View source for MAX-3SAT

{{Use dmy dates|date=July 2013}}
[[File:magdeburg-reverberation chamber.jpg|thumb|330px|A look inside the (large) Reverberation Chamber at the Otto-von-Guericke-University Magdeburg, Germany. On the left side is the vertical ''Mode Stirrer'' (or ''Tuner''), that changes the electromagnetic boundaries to ensure a (statistically) homogeneous field distribution.]]
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.<ref>Mendes, H.A.: ''A new approach to electromagnetic field-strength measurements in shielded enclosures.'', Wescon Tech. Papers, Los Angeles, CA., August, 1968.
</ref> A reverberation chamber is [[Faraday cage|screened room]] with a minimum of [[absorption (electromagnetic radiation)|absorption]] of [[Electromagnetic radiation|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 spatial distribution of the electrical and magnetic field strength is strongly inhomogeneous ([[standing waves]]). To reduce this inhomogeneity, 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 [[International Electrotechnical Commission|IEC]] standard 61000-4-21.<ref>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])</ref> 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 <math>s</math>) means that quantities are taken for different chamber positions, and ''ensemble domain'' (subscript <math>e</math>) refers to different boundary or excitation conditions (e.g. tuner positions).

===General===
* <math>\vec{E}</math>: [[vector (geometric)|Vector]] of the [[electric field]].
* <math>\vec{H}</math>: [[vector (geometric)|Vector]] of the [[magnetic field]].
* <math>E_T,\, H_T</math>: The total electrical or magnetical [[field strength]], i.e. the [[magnitude (mathematics)|magnitude]] of the field [[vector (geometric)|vector]].
* <math>E_R,\, H_R</math>: [[Field strength]] ([[magnitude (mathematics)|magnitude]]) of one [[rectangular]] [[Vector component|component]] of the electrical or magnetical field [[vector (geometric)|vector]].
* <math>Z_0=\frac{|\vec{E}|}{|\vec{H}|}=120\cdot \pi\, \Omega</math>: [[Characteristic impedance]] of the free space
* <math>\eta_{\rm Tx}</math>: [[wikt:Efficiency|Efficiency]] of the transmitting [[antenna (radio)|antenna]]
* <math>\eta_{\rm Rx}</math>: [[wikt:Efficiency|Efficiency]] of the receiving [[antenna (radio)|antenna]]
* <math> P_{\rm fwd}, \, P_{\rm bwd}</math>: [[Power (physics)|Power]] of the forward and backward running [[waves]].
* <math>Q</math>: The [[quality factor]].

===Statistics===
* <math>{}_s\langle X \rangle_N</math>: spatial [[mean]] of <math>X</math> for <math>N</math> objects (positions in space).
* <math>{}_e\langle X \rangle_N</math>: ensemble [[mean]] of <math>X</math> for <math>N</math> objects (boundaries, i.e. tuner positions).
* <math>\langle X \rangle</math>: equivalent to <math>\langle X \rangle_\infty</math>. Thist is the [[expected value]] in [[statistics]].
* <math>{}_s\lceil X \rceil_N</math>: spatial maximum of <math>X</math> for <math>N</math> objects (positions in space).
* <math>{}_e\lceil X \rceil_N</math>: ensemble maximum of <math>X</math> for <math>N</math> objects (boundaries, i.e. tuner positions).
* <math>\lceil X \rceil</math>: equivalent to <math>\lceil X \rceil_\infty</math>.
* <math>{}_s\!\dagger\!(X)_N</math>: max to mean ratio in the spatial domain.
* <math>{}_e\!\dagger\!(X)_N</math>: 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 resonator]]s briefly.

For rectangular cavities, the [[resonance frequency|resonance frequencies]] (or [[eigenfrequency|eigenfrequencies]], or
[[Natural Frequency|natural frequencies]]) <math>f_{mnp}</math> are given by

<math>
f_{mnp} = \frac{c}{2}\sqrt{\left(\frac{m}{l}\right)^2+\left(\frac{n}{w}\right)^2+\left(\frac{p}{h}\right)^2},
</math>

where <math>c</math> is the [[speed of light]], <math>l</math>, <math>w</math> and <math>h</math> are the cavity's length, width and height, and <math>m</math>, <math>n</math>, <math>p</math> are non-negative [[integer]]s (at most one of those can be [[0 (number)|zero]]).

With that equation, the number of [[normal mode|modes]] with an [[eigenfrequency]] less than a given limit <math>f</math>, <math>N(f)</math>, can be counted. This results in a [[step function|stepwise function]]. In principle, two modes—a transversal electric mode <math>TE_{mnp}</math> and a transversal magnetic mode <math>TM_{mnp}</math>—exist for each [[eigenfrequency]].

