EuCAP 2006 - European Conference on Antennas & Propagation

 
Session: Session 4A10A - Antenna Measurements (12j)
Type: Oral Antenna
Date: Thursday, November 09, 2006
Time: 08:30 - 12:30
Room: Gallieni B
Chair:
Co-chair:
Remarks:


Seq   Time   Title   Abs No
 
1   08:30   Evaluation of Diversity and MIMO Performance of Antennas from Phaseless Radiation Patterns
Krogerus, J.; Suvikunnas, P.; Icheln, C.; Vainikainen, P.
Helsinki University Of Technology, FINLAND

A convenient way to evaluate the performance of multielement antenna configurations in small mobile terminals is mathematically combining simulated or measured 3-D radiation patterns of the antenna with the measured 3-D signal distribution of the propagation environment. This method is called the Experimental Plane-Wave Based Method (EPWBM) [1].

Often, the antenna 3-D radiation patterns are available only without phase information e.g. if an amplitude-only receiver (e.g. a spectrum analyzer) is used in the measurement of an active device (e.g. a mobile phone). In this study we investigate the significance of phase information of radiation patterns for the performance estimates of diversity and MIMO antenna configurations.

We carried out performance evaluations of several two-antenna mobile terminal prototypes with the EPWBM. The 3-D gain patterns of the prototypes were measured and simulated in free space and some of them also beside a phantom head. Several radio channel distributions measured with a spherical antenna array in outdoor macrocell, microcell, and indoor microcell environments were used in the analysis. With the EPWBM, the prototype antennas' radiation patterns were virtually "driven" through the characterized signal environments. Instantaneous values and distributions of some important parameters e.g. maximal ratio combined (MRC) signal power, were calculated over the measured routes. The MRC signal power at the 10 % probability level (Pmrc,10%) was used as the parameter describing the diversity performance of a multi-antenna prototype on a test route. The average of Pmrc,10% for a representative set of radio channel routes was used as the figure of merit for each prototype. We analyzed three different scenarios for the 3-D radiation patterns: a) original measured complex radiation pattern, b) radiation pattern with zero phase values (i.e. phaseless), and c) random-phased radiation patterns. We compared Pmrc,10% averaged over the set of selected test routes between these three scenarios of antenna radiation pattern information. Fig.1 presents this comparison for five example prototype diversity configurations (P1-P5) at the 2-GHz band.

In the diversity analysis the error in Pmrc,10% (average over all the test routes) obtained with phaseless patterns was -1.5 dB, when compared with the reference results obtained with original complex radiation patterns. Hence, the use of phaseless radiation patterns leads to unacceptable underestimation of performance of multielement antenna systems. On the other hand, the proposed, new way of substituting the missing phase information in radiation patterns with random phases yielded a much smaller error of only 0.6 dB.


Figure 1. Example of results: MRC signal power at 10% probability level, average of eight radio channel routes.

[1] P. Suvikunnas et al., "Evaluation of the Performance of Multi-Antenna Terminals Using a New Approach," Accepted in Dec. 2005 to IEEE Trans. Instr. and Meas.

 
 
2   08:50   Development of a Hologram-Based CATR for Testing a Very High Gain Antenna at 650 GHz
Ala-Laurinaho, J.1; Koskinen, T.1; Hakli, J.1; Karttunen, A.1; Lonnqvist, A.1; Noponen, E.1; Mallat, J.1; Vaaja, M.1; Viikari, V.1; Raisanen, A. V.1; Heinonen, J.2; Hautala, P.2; Lemanczyk, J.3
1MilliLab, SMARAD Radio Laboratory, Helsinki University of Technology, FINLAND;
2Laboratory of Machine Design, Helsinki University of Technology, FINLAND;
3ESA/ESTEC, NETHERLANDS

MilliLab / Radio Laboratory at Helsinki University of Technology (TKK) develops compact antenna test range (CATR) based on a binary amplitude hologram for testing high gain antennas at submillimetre wavelengths. The transmission-type hologram is used as a collimating device to produce a plane wave, which is needed in antenna testing. The previous project for the European Space Agency (ESA) culminated in the successful test campaign of the ADMIRALS RTO (Representative Test Object) 1.5 m antenna at 322 GHz during summer 2003. Within a new ESA project, the aim is to test a 1.5 m antenna at a frequency of 650 GHz during autumn 2006. The potential test objects are the telescope of the Planck RFQM (Radio Frequency Qualification Model) and the ADMIRALS RTO. The Planck RFQM is the first option, but the final decision of the test object will be done in the near future. For these antenna tests, the Radio Laboratory is designing and constructing a 650 GHz CATR based on 3.16 m hologram.

