Electromagnetic reverberation chamber

The notation is mainly the same as in the IEC standard 61000-4-21.[2] 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 ${\displaystyle s}$) means that quantities are taken for different chamber positions, and ensemble domain (subscript ${\displaystyle e}$) refers to different boundary or excitation conditions (e.g. tuner positions).

• ${\displaystyle {\vec {E}}}$: Vector of the electric field.
• ${\displaystyle {\vec {H}}}$: Vector of the magnetic field.
• ${\displaystyle E_{T},\,H_{T}}$: The total electrical or magnetical field strength, i.e. the magnitude of the field vector.
• ${\displaystyle E_{R},\,H_{R}}$: Field strength (magnitude) of one rectangular component of the electrical or magnetical field vector.
• ${\displaystyle Z_{0}={\frac {|{\vec {E}}|}{|{\vec {H}}|}}\approx 120\cdot \pi \,\Omega }$: Characteristic impedance of the free space
• ${\displaystyle \eta _{\rm {Tx}}}$: Efficiency of the transmitting antenna
• ${\displaystyle \eta _{\rm {Rx}}}$: Efficiency of the receiving antenna
• ${\displaystyle P_{\rm {fwd}},\,P_{\rm {bwd}}}$: Power of the forward and backward running waves.
• ${\displaystyle Q}$: The quality factor.

• ${\displaystyle {}_{s}\langle X\rangle _{N}}$: spatial mean of ${\displaystyle X}$ for ${\displaystyle N}$ objects (positions in space).
• ${\displaystyle {}_{e}\langle X\rangle _{N}}$: ensemble mean of ${\displaystyle X}$ for ${\displaystyle N}$ objects (boundaries, i.e. tuner positions).
• ${\displaystyle \langle X\rangle }$: equivalent to ${\displaystyle \langle X\rangle _{\infty }}$. Thist is the expected value in statistics.
• ${\displaystyle {}_{s}\lceil X\rceil _{N}}$: spatial maximum of ${\displaystyle X}$ for ${\displaystyle N}$ objects (positions in space).
• ${\displaystyle {}_{e}\lceil X\rceil _{N}}$: ensemble maximum of ${\displaystyle X}$ for ${\displaystyle N}$ objects (boundaries, i.e. tuner positions).
• ${\displaystyle \lceil X\rceil }$: equivalent to ${\displaystyle \lceil X\rceil _{\infty }}$.
• ${\displaystyle {}_{s}\!\dagger \!(X)_{N}}$: max to mean ratio in the spatial domain.
• ${\displaystyle {}_{e}\!\dagger \!(X)_{N}}$: max to mean ratio in the ensemble domain.

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) ${\displaystyle f_{mnp}}$ are given by

${\displaystyle f_{mnp}={\frac {c}{2}}{\sqrt {\left({\frac {m}{l}}\right)^{2}+\left({\frac {n}{w}}\right)^{2}+\left({\frac {p}{h}}\right)^{2}}},}$

where ${\displaystyle c}$ is the speed of light, ${\displaystyle l}$, ${\displaystyle w}$ and ${\displaystyle h}$ are the cavity's length, width and height, and ${\displaystyle m}$, ${\displaystyle n}$, ${\displaystyle p}$ 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 ${\displaystyle f}$, ${\displaystyle N(f)}$, can be counted. This results in a stepwise function. In principle, two modes—a transversal electric mode ${\displaystyle TE_{mnp}}$ and a transversal magnetic mode ${\displaystyle TM_{mnp}}$—exist for each eigenfrequency.

The fields at the chamber position ${\displaystyle (x,y,z)}$ are given by

• for the TM modes (${\displaystyle H_{z}=0}$)

 ${\displaystyle E_{x}=-{\frac {1}{j\omega \epsilon }}k_{x}k_{z}\cos k_{x}x\sin k_{y}y\sin k_{z}z}$
 ${\displaystyle E_{y}=-{\frac {1}{j\omega \epsilon }}k_{y}k_{z}\sin k_{x}x\cos k_{y}y\sin k_{z}z}$
 ${\displaystyle E_{z}={\frac {1}{j\omega \epsilon }}k_{xy}^{2}\sin k_{x}x\sin k_{y}y\cos k_{z}z}$
 ${\displaystyle H_{x}=k_{y}\sin k_{x}x\cos k_{y}y\cos k_{z}z}$
 ${\displaystyle H_{y}=-k_{x}\cos k_{x}x\sin k_{y}y\cos k_{z}z}$
 ${\displaystyle k_{r}^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2},\,k_{x}={\frac {m\pi }{l}},\,k_{y}={\frac {n\pi }{w}},\,k_{z}={\frac {p\pi }{h}}\,k_{xy}^{2}=k_{x}^{2}+k_{y}^{2}}$

• for the TE modes (${\displaystyle E_{z}=0}$)

 ${\displaystyle E_{x}=k_{y}\cos k_{x}x\sin k_{y}y\sin k_{z}z}$
 ${\displaystyle E_{y}=-k_{x}\sin k_{x}x\cos k_{y}y\sin k_{z}z}$
 ${\displaystyle H_{x}=-{\frac {1}{j\omega \mu }}k_{x}k_{z}\sin k_{x}x\cos k_{y}y\cos k_{z}z}$
 ${\displaystyle H_{y}=-{\frac {1}{j\omega \mu }}k_{y}k_{z}\cos k_{x}x\sin k_{y}y\cos k_{z}z}$
 ${\displaystyle H_{z}={\frac {1}{j\omega \mu }}k_{xy}^{2}\cos k_{x}x\cos k_{y}y\sin k_{z}z}$

Due to the boundary conditions for the E- and H field, some modes do not exist. The restrictions are:[3]

• 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 ${\displaystyle N(f)}$, ${\displaystyle {\overline {N}}(f)}$, is given by

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

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.

