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Statistical Response of Nonlinear Equipment in a Reverberation Chamber

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Statistical Response of Nonlinear Equipment in a Reverberation Chamber Statistical Response of Nonlinear Equipment in a Reverberation Chamber Ian Flintoft, Andrew Marvin and Linda Dawson University of York, Department of Electronics Heslington, York, UK, Email: idf1,acm,linda @ohm.york.ac.uk { } Abstract—The statistics of the fields scattered by nonlinear “key-hole” effect in communication paths [3] that has been equipment in a reverberation chamber are investigated using simulated using coupled RCs [4]. both a Monte-Carlo simulation of a simple statistical model and The presence of electronic devices in an environment can measurements of real equipment. The probability distributions of the scattered fields of the second and third harmonic are potentially be detected by using a harmonic radar system [5], predicted and compared to measurement. The implications of [6]. In order to quantify the effectiveness of such a surveillance the results for the immunity assessment of digital hardware and system it is necessary to know the statistical variation of harmonic cross-section measurements in reverberation chambers the harmonic scattering cross section of a wide range of are discussed. potential target devices. Due to the complexity of the harmonic Index Terms—immunity testing, reverberation chamber, non- linear scattering scattering cross-section an approach based on the total radiated power of the device at the harmonic frequencies under an im- pressed incident power density is attractive. The reverberation I. INTRODUCTION chamber provides an ideal measurement environment for such In this paper we investigate the effect of nonlinear scat- a characterisation of target devices. In a RC the non-linear tering on the statistics of the electromagnetic fields inside a scattering cross-section can be defined as reverberation chamber (RC). The nonlinearity will be shown Prad (nf1) A (nf1) = h i, (1) to generate field statistics at the harmonics of the illuminting h i Sinc (f1) frequeny that are significantly different from the Rayleigh dis- h i where P (nf ) is the total radiated power from the tribution that is usually found in the normal operation of a RC. h rad 1 i This has implications for a number of applications in which device-under-test (DUT) at the harmonic frequency, nf1, and S (f ) is the average power density illuminating the DUT a RC may be used to test or measure nonlinear equipment, h inc 1 i including EMC immunity testing of digital hardware. at the incident frequency f1. It has been shown that the immunity of digital hardware It is well known that the distribution of the magnitude to external fields can be quantified by an analysis of the re- of the electric field strength in a RC follows a Rayleigh distribution [7], p ( E σ), where radiated energy [1]. The level of the modulation products ray | | | scattered from a digital device and their dependence on the x x2 p (x σ)= exp − (2) level of the incident field can provide information on the extent ray | σ2 2σ2 to which interference has penetrated into the equipment’s electronics. The complexity of the scattering cross-section of For the applications discussed above it is important to address electrically large equipment makes such measurements very the implication of either the non-linearity or the “pinch-point” difficult and time consuming in an anechoic environment. The on the statistics of the fields in the RC. reverberation chamber offers a potentially more robust and II. SIMPLIFIED NONLINEAR SCATTERING MODEL practical testing technique. It is also worth noting that the simpler case of linear In order to investigate the statistical aspects of nonlinear scattering from an object has some correspondence to the scattering in RC a simple model of the process has been nested reverberation chamber measurement of the shielding constructed which incorporates the essential features of the effectiveness of a material or enclosure [2]. The shielding interactions involved. The model is not intended to represent aperture or material obstacle in this case provides a linear the full compexity of the scattering in a real RC, in particular “pinch-point” in the coupling path between the two antennas, the vector nature of the fields and the distributed nature of causing the single path to be decomposed into two statistically the scattering is not accounted for. An outline of the complete independent paths. This is the same phenomenon as the model with all the signal paths is illustrated in Fig. 1. The signal injected into the RC from the transmitter is This work was supported by funding from the EU Framework 6 CRAFT assumed to be a sinusoidal waveform Project Number 032585: SAFETALK: Development