# Method of Moments for Antenna Calculations

### What is a "moment"?

If one looks carefully at the formulae for radiation from a wire antenna, using the precept that "accelerated charge radiates", one comes to the conclusion that it is the rate of change of (current times element of length) that contributes to the far field.

For a given antenna structure the conductors can be broken into "segments", and the currents on the segments can then be determined. The "moment" is numerically the size of the current times the vector describing the little segment (length and orientation). One matches the currents at the ends of the segments. A set of "basis functions" may be assumed into which the current distributions are decomposed.

In the simplest case, the basis functions are rectangular approximations to the Dirac delta function. Because the widths of the rectangular sections are non-zero, only a finite (reasonably small) number of them are needed to cover the antenna wire structure. The next more complicated basis functions are triangular in shape. This gives a smoother approximation to the current distribution, in which the current distribution is piecewise-linear between the matching points. In general, the "n-th moment" is obtained by integrating the product of the Green's function with the n-th basis function. In the case of rectangular basis functions this gives the interpretation of "moment" set out simply above. However, for more complicated basis function shapes the "moment" is a more abstract concept. For example, we could start with an infinite set of basis functions which were the sinusoidal and co-sinusoidal Fourier components with spatial period equal to the size of the antenna. We then truncate this infinite Fourier series to make the calculation tractable in a sensible time on a computer.

The "method of moments" starts from deriving the currents on each segment, or the strength of each moment, by using a coupling Green's function. This Green's function incorporates electrostatic coupling between the moments for if the spatial change of the currents is known accurately then one can compute the build up of charges at points on the structure. It is usual to approximate antennas having area by wire grid approximations, which also have to be chosen extremely carefully. As one always is presented with a computed result for a simulation, even if the model is in error, one can see that replacement of areas of metal by wire grids requires physical insight into the processes involved, rather than blind application of an algorithm.

One of the puzzles that this author has noticed, is that in the classic text books (eg; Antennas and Radiowave Propagation by R E Collin, McGraw-Hill 1985, ISBN 0-07-066156-1) the appropriate Green's function might be thought to include near-field interactions, and yet the maths seems to state that only the far field, or radiation-field, Green's function is used in the calculations. Why is it allowed to neglect the near field interactions in calculating the current distribution on the antenna structure?

The answer to this conundrum may be found in the fact that the Green's function is given for the magnetic vector potential, which falls off strictly as 1/R, where R is the distance from the excitation point on the antenna structure to the field point, even when that lies in the near field region. For a given field point, as the integral runs over all the points on the structure, R will vary slightly, which may not make much difference to the magnitude of the vector potential, but will certainly affect its phase. The local fields to the antenna are found from differentiating the vector potential A, and the space derivative of a function falling off as 1/R gives rise to the local fields, or "near fields", which fall off faster than 1/R. So the Green's function itself is of universal applicability, and the near field terms are derived quantities. A further puzzle lies in how the near fields from one current element set up the adjacent current elements. This is not easy to visualise.

When the source current distribution across the antenna structure has been determined, it may then easily be integrated to find the total far field.

A Green's function is a kind of 3-dimensional version of the impulse response function familiar in linear electronic circuit analysis. One sets up a structure for the space under consideration, specifying where the excitations can be and what the boundary conditions are. One then excites the structure with a single little region of excitation, with all the other possible excitations set to zero. The Green's function is the response of all the other regions in the problem to this excitation. Since the system is assumed linear, the principle of superposition applies and the total response to an arbitrary set of excitations can be obtained for the problem by direct summation or integration over all the excitations.

### External problem and Internal problem

The procedure whereby the far fields are calculated from a known current distribution is termed the External problem, and is computationally straightforward.

The procedure which establishes the current distribution on the structure, in terms of the geometry of the structure, any loading by objects or dielectrics in the near field, and the feed terminal current or voltage, is termed the Internal problem. This is computationally much harder.

#### The Internal problem

One of the problems in analysing antennas is that the excitation is often only specified as a single terminal voltage or current. Since the current distribution on the entire structure has to be known to integrate and find the far field response, and the fields affect the currents as well as the currents affecting the fields, a certain amount of obscuration of the mechanism of this calculation is endemic in the mathematics used to solve the equations. For example, one can easily calculate the far field of a Yagi-Uda antenna if one knows the amplitudes and phases of the rod currents, but finding these from a knowledge of the driven element voltage is a far harder proposition.

