is attached to the ground on one or both sides, there will be a residual compo­nent of current at one or both ends. In this case, a current pulse is automatically added to the end point in the formu­lation.

7′ — 7

^mn — ^ mn

where

[Vm ] = [Z’mnJIn

Lumped Parameter Loading. If an additional complex load is added to the perfectly conducting wire (Fig. 6), there will be an additional voltage drop created at that point if the location of the load (Zl = Ri + jXl) is at a point of nonzero pulse func­tion. The impedance matrix is modified to to meet the following requirements: (a) the segmentation den­sity, (b) thin-wire criteria, (c) small radius calculation, (d) step changes in wire radius, (e) spacing of wires, (f) loop an­tenna, and (g) monopoles and antennas above ground.

Zmn +Zlt Zmn

Z’ =

mn

[Vm ] — [ZLJUn ]

where Z’mn is the modified impedance matrix and is given by

for m — n for m — n

Validation of the MININEC Code

Extensive work has been reported on the validation of the MININEC. Numerous validation runs have been carried out

Operation of Currents-LU Decomposition

The operation is oriented around the Menu shown below. Here we describe the DOS version (19,20), but the Windows version is also available (21-23).

MENU

1 – COMPUTE/DISPLAY CURRENTS

2 – CHANGE EXCITATION

3 – CHANGE FREQUENCY

4 – CHANGE LOADING

5 – LOAD GEOMETRY

6 – SELECT OUTPUT DEVICE

7 – RETURN TO SYSTEM SUPERVISOR 0 – EXIT TO DOS

SELECTION (1-7 OR 0)?

(a)

Some Examples Using MININEC

Tee Antenna. Figure 7 shows the geometry of the Tee an­tenna fed from the base by a coaxial line. The impedance cal­culations of the Tee antenna using different computer pro­grams including CURLU in MININEC and have been compared with measurements (25).

Near and Far Fields. The near – and far-field programs (FIELDS) calculate near and far fields using the current dis­tribution on the structure obtained by integral equation for­mulations. The current distribution can be computed using three programs: CURLU, CURTE, and CURRO. The current distribution can be computed using perfect and imperfect grounds, although the real ground corrections are applied to the far field only. The real ground correction is included in the form of reflection coefficients for parallel and perpendicular polarizations. For details, the reader is referred to Chap. 8 of Ref. 7. The menu is given below. User input(UI) means the user is expected to respond at that point.

Overlapping pulse

Wire 2

THE METHOD OF MOMENTS SOLUTION

Wire 1′

(b)

THE METHOD OF MOMENTS SOLUTION

Figure 6. Overlap scheme used at a multiple junction of wires. (a) Wire 1 with no end connections. (b) Wire 2 overlaps onto wire 1. (c) Wire 3 overlaps onto wire 1.

100 200 300 400

Radial distance (m)

Figure 9. Monopole near fields: Electric fields Ez and Er versus ra­dial distance.

0

THE METHOD OF MOMENTS SOLUTION

THE METHOD OF MOMENTS SOLUTION

103

102

4 8 12 16

Horizontal distance (m)

Figure 8. Monopole near fields: Ez versus horizontal distance.

0

£

>

33

ф

10[5]

20

500

10"

10"

10"

0

MENU

1 – COMPUTE NEAR FIELDS

2 – COMPUTE FAR FIELDS

3 – SELECT/CHANGE ENVIRONMENT

4 – SELECT/CHANGE CURRENTS FILE

5 – SELECT OUTPUT DEVICE

6 – RETURN TO SYSTEM SUPERVISOR 0 – EXIT TO DOS

SELECTION (1-6 OR 0)? User Input

NAME OF INPUT CURRENT FILE? User Input (UI) ELECTRIC OR MAGNETIC NEAR FIELDS (E/M)? User Input

X-COORDINATE Y-COORDINATE

INITIAL VALUE? UI INITIAL VALUE?

INCREMENT? UI INCREMENT?

NO. OF PTS? UI NO. OF PTS?

Z-COORDINATE UI INITIAL VALUE? UI UI INCREMENT? UI

UI NO. OF PTS? UI

PRESENT POWER LEVEL IS : CURRENT VALUE CHANGE POWER LEVEL (Y/N)? UI NEW POWER LEVEL (WATTS)? UI

Once the parameters are specified, the near – and far-field results are printed out in words. Figures 8 through 15 show the near-field characteristics of the monopole.