Comparison of the exact and the smoothed number of modes for the Large Magdeburg Reverberation Chamber.

Based on ${\displaystyle {\overline {N}}(f)}$ the mode density ${\displaystyle {\overline {n}}(f)}$ is given by

${\displaystyle {\overline {n}}(f)={\frac {d{\overline {N}}(f)}{df}}={\frac {8\pi }{c}}lwh\left({\frac {f}{c}}\right)^{2}-(l+w+h){\frac {1}{c}}.}$

An important quantity is the number of modes in a certain frequency interval ${\displaystyle \Delta f}$, ${\displaystyle {\overline {N}}_{\Delta f}(f)}$, that is given by

${\displaystyle {\begin{matrix}{\overline {N}}_{\Delta f}(f)&=&\int _{f-\Delta f/2}^{f+\Delta f/2}{\overline {n}}(f)df\\\ &=&{\overline {N}}(f+\Delta f/2)-{\overline {N}}(f-\Delta f/2)\\\ &\simeq &{\frac {8\pi lwh}{c^{3}}}\cdot f^{2}\cdot \Delta f\end{matrix}}}$

The Quality Factor (or Q Factor) is an important quantity for all resonant systems. Generally, the Q factor is defined by ${\displaystyle Q=\omega {\frac {\rm {maximum\;stored\;energy}}{\rm {average\;power\;loss}}}=\omega {\frac {W_{s}}{P_{l}}},}$ where the maximum and the average are taken over one cycle, and ${\displaystyle \omega =2\pi f}$ is the angular frequency.

The factor Q of the TE and TM modes can be calculated from the fields. The stored energy ${\displaystyle W_{s}}$ is given by

${\displaystyle W_{s}={\frac {\epsilon }{2}}\iiint _{V}|{\vec {E}}|^{2}dV={\frac {\mu }{2}}\iiint _{V}|{\vec {H}}|^{2}dV.}$

The loss occurs in the metallic walls. If the wall's electrical conductivity is ${\displaystyle \sigma }$ and its permeability is ${\displaystyle \mu }$, the surface resistance ${\displaystyle R_{s}}$ is

${\displaystyle R_{s}={\frac {1}{\sigma \delta _{s}}}={\sqrt {\frac {\pi \mu f}{\sigma }}},}$

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

The losses ${\displaystyle P_{l}}$ are calculated according to

${\displaystyle P_{l}={\frac {R_{s}}{2}}\iint _{S}|{\vec {H}}|^{2}dS.}$

For a rectangular cavity follows[4]

• for TE modes:

 ${\displaystyle Q_{\rm {TE_{mnp}}}={\frac {Z_{0}lwh}{4R_{s}}}{\frac {k_{xy}^{2}k_{r}^{3}}{\zeta lh\left(k_{xy}^{4}+k_{x}^{2}k_{z}^{2}\right)+\xi wh\left(k_{xy}^{4}+k_{y}^{2}k_{z}^{2}\right)+lwk_{xy}^{2}k_{z}^{2}}}}$
${\displaystyle \zeta ={\begin{cases}1&{\mbox{if }}n\neq 0\\1/2&{\mbox{if }}n=0\end{cases}},\quad \xi ={\begin{cases}1&{\mbox{if }}m\neq 0\\1/2&{\mbox{if }}m=0\end{cases}}}$

• for TM modes:

${\displaystyle Q_{\rm {TM_{mnp}}}={\frac {Z_{0}lwh}{4R_{s}}}{\frac {k_{xy}^{2}k_{r}}{w(\gamma l+h)k_{x}^{2}+l(\gamma w+h)k_{y}^{2}}}}$
${\displaystyle \gamma ={\begin{cases}1&{\mbox{if }}p\neq 0\\1/2&{\mbox{if }}p=0\end{cases}}}$

Using the Q values of the individual modes, an averaged Composite Quality Factor ${\displaystyle {\tilde {Q_{s}}}}$ can be derived:[5] ${\displaystyle {\frac {1}{\tilde {Q_{s}}}}=\langle {\frac {1}{Q_{mnp}}}\rangle _{k\leq k_{r}\leq k_{r}+\Delta k}}$ ${\displaystyle {\tilde {Q_{s}}}={\frac {3}{2}}{\frac {V}{S\delta _{s}}}{\frac {1}{1+{\frac {3c}{16f}}\left(1/l+1/w+1/h\right)}}}$

${\displaystyle {\tilde {Q_{s}}}}$ 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 ${\displaystyle Q_{a}}$ is given by ${\displaystyle Q_{a}={\frac {16\pi ^{2}Vf^{3}}{c^{3}N_{a}}},}$ where ${\displaystyle N_{a}}$ 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:

${\displaystyle {\frac {1}{Q}}=\sum _{i}{\frac {1}{Q_{i}}}}$

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 ${\displaystyle {\rm {BW}}_{Q}}$ is a measure of the frequency bandwidth over which the modes in a reverberation chamber are correlated. The ${\displaystyle {\rm {BW}}_{Q}}$ of a reverberation chamber can be calculated using the following:

${\displaystyle {\rm {BW}}_{Q}={\frac {f}{Q}}}$

Using the formula ${\displaystyle {\overline {N}}_{\Delta f}(f)}$ the number of modes excited within ${\displaystyle {\rm {BW}}_{Q}}$ results to

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

Related to the chamber quality factor is the chamber time constant ${\displaystyle \tau }$ by

${\displaystyle \tau ={\frac {Q}{2\pi f}}.}$

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