**K.E. Schmidt
Department of Physics and Astronomy
Arizona State University
Tempe, AZ 85287
**

This note was published in *IEEE Trans. on Antennas and Propagation*,
**44**, 1298 (1996).

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By including the point charges at the ends of the nonplanar skew monopoles with sinusoidal current distributions, the expressions given by Richmond and Geary are simplified. The simplified expressions satisfy reciprocity.

I(x) = ,
| (1) |

Since the current
at the *x*_{b} end is not zero, there is necessarily a point charge
there which contributes to the electric field. Richmond and Geary
neglected this contribution because it cancels with a similar
term of opposite sign when monopoles are
connected together to produce the continuous physical current.
In
this note, I show that adding these point charge contributions
cancels one of the terms in the expression of Richmond and Geary.
The result is that the *F*_{ik} terms in Eq. 13 of reference 1 may
be dropped since they cancel when summed over the terms making up
the mutual impedance of the dipole. An added advantage
is that the resulting monopole-monopole mutual impedance satisfies
reciprocity so no additional bookkeeping is needed to keep track
of source and observation monopoles.
To demonstrate these statements I will follow Richmond and Geary
and set up the system as they do, but keep the point charge
terms in the fields.

= | , | ||

= | , | (2) |

cos() = , | (3) |

= | , | ||

d
| = | ( - ) . | (4) |

= + | (5) |

z_{1}
| = | ( - ) , | |

z_{2}
| = | ( - ) , | |

t_{1}
| = | ( - ) , | |

t_{2}
| = | ( - ) . | (6) |

d_{1}
| = |
z_{2} - z_{1},
| |

d_{2}
| = |
t_{2} - t_{1}.
| (7) |

Richmond and Geary calculate the mutual impedance in the usual way by integrating the dot product of the current density and the electric field. The point charge contributions can be calculated similarly. An alternative method that displays reciprocity immediately and also simplifies the derivation of the expressions is given by writing the mutual impedance in terms of the scalar and vector potentials. The mutual impedance is

Z = dtdz(- R)/R,
| (8) |

q(z) = - + (z - z_{2}),
| (9) |

Eq. 8 can be written in terms of exponential integrals. As in Richmond and Geary an E function is defined to be

E(( + j)) = exp(j)dw
| (10) |

The mutual impedance expression Eq. 8 is evaluated as
combinations of two kinds of integrals,

- s_{t} = exp(s_{t}z_{2}cos())E((R_{2} - s_{t}(t - z_{2}cos())))
| (11) |

dtdzexp(-(R+s_{t}t+s_{z}z))/R=[
| |||

- exp(- (s_{t} + s_{z}cos())t)E((R + s_{z}(z - tcos()))) - exp(- (s_{z} + s_{t}cos())z)E((R + s_{t}(t - zcos())))
| |||

+ E((R + s_{t}t + s_{z}z + j)) + E((R + s_{t}t + s_{z}z - j))],
| (12) |

*R*_{2} in Eq. 11 is the distance between the *z*_{2} point
and the *t* point.
The right hand side of Eq. 12
is evaluated at the four limits of the integration.
The E function is defined above as being the integral from *t*_{1} to *t*_{2} , and
this definition can be used for the last three terms of Eq. 12.
In the first term,
the roles of z and t must be interchanged; the first E function is
the integral from *z*_{1} to *z*_{2} and the result at *t*_{1} is subtracted
from the result at *t*_{2} . This reversal of roles comes from an integration
by parts in deriving Eq. 12.

The mutual impedance
of two monopoles
is,

Z = [
| |||

+ |
s_{t}s_{z}exp((s_{t}t_{1} + s_{z}z_{1}))(- 1)^{i}E((R_{i} + s_{t}t + s_{z}z_{i} + js_{b}))],
| (13) |

= d (cos() + s_{z}s_{t})/sin()
| (14) |

The Richmond-Geary expression, with the additional *F*_{ik} terms,
is obtained
by dropping
the
function term in *q*(*z*) .