In this paper, the development and construction of the CATR for 650 GHz antenna tests is described. The design of the hologram is based on the use of an in-house developed FDTD analysis program. The design of the hologram is described and simulation results of the quiet-zone field are presented. The high test frequency introduces challenges for the RF-instrumentation, as it has to ensure sufficient dynamic range for accurate antenna radiation pattern and quiet-zone field measurements. The RF-instrumentation based on the ABmm MVNA-8-350 millimetre wave vector network analyser and the associated transmitter and receiver is described and the dynamic ranges are evaluated. The design and construction of the mechanical systems for the CATR are described. This includes the modification of the antenna positioner for the test object, the construction of the high quality scanner for the quiet-zone field verification, the construction of the stiff frame and the pedestal for the hologram, and the construction of the positioner for the feed of the CATR. The manufacturing of the hologram is described, consisting of the hologram pattern etching on the copper plated polyester film and the joining of the three pieces with high accuracy.

The CATR is constructed in the large test hall of the High Voltage Institute at TKK. The walls are metallic, which requires the elimination of the stray radiation with large amount of absorbers. The layout of the hologram CATR and absorber placement are presented.

 
 
3   09:10   Studies on an Amplitude Hologram as a Submillimeter-Wave Collimator at Circular Polarization
Koskinen, T.; Ala-Laurinaho, J.; Hakli, J.; Raisanen, A.V.
MilliLab, SMARAD Radio Laboratory, Helsinki University of Technology, FINLAND

An amplitude hologram is a slot pattern that consists of numerous slightly curved, vertically oriented, slots and metal strips. The pattern is generated with a computer and etched on a thin metal-plated dielectric film. We use holograms as transmission-type collimators to form a plane-wave field needed in compact antenna test ranges (CATR) or in radar cross-section (RCS) ranges. Especially, we are interested in using holograms at submillimeter wavelengths, where the use of shaped reflectors becomes cumbersome and expensive due to a very stringent surface accuracy requirement. Simple and inexpensive manufacturing makes the use of amplitude holograms very attractive at submillimeter wavelengths. In addition, due to the transmission-type operation, holograms have a much lower surface accuracy requirement than reflectors have. We have used holograms in CATRs for antenna testing at 119 and 322 GHz. Also, we have carried out promising demonstrations at 650 GHz.

Currently, operation of holograms is limited to vertical polarization, i.e., the illuminating electric field and the generated plane wave are vertically polarized. This is because holograms are illuminated with a corrugated horn antenna, which has a Gaussian beam with a rather high edge illumination (up to -2-4 dB). Therefore, the amplitude of the transmitted field has to be tapered on the edges of the hologram to avoid harmful edge diffraction. With a vertically polarized illumination, the amplitude tapering can be done by narrowing the slots towards the edges of the hologram pattern, which attenuates the transmitted field. However, this is not possible at horizontal polarization, where the transmission does not clearly depend on the slot width, and therefore narrowing of slots does not work properly.

Narrowing of slots is not needed if the amplitude tapering is included in the illuminating field. This can be done by a dual reflector feed system (DRFS), which transforms the horn antenna radiation to an illumination, which has a flat amplitude profile in the middle, and which is strongly tapered on the edges of the beam (down to 17 dB). The phase of the illumination remains spherical. A demonstrative DRFS has been constructed and it has already been used to illuminate holograms at 310 GHz. A hologram pattern designed for the DRFS illumination has no narrow slots on the edges, and therefore it can operate also at horizontal polarization.

In this paper, we study a hologram that operates optimally at vertical and horizontal polarizations at the same time. We study what are the requirements to make the hologram to operate identically at both linear polarizations. As a demonstration, we have designed a hologram of 60 cm in diameter that operates with the DRFS illumination at 310 GHz. According to simulations, the operation of this hologram is almost identical at both polarizations, and therefore, we may assume that the hologram can also be used at circular polarization. The hologram is now under manufacturing and will be tested during the spring 2006.

