Electromagnetic Reverberation Chamber

An electromagnetic reverberation chamber (also known as a reverb chamber (RVC) or mode-stirred chamber (MSC)) is an environment for electromagnetic compatibility

Electromagnetic compatibility

Electromagnetic compatibility is the branch of electrical sciences which studies the unintentional generation, propagation and reception of electromagnetic energy with reference to the unwanted effects that such energy may induce...

(EMC) testing and other electromagnetic investigations. Electromagnetic reverberation chambers have been introduced first by H.A. Mendes in 1968. A reverberation chamber is screened room

Faraday cage

A Faraday cage or Faraday shield is an enclosure formed by conducting material or by a mesh of such material. Such an enclosure blocks out external static and non-static electric fields...

with a minimum of absorption

Absorption (electromagnetic radiation)

In physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...

of electromagnetic

Electromagnetic radiation

Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

. Due to the low absorption very high field strength

Field strength

In physics, the field strength of a field is the magnitude of its vector value.In theoretical physics, field strength is another name for the curvature form...

can be achieved with moderate input power. A reverberation chamber is a cavity resonator with a high Q factor

Q factor

In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....

. 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

Microwave oven

A microwave oven is a kitchen appliance that heats food by dielectric heating, using microwave radiation to heat polarized molecules within the food...

Preface

The notation is mainly the same as in the IEC

International Electrotechnical Commission

The International Electrotechnical Commission is a non-profit, non-governmental international standards organization that prepares and publishes International Standards for all electrical, electronic and related technologies – collectively known as "electrotechnology"...

standard 61000-4-21. For statistic quantities like mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

and maximal values, a more explicit notation is used in order to emphasize the used domain. Here, spatial domain (subscript

) means that quantities are taken for different chamber positions, and ensemble domain (subscript

) refers to different boundary or excitation conditions (e.g. tuner positions).

General

: Vector of the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

.
: Vector of the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

.
: The total electrical or magnetical field strength
Field strength
In physics, the field strength of a field is the magnitude of its vector value.In theoretical physics, field strength is another name for the curvature form...

, i.e. the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

of the field vector.
: Field strength
Field strength
In physics, the field strength of a field is the magnitude of its vector value.In theoretical physics, field strength is another name for the curvature form...

(magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

) of one rectangular component of the electrical or magnetical field vector.
: Characteristic impedance
Characteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...

of the free space
: Efficiency of the transmitting antenna
Antenna (radio)
An antenna is an electrical device which converts electric currents into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver...
: Efficiency of the receiving antenna
Antenna (radio)
An antenna is an electrical device which converts electric currents into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver...
: Power
Power (physics)
In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

of the forward and backward running waves
WAVES
The WAVES were a World War II-era division of the U.S. Navy that consisted entirely of women. The name of this group is an acronym for "Women Accepted for Volunteer Emergency Service" ; the word "emergency" implied that the acceptance of women was due to the unusual circumstances of the war and...

.
: The quality factor.

Statistics

: spatial mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

of for objects (positions in space).
: ensemble mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

of for objects (boundaries, i.e. tuner positions).
: equivalent to . Thist is the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

.
: spatial maximum of for objects (positions in space).
: ensemble maximum of for objects (boundaries, i.e. tuner positions).
: equivalent to .
: max to mean ratio in the spatial domain.
: max to mean ratio in the ensemble domain.

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)

are given by

where

is the speed of light

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

and

are the cavity's length, width and height, and

are non negative integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s (at most one of those can be zero

0 (number)

0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

).

With that equation, the number of modes

Normal mode

A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...

with an eigenfrequency less than a given limit

, can be counted. This results in a stepwise function

Step function

In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...

. In principle, two modes—a transversal

Transversal

In geometry , when two coplanar lines exist such that a third coplanar line passes thru both lines. This third line is named the Transversal....

electric mode

and a transversal

Transversal

In geometry , when two coplanar lines exist such that a third coplanar line passes thru both lines. This third line is named the Transversal....

magnetic mode

—exist for each eigenfrequency.

The fields at the chamber position

are given by

for the TM modes ()

for the TE modes ()

Due to the boundary conditions for the E- and H field , some modes does 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

Approximation

An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...

, is given by

The leading term is proportional

Proportionality (mathematics)

In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

to the chamber volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

and to the third power of the frequency

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

. This term is identical to Weyl's formula.

Based on

the mode density

is given by

An important quantity is the number of modes in a certain frequency interval

Interval

Interval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement...

, that is given by

Quality Factor

The Quality Factor

Q factor

(or Q Factor) is an important quantity for all resonant systems. Generally, the Q factor is defined by

where the maximum and the average are taken over one cycle, and

is the angular frequency

Angular frequency

In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

.

The factor Q of the TE and TM modes can be calculated from the fields. The stored energy

is given by

The loss occurs in the metallic walls. If the wall's electrical conductivity is

and its permeability

Permeability (electromagnetism)

In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

, the surface resistance

where

is the skin depth of the wall material.

The losses

are calculated according to

For a rectangular cavity follows

for TE modes:

for TM modes:

Using the Q values of the individual modes, an averaged Composite Quality Factor

can be derived:

includes only losses due to the finite conductivity of the chamber walls and is therefore an upper limit. Other losses are dielectric

Dielectric

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing 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

is given by

where

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:

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

is a measure of the frequency bandwidth over which the modes in a reverberation chamber are
correlated. The

of a reverberation chamber can be calculated using the following:

Using the formula

the number of modes excited within

results to

Related to the chamber quality factor is the chamber time constant

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

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