As an aside, the expression for the mutual impedance between an electric current monopole and a magnetic current monopole[5] can also be derived using just Eq. 12. This impedance is proportional to the volume integral of the magnetic current dotted with the magnetic field from the electric current. Writing the magnetic field as the curl of the vector potential and using the identity

( x ) = sin()
| (15) |

sin()dtdzI_{m}(t)I(z)exp(- R)/R,
| (16) |

The efficiency improvement can be estimated
by assuming that the computation is dominated by
the calculation of the exponential integrals. Dropping
the *F*_{ik} terms will reduce the number of exponential integrals needed
to calculate the mutual impedance of an isolated dipole from 36 to 32 per
monopole pair,
giving roughly a 10 percent increase in efficiency.
For a large structure, each monopole needs to be evaluated with
the current equal to one at each end in turn except for those monopoles at the
ends of wires. In this case, the number of
exponential integrals required per monopole pair is reduced from
40 to 32 or a 20 percent savings. For the special case where the monopoles are
skew coplanar, d is zero and the
number of exponential integrals required drops from 24 to 16 for
about a 30 percent reduction in computational cost.

**1**- J. H. Richmond and N. H. Geary,``Mutual Impedance of Nonplanar-Skew Sinusoidal Dipoles,'' IEEE Trans. Antennas Propagat. vol. AP-18, pp. 412-414, 1970.
**2**- F. L. Whetten, K. Liu, and C. A. Balanis, ``An Efficient Numerical Integral in Three-Dimensional Electromagnetic Field Computations,'' IEEE Trans. Antennas Propagat. vol. AP-38, pp. 1512-1514, 1990.
**3**- D. E. Amos, ``Computation of Exponential Integrals of a Complex Argument'', ACM Trans. Math. Softw. vol. 16, pp. 169-177, 1990.
**4**- D. E. Amos, ``Algorithm 683, A Portable FORTRAN Subroutine for Exponential Integrals of a Complex Argument'', ACM Trans. Math. Softw. vol. 16, pp. 178-182, 1990.
**5**- K. Liu, C. A. Balanis, and G. C. Barber,``Exact Mutual Impedance Between Sinusoidal Electric and Magnetic Dipoles'', IEEE Trans. Antennas Propagat. vol. AP-39, pp. 684-686, 1991.

Table I. The mutual impedance in ohms calculated for two v-dipoles
in free space. The
planes of the dipoles are parallel and the
midpoints of the dipoles are separated by .01 wavelength.
Each dipole consists of two monopoles of equal length (.1 wavelength
for one set and .2 wavelength for the other) at
an angle of 90 degrees. The four monopole mutual impedances (labeled
11, 12, 21, and 22) are shown, as well
as the final dipole-dipole result *Z*_{tot} .
Four calculations are
compared. These use
Eq. 13 without the first term point-charge point charge interaction;
Eq. 13;
and lastly the Richmond and Geary
expressions. RG1 is the Richmond and Geary result from
integrating the E field from z with
the current from t. RG2 corresponds to integrating the field from t
with the current from z.

Z | Eq. 13 w/o charge | Eq. 13 | RG1 | RG2 |

11 | 34.60 , 116.18 | 4.64, -360.02 | 5.32, -15.33 | 7.69, 61.32 |

12 | -29.92 , -153.15 | 0.04, 323.04 | -0.63, -21.64 | -3.01, -98.30 |

21 | -29.92 , -153.15 | 0.04, 323.04 | -0.63, -21.64 | -3.01, -98.30 |

22 | 34.60 , 116.18 | 4.64, -360.02 | 5.32, -15.33 | 7.69, 61.32 |

Z_{tot} |
9.36 ,-73.95 | 9.36, -73.95 | 9.36, -73.95 | 9.36, -73.95 |