antennas transmission lines and waveguides are devices...
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Antennas
Transmission lines and waveguides are devices used to transmitsignals in the form of guided electromagnetic waves from a source(generator) to a load. Antennas may be used to transmit signals from asource to a load in the form of directed but unguided waves.
Guided Wave Source-Load Connection
Examples - power transmission lines, telephone lines, cable television
Unguided Wave Source-Load Connection
Examples - broadcast television and radio, radar, wirelesscommunications
Antenna - a device which radiates and/or receives electromagnetic wavesefficiently (antennas also match the transmission line or waveguideto the surrounding medium, or vice versa).
Major Classes of Antennas
Wire Antennas - monopole, dipole, loop, helical
Aperture Antennas - horn, slot, microstrip patch
Reflector Antennas - parabolic dish, corner reflector
Antenna Characteristics
Antenna impedance - an antenna must be matched to the connectingtransmission line or waveguide for efficient radiation.
Radiated power - the amount of power radiated by a transmit antennawill limit the separation distance between the transmit andreceive antennas.
Directivity - the direction in which the antenna radiates the powerwill dictate how the transmit and receive antennas should bepositioned (the radiation pattern of the antenna defines theantenna radiated power as a function of direction).
Efficiency (losses) - the amount of power dissipated by the antennashould be small in comparison to the amount of power radiatedin order to minimize the source power requirements.
Electromagnetic Interference (EMI) - unwanted radiation.
Electromagnetic Compatibility (EMC) - designing to minimize theproblems with EMI.
Fundamentals of Antenna Radiation
Antenna as the termination of a transmission line
The open-circuited transmission line does not radiate effectively becausethe transmission line currents are equal and opposite (and very closetogether). The radiated fields of these currents tend to cancel one another.The current on the arms of the dipole antenna are aligned in the samedirection so that these radiated fields tend to add together making thedipole and efficient radiator.
Antenna as the termination of a waveguide
The open-ended waveguide will radiate, but not as effectively as thewaveguide terminated by the horn antenna. The wave impedance insidethe waveguide does not match that of the surrounding medium creating amismatch at the open end of the waveguide. Thus, a portion of theoutgoing wave is reflected back into the waveguide. The horn antenna actsas a matching network, with a gradual transition in the wave impedancefrom that of the waveguide to that of the surrounding medium. With amatched termination, the reflected wave is minimized and the radiatedfield is maximized.
Antenna Radiation Fields in Termsof Potential Functions
The electromagnetic fields radiated by an antenna can be determinedby a direct solution of Maxwell’s equations. It has been shown previouslythat a direct solution of Maxwell’s equations involves difficultintegrations. In many cases, the antenna radiated fields can be determinedmore easily using potential functions (electric scalar potential - V,magnetic vector potential - A). The sources of the antenna radiation fieldsare time-varying current (J) and charge (v) on the antenna. The antennacurrent and charge are related by the continuity equation:
Potentials are also employed in the solution of electrostatic andmagnetostatic fields produced by static charge and steady current,respectively. These static solutions are simply a special case of the generalcase of time-varying fields.
Electrostatic Fields (electric scalar potentialV )
Magnetostatic Fields (magnetic vector potentialA )
Electromagnetic Fields (both A and V are required)
The electromagnetic fields are found to satisfy wave equations. For time-harmonic fields (time-harmonic currents and charges), we may use phasorsto simplify the analysis.
Note that the time-harmonic wave equation solutions for the electric scalarpotential and the magnetic vector potential reduce to the correspondingstatic solution when k 0 (zero frequency).
The electric scalar potential and magnetic vector potential are relatedby the Lorentz gauge:
For time-harmonic fields, the Lorentz gauge can be written in terms ofphasors as
Solving for the electric scalar potential in terms of the magnetic vectorpotential gives
This equation defines the electric scalar potential in terms of the magneticvector potential and allows for the fields radiated by an antenna to bedefined in terms of the magnetic vector potential only.
