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# Electromagnetic reverberation chamber

For Reverberation Chambers used for sound, see Reverberation, Acoustic Reverberation Chambers (Echo Chambers).

An electromagnetic reverberation chamber (also known as a reverb chamber (RVC) or mode-stirred chamber (MSC)) is an environment for electromagnetic compatibility (EMC) testing and other electromagnetic investigations. Electromagnetic reverberation chambers have been introduced first by H.A. Mendes in 1968 (Mendes 1968). A reverberation chamber is screened room with a minimum of absorption of electromagnetic energy. Due to the low absorption very high field strength can be achieved with moderate input power. A reverberation chamber is a cavity resonator with a high Q factor. Thus, the spacial distribution of the electrical and magnetical field strength is strongly inhomegenious (standing waves). To reduce this inhomogenity, one or more tuners (stirrers) are used. A tuner is a construction with large metallic reflectors that can be moved to different orientations in order to achieve different boundary conditions. The Lowest Usable Frequency (LUF) of a reverberation chamber depends on the size of the chamber and the design of the tuner. Small chambers have a higher LUF than large chambers.

The concept of a reverberation chambers is comparable to a microwave oven.

## Glossary/Notation

### Preface

The notation is mainly the same as in the IEC standard 61000-4-21(IEC61000-4-21 2003). For statistic quantities like mean and maximal values, a more explicit notation is used in order to emphasize the used domain. Here, spacial domain (subscript s) means that quantities are taken for different chamber positions, and ensemble domain (subscript e) refers to different boundary or excitation conditions (e.g. tuner positions).

### General

• $\vec{E}$: Vector of the electric field.
• $\vec{H}$: Vector of the magnetic field.
• $E_T,\, H_T$: The total electrical or magnetical field strength, i.e. the magnitude of the field vector.
• $E_R,\, H_R$: Field strength (magnitude) of one rectangular component of the electrical or magnetical field vector.
• $Z_0=\frac{|\vec{E}|}{|\vec{H}|}=120\cdot \pi\, \Omega$: Characteristic impedance of the free space
• ηTx: Efficiency of the transmitting antenna
• ηRx: Efficiency of the receiving antenna
• $P_{\rm fwd}, \, P_{\rm bwd}$: Power of the forward and backward running waves.
• Q: The quality factor.

### Statistics

• ${}_s\langle X \rangle_N$: spacial mean of X for N objects (positions in space).
• ${}_e\langle X \rangle_N$: ensemble mean of X for N objects (boundaries, i.e. tuner positions).
• $\langle X \rangle$: equivalent to $\langle X \rangle_\infty$. Thist is the expected value in statistics.
• ${}_s\lceil X \rceil_N$: spacial maximum of X for N objects (positions in space).
• ${}_e\lceil X \rceil_N$: ensemble maximum of X for N objects (boundaries, i.e. tuner positions).
• $\lceil X \rceil$: equivalent to $\lceil X \rceil_\infty$.
• ${}_s\!\dagger\!(X)_N$: max to mean ratio in the spacial domain.
• ${}_e\!\dagger\!(X)_N$: max to mean ratio in the ensemble domain.

## Theory of Operation

### Cavity Resonator

A reverberation chamber is cavity resonator -- usually a screened room -- that is operated in the overmoded region. To understand what that means we have to investigate cavity resonators briefly.

For rectangular cavities, the resonance frequencies (or eigenfrequencies, or natural frequencies) fmnp are given by $f_{mnp} = \frac{c}{2}\sqrt{\left(\frac{m}{l}\right)^2+\left(\frac{n}{w}\right)^2+\left(\frac{p}{h}\right)^2},$

where c is the speed of light, l, w and h are the cavity's length, width and height, and m, n, p are non negative integers (at most one of those can be zero).

With that equation, the number of modes with an eigenfrequency less than a given limit f, N(f), can be counted. This results in a stepwise function. In principle, two modes -- a transversal electric mode TEmnp and a transversal magnetic mode TMmnp -- exist for each eigenfrequency.

