An empirical statistical detection of non-ideal field distribution in a reverberation chamber confirmed by a simple numerical model based on image theory

Abstract

In this article, the effect of the size of the antennas on the distribution of the electric field observed in a reverberation chamber is analyzed. Our hypothesis is that the more the antenna is voluminous, the more the behavior of the chamber appears to be ideal through measurement analysis. A simple numerical model is presented and used to verify the phenomenon and give credence to the assumption.

Appendices

Appendix 1: About the loss coefficient R

In order to determine the loss coefficient R, we measured a CIR in our RC. And we tried to find the value of the loss coefficient that would minimize the error between the cumulated power measured and the cumulated power simulated. We simulated many CIRs with different loss coefficient values. We found that with R = 0.998, we simulated correctly the CIR of our empty RC. This process can be repeated for any loading.

If we consider that in an RC, the CIR follows an exponential function, the power measured can be written:

$$ P_{\rm m}(t)=P_{{\rm m}_{0}}e^{-t/\tau}, $$

(11)

where τ is the time constant of the power measured.

In the numerical model, if we consider the 4i ² + 2 currents of order i, the fraction of the magnitude of the E-field of each source that reaches the initial cavity is approximately 4π/(4i ² + 2). Therefore, if we make the assumption that the incoming E-field from the ith order currents arrives before the E-field from the (i + 1)th order, we can estimate the E-field from the 4i ² + 2 currents of order i:

$$ E(i)\approx E_0R^i $$

(12)

The E-field from the currents of order i arrive approximately at the instant t = iD/c, where D is a typical physical dimension of the chamber.³ We can write the power received :

$$ P(t)\approx P_0R^{2t\frac{c}{D}}\approx E_0e^{2t\frac{c}{D}\ln{R}}, $$

(13)

and:

$$ \tau\approx -\frac{D}{2c\ln{R}}. $$

(14)

We already determined R for our empty RC, and the estimation of τ from a measurement gives a value of D around 3 m. It is almost the smallest dimension of our RC.

Therefore, with a given loading in the chamber, we can measure the corresponding time to live τ _m and estimate the correct value of R by using:

$$ R \approx 1-\frac{D}{2c\tau_{\rm m}} $$

(15)

The value of D should be extracted from measurement. One should note that the quality factor at the frequency f ₀ is given by Q = 2πf ₀ τ. A measurement of Q can be used to determine R.

$$ R \approx 1-\frac{\pi D}{\lambda_0 Q} \label{RQ} $$

(16)

Appendix 2: From time domain to frequency domain, effective quality factor and effect of the length of the channel impulse response

Appendix 1 explains how the loss coefficient can be extracted from a CIR measurement or the quality factor. When we use a CIR measurement, the bandwidth of the pulse emitted in the chamber is around 1.5 GHz, between 1 and 2.5 GHz. It means that the estimated loss coefficient is accurate for the 1–2.5-GHz band. However, our approach exposed in Appendix 1 takes into account the time constant τ to reproduce the loss in the chamber, and hopefully, the time constant τ does not vary drastically within the considered band. Measurements of the Q factor in our chamber show that between 200 MHz and 2 GHz, its value is between 3,000 and 40,000. By using τ = 3 μs, the equivalent quality factor (noted Q _r ) from 200 MHz to 2 GHz is between 4,000 and 40,000. Figure 12 shows that the quality factor simulated in time domain with τ = 3 μs is in agreement with the quality factor measured in our chamber.

Fig. 12

Q measured, Q_r, and $Q_\textrm{ef\/f}$ with L_T = 3μs and 500 ns

In frequency domain, however, the length L _T of the simulated time window reduces the quality factor. The truncation of the channel impulse response implies an artificial reduction of the quality factor. If we note Q _r = 2πf ₀ τ the real quality factor and $Q_{L_{\rm T}}=2\pi f_0L_{\rm T}$ the quality factor of our truncated CIR, we can estimate the effective quality factor of our simulation in frequency domain by using:

$$ Q_\textrm{ef\/f}=\frac{Q_{\rm r}Q_{L_{\rm T}}}{Q_{\rm r}+Q_{L_{\rm T}}}=\frac{2\pi f_0\tau L_{\rm T}}{\tau+L_{\rm T}} $$

(17)

If the typical time constant measured in our chamber is τ = 3 μs and the length of the CIR is L _T = 3 μs, the effective quality factor is half the real quality factor. With 500 ns, the effective quality factor is a seventh of the real quality factor. Figure 12 sums up the different quality factors used in this appendix.

As a result, the shorter is the CIR, the more the resonances are dampened and the length of the CIR may affect the statistics of the E-field. Figure 13 shows the effect of the length of the time window L _T on the statistical behavior of the simulated RC. This figure shows that if the length of the time window is between 1 and 2 μs, the statistics of the component of the E-field are very similar to those measured (see Fig. 8a). With longer time windows, we could think that the rejection rate of the simulated E-field would be greater at a given frequency. With shorter time windows, Fig. 13 shows clearly that the rejection rate is lower than the measurements. Moreover, by using a 3-μs time window, the Weibull parameters obtained by simulation match the parameters measured in the chamber. By using a 3-μs-long time window, the effective quality factor is in the same order of magnitude of the real quality factor and thus does not affect considerably the statistics of the simulated fields. As a result, it seems that the loss in the cavity are correctly reproduced between 100 MHz and 2 GHz.

Fig. 13

Effect of the length of the simulated time window on the statistical behavior of one rectangular component of the E-field simulated (AD GoF test for the Rayleigh distribution with N = 150)

A possible update of the model would be a frequency-dependent loss coefficient by using Eq. 16. An alternative approach would be to simulate a limited number of frequencies with the corresponding R coefficient and to use the response of the chamber by convoluting the CIR by a CW signal and stay in the time domain.