# Far-Field Reconstruction from Transient Near-Field Measurement Using Cylindrical Modal Development

Authors: R. Rammal; M. Lalande; E. Martinod; N. Feix; M. Jouvet; J. Andrieu; B. Jecko
Journal: International Journal of Antennas and Propagation
 (2009)

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## Abstract

The aim of this work is to get far field radiation patterns for any radiating source from transient acquisition, in a large frequency range. An outdoor transient Ultra-Wideband near-field measurement base will be installed, a single time pulse radiated by the source will cover the desired spectrum, and the accurate determination of far field radiations will be accomplished by means of cylindrical waves' modal development. This method uses simplified test equipments, easy to be installed, and it reduces measurement costs.


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## Body

## 1. Introduction

Interested in measuring radiations of antennas placed in their context of use, such as antennas placed inside and outside a vehicle, or radiations of electronic systems integrated in many functional parts of a vehicle, the near field (NF) to far field (FF) transformation technique presents many advantages [1, 2]. Such measurements consider the vehicle as a radiating source; hence the FF distance is too large to be obtained in closed areas. Anechoic rooms provide a good solution for small antennas, but for large dimensions, costs become huge.The proposed method is based on transient metrology [3]; it uses an outdoor time-based measurement base [4]. The system consists of a transient pulse generator connected to the transmitting antenna (antenna to be characterized), and a transient probe as a receiving antenna, connected to a performing data acquisition system (real time scope). This system allows getting field values at the wanted positions by generating a very narrow pulse, which does not disturb the electromagnetic environment, and in which spectral content corresponds to a large frequency spectrum.The transient outdoor measurement method presents many advantages [5]. In fact, absorbent materials are not efficient for measurements in low-frequency domain. Besides, measuring in frequency domain needs expensive equipments, long time, and different types of absorbent materials for a wideband frequency analysis. The cost of anechoic room increases faster when increasing the antenna dimensions. These problems are solved using outdoor transient measurement base. Simple installation and unlimited dimensions decrease the measurement costs. The generated time pulse permits to get values at any frequency in the spectrum covered by the signal, using time windowing and appropriate Fourier Transform techniques.In the following, we will characterize an Ultra-Wideband (UWB) antenna in order to validate the proposed method. The principle of the method and the used antenna will be presented in the first part, near field to far field transformation in cylindrical coordinates in second part, in order to explain the experimental setup and results in the last part.


## 2. Principle of the Method

The UWB transient measurement technique in outdoor is well suited to the characterization in near field. In fact, far field ranges of large antenna (as well as a radiating antenna placed on a vehicle) require a huge infrastructure and an important radiated peak power. In such cases, the elimination of the influence of the environment could be difficult.Starting from near field transient measurements, the fields’ calculation for all frequencies is deduced from the Fourier transform, and the construction of the far field can then be made by the application of specific algorithms.Near Field to Far Field transformations are based on two principles. The first is to measure the values of E and H fields on a closed surface (Huygens), and then to seek secondary radiating sources using integral equations.  The second is to perform a modal development of the tangential near electric field, as a solution of the vector Helmotz equation of propagation, which verifies the boundary conditions on the measurement surface at Near Field as well as at infinity. The theoretical formulation of this transformation exists in three coordinate systems: planar [6], cylindrical [1, 2, 7], and spherical [8]. The antenna’s nature determines the required geometry.Figure 1
Synoptic of the validation.The modal development method [1, 2] permits the construction of the far field’s radiation pattern as function of the tangential Electric near field. Cylindrical coordinates system is chosen for simplicity in installations using a vertical mast and a turntable. This method is adapted for any radiating source. The method’s validation (Figure 1) necessitates a radiating source with a “known” FF radiation, in order to compare it with what we will construct from the measured NF data.The radiating source is a transmitting antenna which radiates ultra-short pulses with very low dispersion. The chosen antenna is the UWB SCISSOR antenna (Figure2) [4, 9], directive in both E and H planes. This antenna measures 60*100 cm.Figure 2
Scissor antenna.The scissor antenna is simulated in CST MICROWAVE STUDIO software (Figure3).Figure 3
Simulated scissor antenna.To analyze the radiation of this antenna, we used a transient solver, with a mesh dimensions going to a maximum ofλ/20. The excitation transient signal feeding the antenna is a Gaussian having a rise time of 280 ps. Its spectrum covers the frequency band 30 MHz–1.5 GHz. The potential difference at the antenna’s entry has a value of 7 volts. The simulation results of the FF patterns at a distance of 10 meters for a working frequency of 1 GHz for E plane and H plane are shown, respectively, in spherical coordinates system on Figures 4 and 5.Figure 4
Normalized FFE plane 1 GHz.Figure 5
Normalized FFH plane 1 GHz.


