2.2. RGFMM Method
 The investigation is carried out using the RGFMM simulation method which is based on the theory initially described by Bialkowski  and Morgan and Schwering . First, it is explained here for the case of a single monopole antenna. This method requires the presence of an upper ground plane for the simulated monopole as shown in Figure 2. When the upper ground plane is greater than one free space wavelength, the input impedance and radiation pattern of the monopole becomes approximately the same as in free space [Morgan and Schwering, 1994]. The reason is that a monopole antenna predominantly radiates in the direction towards the horizon. As a result, when the upper ground plane is moved up, its interaction with the monopole becomes negligible. The convenience of this approach is its short computation time for each frequency point. For an array of monopoles the computation time is increased but only by the factor of approximately of N, where N is the number of monopoles forming the array.
 For the method as applied to an array, it is assumed that all the monopoles are excited from the gaps, g0 (central element) and g1 (peripheral elements), located at the base. The dimensions of the central element are as follows: total monopole height (l = B1 + B2), monopole wire radius (a), excitation gap height (g0), top hat radius (Ra), top hat thickness (B2). The peripheral elements are evenly positioned on a circular radius, ro, and the dimensions of excitation gaps (g1), top hat radius (Rc), top hat thickness (B5) are identical for all these elements, however they can differ from the central element. All these dimensions are shown in Figure 1. There are two extra parameters introduced to the structure when method is being employed. As shown in Figure 2, B is the distance between the bottom and top ground plane and B3 is the distance between the top of the monopole and the top ground plane.
 A single monopole (such as the central monopole) is excited at the base using voltage, V0, which result in an electric field in the gap that can be written as:
where g0 is the height of the gap. This equation assumes that the electric field is uniform and oriented in the y direction. The field variation due to edges is ignored. This assumption is used in many other similar problems concerning excitation of wire antennas and generates accurate results for the input admittance/impedance.
 The input admittance of the monopole looking from the gap is given by
where Sa is the surface surrounding the gap aperture, Ho is the magnetic field in the aperture, and is a unit vector normal to the aperture surface.
 The Yin can be determined once the magnetic field Ho is known. Here the task of determining Ho is accomplished as follows. First, the field in the close proximity of the monopole is assumed axially symmetric. Second, for a single monopole, the volume inside the parallel plate guide is divided into cylindrical regions, I, II, III. Next, the electric and magnetic fields in each region is expressed in terms of cylindrical functions with unknown expansion coefficients. Since the electric currents flowing along the probe have principally y and r components, it is expected that the produced field will have an Ey component, but no Hy component (Hy = 0). The fields around the probe are then TMy.
 In region III, the electric field should satisfy the boundary condition ∂Ey/∂y = 0 at y = 0 and at y = B since those points are at the surfaces of the ground plane. The electric field at region III due to a single monopole is given by
where H0(2)(Γnr) is the zeroth order Hankel function of the second kind, which represents outwardly traveling radial wave. This field is normalized to a radius of r = Ra, and ɛ0n is the Neumann factor (ɛ0n = 1 if n = 0, ɛ0n = 2 if n > 0) which is introduced to make the equation easier to solve. Fn is a set of unknowns which must be determined. Γn is the propagation constant and is given by
are the eigenvalues for region III, and
where k is the free space wave number for region III, ω is the radian frequency, ɛ is the permittivity and μ is the permeability of region III.
 For the electric field in region II, the boundary conditions that must be observed are ∂Ey/∂y = 0 at y = B − B3 and at y = B, as well as being finite at r = 0. Thus it can be written as
where Dn is a set of unknowns to be determined and the propagation constant Γ2n are given by
are the eigenvalues for region II, and
where k2 is the free space wave number, and ɛ2 is the permittivity of region II.
 The electric field in region I is slightly different from the fields from other regions. It is a superposition of the field from the voltage source applied to the gap and the field which is induced due to secondary sources which are generated by reflections of the primary wave from the surrounding structure.
 The electric field induced due to the other sources can be written in the form
which satisfies the boundary condition that Ey1i = 0 on the conducting surface of the probe in region I. The An is a set of unknowns to be determined, and the propagation constant Γ1n is given by
are the eigenvalues for region I, and
where k1 is the free space wave number, and ɛ1 is the permittivity of region I.