 
 
4   09:30   Log-Periodic Dipole Antenna with Low Cross-Polarization
Pivnenko, S.
Technical University of Denmark, DENMARK

The log-periodic principle established in late fifties gave rise to development of several types of wideband antennas [1]. Different log-periodic antennas are used now in many applications where, particularly, the wide bandwidth is one of the important requirements. However, there is a drawback in most of these antennas, namely, the cross-polarization level is not acceptable for certain applications. For example, typical co-polar to cross-polar ratio of commercially available log-periodic antennas is about 25 dB, but for some applications larger values are desired.

One of such applications is near-field antenna measurements, where a relatively narrow-band, less than octave, open-ended waveguide is often used as a probe. Several probes are thus necessary to cover a wide frequency range and their interchange is unavoidable, if the measurement frequency is to be changed. A wideband probe covering the whole range would simplify the manipulations and speed-up the measurements. The requirements for the cross-polarization level of the near-field probes are quite high and the available log-periodic antennas cannot be applied here.

The log-periodic antennas were studied in this work with the aim to improve their cross-polarization level. It was found that some modifications of the traditional design lead to essential improvement of the co-to-cross polarization ratio, particularly, the values up to 40 dB were obtained. The purpose of this paper is to introduce an improved design of a log-periodic dipole antenna with low cross-polarization level. Some recommendations regarding improvement of the polarization characteristics of log-periodic antennas in general are also given.

During the studies, it was also found that the log-periodic antennas can be attributed to the class of so-called first-order (µ = ±1) probes, which is an important requirement for probes in spherical near-field antenna measurements [2]. This requirement implies that the radiation pattern of the probe has only first-order (sin, cos) azimuthal dependence and thus complete information about the pattern can be obtained from only two orthogonal cuts. As a result of this feature the spectrum of spherical modes of such pattern contains only the modes with azimuthal indices µ = ±1. The log-periodic dipole antennas do not satisfy this requirement completely since in their spectra some power is also contained in the spherical modes with azimuthal indices µ = 0, 2, and 3, but their level is relatively low. The influence of these modes can be compensated by application of recently developed iterative probe correction technique [3].

[1] C. E. Smith (Ed.), Log Periodic Antenna Design Handbook, Smith Electronics, Inc., Cleveland, USA, 1966.
[2] J. E. Hansen (Ed.), Spherical Near-Field Antenna Measurements, Peter Peregrinus, Ltd., London, 1988.
[3] T. Laitinen, S. Pivnenko, O. Breinbjerg, "Iterative probe correction technique for spherical near-field antenna measurements," IEEE Antennas and Wireless Propagation Letters, Vol. 4, 2005, pp. 221-223.

 
 
5   09:50   Reduction of Truncation Errors in Planar Near-Field Antenna Measurements Using the Method of Alternating Projections
Martini, E.1; Breinbjerg, O.1; Maci, S.2
1Technical University Of Denmark, DENMARK;
2University Of Siena, ITALY

Planar near-field scanning is a widely adopted technique for antenna measurements. However, due to the limited size of the scanning area, the resulting pattern is reliable only within an angular region depending on the geometry of the antenna under test (AUT) and on the extension and position of the scan area (A. D. Yaghjian, NBS Tech. Report, Oct. 1975). Thus, a large scanning area is required to enlarge the angular region in which the resulting pattern is reliable. However, long measurement times and complex facilities are then required. A method allowing one to estimate the near-field outside the physically available scan region without requiring additional measurements would significantly speed up and simplify the process of measurement acquisition. However, without any additional a priori information the problem of extrapolating near-field data is ill-posed (J-C. Bolomey, O. M. Bucci, L. Casavola, G. D'Elia, M. D. Migliore, and A. Ziyyat, IEEE Trans. Antennas Propagat., 52, 593-602, 2004).

The method of alternating projections is an iterative algorithm that uses the measured data and prior knowledge to recover a signal, and it has been successfully applied to computer-aided tomography and image restoration problems (D. C. Youla, IEEE Trans. Circuits Syst. CAS 25, 695-702, 1978). A special case of this recursive method is the Gerchberg-Papuolis algorithm (R. W. Gerchberg, Opt. Acta 21, 709, 1974 and A. Papoulis, IEEE Trans. Circuits Syst., CAS-22, 735-742, 1975, respectively), which iteratively computes the transform of a band-limited function, the iterations simply involving the computation of fast Fourier transforms. It has been proved that the mean square error monotonically decreases for an increasing number of iterations and that the procedure is very effective also against noisy data. This algorithm can be directly applied to the extrapolation of the near-field data, since on the measurement plane the spectrum is limited to the visible region. However, the convergence has shown to be generally quite slow.

