# Source Code Description: Subroutine GGMM

GGMM

PURPOSE: to calculate the mutual impedance between two filamentary monopoles with sinusoidal current distribution.

METHOD: As stated in subroutine SGANT, the mutual impedance of coupled dipoles may be expressed as sum of four monopole-monopole impedances. This subroutine calculates the mutual impedance with closed-form expressions in terms of exponential integrals.

For skew monopoles it can be shown that the monopole-monopole mutual impedance is given by:

zij = (-1)i+j B [et n (Fj1 - e- z m G12 + ez m G22 )
- e- t n (Fj2 - e- z m G11 + ez m G21 ) ]

where m = 2/i, n = 2/j and

B = h / [16 p sinh( d1) sinh( d2)]

The functions Fik are defined by:

Fik = 2 sinh(d1) e q zi cos(y) E ( Ri+ qzi cos(y) - qt )

where q = (-1)k , d1 and d2 are the lengths of the monopoles being considered. The functions Gik are defined as follows:

Gik = E (R2 + qz2 +q't - jq'') + E (R2 + qz2 + q't + jq'')
- E (r1 + qz1 + q't - jq'') - E (R1 + qz1 + q't + jq'')

where q = (-1)i , q' = (-1)k , and q'' = qb + q'c with b = c cos(y) and c = d/sin(y) . The angle y is the angle formed by the apparent intersection of the two monopoles. This will be discussed later in detail.

In the above equation for Gik , t denotes the position of an observation point somewhere on monopole 2. R1 and R2 are the distances from the endpoints of monopole 1 to this observation point. Finally, the E functions are defined as follows:

E(a + jq'') = ejgq''

where a and q'' are real quantities with dimensions of length, a is a function of t, a1 = a(t1) , a2 = a(t2) and . The integral above is evaluated by subroutine EXPJ.

To explain the input data for GGMM, refer to the above figure. If the monopoles are parallel, then the new coordinate system is defined such that the new z axis is parallel to the monopoles. The coordinate origin may be selected arbitrarily. S1 and S2 denote the z coordinates of the endpoints of the test monopole, T1 and T2 are the coordinates of the endpoints of the expansion monopole, and D is the perpendicular distance (displacement) between the monopoles. The mutual impedance of parallel monopoles is calculated in the last part of GGMM below statement 5.

For skew monopoles, let the test monopole s lie in the xy plane and the expansion monopole t in the plane z = D. (D is the perpendicular distance between the parallel planes.) If the monopoles are viewed along a line of sight parallel with the z axis, the extended axes of the two monopoles will appear to intersect at a point on the xy plane. Let s measure the distance along the axis of the test monopole with the origin at the apparent intersection. S1 and S2 denote the s coordinates of the endpoints of the test monopole. Similarily, let t measure the distance along the axis of the expansion monopole with the origin at the apparent intersection. T1 and T2 denote the t coordinates of the endpoints of the expansion monopole. Let s snd t be unit vectors parallel with the positive s and t axes, respectively. {Note: for simplicity of web-page creation, vectors have been represented as bold underlined on this particular page.} Then CPSI = s. t = cos(y) . The monopole lengths are ds and dt .

The output data from GGMM are the impedances P11, P12, P21, and P22. In defining these impedances, the reference direction is from S1 to S2 for the current on monopole s, and from T1 to T2 for the current on monopole t. In the impedance Pij, the first subscript is 1 or 2 if the test dipole has terminals at S1 or S2 on monopole s. The second subscript is 1 or 2 if the expansion dipole has terminals at T1 or T2 on monopole t. The monopole lengths ds and dt are assumed positive in defining the input data CGDS, SGD1 and SGD2.

For parallel monopoles, CSPS = 1 or -1. S1, S2, T1, and T2 are cartesian coordinates for parallel monopoles and spherical coordinates for skew monopoles. For skew monopoles, the radial coordinates S1, S2, T1, and T2 tend to infinity as the angle y tends to zero or n. Therefore, if the monopoles are within 4.5 degrees of being parallel, they are approximated by parallel dipoles.

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