Field in terms of Vs and As
Field in terms of As only
Infinitesimal Current Element (Hertzian Dipole)
The infinitesimal current element (infinitesimal dipole) is a basicbuilding block in the analysis of wire antennas. The infinitesimal currentelement is an electrically short current segment (its length l is very shortrelative to wavelength) of constant amplitude current. Note that theinfinitesimal current element is a non-physical antenna since the currentmust go to zero on the ends of an wire antenna. However, we can use theinfinitesimal dipole in the analysis of wire antennas by subdividing thewire antenna into many very short segments of current over which thecurrent amplitude is near constant. The overall fields radiated by the wireantenna can then be determined by properly weighting each currentelement and summing theindividual contributions tothe radiated fields. Weassume that the axial currentalong the infinitesimal dipoleis uniform. We also assumethat the thickness of theinfinitesimal dipole is
negligible such that it is treated as a current filament.
In order to determine the magnetic vector potential of the infinitesimaldipole, we must specialize the general integral for As to the geometry ofthe infinitesimal dipole.
If we assume that the potential is determined in the far field of the antenna(radiation field), then we may assume r >> z.
The expression for the distance R can be simplified further through abinomial series approximation.
Using the binomial series with z << r, the square root in the expression forR may be approximated as
which yields
The magnetic vector potential integral for the infinitesimal dipole thenbecomes
Note that the approximation for the distance R appears as a magnitudeterm (in the integrand denominator) and a phase term (in the complexexponential). For the magnitude term, the approximation for R can bereduced further to R r given z >> r. However, this reduction cannot beutilized in the phase term, since e-jkzcos may still contribute a phase shiftwhich is still significant even though z >> r. Using this approximation,the magnetic vector potential may be written
Evaluation of the integral in the expression for As gives
The resulting magnetic vector potential for the infinitesimal dipole is
Given that the radiated far fields of the infinitesimal dipole should exhibitspherical symmetry, we should define As in terms of spherical coordinates(transform the az unit vector into spherical coordinates).
The magnetic field radiated by the infinitesimal dipole is found bydifferentiating the magnetic vector potential according to
The first and fourth terms in the curl of As are zero since As = 0 while thesecond and third terms are zero since Ars and As are both independent of. Evaluation of the non-zero terms yields
The magnetic field of the infinitesimal dipole is
There are three different techniques that we can use to find the electricfield:
Es in terms of Vs and As (requires both Vs and As).
Es in terms of As (requires only As but with a complicateddifferentiation).
Es from the source free Maxwell’s equations (requires only Hs).
Using option ,
Since the magnetic field of the infinitesimal dipole contains only an Hs
component, the derivatives involving Hrs and Hs are zero.
Note that the electric and magnetic fields of the infinitesimal dipoleinclude terms that vary as r1, r2 and r3. The fields close to the antenna(near field) are dominated by the terms that vary as r3 while the fields farfrom the antenna (far field) are dominated by the terms that vary as r1.
Infinitesimal Dipole Near Field
Infinitesimal Dipole Far Field
Note the plane wave -like relationship for the far field electric andmagnetic fields of the infinitesimal dipole. The electric and magneticfields are orthogonal and transverse to the direction of propagation of theoutward traveling waves [given by E × H (a × a = ar)].
Antenna Radiated Power and Radiation Resistance
The total power radiated by a given antenna can be determined if thepower density of the radiated fields is known. The time average vectorpower density (avg) for any electromagnetic field is given by
where S defines the phasor Poynting vector. This time average vectorpower density not only gives the power density at a point but also thedirection of power flow. In order to determine the total power radiated byan antenna (Prad), we simply integrate the power density over anyconvenient surface (S) that encloses the antenna. The most convenientchoice is usually a spherical surface of radius r located in the far field ofthe antenna.
The time average vector power density onthis spherical surface in the far field of theantenna will vary as 1/ro
2 (Es and Hs varyas 1/r).
The antenna radiation resistance is a parameter that provides a wayof comparing the total radiated power of one antenna to another. If theantenna terminal current is defined as Is, the radiation resistance for theantenna is the resistance that would dissipate the same time average poweras that radiated by the antenna, given a resistor current of Is. Thus, if theradiation resistance of antenna A is larger than that of antenna B, thenantenna A radiates more power than antenna B given the same terminalcurrent for both antennas.
The total radiated power of the antenna can be defined in terms of rms orpeak current at the antenna terminals.
