#### 2.2. RGFMM Method

[6] The investigation is carried out using the RGFMM simulation method which is based on the theory initially described by *Bialkowski* [1984] and *Morgan and Schwering* [1994]. 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.

[7] For the method as applied to an array, it is assumed that all the monopoles are excited from the gaps, *g*_{0} (central element) and *g*_{1} (peripheral elements), located at the base. The dimensions of the central element are as follows: total monopole height (*l* = *B*_{1} + *B*_{2}), monopole wire radius (*a*), excitation gap height (*g*_{0}), top hat radius (*R*_{a}), top hat thickness (*B*_{2}). The peripheral elements are evenly positioned on a circular radius, *r*_{o}, and the dimensions of excitation gaps (*g*_{1}), top hat radius (*R*_{c}), top hat thickness (*B*_{5}) 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 *B*_{3} is the distance between the top of the monopole and the top ground plane.

[8] A single monopole (such as the central monopole) is excited at the base using voltage, *V*_{0}, which result in an electric field in the gap that can be written as:

where *g*_{0} 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.

[9] The input admittance of the monopole looking from the gap is given by

where *S*_{a} is the surface surrounding the gap aperture, *H*_{o} is the magnetic field in the aperture, and is a unit vector normal to the aperture surface.

[10] The *Y*_{in} can be determined once the magnetic field *H*_{o} is known. Here the task of determining *H*_{o} 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 *E*_{y} component, but no *H*_{y} component (*H*_{y} = 0). The fields around the probe are then TM^{y}.

[11] In region *III*, the electric field should satisfy the boundary condition ∂*E*_{y}/∂*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 *H*_{0}^{(2)}(Γ_{n}*r*) 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* = *R*_{a}, and ɛ_{0n} is the Neumann factor (ɛ_{0n} = 1 if *n* = 0, ɛ_{0n} = 2 if *n* > 0) which is introduced to make the equation easier to solve. *F*_{n} is a set of unknowns which must be determined. Γ_{n} is the propagation constant and is given by

where

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*.

[12] For the electric field in region *II*, the boundary conditions that must be observed are ∂*E*_{y}/∂*y* = 0 at *y* = *B* − *B*_{3} and at *y* = *B*, as well as being finite at *r* = 0. Thus it can be written as

where *D*_{n} is a set of unknowns to be determined and the propagation constant Γ_{2n} are given by

where

are the eigenvalues for region *II*, and

where *k*_{2} is the free space wave number, and ɛ_{2} is the permittivity of region *II*.

[13] 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.

[14] The electric field induced due to the other sources can be written in the form

which satisfies the boundary condition that *E*_{y1i} = 0 on the conducting surface of the probe in region *I*. The *A*_{n} is a set of unknowns to be determined, and the propagation constant Γ_{1n} is given by

where

are the eigenvalues for region *I*, and

where *k*_{1} is the free space wave number, and ɛ_{1} is the permittivity of region *I*.

[15] The electric field due to the gap source has a similar form to that of *E*_{y3} 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

where

The final superposition of the electric fields in region *I* yields the equation

For the field matching analysis, expressions are required for the *H*_{ϕ} magnetic field components. These may be determined by applying the relationship for the TM^{y} modes [*Harrington*, 1961], which for the *n*th 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* = *R*_{a}. The matching conditions are

[16] 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 *Y*_{in} can be determined by substituting them into equation (Yin).

#### 2.3. Analysis Method for Circular Array of Monopoles

[17] 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*).

[18] 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 *V*_{0},*V*_{1},…, *V*_{N} are the voltages of the monopoles 0, 1, 2,…, *N* and *I*_{0}, *I*_{1},…, *I*_{N} are the resulting currents at the feeding points.

[19] In order to obtain *Y*_{00} and *Y*_{i0}, an excitation of the central monopole is applied while the remaining monopoles are short-circuited. Under such symmetrical excitation, the voltages *V*_{i} = *V*_{1} and the currents *I*_{i} = *I*_{1}, where *i* = 1, 2, …, *N*. Thus the admittance-matrix equation becomes

which can also be written as

Using the fact that *Y*_{0i} = *Y*_{01} for *i* = 1, 2,…, *N*, *Y*_{00} and *Y*_{i0} can be obtained:

[20] Equations (16)–(17) indicate that in order to determine *Y*_{00} and *Y*_{i0} 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.

[21] 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.

[22] 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).

[23] Assuming that *I*_{s1} is the current calculated for the peripheral probe number 1 when the *s*-type of voltage excitation *V*_{sm} is applied, the mutual admittance *Y*_{m1} 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*.

[24] 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.