We can use the reactance theorem to generate a moment method solution to the currents on a thin-wire antenna. Thin-wire solutions assume that there are no circumferential currents and reduces the problem to filamentary currents. An electric field integral equation (EFIE) satisfies the boundary condition of Eq. (1),
 (1)a zero tangential field at the surface of the wires, but it does not seem explicit in the derivation. The reactance theorem produces an impedance matrix whose inversion yields the coefficients of the current expansion. Similar to many other methods, the Green’s function has been solved explicitly to reduce run time. This method uses overlapping sinusoidal currents on V-dipoles as basis function currents and uses the Green’s function to calculate the radiation from one V-dipole at the location of a second V-dipole. Both the radiating and receiving dipoles use the same expansion function. Galerkin’s method uses the same weighting (or testing) function as the basis function and yields the most stable solutions. The reactance equation (2) calculates the mutual impedance between the two dipoles when each has unity current.
 (2)We compute self-impedance by spacing a second V-dipole one radius away and by using the reactance theorem to calculate mutual impedance, a technique equivalent to the induced EMF method.
The scalar (dot) product between the incident vector electric field and the current density along the dipole reduces the vectors to scalars that can be integrated. The current density acts as the testing or weighting function for the method of moments. Performing the integration means that the current density only satisfies the zero tangential electric field boundary condition in an average sense. If series impedances are placed in the V-dipole, their impedance is added to the diagonal elements of the mutual impedance matrix. To excite the structure, we place a delta voltage source in series with the Vdipole terminals. The solution for the currents can be found by inverting the matrix equation and using the voltage excitation vector starting with the matrix equation
 (3)
After computing the matrix inverse and specifying the input voltage vector, the complex current values are found on the structure:
 (4)
Given the input voltage and the solution for the currents, the input impedance can be calculated. Similarly, the far- and near-field patterns can be calculated by using Eqs. (5) and (6) of the dyadic Green’s function.
 (5) (6)The code must satisfy Kirchhoff’s current law at the junction between groups of Vdipoles, which adds a constraint to the currents. Because an overlapping sinusoidal basis function closely follows the actual currents normally excited on dipoles, the segments can be on the order of a quarter-wavelength long or more and yield acceptable results. Basis functions that closely follow expected current distributions are sometimes called entire domain functions. These reduce the size of the matrix to be inverted but require more complicated calculations for matrix terms and radiation. Although the concept of a V-dipole was expanded to a V rectangular plate [8], the method is only a subset of general integral equation solutions. This approach generates a simple impedance matrix formulation easily understood from an engineering point of view. |