As an example, the radiation resistance of the infinitesimal dipolemay be determined given the previously determined expressions for the farfields of this antenna. Note that the antenna far fields are dependent on theconstitutive parameters of the medium surrounding the antenna (,). Thefar fields of the infinitesimal dipole (centered at the coordinate origin andaligned along the z axis) in air are
The resulting time average power density is
Note that the time average power density radiated by the antenna varies as1/r2 in the far field. This characteristic is common to all antennas.
The total radiated power for the infinitesimal dipole is
The radiation resistance of the infinitesimal dipole is
Given that the infinitesimal dipole model over estimates the current on anactual electrically short antenna, the value of radiation resistance obtainedfor the infinitesimal dipole is larger than that of a equivalent lengthantenna. For l = 0.1, the infinitesimal dipole radiation resistance(which is larger than a dipole of the same length) is 7.9 . Compared tolonger dipoles, this is a very small value of radiation resistance. Thus, anelectrically short dipole is not an efficient radiator.
Dipole Antennas
A dipole antenna can be formed by bending apart the conductors ofan open circuited two-wire transmission line. The alignment of the currenton the two arms of the dipole antenna enhances the radiation propertiesover that of the two-wire line which has closely spaced conductors carryingequal currents in opposite directions.
The current on a center-fed (feed point at z = 0) dipole antenna of lengthl can be defined by
The electrically short dipole current distribution looks basicallytriangular in shape. This is caused by the fact that the dipole current,which is assumed to follow the sinusoidal shape defined in the dipolecurrent distribution, is defined by a sine function of small argument, wherethe sine function is almost linear. The current at the feed point of theantenna reaches a maximum for an overall antenna length of /2 (half-wave dipole). Increasing the length of the dipole alters the currentdistribution according to the sine function characteristics. The dipole oflength l = (full-wave dipole) has a very high input impedance(theoretically infinite) since the current goes to zero at the feed point. Theradiation characteristics of the dipole vary with the length of the dipole
since the current distribution varies with the antenna length.
Dipole Far Field
The far field of the infinitesimal dipole may be used to determine thefar field of a dipole antenna of arbitrary length l. The dipole antenna isassumed to be oriented along the z axis and centered at the coordinateorigin. Each differential element of current on the dipole can be modeledas an infinitesimal dipole such that
The incremental far field electric field (dEs) due to the differential elementof current Is(z)dz is found by replacing the Iol term in the infinitesimaldipole expression by Is(z)dz .
Inserting the approximation for R gives
where the zcos term is included in the phase term but not in themagnitude term. The overall electric field of the dipole is found bysumming the contributions from all the differential lengths (integrating)which make up the dipole.
The result of the integration is
where F() is the pattern function defined by
The pattern function shows how the antenna radiates as a function of theangles and (the dipole far field is independent of ). The far field ofthe dipole antenna exhibits the same plane wave characteristic as theinfinitesimal dipole such that
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For the special case of a half wave dipole in air,
The power density in the half-wave dipole far field in air is
The total radiated power by the half-wave dipole is
The half-wave dipole is found to have a much higher radiation resistancethan that of the electrically short dipole (which was approximated by theinfinitesimal dipole) and thus is a much better radiator.
Monopole
A monopole antenna is another simple and efficient wire antennawhich is formed by driving a wire with a voltage between the wire and aconducting ground plane. Using image theory, the monopole antenna overa perfectly conducting ground plane may be shown to be equivalent to adipole antenna in a homogeneous region. The equivalent dipole is twicethe length of the monopole and is driven with twice the antenna sourcevoltage (a quarter-wave monopole is equivalent to a half-wave dipole).These equivalent antennas generate the same fields in the region above theground plane.
Antenna Impedance
The input impedance of an antenna is defined just like any othernetwork by taking the ratio of phasor voltage to current at the antennainput terminals. The resulting complex antenna impedance (ZA) consistsof a real component [the antenna resistance (RA)] and an imaginarycomponent [the antenna reactance (XA)]. The resistance of the antenna isassociated with power “loss” in the form of (1) antenna radiation [antennaradiation resistance (Rrad)] and (2) ohmic losses in the antenna [antennaloss resistance (Rloss.)]. The reactance of the antenna is associated withenergy storage in the near field of the antenna.
