# Antenna measurement

Antenna measurement

Antenna measurement techniques refers to the testing of antennas to ensure that the antenna meets specifications or simply to characterize it. Typical parameters of antennas are gain, radiation pattern, beamwidth, polarization, and impedance.

The antenna pattern is the response of the antenna to a plane wave incident from a given direction or the relative power density of the wave transmitted by the antenna in a given direction. For a reciprocal antenna, these two patterns are identical. A multitude of antenna pattern measurement techniques have been developed. The first technique developed was the far-field range, where the antenna under test (AUT) is placed in the far-field of a range antenna. Due to the size required to create a far-field range for large antennas, near-field techniques were developed, which allow the measurement of the field on a surface close to the antenna (typically 3 to 10 times its wavelength). This measurement is then predicted to be the same at infinity. A third common method is the compact range, which uses a reflector to create a field near the AUT that looks approximately like a plane-wave.

Far-Field Range (FF)

The far-field range was the original antenna measurement technique, and consists of placing the AUT a long distance away from the instrumentation antenna. Generally, the far-field distance, "d", is considered to be

:$d =$2D^2}over{lambda,

where D is the antenna diameter and $\left\{lambda\right\}$ is the wavelength of the radio wave. Separating the AUT and the instrumentation antenna by this distance reduces the phase variation across the AUT enough to obtain a reasonably good antenna pattern.

IEEE suggests the use of their antenna measurement standard, document number IEEE-Std-149-1979 for far-field ranges and measurement set-up for various techniques including ground-bounce type ranges.

Near-Field Range (NF)

Planar Near-Field Range

Planar near-field measurements are conducted by scanning a small probe antenna over a planar surface. These measurements are then transformed to the far-field by use of a Fourier Transform, or more specifically by applying a method known as stationary phase ["Asymptotic Behavior of Monodromy", Springer Berlin / Heidelberg, 1991, ISBN 978-3-540-55009-9 ] to the Laplace Transform . Three basic types of planar scans exist in near field measurements.

• rectangular planar scanning

The probe moves in the Cartesian coordinate system and its linear movement creates a regular rectangular sampling grid with a maximum near-field sample spacing of Δx = Δy = λ /2.

• polar planar scanning

More complicated solution to the rectangular scanning method is the plane polar scanning method.

• bi-polar planar scanning

The bi-polar technique is very similar to the plane polar configuration. Bi-polar method ..

Cylindrical Near-Field Range

Cylindrical near-field ranges measure the electric field on a cylindrical surface close to the AUT. Cylindrical harmonics are used transform these measurements to the far-field.

pherical Near-Field Range

Spherical near-field ranges measure the electric field on a spherical surface close to the AUT. Spherical harmonics are used transform these measurements to the far-field

Free-Space Ranges

Compact Range

A Compact Antenna Test Range (CATR) is a facility which is used to provide convenient testing of antenna systems at frequencies where obtaining far-field spacing to the AUT would be infeasible using traditional free space methods. The CATR uses a source antenna which radiates a spherical wavefront and one or more secondary reflectors to collimate the radiated spherical wavefront into a planar wavefront within the desired test zone. One typical embodiment uses a horn feed antenna and a parabolic reflector to accomplish this.

The CATR is used for microwave and millimeter wave frequencies where the 2 D2/λ far-field distance is large, such as with high-gain reflector antennas. The size of the range that is required can be much less than the size required for a full-size far-field anechoic chamber, although the cost of fabrication of the specially-designed CATR reflector can be expensive due to the need to ensure precision of the reflecting surface (typically less than λ/100 RMS surface accuracy) and to specially treat the edge of the reflector to avoid diffracted waves which can interfere with the desired beam pattern.

Elevated Range

A means of reducing refection from waves bouncing off the ground.

lant Range

A means of eliminating symmetrical wave reflection.