## 3. Near Field to Far Field Transformation Technique

This method deals with the reconstruction of Far-Field (Findex) radiation pattern from Near-Field (Nindex) measurements on a cylinder. The tangential NF components lead to FF pattern using an expansion in terms of modes, as a linear combination of elementary solutions of the vector wave equation outside the antenna [1, 2, 7, 10, 11].The measurement cylinder is chosen around the tested antenna (Figure6), in the NF zone with a radius “a.” The tangential components of the electric NF,  radiated by the antenna, can be written as a superposition of elementary cylindrical waves:
(1)EzN(ϕ,z)=∑n=-∞+∞∫-∞+∞bn(kz)Λ²kHn(2)(Λa)ejnϕe-jkzzdkz,EϕN(ϕ,z)=∑n=-∞+∞∫-∞+∞(bn(kz)n·kzkaHn(2)(Λa)-an(kz)∂Hn2(Λr)∂r|r=a)ejnϕe-jkzzdkz,
where “a” is the NF measurement cylinder radius, k=2π/λ is the wave vector, and “n” is the mode number.Figure 6
Experimental setup.Hn(2)(Λa) is the Hankel function of second kind, of order n, with Λ=k²-kz².From these equations, we can get the modal coefficientsan(kz) and bn(kz), as functions of two-dimension spatial integrals of EzN(ϕ,z) and EϕN(ϕ,z):(2)bn(kz)=kΛ2Hn(2)(Λa)εv(n,kz),an(kz)=1∂Hn(2)(Λr)/∂r|r=a×[bn(kz)·n·kzk·a·Hn(2)(Λa)-εH(n,kz)],
where εH(n,kz) and εv(n,kz) are the two-dimensional transformation of the measured NF according to Φ and z directions, respectively:(3)εH(n,kz)=14π²∫-∞+∞∫-π+πEϕN(ϕ,z)e-jnϕe+jkzzdϕdz,εv(n,kz)=14π²∫-∞+∞∫-π+πEzN(ϕ,z)e-jnϕe+jkzzdϕdz.
Using these coefficients,  we can reconstruct the FF radiation pattern in spherical coordinates system as follows:(4)EθF(θ,ϕ)=-j2ksinθ·e-jkRR∑n=-∞+∞jnbn(kcosθ)ejnϕ,EϕF(θ,ϕ)=-2ksinθ·e-jkRR∑n=-∞+∞jnan(kcosθ)ejnϕ,
where “R” is the FF distance.These equations relate the modal coefficients of the tangential fields on the NF measurement cylinder.So, using the tangential components of the electric field, we can get the modal coefficients of the cylindrical wave expansionan(kz) and bn(kz) by applying a two-dimensional Fast Fourier Transform (FFT_2D).It is very important that acquisition steps respect some criteria since we deal with Fourier Transform. Sample spacing must be such as to make aliasing error negligible. The standard sampling criteria adopted in cylindrical NF-FF transformation [10, 11] are(5)Δz≤λ2,Δϕ≤πkρ=λ(2ρ),
where “ρ” is the radius of the smallest sphere enclosing the antenna under test. The first condition fixes the height of each sample Δz, and the second fixes an angular increment Δϕ. It is interesting to measure on a cylinder having the smallest radius in the NF region, in order to minimize the number of acquired points.


