INTERMED RUSSIAN SLAV 5
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Date Created: 10/22/15
Notes 5 ECE594I Fall 2008 ER Brown Free Space Power Coupling for Two Special Cases Radar and Radiometry Friis39 Transmission Formulation Marconi was the pioneer for a new generation of electrical engineers working in the area of wireless One of the truly brilliant amongst these was Friis working at Bell Laboratories in the 1920s and 30s Among other things Friis was the rst to take advantage of the inherent nature of antennas as passive reciprocal components and treat the freespace propagation between a transmit antenna and receive antenna as a twoport link This was done rst and foremost for the wireless communications link which we review here rst to set the stage for the following RF and THz sensor ie radar and radiometer link formulation The rst step in Friis formulation is the concept of an effective aperture Aeff for the receiving antenna Pm 142731quot 9 1h 39 8p where Prec is the power available to the antenna for delivery to a load SW 9 l is the average Poynting vector for incoming radiation along the direction 9pm in the spherical coordinates centered at the receiving antenna and Sp is the polarization coupling ef ciency Note that this expression applies only when S W 9 r is aligned with the direction of the beam pattern maximum When there is misalignment another factor is required which is the just the receive beampattern Pm Ag F9 Smt9n 39gp 1 2 Next we suppose that this received Poynting vector is generated by a second transmitting antenna We can relate the received power to the properties of the transmitting antenna by Smt9n E SU9 i 2 rr E Pmle 39Fl62 i TU 47rr PmGl 39F262 i 47239r2 3 Notes 5 ECE594I Fall 2008 ER Brown where the subscript quottquot is for transmitting Pmd is the total radiated power Pin is the power used to drive the transmitting antenna in the matched case equal to Prad 91 and 11 are the spherical angles in the spherical coordinate system centered at the transmitting antenna 139 r is the power transmission function including all attenuation effects and r is the distance ie the range between transmitter and receiver In writing 3 it is understood that F1 is taken in the direction 9141 pointing towards the receiver which is not necessarily the direction of the maximum of Ft Substitution of 3 into 2 yields the relationship P z Aw PmG E6 F6 r2 139 r g 4 4m p This can be simplified further in terms of the ostensibly known parameters of the receiving antenna using the relationships G G out PrecirEP r P Dr rec 4nAeff m2 5 where Pam is the power delivered to the load of the receiving antenna Substitution of 4 into 5 yields 2 P B 66 F 6 1 F 6 rm 8 6 the expression commonly known as Friis formula after its originator It effectively treats the antenna combination like a twoport circuit with the pattern angular dependence and polarization dependence included explicitly The term M 411r2 is called the freespace loss factor which is of considerable practical and historical importance Several theoreticians of the 19Lh century believed that radiation would decay faster than lr2 from a source It was Hertz s observation of this lr2 dependence of radiation that encouraged the technology of quotwirelessquot 60 Notes 5 ECE594I Fall 2008 ER Brown Friis transmission for Radar For radar systems the transmitter in systems engineering often shortened to TX and the receiver often shortened to RX have in addition to free space a body between them ie the radar quottargetquot that scatters electromagnetic radiation from the TX to the RX To rst order some bodies particularly round metallic ones absorb practically none of the incident power and instead scatter it isotropically Conceptually we can then think of the body as a passive RXTX combination that receives a power according to 1 above and transmits it isotropically so that Pine Aeff l M l5 cl mab and 7 lta gtl PM Mm 8 where 6 is the target scattering cross section and r is the distance between the scatterer and the observation point We now assume that Sim originates from a TX antenna and Sscm radiates back to a second RX antenna to create an echo of receivedaperture power Prec In this case l 9 l W 7a and 9 472732 Prec AeffFr era rl gscatteal l 8p 10 where r1 is the distance between TX and the scatterer and Sp is the fraction of the scattered power that has the same polarization characteristics as the RX antenna As in 5 above we assume to know the RX properties so that G G Pout PrecirEP r 11 Dr rec 2 47TAeff N 61 Notes 5 ECE594I Fall 2008 ER Brown where Pam is the power delivered from the RX antenna to its load By substitution of 9 into 7 7 into 8 8 into 10 and 10 into 11 we nd the relation 2 PM T007003 O KWGrGtFl6r9 rFl6H tgp 12 1 2 This is the famous quotbistaticquot two stationary point radar transmission equation In the special quotmonostaticquot case that the transmitter and receiver share a common antenna r1 r2 Gr Gt Fr Ft 1r1 39crz 1r and 12 reduces to i 4727 PM more j 62 F6 12 81 13 4727 Like Friis formula for communications this treats the radar problem like a twoport equivalent circuit But physically it differs from Friis with the additional r392 factor leading to an overall r394 dependence of PM on Pin This result is of great practical importance because it generally means that radar systems must transmit much higher power levels than communications systems to achieve a satisfactory received power for signal processing 62 Notes 5 ECE594I Fall 2008 ER Brown Example One application for THz radar systems is shortrange concealed object detection and imaging This example calculates a the received power and b the backgroundlimited signal tonoise ratio SNR for a 600 GHz bistatic coherent radar located 1 m from the target a To get the received power we make the following practical assumptions 1 the transmit and receive feedhorns are located sidebyside in close enough proximity that their antenna patterns in space overlap