Measured Antenna Parameters

Except for polarization, the SWR is the most easily measured of the parameters above. Impedance can be measured with specialized equipment, as it relates to the complex SWR. Measuring radiation pattern requires a sophisticated setup including significant clear space (enough to put the sensor into the antenna's far field, or an anechoic chamber designed for antenna measurements), careful study of experiment geometry, and specialized measurement equipment that rotates the antenna during the measurements.

The radiation pattern is a graphical depiction of the relative field strength transmitted from or received by the antenna. As antennas radiate in space often several curves are necessary to describe the antenna. If the radiation of the antenna is symmetrical about an axis (as is the case in dipole, helical and some parabolic antennas) a unique graph is sufficient.

Each antenna supplier/user has different standards as well as plotting formats. An antenna radiation pattern allows to easily see sidelobes and backlobes. Each format has its own advantages and disadvantages. Radiation pattern of an antenna can be defined as the locus of all points where the emitted power per unit surface is the same. The radiated power per unit surface is proportional to the squared electrical field of the electromagnetic wave. The radiation pattern is the locus of points with the same electrical field. In this representation, the reference is usually the best angle of emission. It is also possible to depict the directive gain of the antenna as a function of the direction. Often the gain is given in decibels.

The graphs can be drawn using cartesian (rectangular) coordinates or a polar plot. This last one is useful to measure the beamwidth, which is, by convention, the angle at the -3dB points around the max gain. The shape of curves can be very different in cartesian or polar coordinates and with the choice of the limits of the logarithmic scale. The four drawings below are the radiation patterns of a same half-wave antenna.

Efficiency

"Efficiency" is the ratio of power actually radiated to the power put into the antenna terminals. A dummy load may have an SWR of 1:1 but an efficiency of 0, as it absorbs all power and radiates heat but not RF energy, showing that SWR alone is not an effective measure of an antenna's efficiency. Radiation in an antenna is caused by radiation resistance which can only be measured as part of total resistance including loss resistance. Loss resistance usually results in heat generation rather than radiation, and reduces efficiency. Mathematically, efficiency is calculated as radiation resistance divided by total resistance.

Bandwidth

IEEE defines bandwidth as "The range of frequencies within which the performance of the antenna, with respect to some characteristic, conforms to a specified standard." ["IEEE standard definitions of terms for antennas.," IEEE Std 145-1993 , pp. 6, 21 Jun 1993 [http://ieeexplore.ieee.org/servlet/opac?punumber=2785] ] In other words, bandwidth depends on the overall effectiveness of the antenna through a range of frequencies, so all of these parameters must be understood to fully characterize the bandwidth capabilities of an antenna. However, in practice, bandwidth is typically determined by looking only at SWR, i.e., by finding the frequency range over which the SWR is less than a given value. Bandwidth over which an antenna exhibits a particular radiation pattern is also important, for in practical use the performance of an antenna at the extremes of an assigned frequency band is important.

Directivity

Antenna directivity is usually measured in dBi, or decibels above isotropic. This number is obtained by measuring the gain in the strongest lobe, and comparing it to the total gain (as if all power was radiated uniformly in all directions):

dBi = 10 * log10(Directional Power / Total Power)

Gain

Gain as a parameter measures the directionality of a given antenna. An antenna with a low gain emits radiation in all directions equally, whereas a high-gain antenna will preferentially radiate in particular directions. Specifically, the Gain, Directive gain or Power gain of an antenna is defined as the ratio of the intensity (power per unit surface) radiated by the antenna in a given direction at an arbitrary distance divided by the intensity radiated at the same distance by an hypothetical isotropic antenna:::$G=\left\{left\left(\left\{P over S\right\} ight\right)_\left\{ant\right\} over left\left(\left\{P over S\right\} ight\right)_\left\{iso,!$We write "hypothetical" because a perfect isotropic antenna cannot exist in reality (the electric and magnetic field would not satisfy Maxwell's equations for electromagnetic fields). Gain is a dimensionless number (without units).