of a continuous sweep RF harmonic spectral analysis detector. vsource(t)= v cos(2πf1t) (3) 978-1-4244-1699-8/08/$25.00 ©2008 IEEE vrerad r The statistical distribution of the scattered signal is de- 1 termined by the terms is square brackets in (8)-(10). The v v v source r inc r scat scattered signal at f1 contains the component [r0r1], a product 0 2 of two independent Rayleigh random processes. The overall vrec distribution of this component is therefore a Double-Rayleigh r3 distribution, p (x σ σ ), with scale parameter σ σ , where dray | 0 1 0 1 x x rd v p x σ K (12) direct dray( )= 2 0 | σ σ Fig. 1. Schematic illustration of the simplified nonlinear scattering model. and K0 is the modified Bessel function of zeroth order [8]. The scattered second harmonic has a distribution deter- 2 mined by [r0r2]. Ignoring the outbound coupling path, r2, 2 with amplitude v and frequency f1. The signal received from the distribution would be that of [r0] which is an exponential 2 the RC, vrec(t), is composed of two components, distribution p (x 2σ ) given by exp | 0 v t v t v t , (4) 1 x rec( )= direct( )+ scat( ) p (x σ)= exp . (13) exp | σ −σ where v (t) represents energy which is directly coupled direct Clearly the outbound coupling path statistics will alter this from the source to the receiver, without interaction with the distribution. nonlinear scatterer, and v (t) represents energy that has scat Using standard technqiues for determining the overall distri- been scattered by the device. The directly coupled energy is bution of combined random processes, integral representations modelled by of the probability density functions (PDFs) of the second v = r v , (5) direct d source and third harmonic can be obtained [9]. The distribution where rd is a random processes with a Rayleigh probability of the second harmonic is found to be p2nd(x σ2f1 ) where 2 | distribution p (x σ ) with scale parameter σ . Similarly the σ2f1 = σ σ2 and ray | d d 0 signal incident on the nonlinear device is modelled by another 2 1 ∞ t x/(2√2σt) independent Rayleigh distribution, p2nd(x σ)= e− e− dt. (14) | √2σ Z0 vinc = r0vsource, (6) Similarly the PDF of the third harmonic is p3rd(x σ3f1 ) where 3 | σ3f1 = σ σ3 and with scale parameter σ0 and probability density function 0 1 p x σ 3 ray( 0). 2 2 3 | 1 4 ∞ 1 t (x/4σt) / The nonlinear device is modelled as reradiating a signal p (x σ)= t 3 e− e− dt. (15) 3rd | 3 xσ2 that is related to the incident signal by a power series (a soft Z0 nonlinearity) of which we examine only the first three terms These inegrals are not known analytically but can be evalutated in this paper: numerically. It is also possible to further analyse the scattered fields at 2 3 vrerad = avinc + bvinc + cvinc, (7) the second and third harmonics by a Monte-Carlo simulation with coefficients a, b and c. Substituting (6) in to (7) the rera- of the model. This was accomplished by drawing samples for the independent Rayleigh distributions r , r , r and r over diated signal at f1, f2 =2f1 and f3 =3f1 can be identified. d 0 2 3 Each of these frequency components is then assumed to give Nt trails and evaluating the model for each set of samples. Nt rise to a scattered field that couples to the receiver via inde- corresponds to the number of stirrer positions in the real RC measurement. pendent Rayleigh distributed random processes r1, r2 and r3 with scale parameters σ1, σ2 and σ3 respectively. The scattered III. MEASUREMENTS field components at the incident and harmonic frequencies, A series of control and device measurements were made in a now written in the frequency-domain, are therefore RC using the configuration shown in Fig. 2. The reverberation 3 3 3 3 chamber used had dimensions 2.5m 3m 4.5m and was Vscat(f1) = av[r0r1]+ ca v [r r1] (8) 4 0 tuned mechanical by a stepped stirrer.× For the× measurements 1 2 2 presented here the statistical averages were taken over 256 Vscat(f2) = bv [r0r2] (9) 2 positions of the paddle, equally spaced over one full rotation. 1 3 3 Energy was injected into the chamber from a signal genera- Vscat(f3) = cv [r0r3]. (10) 4 tor and power amplifier via a horn antenna. For the nonlinear The total received signal at the frequency of the incident field scattering measurements the energy was injected at 900MHz, is given by with a level of 30 dBm, and a low-pass filter (LPF) was inserted between the power amplifier and antenna to suppress V (f )= V (f )+ V (f ), (11) rec 1 direct 1 scat 1 the harmonics from the signal generator and amplifier that where Vdirect(f1)= rdVsource(f1)= vrd. enter the RC. LPF 0.5 Monte-Carlo Data Double-Rayeigh MLE Fit f = 900 MHz 0.45 0 ) -1 0.4 0.35 0.3 0.25 vinc 0.2 DUT vscat 0.15 vsource 0.1 vdirect Probability density function (V 0.05 0 vrec PC Spectrum 0 2 4 6 8 10 12 Analyser Received voltage (V) HPF (a) Fundamental.
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