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.
Fig. 2. Experimental arrangement of the reverberation chamber. 25 Monte-Carlo Data Exponential MLE Fit Exact PDF )
-1 20 The scattered energy was received using another horn an- tenna and a spectrum analyser. A high-pass filter (HPF) was used to attenuate the energy at the incident frequency that 15 was coupled to the spectrum analyser and thus reduce any harmonic generation in the receiver itself. The HPF had 70dB 10 of attenuation at the injection frequency of 900MHz. The whole measurement was controlled by a personal computer 5 that also processed the data. Probability density function (V The nonlinear characteristics of the signal generator, power 0 amplifier and spectrum analyser were quantified independently 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 using standard techniques before the RC measurements so that Received voltage (V) the nonlinear behaviour of the measurement system was fully (b) Second harmonic. understood. 800 Monte-Carlo Data ESULTS Exponential MLE Fit IV. R 700 Exact PDF ) -1 A. Monte-Carlo Simulations 600
The results of a Monte-Carlo simulation of the simple model 500 presented in section II are shown in Fig. 3. For this simulation the source strength was v = 1 and the nonlinear coefficients 400 were a = 1, b = 0.1 and c = 0.01. The Rayleigh scale 300 parameters of the propagation paths were taken to be σ0 =1, 200 σ1 = 1, σ2 = 0.5 and σ3 = 0.25. The statistics were taken Probability density function (V over Nt = 3000 stirrer positions. 100
Fig. 3(a) shows the estimated probability density function 0 for the scattered voltage at the incident frequency, Vscat(f0). 0 0.005 0.01 0.015 0.02 0.025 Note that this is the scattered signal only and does not include Received voltage (V) the directly coupled signal given in (5), which will generally (c) Third harmonic be much larger in magnitude. The Monte-Carlo data has been Fig. 3. Probability density functions of the scattered voltage in the Monte- Carlo simulation of the simple statistical model at the fundamental, second fitted to a Double-Rayleigh distribution using a maximum and third harmonic frequencies. likelihood estimate (MLE). As can be seen the data is approx- imately Double-Rayleigh distributed as expected, since the 3 more complex [r0r1] term in (8) is much smaller in magnitude than the Double-Rayleigh distributed [r0r1] term for the set of Fig. 3(b). The distribution in this case is approximately expo- nonlinear coefficients chosen in this example. nential as shown by the MLE fit. However, as the analysis in The distribution for the scattered voltage at the second section II indicates the true distribution will not be exponential harmonic of the incident frequency, Vscat(2f0), is shown in due to the contribution of the random process representing the TABLE I 7000 MEANS (µ) AND STANDARD DEVIATIONS (σ) OF THE RECEIVED VOLTAGES Measured Data (IN MV) FORTHEREVERBERATIONCHAMBERMEASUREMENTS. Rayleigh MLE Fit 6000 ) -1
Frequency Empty RC Empty RC DUT in RC 5000 (MHz) Direct Scattered Scattered Injection Harmonics Harmonics 4000 µ σ µ σ µ σ 3000 900 0.23 0.14 0.27 0.16 0.27 0.15
1800 0.20 0.10 0.02 0.03 0.13 0.15 2000 2700 0.17 0.09 0.05 0.06 0.05 0.06 Probability density function (V 1000
0 coupling path of the scattered signal in the receiver. The exact 0 0.0002 0.0004 0.0006 0.0008 0.001 PDF predicted by (14) is also shown in the figure; it was Received voltage (V) evaluated numerically using the known scale parameters, σ0 (a) 900 MHz and σ2, of the coupling paths r0 and r2 7000 The distribution for the third harmonic is shown in 3(c). Measured Data Rayleigh MLE Fit Again the distribution is compared to a MLE fit to an expo- 6000 ) nential distribution and to the exact PDF given by (15). In -1 this case it is clear that the actual distribution is more sharply 5000 concentrated than the exponential distribution at lower values 4000 of the received voltage, but with a much longer tail to maintain the normalisation of the PDF. This long tail to the distribution 3000 is an important feature of both the second and third harmonics. 2000
