International Journal of Antennas and Propagation

Volume 2017, Article ID 8715020, 8 pages

https://doi.org/10.1155/2017/8715020

## A Fast Finite-Difference Time Domain Simulation Method for the Source-Stirring Reverberation Chamber

^{1}Department of Information and Communication Engineering, Harbin Engineering University, Harbin, Heilongjiang 150001, China^{2}College of Engineering and Computational Sciences, Colorado School of Mines, Golden, CO 80401, USA

Correspondence should be addressed to Chongyi Yue; nc.ude.uebrh@iygnohceuy

Received 19 December 2016; Accepted 13 February 2017; Published 8 March 2017

Academic Editor: Safieddin Safavi-Naeini

Copyright © 2017 Wenxing Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Numerical analysis methods are often employed to improve the efficiency of the design and application of the source-stirring reverberation chamber. However, the state of equilibrium of the field inside the chamber is hard to reach. In this paper, we present a fast simulation method, which is able to significantly decrease the simulation time of the source-stirring reverberation chamber. The mathematical model of this method is given in detail and home-made FDTD code is employed to conduct the simulations and optimizations as well. The results show that the implementation of the method can give us the accurate frequency response of the source-stirring chamber and make the simulation of source-stirring chamber more efficient.

#### 1. Introduction

Reverberation chamber (RC) is a kind of test facility for EMC, EMI, and antenna measurements [1–4]. As a new type of RC, the source-stirring reverberation chamber (SSRC) was proposed recently and attracted much notice [5–8]. The FDTD method is a suitable simulation tool of RCs [9–11], especially for SSRCs since they are often designed to cover a broad frequency range, and there are no irregularly shaped mechanical stirrers in SSRCs [12, 13]. However, SSRCs are generally made of good conductors that cause little loss in the chamber. It often takes a long time for the signal inside to reach the condition of equilibrium, which turns to a huge CPU time for the simulation of SSRCs. Previous studies showed that it costs at least 24 hours for simulating a medium size chamber for just one stirring step on personal computers [14]. Thus accelerating the speed of simulation is one of the challenges in the study of SSRCs.

One way to improve the computational efficiency is introducing additional losses to the chamber [15]. Adding some lossy objects or materials to the chamber can make the electromagnetic signal in RCs attenuate more quickly, but excessive loads in the chamber could degrade the field uniformity and field level, which makes the performance of the chamber unacceptable. Another way to accelerate the simulation of RCs is adding artificial loss to the volume inside the chamber [16], but parameters such as conductivity of the medium inside the cavity used in this method are frequency-dependent. If the wide-band result is required, the FDTD iteration has to be conducted repeatedly with the parameters averaged in different narrow bands.

Instead of introducing additional losses in the working volume or on the wall, another method can also be used to accelerate simulation of SSRCs. This method allows us to obtain the frequency response of a nonideal lossy chamber by simulating an ideal lossless chamber and conducting a postprocessing technique on the simulation results. We can obtain a wide-band frequency response of the chamber by running the FDTD iteration just once and subsequently applying the postprocessing at different narrow bands. Although the postprocessing technique was presented in [17, 18], there are still some problems. For example, different damping functions are used in these two previous studies and both got the desired results, which is contradictory to each other. In this paper, we give the right form of the damping function. Besides, we first time use the postprocessing technique to simulate the frequency response in a source-stirring chamber and the simulation configuration is specified in detail. In addition, we further discuss and optimize the postprocessing technique in terms of the time window and the coefficient in the damping function.

This paper is organized in four sections. The mathematical model of the postprocessing technique and the quality factor ( factor) of the chamber is analyzed theoretically in Section 2. The simulation configuration of FDTD method and numerical results of chambers are shown in Section 3. The discussion and optimization of the fast simulation method are described in Section 4.

#### 2. Mathematical Model

##### 2.1. The Postprocessing Technique

Suppose a pulse excites the ideal lossless chamber, the chamber can seem to be a system, and an arbitrary scalar electric field sampled in the working volume of the chamber is the response of the pulse. The corresponding frequency response of the chamber is , which can be written aswhere is the index of the resonant frequencies and is the set of nature numbers. is the coefficient of each resonant frequency and denotes the resonant frequencies of the chamber. The resonant frequencies can be also calculated analytically usingwhere , , and represent the dimensions of the chamber.

Because it is a lossless cavity, the chamber response will never vanish once the chamber is excited by the pulse and we can not get the frequency response of an endless temporal signal. The response should be cut by a time window of duration and multiplied by a damping function associated with the factor of the lossy chamber, which can be written aswhere is a rectangular time windowand is a damping exponential functionIn (5), is the center frequency of the narrow bandwidth where we apply the postprocessing technique and value of factor is the average value of the quality factor of chamber in the desired frequency band. The Fourier transform of iswhere is the Fourier transform of the damping function that can be written as

Suppose equals , which can be indicated in the time domain as , which implies that . If we use as the criterion, we obtain the restriction of duration of the time window, which can be written asand with this restriction, we rewrite (6) asand substituting (7) for in (9), we get

Since the factor of the SSRC varies with frequencies and the value of is calculated using the desired center frequency , this equation is only valid over a narrow frequency band centered at .

##### 2.2. Calculation of the Factor

factor is an important parameter measuring the ability of RCs to store the energy. The value of the factor of the lossy chamber depends on the inherent loss mechanism of the chamber and the additional loads [15]. It is defined concisely as where is the operating frequency, is the energy stored in the chamber, and is dissipated power. For a lossy RC without additional loadings, the losses in the chamber walls, antennas, and equipment under test are the dominant loss mechanisms, and they can be approximately calculated aswhere , , and are the dimensions of the chamber, and are the volume and surface of the cavity, and represents the skin depth of the cavity wall aswhere and are frequency and permeability of the wall. is the equivalent conductivity of the chamber wall, which means the loss of the chamber wall accounts for all the other loss mechanisms.

#### 3. Simulation of the Chambers

##### 3.1. Setup and Configuration of the Simulations

Simulations of two chambers are conducted. One is the chamber with PEC walls (lossless chamber), which is also the prototypical chamber employing the postprocessing technique. The other one is a cavity with the walls of good conductors (lossy cavity), which will be used to assess the feasibility of the fast simulation method. In other words, we get and process the temporal signal from the lossless chamber, then transform it to frequency domain, and compared it with the reference frequency signal obtained from the lossy cavity.

The conductivity of the walls of the two chambers is set as and , respectively. Except for the conductivity, these two chambers have the same configurations in terms of the dimensions, which are specified as follows: The dimensions of the chambers are 3 m × 4 m × 2.5 m and the thickness of the chamber wall is 0.025 m. For the sake of simplicity, a dipole antenna operating from 400 MHz to 480 MHz is used as the transmitting antenna. In the Cartesian coordinate system, the dipole is centered at (0.5, 1, 1) along -axis and excited by a Gaussian pulse, which is indicated by a red dot in Figure 1. Two observation points P_{1} (1.5, 2.5, 1.8) and P_{2} (2, 3, 1), shown as green diamonds in Figure 1, are selected to record the electric field during the simulation.