The fields at the chamber position (x,y,z) are given by

• for the TM modes (Hz = 0) $E_x=-\frac{1}{j\omega\epsilon} k_x k_z \cos k_x x \sin k_y y \sin k_z z$ $E_y=-\frac{1}{j\omega\epsilon} k_y k_z \sin k_x x \cos k_y y \sin k_z z$ $E_z= \frac{1}{j\omega\epsilon} k_{xy}^2 \sin k_x x \sin k_y y \cos k_z z$
Hx = kysinkxxcoskyycoskzz
Hy =  − kxcoskxxsinkyycoskzz $k_r^2=k_x^2+k_y^2+k_z^2,\, k_x=\frac{m\pi}{l},\, k_y=\frac{n\pi}{w},\, k_z= \frac{p\pi}{h}\, k_{xy}^2=k_x^2+k_y^2$

• for the TE modes (Ez = 0)
 Ex = kycoskxxsinkyysinkzz
Ey =  − kxsinkxxcoskyysinkzz $H_x=-\frac{1}{j\omega\mu} k_x k_z \sin k_x x \cos k_y y \cos k_z z$ $H_y=-\frac{1}{j\omega\mu} k_y k_z \cos k_x x \sin k_y y \cos k_z z$ $H_z= \frac{1}{j\omega\mu} k_{xy}^2 \cos k_x x \cos k_y y \sin k_z z$


Due to the boundary conditions for the E- and H field , some modes does not exist. The restrictions are (Cheng 1989):

• For TM modes: m and n can not be zero, p can be zero
• For TE modes: m or n can be zero (but not both can be zero), p can not be zero

A smooth approximation of N(f), $\overline{N}(f)$, is given by $\overline{N}(f) = \frac{8\pi}{3}lwh\left(\frac{f}{c}\right)^3 - (l+w+h)\frac{f}{c} +\frac{1}{2}.$

The leading term is proportional to the chamber volume and to the third power of the frequency. This term is identical to Weyl's formula.

Based on $\overline{N}(f)$ the mode density $\overline{n}(f)$ is given by $\overline{n}(f)=\frac{d\overline{N}(f)}{df} = \frac{8\pi}{c}lwh\left(\frac{f}{c}\right)^2 - (l+w+h)\frac{1}{c}.$

An important quantity is the number of modes in a certain frequency interval Δf, $\overline{N}_{\Delta f}(f)$, that is given by $\begin{matrix} \overline{N}_{\Delta f}(f) & = & \int_{f-\Delta f/2}^{f+\Delta f/2} \overline{n}(f) df \\ \ & = & \overline{N}(f+\Delta f/2) - \overline{N}(f-\Delta f/2)\\ \ & \simeq & \frac{8\pi lwh}{c^3} \cdot f^2 \cdot \Delta f \end{matrix}$

### Quality Factor

The Quality Factor (or Q Factor) is an important quatity for all resonant systems. Generally, the Q factor is defined by $Q=\omega\frac{\rm maximum\; stored\; energy}{\rm average\; power\; loss} = \omega \frac{W_s}{P_l},$ where the maximum and the average are taken over one cycle, and ω = 2πf is the angular frequency.

The factor Q of the TE and TM modes can be calculated from the fields. The stored energy Ws is given by $W_s = \frac{\epsilon}{2}\iiint_V |\vec{E}|^2 dV = \frac{\mu}{2}\iiint_V |\vec{H}|^2 dV.$

The loss occurs in the metallic walls. If the wall conductivity is σ and the wall permeability is μ, the surface resistance Rs is $R_s = \frac{1}{\sigma\delta_s} = \sqrt{\frac{\pi\mu f}{\sigma}},$

where $\delta_s=1/\sqrt{\pi\mu\sigma f}$ is the skin depth of the wall material.

The losses Pl are calculated according to $P_l = \frac{R_s}{2}\iint_S |\vec{H}|^2 dS.$

For a rectangular cavity follows (Chang 1989)

• for TE modes: $Q_{\rm TE_{mnp}} = \frac{Z_0 lwh}{4R_s} \frac{k_{xy}^2 k_r^3} {\zeta l h \left(k_{xy}^4+k_x^2k_z^2 \right) + \xi w h \left(k_{xy}^4+k_y^2k_z^2 \right) + lw k_{xy}^2 k_z^2}$ $\zeta= \begin{cases} 1 & \mbox{if }n\ne 0 \\ 1/2 & \mbox{if }n=0 \end{cases},\quad \xi= \begin{cases} 1 & \mbox{if }m\ne 0 \\ 1/2 & \mbox{if }m=0 \end{cases}$