perfectly 2 the transmit power is Pin 1 mW at 600 GHz 7 05 mm a very small number by microwave radar standards but about all that is available today from affordable solidstate sources at 650 GHz centered at one of the THz windows 3 the antenna gain is 100 20dB easy to achieve from standard gain feedhorns without any other optics 4 the target RC8 is 6 10394 m2 eg barrel ofa hand gun and 5 the polarization is scrambled so that Sp 05 most concealed objects of interest such as guns have complicated shapes so are bound to scramble the incident polarization upon scattering The maximum antenna output power occurs when the peak ofthe antenna pattern is aimed at the target ie F0 10 If we assume zero attenuation between Tx and the scatterer we get Pom 54x103914 W down over 10 orders of magnitude compared to the transmit power And this is the bestcase scenario since in the THz region the atmosphere and the materials concealing the object always attenuate significantly While appearing hopelessly low this level of received power is not unusual in radar systems and remote sensors of all types and can lead to useful detection if the receiver is designed correctly b The backgroundlimited SNR can be calculated assuming a terrestrial environment single spatial mode operation of the receive feedhorn and bandlimited receiver of bandwidth Af In this case the average thermal power and rms power uctuation collected by the receiver is just PN kBTAf where the background temperature is generally N300 K anywhere in the THz region One advantage of a coherent radar over an incoherent one more on this later is that the bandwidth can be made much more narrow just great enough to accommodate the modulation bandwidth of the coherent waveform from the transmitter We will address waveforms more later on but suffice it to say that modulation is generally used sometimes amplitude modulation sometimes phase or frequency modulation to get other information from the radar about the target such as its range Another reason to modulate the transmit power is to help mitigate the effect of other scatterers within the fieldofview of the radar that collectively get called clutter For the present purpose let s assume the receiver bandwidth is 1 MHz The rms noise power is then PN 41x103915 W leading to a backgroundlimited signaltonoise ratio of 13 or 11 dB As we will see later insertion losses and physical noise in the receiver will generally make the backgroundlimited SNR significantly lower than this and clutter makes it worse yet But the backgroundlimited value forms a bestcase scenario this is always good to know in systems engineering The radartransmission equation is often reformulated because the Rx output power is often known relative to a lowerlimit Pain dictated by noise in the receiver electronics or from environmental effects Then the interesting question is the maximum range at which a target of cross section 6 can be measured Rearranging Eq 13 and assuming 139 is independent ofr we find 63 Notes 5 ECE594I Fall 2008 ER Brown Pm all 2 2 14 r E 74 31G Ft9 8p 14 which is one form of the quotradar range equationquot very useful expression for predicting the performance of radar systems under ideal conditions In many practical cases r is limited to a maximum possible value by the fact that Pam is masked by receiver noise as shown next Another important application of 13 is the minimum transmit power required to achieve a certain minimum receiver power Pmin This can be calculated as 71 A 2 2 2 l 6 F6 1 st 15 O39 2 I Pm Pmin 170quot a 4727 In some literature this is called the minimum required transmit power Received Power in Radiometgy Antenna Back Projection Antenna reciprocity is a very important concept in RF sensors since it allows us to think about antennas interchangeably in transmit or receive modes knowing just the radiation pattern and its properties In radiometry we are usually concerned with detecting thermal radiation emitted from a target or environmental radiation re ected from a target at a distance great enough that the angle subtended by the target at the sensor QT may be greater or less than the diffractionlimited beam able 23 An interesting application of the antenna theorem comes in using Planck s radiation law to estimate the portion of the thermal emission from targets received by passive radiometers We start by rewriting the blackbody spectral density lnction this time for just one polarization and one direction hence the 4x reduction compared to that derived earlier since different polarizations constitute different spatial modes A27z39hv3 hv dv 62ehvkBT 1 12 eakg 1 And now our interpretation of A is the antenna effective aperture Application of the diffraction limited antenna theorem yields 64 Notes 5 ECE594I Fall 2008 ER Brown dP 27239 hv E 38ehvkgf 1 We recognize the factor 21193 as the number of spatial modes contained in one hemisphere if decomposed into the diffractionlimited solid angle 93 of the antenna Rarely if ever does a target occupy 211 steradians in space with respect to the sensor And generally the antennas in RF sensors operate with just one spatial mode So to estimate the received power we use reciprocity to back project the diffractionlimited beam to the target recognizing that the number of diffractionlimited solid angles lled by the target is one if QT gt QB but less than one of QT lt 23 Mathematically this leads to the received power spectral density E N ehvkBT 1 where D is the Heaviside unit step function This has the expected behavior that if the angle dP Q I lV N STEOlga QT QT QB B subtended by the target is larger than the beam solid angle and in the RayleighJeans limit E kBT dv Sometimes this is called the overfilled condition But if the target angle is smaller than the beam solid angle then the received power is less than expected from thermodynamic equipartition by the spatial fill factor SETQB ampaampm e 8 1 QB 65
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