The gain of an antenna is a passive phenomenon - power is not added by the antenna, but simply redistributed to provide more radiated power in a certain direction than would be transmitted by an isotropic antenna. If an antenna has a greater than one gain in some directions, it must have a less than one gain in other directions since energy is conserved by the antenna. An antenna designer must take into account the application for the antenna when determining the gain. High-gain antennas have the advantage of longer range and better signal quality, but must be aimed carefully in a particular direction. Low-gain antennas have shorter range, but the orientation of the antenna is inconsequential. For example, a dish antenna on a spacecraft is a high-gain device (must be pointed at the planet to be effective), while a typical WiFi antenna in a laptop computer is low-gain (as long as the base station is within range, the antenna can be in an any orientation in space).

As an example, consider an antenna that radiates an electromagnetic wave whose electrical field has an amplitude $scriptstyle\left\{E_ heta\right\}$ at a distance $scriptstyle\left\{r\right\}$. This amplitude is given by::: $E_ heta= \left\{AI over r\right\}$where:
* $scriptstyle\left\{I\right\}$ is the current fed to the antenna and
* $scriptstyle\left\{A\right\}$ is a constant characteristic of each antenna.

For a large distance $scriptstyle\left\{r\right\}$. The radiated wave can be considered locally as a plane wave. The intensity of an electromagnetic plane wave is::: $\left\{Pover S\right\}=\left\{cvarepsilon_circover2\right\}\left\{E_ heta\right\}^2=\left\{1over 2\right\}$E_ heta}^2over Z_circ},!where $scriptstyle\left\{Z_circ=sqrt$mu_circ over varepsilon_circ= 376.730313461, Omega},! is a universal constant called vacuum impedance.and:: $left\left(\left\{Pover S\right\} ight\right)_\left\{ant\right\}=\left\{1over 2Z_circ\right\} \left\{A^2I^2over r^2\right\},!$If the resistive part of the series impedance of the antenna is $scriptstyle\left\{R_s\right\}$, the power fed to the antenna is $scriptstyle$1over 2}R_sI^2}. The intensity of an isotropic antenna is the power so fed divided by the surface of the sphere of radius $scriptstyle\left\{r\right\}$:: $left\left(\left\{P over S\right\} ight\right)_\left\{iso\right\}=$1over 2}R_sI^2 over 4pi r^2 },!The directive gain is:::$G=$1over 2Z_circ} {A^2I^2over r^2} over 1over 2}R_sI^2 over 4pi r^2 } } ={A^2 over 30 R_s},!

For the commonly utilized half-wave dipole, the particular formulation works out to the following, including its decibel equivalency, expressed as dBi (decibels referenced to isotropic radiator):

&=60operatorname{Cin}(2pi)=60left [ln(2pigamma)-operatorname{Ci}(2pi) ight] =120int_{0}^{frac{pi}{2frac{cosleft(frac{pi}{2}cos heta ight)^2}{sin heta}d heta,\&=15left [2pi^2-frac{1}{3}pi^4+frac{4}{135}pi^6-frac{1}{630}pi^8+frac{4}{70875}pi^{10}ldots-(-1)^nfrac{(2pi)^{2n{n(2n)!} ight] ,\&=73.1296ldots;Omega;end{align},! :::"(In most cases 73.13, is adequate)"

::::}^{int_{0}^{frac{pi}{2frac{cosleft(frac{pi}{2}cos heta ight)^2}{sin heta}d heta,\&approxfrac{120}{73.1296}approx 1.6409224approx 2.15088,mathrm{dBi};end{align},!::::"(Likewise, 1.64 and 2.15 dBi are usually the cited values)"

Sometimes, the half-wave dipole is taken as a reference instead of the isotropic radiator. The gain is then given in dBd (decibels over dipole)::: 0 dBd = 2.15 dBi