B. Reverberation Chamber Measurements Probability density function (V 1000 Initially, in order to verify the normal chamber statistics, 0 energy was directly injected into the chamber at 900MHz, 0 0.0002 0.0004 0.0006 0.0008 0.001 1800MHz and 2700MHz at the same power level of 30dBm. Received voltage (V) For these measurements the HPF shown in Fig. 2 was removed (b) 1800 MHz from the measurement system. The distribution of the received 7000 voltage in these cases are shown in Fig. 4, compared to Measured Data MLE fits to a Rayleigh distribution. As expected, at all three Rayleigh MLE Fit 6000 ) frequencies the voltage distribution is Rayleigh in form. The -1 means and standard deviations of the distributions are given 5000 in columns two and three of Table I respectively. To provide a baseline for the device measurement and 4000 to identify any nonlinear characteristics of the measurement 3000 system itself, a series of measurements were performed by injecting energy into the empty RC at 900MHz. The received 2000
voltage distributions at 900MHz, 1800MHz and 2700MHz Probability density function (V 1000 were determined and the results are shown in Fig. 5 and columns four and five of Table I. Note that the cable loss 0 was not calibrated out of these measurement data, which 0 0.0002 0.0004 0.0006 0.0008 0.001 Received voltage (V) accounts for part of the gradual reduction in the mean received (c) 2700 MHz voltage with frequency seen column 2 of Table I. Recall Fig. 4. Probability density functions of the received voltage for the empty also that in the scattering measurements the fundamental is reverberation chamber with direct injection of energy at 900 MHz, 1800 MHz measured through the high-pass filter which has an attenuation and 2700 MHz. of approximately 70dB at 900MHz. The received voltage at 900MHz is therefore similar in magnitude to that of the harmonics. simple statistical model p (x σ ) is also shown, using 2nd | 2nd As can be seen, the second and third harmonic of the an empirical estimate for the scale parameter σ2nd. This is incident frequency are present even without a nonlinear device also a reasonable fit to the measured data for the second in the chamber. The distribution of the second harmonic is harmonic. However, for the third harmonic, shown in Fig. 5(c), exponential-like for the measurement conditions applied as the exponential distribution is clearly a much better fit to the shown by the MLE fit in Fig. 5(b). The exact PDF for the data than the exact PDF, p (x σ ), of the simple model. 3rd | rdd 5000 5000 Measured Data Measured Data Rayleigh MLE Fit Rayleigh MLE Fit ) )
-1 4000 -1 4000
3000 3000
2000 2000
1000 1000 Probability density function (V Probability density function (V
0 0 0 0.0002 0.0004 0.0006 0.0008 0.001 0 0.0002 0.0004 0.0006 0.0008 0.001 Received voltage (V) Received voltage (V) (a) 900 MHz (a) 900 MHz
35000 35000 Measured Data Measured Data Exponential MLE Fit Exponential MLE Fit 30000 Exact PDF 30000 Exact PDF ) ) -1 -1
25000 25000
20000 20000
15000 15000
10000 10000
Probability density function (V 5000 Probability density function (V 5000
0 0 0 0.0002 0.0004 0.0006 0.0008 0.001 0 0.0002 0.0004 0.0006 0.0008 0.001 Received voltage (V) Received voltage (V) (b) 1800 MHz (b) 1800 MHz
20000 20000 Measured Data Measured Data Exponential MLE Fit Exponential MLE Fit Exact PDF Exact PDF ) ) -1 -1 15000 15000
10000 10000
5000 5000 Probability density function (V Probability density function (V
0 0 0 0.0002 0.0004 0.0006 0.0008 0.001 0 0.0002 0.0004 0.0006 0.0008 0.001 Received voltage (V) Received voltage (V) (c) 2700 MHz (c) 2700 MHz Fig. 5. Probability density functions of the received voltage for the empty Fig. 6. Probability density functions of the received voltage for the reverberation chamber with energy injected at 900 MHz. reverberation chamber with a nonlinear DUT inside and energy injected at 900 MHz.