In this work, a more effective application of the alternating projections algorithm is proposed; it exploits the a priori knowledge on the AUT geometry and position and it exhibits a quite fast convergence. The procedure is based on the use of the reliable part of the spectrum and on the requirement that the tangential electric field is basically concentrated on the antenna aperture. Accordingly, the Gerchberg algorithm is applied to continue the spectrum outside the "reliable region" by exploiting the condition of spatial-limitedness of its Inverse Fourier Transform. First, the spectrum is calculated from the truncated measurements and back propagated to the AUT plane. Then, its reliable part is Inverse Fourier transformed to yield an estimate of the field distribution on the AUT plane; this is subsequently modified by setting all the samples outside the antenna aperture to zero. Thus modified, the field distribution is Fourier transformed and the generated spectrum is corrected so that the previous values are restored where the original spectrum is reliable. This process is then iterated until convergence is reached.

The proposed technique has been applied to theoretical as well as measured near field data and in both cases it has turned out to be very effective.

 
 
6   10:40   Fast 3-D Total Radiated Sensitivity Measurement Technique for Mobile Terminals
Pannetrat, S.; Derat, B.; Leray, C.
SAGEM Communication, FRANCE

3-D pattern measurement in anechoic rooms is now widely used for evaluating mobile terminals radiated performances. The Total Radiated Power (TRP) and Total Radiated Sensitivity (TRS) are the two principal characteristics. Standard techniques to evaluate the "full TRS" are firstly based on the acquisition of a large number of points (more than 200), which leads to a significant far-field assessment duration. Secondly, the iterative sensitivity process is highly time consuming. Starting from these observations, this paper presents an alternative fast 3-D measurement method that is now considered in standardization processes. Experimental results presented in this paper highlight the good accuracy of the method.

1.Importance of the number of frames on the accuracy of TRS measurements

The TRS prohibitive time duration is a "well-known" issue in the area of mobile radiated performances evaluation. To obtain faster TRS assessment using current tools generally requires to reduce the number of measurement points and, above all, the number of frames, transmitted during the iterative sensitivity process. However, reducing the number of frames can have a significant impact on the repeatability of measurements, due to the necessary increase of the BER target, leading to a much lower accuracy.

2.Fast 3-D TRS measurement technique

The idea of the fast 3-D TRS assessment technique presented in this paper is completely different, because the number of points is kept unchanged, whereas the communication frames number can be largely increased. The main idea to accelerate the measurement process is to use differently the RxLev or CPICH RSCP parameters. In fact, assessing these parameters at each point allows to obtain a full 3-D pattern of the power received by the mobile station. Though much quicker to assess than sensitivity, it can be considered that RxLev / CPICH RSCP and sensitivity are directly proportional in every position. Hence, measuring 3-D RxLev or CPICH RSCP patterns and only one point of sensitivity is sufficient to estimate a full 3-D sensitivity pattern. Moreover the point of sensitivity can be evaluated thanks to an accelerated algorithm, which is proposed in this paper. This algorithm relies on known relationships between the BER level and signal-to-noise ratio. Since the sensitivity has to be here evaluated at one point only, the number of frames considered can be largely increased (600), leading to a significant improvement in measurement repeatability.

3.Experimental validation of the Fast 3-D TRS technique

First, full TRS measurements of 10 mobile phones, in various frequency bands, are compared to fast 3-D TRS assessments. The good accuracy of the method is demonstrated. Second, the repeatability of measurement is investigated, and effectively shows a significant improvement compared to a full TRS technique. Indeed, it is shown that the TRS is repeatable within ±0.6dB, with a 95% confidence interval. Moreover, the typical time duration of measurement, for one frequency, is significantly reduced, typically by a factor of 3.
This fast measurement technique should help in accelerating the design process of mobile phone antennas.