The fields at the chamber position <math>(x,y,z)</math> are given by
* for the TM modes (<math>H_z=0</math>)
  <math>
  E_x=-\frac{1}{j\omega\epsilon} k_x k_z \cos k_x x \sin k_y y \sin k_z z
  </math>
  <math>
  E_y=-\frac{1}{j\omega\epsilon} k_y k_z \sin k_x x  \cos k_y y \sin k_z z
  </math>
  <math>
  E_z= \frac{1}{j\omega\epsilon} k_{xy}^2 \sin k_x x  \sin k_y y \cos k_z z
  </math>
  <math>
  H_x= k_y \sin k_x x  \cos k_y y \cos k_z z
  </math>
  <math>
  H_y= - k_x \cos k_x x  \sin k_y y \cos k_z z
  </math>
  <math>
  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
  </math>
* for the TE modes (<math>E_z=0</math>)
  <math>
  E_x= k_y \cos k_x x  \sin k_y y \sin k_z z
  </math>
  <math>
  E_y=- k_x \sin k_x x  \cos k_y y \sin k_z z
  </math>
  <math>
  H_x=-\frac{1}{j\omega\mu} k_x k_z \sin k_x x  \cos k_y y \cos k_z z
  </math>
  <math>
  H_y=-\frac{1}{j\omega\mu} k_y k_z \cos k_x x  \sin k_y y \cos k_z z
  </math>
  <math>
  H_z= \frac{1}{j\omega\mu} k_{xy}^2 \cos k_x x  \cos  k_y y \sin k_z z
  </math>

Due to the [[boundary condition]]s for the E- and H field, some modes do not exist. The restrictions are:<ref>Cheng, D.K.: ''Field and Wave Electromagnetics'', Addison-Wesley Publishing Company Inc., Edition 2, 1998. ISBN 0-201-52820-7</ref>
* 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 <math>N(f)</math>, <math>\overline{N}(f)</math>, is given by

<math>
\overline{N}(f) = \frac{8\pi}{3}lwh\left(\frac{f}{c}\right)^3 - (l+w+h)\frac{f}{c} +\frac{1}{2}.
</math>

The leading term is [[Proportionality (mathematics)|proportional]] to the chamber [[volume]] and to the third power of the [[frequency]]. This term is identical to [[Weyl]]'s formula.

[[File:cummodes.svg|thumb|Comparison of the exact and the smoothed number of modes for the Large Magdeburg Reverberation Chamber.]]

Based on <math>\overline{N}(f)</math> the ''mode density'' <math>\overline{n}(f)</math> is given by

<math>
\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}.
</math>

An important quantity is the number of modes in a certain frequency [[Interval (mathematics)|interval]] <math>\Delta f</math>, <math>\overline{N}_{\Delta f}(f)</math>, that is given by

<math>
\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}
</math>

===Quality factor===
The [[Q factor|Quality Factor]] (or Q Factor) is an important quantity for all [[resonant]] systems. Generally, the Q factor is defined by
<math>
Q=\omega\frac{\rm maximum\; stored\; energy}{\rm average\; power\; loss} = \omega \frac{W_s}{P_l},
</math>
where the maximum and the average are taken over one cycle, and <math>\omega=2\pi f</math> is the [[angular frequency]].

The factor Q of the TE and TM modes can be calculated from the fields. The stored energy <math>W_s</math> is given by

<math>
W_s = \frac{\epsilon}{2}\iiint_V |\vec{E}|^2 dV = \frac{\mu}{2}\iiint_V |\vec{H}|^2 dV.
</math>

The loss occurs in the metallic walls. If the wall's [[electrical conductivity]] is <math>\sigma</math> and its [[Permeability (electromagnetism)|permeability]] is <math>\mu</math>, the [[surface resistance]] <math>R_s</math> is

<math>
R_s = \frac{1}{\sigma\delta_s} = \sqrt{\frac{\pi\mu f}{\sigma}},
</math>

where <math>\delta_s=1/\sqrt{\pi\mu\sigma f}</math> is the [[skin depth]] of the wall material.

The losses <math>P_l</math> are calculated according to

<math>
P_l = \frac{R_s}{2}\iint_S |\vec{H}|^2 dS.
</math>

For a rectangular cavity follows<ref>Chang, K.: ''Handbook of Microwave and Optical Components'', Volume 1, John Willey & Sons Inc., 1989. ISBN 0-471-61366-5.</ref>
* for TE modes:
  <math>
 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}
 </math>
 <math>
 \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}
 </math>
* for TM modes:
 <math>
 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}
 </math>
 <math>
 \gamma=
  \begin{cases}
    1 & \mbox{if }p\ne 0 \\
  1/2 & \mbox{if }p=0
  \end{cases}
 </math>

Using the Q values of the individual modes, an averaged ''Composite Quality Factor'' <math>\tilde{Q_s}</math> can be derived:<ref>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.</ref>
<math>
\frac{1}{\tilde{Q_s}} = \langle\frac{1}{Q_{mnp}}\rangle_{k\le k_r \le k_r+\Delta k}
</math>
<math>
\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)}
</math>

<math>\tilde{Q_s}</math> 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 <math>Q_a</math> is given by
<math>
Q_a = \frac{16\pi^2 V f^3}{c^3 N_{a}},
</math>
where <math>N_a</math> 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:

<math>
\frac{1}{Q} = \sum_i \frac{1}{Q_i}
</math>

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'' <math>{\rm BW}_Q</math> is a measure of the frequency bandwidth over which the modes in a reverberation chamber are
correlated. The <math>{\rm BW}_Q</math> of a reverberation chamber can be calculated using the following:

<math>{\rm BW}_Q=\frac{f}{Q}</math>

Using the formula <math>\overline{N}_{\Delta f}(f)</math> the number of modes excited within <math>{\rm BW}_Q</math> results to

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

Related to the chamber quality factor is the ''chamber time constant'' <math>\tau</math> by

<math>
\tau=\frac{Q}{2\pi f}.
</math>

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

==See also==
* [[Anechoic chamber]]
* [[Reverberation room]]
* [[Echo chamber]]
*[[Integrating sphere]]

==Notes==
{{Reflist}}

==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.

{{DEFAULTSORT:Electromagnetic Reverberation Chamber}}
[[Category:Electromagnetic radiation]]