Much effort was expended trying to locate the source of the that the nonlinear interaction is taking place with stirred nonlinear interaction giving rise to these results. The statistics energy that is also stirred after reradiation. For example, if the themselves actually provide some diagnostic information about received harmonics were due to spurious harmonics generated the coupling path of the harmonic energy: the fact that the dis- in the power amplifier which coupled into the chamber by a tributions of the second and third harmonic are approximately stray coupling path we would expect the received energy to exponential in form and not Rayleigh distributions indicates have a Rayleigh distribution exactly as for the direct injection case. from the model. The difference is much more significant for This indicates that the source of the nonlinear interaction is the third harmonic than the second harmonic, which is fitted either in the chamber itself or possibly outside the chamber reasonably well by both the exact PDF of the model and but both coupled into and out of the chamber in such a way an exponential distribution. Deficiencies in the model which that stirred energy interacts with it and is reradiated back into could account for this difference include the neglection of the chamber. Possible sources of the nonlinearity therefore the distributed nature of the scattering and representration of include the paddle stepper motor in the chamber or the motor the soft nonlinearity of the equpiment. Further work is being controller which is outside the chamber but connected to the carried out to determine the causes of the discrepancy. motor by an unfiltered cable with little shielding. We have shown that the RC is a feasible environment The statistical distributions obtained when a nonlinear DUT, for nonlinear equipment measurements. However, maintaining a mobile phone, is placed into the RC are shown in Fig. 6. a measurement environment that is sufficiently linear for The corresponding means and standard deviations are given practical nonlinear scattering measurements is a difficult un- in columns six and seven of Table I. The distribution at the dertaking. One principle that can be applied is simplification: injected frequency is again Rayleigh since it is dominated removing as many potential sources of nonlinearity as possible by the directly coupled energy that does not interact with from the measurement system. It is therefore suggested that the DUT. Comparing the distribution at the second harmonic mode tuning by frequency stirring rather than mechanical frequency, Fig. 6(b), with the empty chamber case, Fig. 5(b), stirring, as was used for the results presented in this paper, indicates that the presence of the the DUT is clearly detected may be more appropriate as it allows the removal of the by the measurement. The mean received voltage at 1800MHz joints, power suppliers and controller circuits associated with increases from 0.02mV without the DUT to 0.13mV with the the stepped paddle. The large RC used here also had many DUT, an increase of 650%. The distribution of the second metal-metal joints and interfaces of different material types harmonic with the DUT is again approximately exponential in which again may be the source of nonlinear interactions. A form, but with a large width. As for the empty chamber, both specifically designed RC with more consideration given to this an exponential distribution and the distribution p2nd(x σ2nd) particular application would therefore probably improve the of the simple model are reasonable fits to the data. | dynamic range of the measurement. The situation for the third harmonic is however different. Once these limitations of the current measurement system Comparing Fig. 6(c) with Fig. 5(c), we see that the distribution are addressed a systematic methodology for the measurement at 2700MHz is not noticeably effected by the presence of of the harmonic scattering cross-section given in (1) can be the DUT, with the mean received voltage being 0.05mV developed, including the proper calibration of chamber loading in both cases. This indicates that the “spurious” harmonics effects. This will aid the development of immunity testing of generated by the measurement system itself are larger than any digital hardware in RC and the quantification of the probability harmonics created by the DUT. The third harmonic distribution of detection in harmonmic radar based nonlinear junction is again close to exponential in form and the exact PDF of the detection systems. simple model, p (x σ ), is a poor fit to the data. 3rd | 3rd REFERENCES V. CONCLUSIONS [1] I. D. Flintoft, A. C. Marvin, M. P. Robinson, K. Fischer, and A. J. Rowell, “The re-emission spectrum of digital hardware subject to EMI,” IEEE The distribution of fields scattered by nonlinear objects Trans. Electromag. Compat., vol. 45, no. 4, pp. 576–585, November 2003. inside a RC does not follow the typical Rayleigh distribution [2] C. L. Holloway, D. A. Hill, J. Ladbury, G. Koepke, and R. Garzia, “Shield- ing effectiveness measurements of materials using nested reverberation usually associated with RCs. 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Vainkainen, “The distribution of the product of independent Rayleigh random variables,” IEEE Trans. followed by real equipment. The measured distributions of Antennas Propagat., vol. 54, no. 2, pp. 639–643, February 2006. both the second and third harmonics are exponential in form, [9] J. A. Rice, Mathematical statistics and data analysis, 3rd ed., ser. rather than the more sharply peaked distribution determined Duxbury Advanced Series. Thomson-Brooks/Cole, 2007.