 The electric field due to the gap source has a similar form to that of Ey3 in terms of the outward traveling wave and can be written as
Its exact form may be determined by performing a Fourier analysis of the electric field in the gap along the surface at radius r = a. After accomplishing a Fourier analysis, the electric field due to the voltage in the gap may then be written as
The final superposition of the electric fields in region I yields the equation
For the field matching analysis, expressions are required for the Hϕ magnetic field components. These may be determined by applying the relationship for the TMy modes [Harrington, 1961], which for the nth y harmonic is given by
Using this approach, the ϕ component of the magnetic field is given in the following expression:
Applying the same approach to equation (4), the ϕ-component of the magnetic field in region II may be written as
Similarly for region III
Now, all the electric and magnetic field components for each region have been obtained. They can now be matched at the common interface r = Ra. The matching conditions are
 Determining the unknown field expansion coefficients can be accomplished by using the Galerkin procedure, which produces a set of linear equations for the unknown field expansion coefficients. Gaussian elimination or matrix inversion can then be used to solve the set of linear algebraic equations. Once the unknown coefficients are found, the Yin can be determined by substituting them into equation (Yin).
2.3. Analysis Method for Circular Array of Monopoles
 The field matching method derivations for a single element, described above, can be extended to the case of a circular array. For the case when a single monopole is excited and the remaining monopoles are terminated (for example, short-circuited), the problem can be resolved into N + 1 canonical problems, each featuring a circular symmetry. Using this decomposition, it is sufficient to consider only the regions of the central active element (I below the top hat, II above the top hat), only one of the peripheral elements' regions (IV below the top hat, V above the top hat) and the region external to the monopoles (region III).
 If the central monopole is labelled #0, and the peripheral monopoles as 1, 2, …, N, the Y-matrix equation of the array can be written as
where V0,V1,…, VN are the voltages of the monopoles 0, 1, 2,…, N and I0, I1,…, IN are the resulting currents at the feeding points.
 In order to obtain Y00 and Yi0, an excitation of the central monopole is applied while the remaining monopoles are short-circuited. Under such symmetrical excitation, the voltages Vi = V1 and the currents Ii = I1, where i = 1, 2, …, N. Thus the admittance-matrix equation becomes
which can also be written as
Using the fact that Y0i = Y01 for i = 1, 2,…, N, Y00 and Yi0 can be obtained:
 Equations (16)–(17) indicate that in order to determine Y00 and Yi0 only currents on two monopoles, the central and the peripheral, need to be determined. This requires considering the electromagnetic field solution involving 2 × M unknowns, where M is the number of modes (and field expansion coefficients) required to describe the field surrounding both the central and one of the peripheral antenna elements.
 When one peripheral monopole, for example m = 1 (due to symmetry, for the remaining peripheral probes the solution is obtained by re-numbering them), is excited and the remaining peripheral monopoles m = 2, 3,…, N are short-circuited, the solution to this problem can be obtained through the decomposition of the original excitation into symmetric excitations of the peripheral elements. This decomposition explores the symmetry of the array. The following set of excitation voltages that are applied to peripheral monopoles (while the central monopole is short-circuited) accomplishes this task:
where s = 1, 2,…, N − 1. Owing to symmetry and the form of excitation, only currents on the central monopole and only on one peripheral monopole need to be determined.
 When all these excitations are summed up (using the principle of superposition), then
The above result implies that the superposition of such N (s = 1, 2,…, N − 1) symmetric excitations is equivalent to the excitation when antenna m = 1 is excited with the voltage of 1 while the remaining N − 1 antennas are short-circuited (their excitation voltages are 0).
 Assuming that Is1 is the current calculated for the peripheral probe number 1 when the s-type of voltage excitation Vsm is applied, the mutual admittance Ym1 is given by
It is apparent that using the method of decomposing a voltage excitation into N symmetrical excitations, N problems need to be solved each featuring only 2 × M unknowns, where M is the number of harmonics describing the field in the y direction. The resulting problem of the two probes is solved by forming field equations similar to those for a single probe. The difference lies in the inclusion of an extra term which describes interactions between the two probes. This term is given in terms of outgoing cylindrical wave harmonics and concerns region III.
 Using a computer algorithm, which is developed based on this theory, investigations into the effect of top hat and coating dielectric on the resonant frequency and input impedance of top-hat monopoles can be performed. In particular, an insight into reducing the height of the monopole array can be established.