• for TM modes: $Q_{\rm TM_{mnp}} = \frac{Z_0 lwh}{4 R_s} \frac{k_{xy}^2 k_r} { w(\gamma l+h) k_x^2 + l(\gamma w+h)k_y^2}$ $\gamma= \begin{cases} 1 & \mbox{if }p\ne 0 \\ 1/2 & \mbox{if }p=0 \end{cases}$


Using the Q values of the individual modes, an averaged Composite Quality Factor $\tilde{Q_s}$ can be derived (Liu 1983): $\frac{1}{\tilde{Q_s}} = \langle\frac{1}{Q_{mnp}}\rangle_{k\le k_r \le k_r+\Delta k}$ $\tilde{Q_s} = \frac{3}{2} \frac{V}{S\delta_s} \frac{1}{1+\frac{3c}{16f}\left(1/l + 1/w + 1/h \right)}$ $\tilde{Q_s}$ includes only losses due to the finite conductivity of the chamber walls and is therefore an upper limit. Other losses are dielectric losses e.g. in antenna support structures, loaaes due to wall coatings, and leakage losses. For the lower frequency range the dominant loss is due to the antenna used to couple energy to the room (transmitting antenna, Tx) and to monitor the fields in the chamber (receiving antenna, Rx). This antenna loss Qa is given by $Q_a = \frac{16\pi^2 V f^3}{c^3 N_{a}},$ where Na is the number of antenna in the chamber.

The quality factor including all losses is the harmonic sum of the factors for all single loss processes: $\frac{1}{Q} = \sum_i \frac{1}{Q_i}$

Resulting from the finite quality factor the eigenmodes are broaden in frequency, i.e. a mode can be excited even if the operating frequency does not exactly match the eigenfrequency. Therefore, more eigenmodes are exited for a given frequency at the same time.

The Q-bandwidth BWQ is a measure of the frequency bandwidth over which the modes in a reverberation chamber are correlated. The BWQ of a reverberation chamber can be calculated using the following: ${\rm BW}_Q=\frac{f}{Q}$

Using the formula $\overline{N}_{\Delta f}(f)$ the number of modes excited within BWQ results to $M(f)=\frac{8\pi V f^3}{c^3 Q}.$

Related to the chamber quality factor is the chamber time constant τ by $\tau=\frac{Q}{2\pi f}.$

That is the time constant of the free energy relaxation of the chamber's field (exponential decay) if the input power is switched off.

## References

• a  IEC 61000-4-21: Electromagnetic compatibility (EMC) - Part 4-21: Testing and measurement techniques - Reverberation chamber test methods, Ed. 1.0, August, 2003. (IEC61000-4-21)
• a  Mendes, H.A.: A new approach to electromagnetic field-strength measurements in shielded enclosures., Wescon Tech. Papers, Los Angeles, CA., August, 1968.
• a  Cheng, D.K.: Field and Wave Electromagnetics, Addison-Wesley Publishing Company Inc., Edition 2, 1998. ISBN 0-201-52820-7
• a  Chang, K.: Handbook of Microwave and Optical Components, Volume 1, John Willey & Sons Inc., 1989. ISBN 0471613665
• a  Liu, B.H., Chang, D.C., Ma, M.T.: Eigenmodes and the Composite Quality Factor of a Reverberating Chamber, NBS Technical Note 1066, National Bureau of Standards, Boulder, CO., August 1983.
• Crawford, M.L.; Koepke, G.H.: Design, Evaluation, and Use of a Reverberation Chamber for Performing Electromagnetic Susceptibility/Vulneribility Measurements, NBS Technical Note 1092, National Bureau od Standards, Boulder, CO, April, 1986.
• Ladbury, J.M.; Koepke, G.H.: Reverberation chamber relationships: corrections and improvements or three wrongs can (almost) make a right,

Electromagnetic Compatibility, 1999 IEEE International Symposium on, Volume 1, 1-6, 2-6 Aug. 1999.

• Lehman, T.H.: A Statistical Theory of Electromagnetic Fields in Complex Cavities, Interaction Note 494, May, 1993. (pdf)