Power radiatedby the antenna
The complex power (Ss) associated with the antenna is defined by
where P is the real antenna power and Q is the reactive antenna power.The antenna complex power can be written is terms of the antennaimpedance as
The antenna radiation efficiency (r) is defined as the percentage ofthe real input power that is radiated by the antenna.
Power stored inthe near field of
the antenna
Power dissipatedby the antenna
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XA = 42.5 Ω
XA = 0 Ω (resonant dipole)
Rrad = 73 Ω
Rrad = 72 Ω
The input impedance of a very thin, lossless dipole is shown belowas a function of the antenna length (Rloss=0, ZA=Rrad+ jXA). The impedanceof the half-wave dipole is approximately ZA=(73+j42.5) [a significantreactance for matching purposes]. However, the antenna reactance variesrapidly with length in the vicinity of l= 0.5. The dipole antenna can bemade resonant (XA= 0) by trimming the antenna to a length slightly lessthan 0.5. The impedance of the resonant thin, lossless dipole isapproximately ZA=(72+j0) . The resonant antenna has much bettermatching characteristics when connected to a 75 transmission line thanthe l= 0.5 dipole. In practice, the resonant length of the dipole is actuallya function of the antenna conductor radius. In general, thicker conductorsrequire shorter lengths to obtain a resonant dipole. The range of lengthsencountered for typical resonant dipoles ranges from approximately 0.45to 0.49.
The characteristics of the quarter-wave monopole impedance can bedetermined using the image theory equivalent dipole.
The impedances of the monopole and equivalent dipole are given by
or
Thus, the input impedance of the monopole is exactly one-half that of theequivalent dipole (this is true for any length l). The input impedance of thethin, lossless quarter wave monopole is then
We see from this equation that the radiation resistance of the quarter-wavemonopole is 36.5. The monopole can also be made resonant by reducingthe length to slightly less than one-quarter wavelength.
Small Loop Antenna(Magnetic dipole)
The small loop antenna and the infinitesimal dipole are dual sources.For dual antennas, the electric field of one antenna has the samemathematical form as the magnetic field of its dual antenna, and viceverse. The small loop of radius o is assumed to be centered at thecoordinate origin and lie in the x-y plane. The small loop is assumed tocarry a uniform current Io. The assumption of a near-uniform current in anactual loop antenna is valid up to a loop circumference of approximately0.2.
Evaluation of the magnetic vector potential integral for the small loopyields
Small loop antenna (o 0.1/) Infinitesimal dipole (l 0.1)
Given the magnetic vector potential for the small loop, the electricand magnetic fields are found using the following equations.
The terms in the radiated field expressions which vary as 1/r represent thesmall loop far field.
The far fields of the small loop antenna yield plane wave characteristicsjust like the dipole antenna.
The power density in the small loop antenna far field in air is
The total power radiated by the small loop is
The electrically small loop, just like the short dipole, is not an efficientradiator. For the maximum small loop radius of o= 0.1/, the radiationresistance is Rrad= 0.316.
Antenna Patterns
The time-average power density in the far field radiated by anantenna has been shown to be proportional to the square of the electricfield or magnetic field.
The radiation characteristics of a given antenna can be illustrated conciselythrough simple plots known as antenna patterns.
Antenna Pattern (radiation pattern) - a plot of the antenna radiationcharacteristics [typically normalized to the maximum value].
Field Pattern - a plot of the radiated electric or magnetic fieldmagnitude at a constant radius.
Power Pattern - a plot of the radiated antenna power at aconstant radius.
E-Plane Pattern - a pattern plot containing the direction ofmaximum radiation and the electric field vector.
H-Plane Pattern - a pattern plot containing the direction ofmaximum radiation and the magnetic field vector.
The antenna radiation parameters (field or power) exhibit sphericalsymmetry in the far field where the antenna approximates a point source.The E-plane and H-plane are always perpendicular in the far field of theantenna. Given that all antenna pattern parameters are plotted at a constantradius, the spherical coordinates of interest are the angles (elevationangle) and (azimuth angle).