The way the internal problem is handled in the MoM is to assume a set of unknown coefficients which represent the weightings of the basis functions along the structure. The electric field next to the surface of the antenna conductors is then calculated as a matrix equation for a point set at the segments of the structure specified in the model. At each point the electric field will be a sum over the complete set of basis functions or moments; that is, it is a unique function of the current distribution on the structure.

We then force the electric field at ALL the points on the structure representing the segmentation to be zero (in the case of a perfect conducting antenna surface); or otherwise comply with the boundary condition that we know must exist. The matrix equation may then be routinely inverted to give one, and only one, unique set of weights for the basis functions which then quantitatively determine the current distribution over the structure that uniquely results in the boundary condition(s) being satisfied.

This method therefore rests on two assumptions. First, that the problem is linear, which means that any solution found is unique and represents the ONLY correct solution; and second, that given the Green's function and the structure of the problem, the only extra information needed is to specify the boundary conditions on that structure.

Now, in the more recent versions of NEC (Numerical Electromagnetic Code), it is possible to handle boundary conditions on imperfectly conducting grounds, and tapered wire structures, and wire structures which are connected to ground. One can argue that all structures on which boundary conditions are forced must be considered as "part of the antenna" for the purposes of determining its radiating properties.

Now in array antennas, having multiple feed points, energy can flow across the structure from one feed to another. Thus the current distributions on the structure will depend on the various feeder lengths and terminal impedances. There should be no problem providing these are all incorporated into the model. However, conducting and scattering objects in the near field also couple to the antenna currents and must be included in the description of the antenna, as the moment method algorithm makes no distinction between the various parts of the array antenna. This is particularly important for array antennas mounted on mobiles; consider a helicopter, for example.

There are varieties of techniques which go under the collective title "method of moments". Point-wise matching of the approximate piecewise solution to the exact solution may be made by minimising a "functional" depending on the currents and fields at a collection of discrete points across the structure. There are many ways of decomposing the structure into "elements" and choosing the appropriate "basis functions". While the mathematical approximate method of solving the integral form of Maxwell's equations may be justified, with appropriate considerations of convergence, very little insight into the physics of the model is to be gained from considering the approximate solution techniques.

### A possible question

Does the method of moments calculation, which can be a component of the NEC "Numerical Electromagnetic Code" for antenna radiation calculations, require pre-assumptions about the current patterns on the antenna structure, and if so, does it relax this pattern by successive iterations of the calculations, or does it just solve for the far fields and the amplitude and phase of the current elements by a huge one-pass matrix calculation?

Here is what is said by two of the practitioners.

Practitioner 1

Quote

The way I see it is .... a numerical solution of an integral equation, which is approached by dividing the integration range into discrete steps, thereby turning it into a set of linear equations. The computation is therefore a matrix solution rather than an iteration. Matrices can of course be factorised by iteration, but in NEC etc. the matrix solution is done by Gaussian elimination, Gauss-Seidel etc. with a predetermined number of calculations.

endquote

Practitioner 2

Quote

(Practitioner 1) is right, in that Gaussian elimination etc methods of matrix solution are regarded as one-pass methods, albeit with many steps. The convergence problem (arises) if you do a series of method-of-moments input impedance calculations for the antenna represented by an increasing number of smaller and smaller elements. You then get apparent convergence to the bottom of a bathtub curve, followed by an increase (as the element size is reduced, rather than a decrease) towards the tap end. The usual practice is to assume that the bottom of the bathtub curve is actually the correct value.

The method assumes the current on the structure to be given by a series of basis functions of chosen shape-distribution but of unknown amplitude and phase for each sub-part of the structure. You compute the spatial EM mutual coupling by means of a Greens function between each current component/element. You use Kirchoff's Laws where components are touching. You throw all these circuit equations into a huge matrix solver and hope for a solution to all the unknowns. You can use either an externally applied field as a source (receive case) or a current or voltage source at the input (transmit case). In the transmit case you can then compute the far field radiation from the computed currents on the network.

endquote

### A comment

From the above remarks it seems that the current distributions and the far fields are all computed at once from knowing just the physical structure of the wires and rods of the antenna, and a single driving point excitation (current or voltage). Errors in computation are then due to unwise segmentations, imprecise specification of material properties which affect the boundary conditions, and computational precision and rounding problems. There appears to be no other physical reason to doubt the soundness of this complex procedure.

A succinct introduction to the formal mathematics may be found on the OCW website of the Massachusetts Institute of Technology (MIT) at http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6-635Advanced-ElectromagnetismSpring2003/284D03BE-FBA8-41BB-965F-F26056CC0261/0/Mar10.pdf