Physical background

The electrical field created by an electric charge $scriptstyle\left\{q\right\}$ is::$vec E=\left\{-qover 4pi varepsilon_circ\right\}left \left[\left\{vec e_\left\{r\text{'}\right\}over r\text{'}^2\right\}+\left\{r\text{'}over c\right\}\left\{d over dt\right\}left\left(\left\{vec e_\left\{r\text{'}\right\}over r\text{'}^2\right\} ight\right) +\left\{1over c^2\right\}\left\{d^2 over dt^2\right\}left\left(vec e_\left\{r\text{'}\right\} ight\right) ight\right] ,$where:
* $scriptstyle\left\{c\right\}$ is the speed of light in vacuum.
* $scriptstyle\left\{varepsilon_circ \right\}$ is the permittivity of free space.
* $scriptstyle\left\{r\text{'}\right\}$ is the distance from the observation point (the place where $scriptstyle\left\{vec E\right\}$ is evaluated) to the point where the charge "was" $scriptstyle\left\{r\text{'}over c\right\}$ seconds "before" the time when the measure is done.
* $extstyle\left\{vec e_\left\{r\text{'}$ is the unit vector directed from the observation point (the place where $scriptstyle\left\{vec E\right\}$ is evaluated) to the point where the charge "was" $scriptstyle\left\{r\text{'}over c\right\}$ seconds "before" the time when the measure is done.

The "prime" in this formula appears because the electromagnetic signal travels at the speed of light. Signals are observed as coming from the point where they were emitted and not from the point where the emitter is at the time of observation. The stars that we see in the sky are no longer where we see them. We will see their current position years in the future; some of the stars that we see today no longer exist.

The first term in the formula is just the electrostatic field with retarded time.

The second term is " as though nature were trying to allow for the fact that the effect is retarded" (Feynman).

The third term is the only term that accounts for the far field of antennas.

The two first terms are proportional to $extstyle\left\{1over r^2\right\}$. Only the third is proportional to $extstyle\left\{1over r\right\}$.

Near the antenna, all the terms are important. However, if the distance is large enough, the first two terms become negligible and only the third remains:::$vec E=\left\{-qover 4pi varepsilon c^2_circ\right\}\left\{d^2 over dt^2\right\}left\left(vec e_\left\{r\text{'}\right\} ight\right)=-q10^\left\{-7\right\}\left\{d^2 over dt^2\right\}left\left(vec e_\left\{r\text{'}\right\} ight\right),$

If the charge "q" is in sinusoidal motion with amplitude $scriptstyle\left\{ell_circ\right\}$ and pulsation $scriptstyle\left\{omega\right\}$ the power radiated by the charge is::: $P= \left\{q^2omega^4ell_circ^2 over 12pivarepsilon_circ c^3\right\}$ watts.Note that the radiated power is proportional to the fourth power of the frequency. It is far easier to radiate at high frequencies than at low frequencies. If the motion of charges is due to currents, it can be shown that the (small) electrical field radiated by a small length $scriptstyle\left\{dell\right\}$ of a conductor carrying a time varying current $scriptstyle\left\{I\right\}$ is::$dE_ heta\left(t+ extstyle\left\{rover c\right\}\right)=displaystyle\left\{-dell sin heta over 4pivarepsilon_circ c^2 r\right\}\left\{dIover dt\right\},$The left side of this equation is the electrical field of the electromagnetic wave radiated by a small length of conductor. The index $scriptstyle\left\{ heta\right\}$ reminds that the field is perpendicular to the line to the source. The $scriptstyle\left\{t+\left\{rover c$ reminds that this is the field observed $scriptstyle$rover c seconds after the evaluation on the current derivative. The angle $scriptstyle\left\{ heta\right\}$ is the angle between the direction of the current and the direction to the point where the field is measured.

The electrical field and the radiated power are maximal in the plane perpendicular to the current element. They are zero in the direction of the current.

Only time-varying currents radiate electromagnetic power.