Example (E-plane and H-plane patterns of the infinitesimal dipole)
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Alternatively, the overall radiation pattern can be shown as a three-dimensional plot verses and . For both the electric field and magneticfields, the pattern function is
The resulting three-dimensional electric or magnetic field pattern plot forthe infinitesimal dipole has the shape of a doughnut formed by taking thepreviously shown -dependent plot and rotating it through 0o to 360o in .
The power pattern functions for the infinitesimal dipole in the E-plane and H-plane are
E-Plane Pattern Functionf()=|sin|
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H-Plane Pattern Functionf()=1
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In order to quantify how efficiently an antenna radiates in a givendirection, the antenna radiation pattern is compared to that of an isotropicradiator.
Isotropic radiator - an ideal (point source) antenna which radiatesequally in all directions (produces an isotropic pattern).
Omnidirectional pattern - a radiation pattern that is rotationallysymmetric about an axis (the infinitesimal dipole has anomnidirectional pattern).
Radiation intensity [U(,)] - radiated power normalized to a unitsphere.
The total power radiated by any antenna can be found by integratingthe vector power density over a surface enclosing the antenna such that
H-Plane Power Pattern Functionf()=1
E-Plane Power Pattern Functionf()=sin2
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The electric and magnetic far fields vary as 1/r and the direction of thevector power density (avg), which varies as 1/r2, is radially outward.Integrating the power density over a spherical surface of radius r gives
If we defined avgr2 = U(,) as the radiation intensity, then
where d = sindd defines the differential solid angle (integration overthe unit sphere). The units on the radiation intensity are defined as wattsper unit solid angle (W/rad2). The average radiation intensity is found bydividing the radiation intensity by the area of the unit sphere (4) whichgives
Given an antenna radiating a total power of Prad , the average radiationintensity of the antenna represents the radiation intensity of an isotropicradiator radiating Prad.
The directive gain [Gd(,)]of the antenna is a measure of the abilityof the antenna to concentrate radiated power in a given direction. Thedirective gain is defined as the ratio of antenna radiation intensity functionto that of an isotropic radiator.
For an isotropic radiator (a nonphysical antenna),
The directive gain of a physical antenna is always a function of angle suchthat the antenna radiates more efficiently in one direction over another.The maximum directive gain of a given antenna is defined as thedirectivity [D] of the antenna.
Antenna gain and directivity are normally defined in units of dB.
The power gain of the antenna [Gp(,)], which accounts for the antennalosses, is defined as
where Pin is the power delivered to the antenna terminals. The input poweris defined in terms of the power radiated by the antenna and the powerdissipated due to ohmic losses.
The power gain of a realistic antenna is always less than the directive gaindue to losses.
Antenna Pattern Parameters
Previously determined by integrating the timeaverage power density over a spherical
surface enclosing the antenna.
Example (Antenna Characteristics)
Determine the following values for an infinitesimal dipole: (a.)radiation intensity (b.) directive gain (c.) directivity.
From these results, we see that the infinitesimal dipole lying along thez-axis produces a power density in the x-y plane that is 1.5 times larger thatproduced by an isotropic radiator when both antennas radiate the sametotal power. For a half-wave dipole, the directivity is 1.64 (2.15 dB).
Antenna Arrays
An antenna array is a configuration of multiple antennas. Mostantenna arrays consist of identical antennas. The antennas of an array(array elements) are positioned in a particular geometry in order to achievespecific radiation characteristics for the overall array pattern.
Linear Array - array elements lie along a straight line.
Planar Array - array elements are spread over a planar surface.
Conformal Array - array elements conform to some non-planarsurface (such as an aircraft fuselage).
Phased Array - the main beam of the stationary array can be steeredelectronically by changing the phases of the individual arrayelements.
The basis for the design of an antenna array is the patternmultiplication theorem. Assuming that the array consists of identicalelements, the pattern multiplication theorem states that the overall patternof an array is equal to the product of the array element pattern times afunction known as the array factor (AF). The array factor is a function ofthe geometry of the array (the position of the elements) and the excitationof the elements, but not a function of the type of antennas that constitutethe array.