 
 
7   11:00   Histogram Equalisation as a Method for Making an Objective Comparison between Antenna Patterns Functions
Gregson, S.F.1; McCormick, J.2; Parini, C.G.3
1Nearfield Systems Inc., UNITED STATES;
2Selex Sensors and Airborne Systems, UNITED KINGDOM;
3Queen Mary, University of London, UNITED KINGDOM

Attempts to produce objective, quantitative measures of comparison between data sets using statistical methods have been widely reported in the literature [1,2]. Impartial, repeatable assessments of similarity such as these have been shown to be of value in many fields however, they are of particular utility in the verification and validation of new antenna measurement systems and techniques. More recently [3], techniques have been developed that require the antenna pattern functions to be converted into histograms before the comparison, i.e. the measure of adjacency, is made. The success of such a tactic can be crucially dependent upon the choice of categorising "bins". As the number, level, and size of these bins can be chosen both a priori and freely, it is possible that the resulting histograms will be sparsely populated with the majority of the samples falling within only a few of the categories. Unfortunately, histograms that are grossly uneven can place significant demands on any subsequent measure of adjacency. However, this difficulty can be avoided if the bins are defined in such a way that roughly equal numbers of samples fall within each of the categorising bin. In the limit then, if each of the bins contain exactly the same number of samples for a given data set, the degree of deviation from uniformity that is obtained when forming a histogram for any other data set can be used as a measure of adjacency. This paper describes an efficient method for "equalising" a given histogram and the effectiveness of this strategy is then illustrated with example data.

Reference

[1] S.F. Gregson, "Probe-corrected Poly-planar Near-Field Antenna Measurements", PhD thesis, University of London, 2003
[2] S.F. Gregson, J. McCormick, C.G. Parini, "Advances in the Objective Measure of Comparison Between Antenna Pattern Functions", International Conference on Antennas and Propagation, University of Exeter, 2003
[3] J. McCormick, S.F. Gregson, C.G. Parini, "Quantitative Measure of Comparison between Antenna Pattern Data Sets", IEE Proc.-Microw. Antennas Propag, Vol. 152, No6, December 2005

 
 
8   11:20   User Effect on Total Radiated Power and 3-D Radiation Pattern of Mobile Handsets
Krogerus, J.; Toivanen, J.; Icheln, C.; Vainikainen, P.
Helsinki University of Technology, FINLAND

The user's body typically reduces transmitted and received power of a handheld mobile communications device and alters its radiation and polarization patterns. Measurements with real humans and statistical analysis are needed e.g. for developing realistic phantoms. In previous studies it has been found up to 10 dB differences in antenna performance between individual users [1]. We investigate the reasons for the user-dependent differences. We also consider separate loss contributions of head and of hand.

We carried out measurements for 13 test persons with the Rapid Antenna Measurement System (RAMS), which allows the measurement of a 3-D active uplink (MS-TX) radiation pattern of GSM phones within a few seconds without any movement of the test object. A functional GSM 900/1800 phone was used as the device under test (DUT). The radiated field and total radiated power (TRP) of the DUT can be characterized using spherical wave expansion technique from the signals measured by the 32 dual-polarized probes of the RAMS. The body loss (BL) is the difference of TRP in a use position and in free space. The main measurement set-ups are shown in Fig. 1. Fig.2. presents the comparison of TRP results at GSM900. Examples of radiation patterns are shown in the final paper. We found up to 7 dB variation and standard deviation of about 3 dB in TRP (and BL) at GSM 900 within the test group of 13 users, when the phone was held in a user-preferred way. The mean total BL (head and hand) was 9.2 dB. When the phone was held in the pre-defined talk position, the standard deviation of the total BL was less than 1 dB and the mean total BL was about 7.3 dB. Furthermore, the mean BL of the test persons heads corresponded well with the BL of a phantom head. Thus, individual position of the users hand on the handset is a main source for large variations in the performance between different users, whereas anatomical differences in the heads play a much smaller role.

Figure 1. The main measurement set-ups: a) DUT in free space, b) a pre-defined talk position (head and hand), c) a pre-defined position next to the head (head only), e) a user-selected talk position (head + hand), individual for each user, g) a talk position next to a head phantom (head only).

Figure 2. Comparison of TRP results between different users at 900 MHz band.

[1] J. . Nielsen et al. "Statistics of measured body loss for mobile phones," IEEE Trans. Ant. and Prop., Vol. 49, pp. 1351 1353, Sept. 2001.