## 4. Experimental Setup and Results

For the scissor antenna we used a shadow cylinder of radius 110 cm, centered at the antenna’s feeding point (Figure6). Respecting the sample spacing criteria (5) for the working frequencies 1 GHz and 600 MHz, we will use a step of 10 cm for height and 10 degrees for angles.The radiating source (scissor antenna) will be connected to a transient pulse generator through a matching balun. The generated pulse feeding the antenna is a Gaussian having a rise time of 116 ps (Figure7). Its spectrum covers the frequency band 30 MHz–1.5 GHz (-3 dB) (Figure 8).Figure 7
Generated time pulse.Figure 8
Normalized spectrum.The used probe is a transient electric field sensor AD-80 model (Figure9) made by Prodyn. The output of this sensor is a potential V, from which we can get the measured electric field as follows [12]:Figure 9
Transient E field sensor.Figure 10
Time Windowing.(6)E(t)=1R·Aeq·ε0·∫0tV(u)du,
where V is the output voltage of the sensor, R is the impedance characteristic (100 ohms), Aeq is the sensor equivalent surface of 3×10-4m², and ε0 is the free space permittivity, cutoff frequency 5.5 GHz.The sensor must be oriented in a way so that antenna’s radiations arrive orthogonal to the ground plane of the sensor.In the following, we will measure only one component of electric NF,Ez; so we will be able to construct only the Eθ component in the FF.The measured electric near field is a signal of two parts, one containing the energy of the radiating pulse, and the other is the reflections from the surrounding to be eliminated. Time windowing is a useful operation to be done on this stage (Figure10). It takes into account the propagation time of reflections from ground, walls, and other existing obstacles.By applying Fourier Transform, we can get the field values at 1 GHz and 600 MHz; then applying the appropriate algorithms of the modal development method, we can construct the far field of this antenna.Another measurement is done, forE and H planes, directly in far field. It allows comparing with the constructed one. This measurement is done in time domain with the same probe; Fourier transform permits to obtain needed values for each frequency covered by the excitation signal.The comparison of the constructed FF and the measured one, for a working frequency of 1 GHz and then at 600 MHz, is given in Figures11 to 14.Figure 11
FFEθ at 1 GHz in E plane.These results show accordance between the FF values calculated from measured NF and the measured FF of the scissor antenna, in two different planes. It should be noted that this measurement is done on a prototype non automated measurement base. The validation of the method can be concluded by these comparisons.By comparing the different results of Figures11 and 12 for Eθ in the E plane and Figures 13 and 14 for Eθ in the H plane, when decreasing the working frequency from 1 GHz to 600 MHz, we realize that the diagram shows a difference when the acquired field is not in front of the transmitting antenna (θ=90°, ϕ=0). This is one of the limitations of this NF-FF transformation method in cylindrical form. At low frequencies, the antenna NF radiations are not very directive; therefore they exist over the cylinder with the used height. The ideal case is a cylinder going to infinity.Figure 12
FFEθ at 600 MHz in E plane.Figure 13
FFEθ at 1 GHz in H plane.Figure 14
FFEθ at 600 MHz in H plane.


## 5. Influence of NF Data Reduction Simulation

The FF construction in predefined planes can be achieved with selected NF data on the measurement cylinder.The condition is to take the entire field spot presented on the cylinder. Figures  15, 16, 17, 18, and 19 explain better this idea.In height, we are able to construct the FF values by taking measurements up toz=±1.4m (Figure 15).Figure 15
Reduced height NF cylinder forEz at 1 GHz.Influence of Height Truncation
The constructed FF in the E plane for two different heights is shown in Figure16.Figure 16
FFEθ at 1 GHz in E plane.Influence of Angle Truncation
The same idea works in theH plane. Taking the NF radiations in front of the antenna (Figure 17) allows us to construct the same FF in θ direction in the E plane facing the antenna (Figure 18), and the part of this field in H plane going to Φ=±90 degrees in phi (Figure 19).Figure 17
Choice of data spot on NF cylinder forEz at 1 GHz.Figure 18
FFEθ at 1 GHz in E plane.Figure 19
FFEθ at 1 GHz in H planeFigure19 shows a fall at Φ=70° comparing with the complete FF radiation in this plane. This is due to the angle truncation of NF data on the acquisition cylinder.These results have been validated for all planes defined byθ = constant or Φ = constant.


## 6. Conclusion and Perspective

This measurement base allows the Far Field determination of antennas and radiating sources from a single transient outdoor acquisition. It provides  a notable gain in terms of measurement time and costs, while remaining nondisruptive to the environment with pulses of very short durations.This measurement technique is more efficient in Near Field ranges. Using NF-FF transformation techniques, it allows us to obtain the radiation pattern of any radiating structure over a wide band of frequencies, from a single measurement followed by a mathematical calculation.The next work will be oriented in two ways. First we will improve the modal development in cylindrical coordinates to get more accurate FF patterns,  while the second way will be the determination of FF directly in time domain, without time to frequency conversion [13–16].



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*Source: 798473-2010-01-21.xml*