Pattern Multiplication Theorem
Given the array factor for a particular geometry and element excitation, theoverall array pattern for different types of antennas can be determined bysimply inserting the particular element pattern into the pattern
multiplication theorem.Two Element Array of Infinitesimal Dipoles
The pattern multiplication theorem can be illustrated by analyzing thefields radiated by a two element array of infinitesimal arrays. The twovertical dipoles, which are separated by a distance d, are assumed to liealong the z-axis. The overall fields radiated by the two element array arethe superposition of the fields radiated by the individual elements. Thecurrents on the two elements are assumed to be Is1 = Io0 and Is1 = Io.
In the far field, the lines defining r, r1 and r2 become almost parallel so that
Element Pattern(infinitesimal dipole at the origin)
Array Factor (AF)
The array factor can be simplified in form according to the following.
Array factor for any two elementarray with this geometry and
current excitation
General N-Element Linear Array
The array factor AF is independent of the antenna type assuming allof the elements are identical. Thus, isotropic radiators may be utilized inthe derivation of the general N-element array factor to simplify the algebra.The field of an isotropic radiator located at the origin may be written as(assuming -polarization)
We assume that the elements of the array are uniformly-spaced with aseparation distance d.
In the far field of the array
The current magnitude on each array element is assumed to be Io while thecurrent phase follows a linear progression from element to element. Thephase shift between adjacent elements is assumed to be while the arrayelement located at the origin is used as the phase reference (zero phase).
The far fields of the individual array elements are
The overall array far field is found using superposition.
The array factor for the N-element linear array is
The function is defined as the array phase function and is a function ofthe element spacing, phase shift, frequency and elevation angle. If the
array factor is multiplied by e j, the result is
Subtracting the array factor from the equation above gives
The complex exponential term in the last expression of the above equationrepresents the phase shift of the array phase center relative to the origin.If the position of the array is shifted so that the center of the array islocated at the origin, this phase term goes to zero. The array factor thenbecomes
The characteristics of the array radiation pattern (the product of the arrayfactor and the element pattern) can be altered by changing the elementspacing and phasing.
Array factor for a linear N-elementarray centered at the origin(linear phase progression)
Below are plots of the array factor AF vs. the array phase function as thenumber of elements in the array is increased. Note that these are notplots of AF vs. the elevation angle .
Some general characteristics of the array factor AF with respect to :(1) [AF ]max = N at = 0 (main lobe).(2) Total number of lobes = N1 (one main lobe, N2 sidelobes).(3) Main lobe width = 4/N, minor lobe width = 2/N
The array factor may be normalized so that the maximum value for anyvalue of N is unity. The normalized array factor is
The nulls of the array function are found by determining the zeros of thenumerator term where the denominator is not simultaneously zero.
The peaks of the array function are found by determining the zeros of thenumerator term where the denominator is simultaneously zero.
The m = 0 term,
represents the angle which makes = 0 (main lobe).The phasing of the uniform linear array elements may be chosen such
that the main lobe of the array factor lies along the array axis (end-firearray) or normal to the array axis (broadside array).
End-fire array main lobe at = 0o or = 180o
Broadside array main lobe at = 90o
The maximum of the array factor occurs when the array phase function iszero.
For a broadside array, in order for the above equation to be satisfied with = 90o, the phase angle must be zero. In other words, all elements of thearray must be driven with the same phase. With = 0o, the normalizedarray factor reduces to
End-fire arrays may be designed to focus the main beam of the arrayfactor along the array axis in either the =0o or =180o directions. Giventhat the maximum of the array factor occurs when
in order for the above equation to be satisfied with = 0o, the phase angle must be
For = 180o, the phase angle must be
which gives
The normalized array factor for an end-fire array reduces to
Normalized array factor for anN-element broadside array
Normalized array factor for anN-element end-fire array
Example (Array Multiplication Theorem)
Plot the electric field array pattern (in the x-z plane) of a two elementbroadside array of horizontal infinitesimal dipoles lying parallel to the x-axis. Assume the array element spacing is d = /2.
