Ruze's Equation
Encyclopedia
Ruze's Equation is an equation relating the gain of an antenna
to the RMS
of the random surface errors. The equation is applicable to parabolic reflector
and antennas, and recently extended to phased array
s. The equation is named after John Ruze who introduced the equation in a paper he wrote in 1952. The equation states that the antenna's gain
is inversely proportional to the exponential of the square of the RMS surface errors. Mathematically, the equation for parabolic reflector antennas
can be expressed as:
where
is the surface RMS errors of the reflector, 
is the wavelength
, and
is the gain of the antenna in the absence of surface errors.
The equation is often expressed in decibel
s as:
(dB)
where
s has been extended to phased array
applications. For phased arrays, the equation is slightly modified, differing by a factor of 2 in the exponential, to give
The factor of 2 difference between the equation for the phased array and the equation for reflectors is that the electromagnetic wave goes in only one direction for phased arrays, but it goes back and forth in reflectors (the wave is reflected).
Consequently, when expressed in dB, Ruze's Equation for phased arrays has a different coefficient, namely:
(dB)
where
is the RMS of the z-directed positional errors of the array elements, and as before, 
is the wavelength
.
Antenna gain
In electromagnetics, an antenna's power gain or simply gain is a key performance figure which combines the antenna's directivity and electrical efficiency. As a transmitting antenna, the figure describes how well the antenna converts input power into radio waves headed in a specified direction...
to the RMS
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...
of the random surface errors. The equation is applicable to parabolic reflector
Parabolic antenna
A parabolic antenna is an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves. The most common form is shaped like a dish and is popularly called a dish antenna or parabolic dish...
and antennas, and recently extended to phased array
Phased array
In wave theory, a phased array is an array of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and suppressed in undesired directions.An antenna array...
s. The equation is named after John Ruze who introduced the equation in a paper he wrote in 1952. The equation states that the antenna's gain
Antenna gain
In electromagnetics, an antenna's power gain or simply gain is a key performance figure which combines the antenna's directivity and electrical efficiency. As a transmitting antenna, the figure describes how well the antenna converts input power into radio waves headed in a specified direction...
is inversely proportional to the exponential of the square of the RMS surface errors. Mathematically, the equation for parabolic reflector antennas
Parabolic antenna
A parabolic antenna is an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves. The most common form is shaped like a dish and is popularly called a dish antenna or parabolic dish...
can be expressed as:
where
is the surface RMS errors of the reflector, is the wavelengthWavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
, and
is the gain of the antenna in the absence of surface errors.The equation is often expressed in decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
s as:
(dB)where
Application to phased array
Ruze's equation, which was originally derived for parabolic reflectorParabolic reflector
A parabolic reflector is a reflective device used to collect or project energy such as light, sound, or radio waves. Its shape is that of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis...
s has been extended to phased array
Phased array
In wave theory, a phased array is an array of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and suppressed in undesired directions.An antenna array...
applications. For phased arrays, the equation is slightly modified, differing by a factor of 2 in the exponential, to give
The factor of 2 difference between the equation for the phased array and the equation for reflectors is that the electromagnetic wave goes in only one direction for phased arrays, but it goes back and forth in reflectors (the wave is reflected).
Consequently, when expressed in dB, Ruze's Equation for phased arrays has a different coefficient, namely:
(dB)where
is the RMS of the z-directed positional errors of the array elements, and as before, is the wavelengthWavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
.
Further reading
- The Radio Astronomy tutorial presented by MIT'sMassachusetts Institute of TechnologyThe Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
Haystack ObservatoryHaystack ObservatoryHaystack Observatory is a group of astronomical observatories owned and operated by Massachusetts Institute of Technology. It is located in Westford, Massachusetts . It is the home of the Millstone Hill Observatory....
(Section 6.2.3)