 
 
9   11:40   Phase Retrieval Measurement Of Antenna Surfaces Using Astronomical Sources And Application To The Green Bank Telescope
Nikolic, B.1; Hills, R.E.1; Richer, J.S.1; Prestage, R.M.2; Balser, D.S.2; Chandler, C.2
1University of Cambridge, UNITED KINGDOM;
2NRAO, UNITED STATES

We present a technique for the accurate estimation of large-scale errors in an antenna surface using astronomical sources and detectors. The technique requires several out-of-focus images of a compact source and the signal-to-noise ratio needs to be good but not unreasonably high. The main advantages of this technique compared to alternatives are that it allows measurement over the full range of elevations; requires no extra equipment beyond that used for routine astronomical observing and can requires relatively little time. We present simulations illustrating the expected accuracy of the technique, verification on the largest millimetre-wave telescope and application of the measurements to improve the performance of the telescope.

For a given pattern of surface errors, the expected form of in- and out-of-focus images of compact sources can be calculated directly. We show that it is possible to solve the inverse problem of finding the surface errors from the images in a stable manner using standard numerical techniques. To do this we describe the surface error as a linear combination of a suitable set of basis functions (we use Zernike polynomials). We illustrate the technique and quantify its accuracy, in particular its dependence on of receiver noise and pointing errors, through simulations. The key result is that good measurements of the error on large spatial scales can be obtained if the input images have a signal-to-noise ratio of order 100 or more.

We have applied the technique to the 100-m diameter Green Bank Telescope. Using the facility astronomical receiver operating at a wavelength of 7mm, a set of beam maps takes approximately 25 minutes to complete and we have been able to produce low-resolution surface error maps with illumination-weighted RMS accuracy of around 100 micron from each such set. During the verification stage we introduced known deformations to the primary surface which are, as shown below, recovered by technique with high accuracy.

By making measurements over a wide range of elevations, we have calculated a model for wavefront-errors due to the uncompensated-for gravitational deformation of the telescope. Implementation of this model as a refinement in the active surface system of the telescope has increased the aperture efficiency of the telescope at low elevation and with the model applied, the efficiency is now largely independent of elevation.

We have also demonstrated that the technique can be used to measure and largely correct for the thermal deformations of the antenna, which during daytime observing often exceed the uncompensated gravitational deformations. We conclude that the aberrations induced by gravity and thermal effects are large-scale and the technique presented here is particularly suitable for measuring such deformations in large millimetre and sub-millimetre wave radio telescopes.

Intended deformation of telescope surface introduced through the active surface system.

Phase retrieval measurement of the deformation.

 
 
10   12:00   Precision Blind Pointing Calibration of the NASA-JPL-DSN Large Reflector Antennas at Ka-Band (32-GHz)
Rochblatt, D.; Withington, P.
Jet Propulsion Laboratory, UNITED STATES

The NASA-JPL-Deep Space Network subnet of 34-m Beam Waveguide (BWG) Antennas was recently upgraded with Ka-Band (32-GHz) frequency feeds for Space Research and Communication. For normal telemetry tracking a Ka-Band monopulse system is used which typically yields 1.6-mdeg Mean Radial Error (MRE) pointing accuracy on the 34-m diameter antennas. However, for the monopulse to be able to acquire and lock, for special radio science applications where monopulse can not be used, or as a back-up for the monopulse, high precision open loop blind pointing is required. This paper describes a new 4th order pointing model and calibration technique which was developed and applied to the DSN 34-m BWG antennas yielding 1.8 to 3.0-mdeg MRE pointing accuracy and amplitude stability of 0.2-dB, at Ka-Band, and successfully used for the CASSINI spacecraft occultation experiment at Saturn and Titan, and during MRO spacecraft telemetry tracks yielding 6-mbps data rates.

The 4th order pointing model was devised as a result of noticing systematic error residuals remaining in the data after applying the conventional 1st Order model. The 1st order model which typically has 6 to 8 mathematical terms (Fig 1.), is a physical model originally developed by Peter Stumpff and published in "Astronomical Pointing Theory for Radio Telescopes" in 1972. The 4th order model (Fig 2.) was derived by expanding the Spherical harmonics which are related to the associated Legendre polynomials by the equations 1 and 2 below, to the 4th order and resulting in 59 mathematical terms. Where,
Eq 1.
Eq 2.

Figure 1. Traditional 1st Order Physical Model Figure 2. 4th Order Model