Element Pattern:
Vertical infinitesimal dipole Horizontal infinitesimal dipole
f () = |sin | f () = |cos |
Array Factor:
N =2, = 0o (broadside array), d = /2
Array Pattern:
Array Pattern = |Element Pattern | × |Array Factor|
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Antenna Effective Area
The effective area of an antenna is a measure of the ability of the(receiving) antenna to extract energy from a passing electromagnetic wave.
VAs open circuit voltage induced at the antenna terminals
ZA = Rrad + jXA Antenna impedance(assuming losses are negligible, Rloss = 0)
From circuit theory, maximum power transfer occurs when
Assuming maximum power transfer, the Thevenin equivalent circuit forthe receiving system reduces to the following.
The time average power delivered by the receive antenna to its terminals(received power) is
The antenna effective area (Ae) is defined as the ratio of the poweravailable at the antenna terminals to the power density of the incidentelectromagnetic wave.
For the special case of an infinitesimal dipole located in air, orientedfor maximum response (incident electric field parallel to the dipole), theopen-circuit antenna voltage is found by integrating the electric field of theincident wave along the antenna which gives
The power received by the infinitesimal dipole is then
The average power in the incident wave can be written in terms of theincident electric field as
The effective area of the infinitesimal dipole is
The relationship above holds true for all antennas such that
defines the maximum effective area for the given antenna in terms ofdirectivity. We can replace the directivity value (D) by the directive gainfunction [Gd(,)] to define the effective area of the antenna for anyorientation.
Infinitesimal dipole directivity
Friis Transmission Formula
The Friis transmission formula describes the fundamental antennaperformance in a transmit/receive system. A simplified model of atransmit/receive system is shown below.
Pt Time-average Pt Time-averagetransmitted power received power
Gdt Transmit antenna Gdr Transmit antennadirective gain directive gain
If the transmit antenna were an isotropic radiator, the time-average powerdensity at the receive antenna (a distance r from the transmit antenna)would be
as the transmitted power Pt is spread uniformly over the spherical surfaceof area 4r2. The directive gain of the transmit antenna defines theantenna radiation characteristics relative to an isotropic radiator. Thus, thetransmit antenna produces a power density at the receive antenna that issimply a product of the isotropic radiator power density and the antennadirective gain.
The total power available at the receive antenna terminals (Pr) is theproduct of the power density at the receive antenna and the effective areaof the receive antenna (Aer).
The resulting relationship between the transmitted power and receivedpower defines the Friis transmission formula.
It should be noted that the directive gains of the transmit and receiveantennas are functions of angle. Thus, the Friis transmission formula canaccount for polarization loss between two antennas that are not aligned formaximum gain.
Example (Friis transmission formula)
Determine the maximum power that can be delivered to a receiveantenna with a gain of 30 dB located 1.5 km from the transmit antennawith a gain of 25 dB that transmits 200 W at 1.5 GHz.
(Friis Transmission Formula)
Radar
Radar stands for radio detection and ranging. Radar employs atransmit and receive system to detect and locate a distant object. The radartransmits an electromagnetic wave toward the target which is reflected bythe target and received by the radar.
Monostatic radar - the transmitter and receiver are located in thesame position.
Bistatic radar - the transmitter and receiver are positioned atseparate locations.
Radar cross section () - the radar cross section (RCS) of a scattereris the cross-sectional area of an isotropic scatterer that producesthe same power density at the receiver as does the actual target.
i - time-average power density of the incident field at the targets - time-average power density of the scattered field at the receiver
The total power incident on the surface S, defined by Pi , is
If Pi is then scattered isotropically, the scattered power density at thereceiver is
Solving for the target RCS yields
Given the radar cross section of the target, the so-called radar equationcan be defined.
Assuming the radar radiates a total power of Prad , the incident powerdensity at the target is
where r1 is the distance from the transmit antenna to the target. Theresulting scattered power density at the receiver is
where r2 is the distance from the target to the receiver. The total receivedpower is
The radar equation defines the received power in terms of the radiatedpower.
For a monostatic radar, r1 = r2 = r. Solving for the distance r yields the so-called radar range equation for a monostatic radar.
The corresponding radar range equation for a bistatic radar can be foundby solving for the r1r2 product.
(Radar equation)
(Radar range equation - monostatic radar)