Electromagnetic reverberation chamber

Glossary/notation

Preface

The notation is mainly the same as in the IEC standard 61000-4-21. For statistic quantities like mean and maximal values, a more explicit notation is used in order to emphasize the used domain. Here, spatial 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}|} \approx 120\cdot \pi, \Omega: Characteristic impedance of the free space
• \eta_{\rm Tx}: Efficiency of the transmitting antenna
• \eta_{\rm Rx}: 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: spatial 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: spatial 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 spatial domain.
• {}_e!\dagger!(X)_N: max to mean ratio in the ensemble domain.

Theory

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) f_{mnp} 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 TE_{mnp} and a transversal magnetic mode TM_{mnp}—exist for each eigenfrequency.

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

• for the TM modes (H_z=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 H_x= k_y \sin k_x x \cos k_y y \cos k_z z H_y= - k_x \cos k_x x \sin k_y y \cos k_z z 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 (E_z=0) E_x= k_y \cos k_x x \sin k_y y \sin k_z z E_y=- k_x \sin k_x x \cos k_y y \sin k_z z 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 do not exist. The restrictions are:

• 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 \Delta f, \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 quantity 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 \omega=2\pi f is the angular frequency.

The factor Q of the TE and TM modes can be calculated from the fields. The stored energy W_s 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's electrical conductivity is \sigma and its permeability is \mu, the surface resistance R_s 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 P_l are calculated according to

P_l = \frac{R_s}{2}\iint_S |\vec{H}|^2 dS.

For a rectangular cavity follows

• 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: \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, losses 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 Q_a is given by Q_a = \frac{16\pi^2 V f^3}{c^3 N_{a}}, where N_a 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 {\rm BW}_Q is a measure of the frequency bandwidth over which the modes in a reverberation chamber are correlated. The {\rm BW}_Q 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 {\rm BW}_Q results to

M(f)=\frac{8\pi V f^3}{c^3 Q}.

Related to the chamber quality factor is the chamber time constant \tau 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.

Notes

References

• Crawford, M.L.; Koepke, G.H.: Design, Evaluation, and Use of a Reverberation Chamber for Performing Electromagnetic Susceptibility/Vulnerability 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 August 1999.

References

1. IEC 61000-4-21: ''Electromagnetic compatibility (EMC) – Part 4-21: Testing and measurement techniques – Reverberation chamber test methods'', Ed. 2.0, January, 2011. ([http://webstore.iec.ch/Webstore/webstore.nsf/ArtNum_PK/44777!opendocument&preview=1])
2. Cheng, D.K.: ''Field and Wave Electromagnetics'', Addison-Wesley Publishing Company Inc., Edition 2, 1998. {{ISBN. 0-201-52820-7
3. Chang, K.: ''Handbook of Microwave and Optical Components'', Volume 1, John Wiley & Sons Inc., 1989. {{ISBN. 0-471-61366-5.
4. 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.