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. ==See also==