Monthly Archives: March 2014
SHIMMING
The harmonic errors in the field of an asbuilt 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 xy plane).
In order to compensate for the presence of various unwanted harmonic errors in the center field of the asbuilt 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 asbuilt 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 secondorder 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 fourthorder 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 firstorder 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, highorder 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 highorder 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 higherdegree 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).
The field of an NMR magnet for high resolution spectroscopy must be shimmed to at least 109 over volumes as large as a 10 mm diameter cylinder of 20 mm length. If, as is usually the case, substantial inhomogeneity arises from highorder 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 frequencysensitive. 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 15Segment model
100 200 300 400
Radial distance (m)
500
MININEC compared to NEC
4 
10c 
10 
І10 
>10 
104 
Antenna: Height = h = 75 m Radius = a = 0.3 m 15Segment model 
Antenna: z – Component at wire surface Radiated power = 8.775 mW at f = 1 MHz 
Near fields:
3 
Vertical and radial components of Efields 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 15Segment 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.
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 lifethreatening 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 beltworn 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 beltworn tape recorder using originally reeltoreel 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, highresolution 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 tapebased 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 12lead 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 12lead 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 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 computerbased 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 righthand 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 deltagap 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 currentcarrying 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 centerfed 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 freespace 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.
(48a) (48b) (48c) 
The availability of highspeed 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 welldocumented in the literature (1,3,613). 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) nearfield numerical anomaly, (f) treatment of the junctions of the segments, (g) wiregrid 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 thinwire 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………. pn1 
pn+1 pN1 
n—1 
n+1 ■ 
■ N1 
E {Sm ) ■ 
N 
n 
0 
s™ — s 
— s 
5m+1 
m1 
m 
m 
Sm1/2 + 
m+1/2 
2 
2 
ss 
s 
m1 
m+1 
m 
m 
= joiA(sm ) ■ 
Sm1/2 + 
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) 
dф 
(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 rm1/2 ) ‘ (sn + 1/2tm, n,n + 1/2 
Zm — — 
4п jrne 
m+1 /2,n, n + l Sn + 1 – Sn 
m+1/2 ,n1,n 
n1/2tm, n1/2,n 
) – 
(58) 
+ sn 
+ ■ 
sn s 
n1 
tm1/2,n, n1 tm1/2,n1,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 ’ Snl/2 In terms of global coordinates, 
1 for sn1/2 < 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 
(61) 
Figure 7. Geometry of the Tee antenna. Typical dimensions are shown. 
+ z, 
m,2Nn+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) thinwire 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 
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 CurrentsLU 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 (2123).
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 (17 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 farfield 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 
Wire 1′ 
(b) 
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. 
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103 
102 
4 8 12 16 Horizontal distance (m) 
Figure 8. Monopole near fields: Ez versus horizontal distance. 
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£ > 33 ф 
10[5] 
20 
500 
10" 
10" 
10" 
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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 (16 OR 0)? User Input
NAME OF INPUT CURRENT FILE? User Input (UI) ELECTRIC OR MAGNETIC NEAR FIELDS (E/M)? User Input
XCOORDINATE YCOORDINATE
INITIAL VALUE? UI INITIAL VALUE?
INCREMENT? UI INCREMENT?
NO. OF PTS? UI NO. OF PTS?
ZCOORDINATE 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 farfield results are printed out in words. Figures 8 through 15 show the nearfield characteristics of the monopole.