If the current is sinusoidal, it can be written in complex form, in the same way used for impedances. Only the real part is physically meaningful:::$I=I_circ e^\left\{jomega t\right\}$where:
* $scriptstyle\left\{I_circ\right\}$ is the amplitude of the current.
* $scriptstyle$omega = 2pi f is the angular frequency.
*$scriptstyle\left\{j = sqrt\left\{-1$The (small) electric field of the electromagnetic wave radiated by an element of current is:::$dE_ heta\left(t+ extstyle$rover c)=displaystyle{-dell jomega over 4pivarepsilon_circ c^2} {sin heta over r} e^{jomega t},And for the time $extstyle\left\{t\right\},$:::$dE_ heta\left(t\right)=\left\{-dell jomega over 4pivarepsilon_circ c^2\right\} \left\{sin heta over r\right\} e^\left\{jleft\left(omega t-\left\{omegaover c\right\}r ight\right)\right\},$The electric field of the electromagnetic wave radiated by an antenna formed by wires is the sum of all the electric fields radiated by all the small elements of current. This addition is complicated by the fact that the direction and phase of each of the electric fields are, in general, different.

Calculation of Antenna parameters in reception

The gain in any given direction and the impedance at a given frequency are the same when the antenna is used in transmission or in reception.

The electric field of an electromagnetic wave induces a small voltage in each small segment in all electric conductors. The induced voltage depends on the electrical field and the conductor length. The voltage depends also on the relative orientation of the segment and the electrical field.

Each small voltage induces a current and these currents circulate through a small part of the antenna impedance. The result of all those currents and tensions is far from immediate. However, using the reciprocity theorem, it is possible to prove that the Thévenin equivalent circuit of a receiving antenna is:

$V_a=\left\{sqrt\left\{R_aG_a\right\},lambdacospsioversqrt\left\{pi Z_circE_b$

*$scriptstyle\left\{V_a\right\}$ is the Thévenin equivalent circuit tension.
*$scriptstyle\left\{Z_a\right\}$ is the Thévenin equivalent circuit impedance and is the same as the antenna impedance.
*$scriptstyle\left\{R_a\right\}$ is the series resistive part of the antenna impedance $scriptstyle\left\{Z_a\right\},$.
*$scriptstyle\left\{G_a\right\}$ is the directive gain of the antenna (the same as in emission) in the direction of arrival of electromagnetic waves.
*$scriptstyle\left\{lambda\right\}$ is the wavelength.
*$scriptstyle\left\{E_B\right\}$ is the magnitude of the electrical field of the incoming electromagnetic wave.
*$scriptstyle\left\{psi\right\}$ is the angle of misalignment of the electrical field of the incoming wave with the antenna. For a dipole antenna, the maximum induced voltage is obtained when the electrical field is parallel to the dipole. If this is not the case and they are misaligned by an angle $scriptstyle\left\{psi\right\}$, the induced voltage will be multiplied by $scriptstyle\left\{cospsi\right\}$.
* $scriptstyle\left\{Z_circ=sqrt$mu_circ over varepsilon_circ= 376.730313461 Omega} is a universal constant called vacuum impedance.

The equivalent circuit and the formula at right are valid for any type of antenna. It can be as well a dipole antenna, a magnetic loop, a parabolic antenna, or an antenna array.

From this formula, it is easy to prove the following definitions:

:: Antenna effective length$= displaystyle${sqrt{R_aG_a}lambdacospsioversqrt{pi Z_circ ,is the length which, multiplied by the electrical field of the received wave, give the voltage of the Thévenin equivalent antenna circuit.

:: Maximum available power$=displaystyle$G_alambda^2over 4pi Z_circ}E_b^2} ,is the maximum power that an antenna can extract from the incoming electromagnetic wave.

:: Cross section or effective capture surface$= displaystyle$G_aover4pi}lambda^2} ,is the surface which multiplied by the power per unit surface of the incoming wave, gives the maximum available power.

The maximum power that an antenna can extract from the electromagnetic field depends only on the gain of the antenna and the squared wavelength $scriptstyle\left\{lambda\right\}$. It does not depend on the antenna dimensions.

Using the equivalent circuit, it can be shown that the maximum power is absorbed by the antenna when it is terminated with a load matched to the antenna input impedance. This also implies that under matched conditions, the amount of power re-radiated by the receiving antenna is equal to that absorbed.

References

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