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Microw aves Microwave s in the Electromagne tic Spectrum (300 MHz - 300 GHz) ELF Extremely Low Frequency 3-30 Hz SLF Super Low Frequency 30-300 Hz ULF Ultra Low Frequency 300 Hz - 3 kHz VLF Very Low Frequency 3 kHz - 30 kHz LF Low Frequency 30 kHz - 300 kHz MF Medium Frequency 300 kHz - 3 MHz HF High Frequency 3 MHz - 30 MHz VHF Very High Frequency 30 MHz - 300 MHz UHF Ultra High Frequency 300 MHz - 3 GHz (decimeter waves) SHF Super High Frequency 3 GHz - 30 GHz (centimeter waves) EHF Extremely High Frequency 30 GHz - 300 GHz (millimeter waves) ? (submillimeter waves) 300 GHz - 3000 GHz IR Infared 3000 GHz - 416,000 GHz Microwave P roperties High bandwidth - The microwave frequency range (300 MHz - 300 GHz) is 999 times that of the entire frequency range below it. Effect of the ionosphere - When lower frequency w aves ar e directed upwar d into the atmosphere, they experience significant reflection due to the ionosphere. The lower frequency w aves which pa ss through the ionos phere suffer distortion. Microwaves pass through the ionos phere with little effect and are therefore uti lized i n sat ellite commu nications and space transmi ssions. Line-of-sight transmission/reception - The microwave receive antenna mu st be w ithin the line-of-sight of the transmi t antenna. Long distance communication on earth requires that microwave relay stations be used. Electromagnetic no ise characteristics - The electromagnetic noise level in nature over the 1-10 GHz frequency range is small. This allows for the detection o f very low signal levels using sensi tive receivers. Antenna gain and directivity - The gain of an antenna is directly propor tional to its electrical size. The beamwidth of an antenna is inversely propor tional to the electrical size of its maxi mum dimensi on. Sh orter wavelengths at microwave fr equencies allow for smaller antennas. At higher frequencies (visible light- lasers), the beamwidth gets very small and pointing accuracy of the de tector becomes a problem. Target reflection of electromagnetic waves (radar cross section) - In general, elec trically larg e conducting radar targets reflect m ore energy (shape is also a fact or - stealth design). Thus, the higher frequencies of microwaves are preferred for radar systems. At millimeter waves, the wavelength becomes comparable to the size of raindrops which results in attenuation of the incident waves. Absorption at resonant frequen cies - v arious materials a bsorb microwave energy (dissipated in the form of heat) at specific resonan t frequencies. Applications of M icrowaves Wireless communications Personal Comm unications Systems (PCS) (pagers, cell phones, etc.) Global Posi tioning Sat ellite (GPS) Systems Wireless Loc al Area Comput er Networks (WLANS) Direct Broadcast Satellite (DBS) Television Telephone Microwave/Satellite Links, etc. Remote sensing Radar (active remot e sensing - radiate and re ceive) Military applications (target tracking) Weather radar Ground Penetrating Radar (GPR) Agricultural applications Radiometry (passive remote sensing - receive inherent emissions) Radio astronomy Industrial and home app lications Cooki ng, drying, heating Microwave spectroscopy - m olecular properties of materials can b e determined by passing m icrowaves through a sample of the material and measuring the absorption spectrum. Analysis T echniques in Microwave Theory In general, circuit theory is not applicable to mi crowave prob lems. Circuit th eory is derived from Maxwell’s equations based on certain assumptions about the fields within the circuit elements. Specifically, the circuit elements must be small relative to wave length for circuit equations to be valid. In this sense, microwave compone nts must be mo deled by distributed elem ents, not lumped element s. For this reason, we must use field theory sol utions (Maxwell’s equations) for microwave app lications. Maxwell’s Equat ions Maxwell’s Equat ions ( instantaneous, symmetric form) (Faraday ’s law) (Ampere’s law) (Gauss’ law - electric fields) (Gauss’ law - magne tic fields) E, H, D, B, J, M - instantaneo us vectors [E =E(x,y,z,t), etc.] mñ, ñ - instantaneo us scal ars [ñ = ñ(x,y,z,t), etc.] E - electric field intensity (V/m) H - mag netic field intensity (A/m) D - electric flux density (C/m ) B - magne tic flux density (Wb/m )22 J - electric current de nsity (A /m) M - magne tic current de nsity (V /m)22 mñ - electric charge density (C/m ) ñ - magne tic charge density (Wb/m )33 The quantities of mag netic cur rent density M and magnetic charge density mñ are nonphysical and included in the symmetric forms of Maxwell’s equation for mathematical conve nience. The se magnetic sources may be used t o simplify the mat hemat ics of particular problems i nvolving actual electric cur rents and char ges. The fl ux and field quantities are related by the con stitutive relations: D = åE B = ìH where å is the permittivity (F/m) and ì is the permeability (H/m) of t he medium in which the fields are located. The permittivity and permeability of a given medium ma y be de fined i n terms of the free space (vacuum) oovalues [ å = 8.854×10 F/m, ì = 4 ð×10 H/m] and u nitless relative!12 !7 rrvalues ( ì,å) such that ro ro å = åå ì = ììBM E D J H D B The instantaneous Maxwell’s equations a re valid given any type of time-dependence for the electromagnetic fields. Most applications in microwave engineering involve fields which have a sinusoidal (harmonic) time-dependence. This harmonic time-dependence al lows us t o simplify Maxwell’s equations by w riting them i n terms of phasors just like w e use in circuit anal ysis. For time-harmoni c fields, we may separat e the de pendenc e on time and space. The real-valued i nstantaneous electric field E(x,y,z,t) may be written as ~~~~~~~~~~ Re al vector [magnitude/direction] Ewhere a is a unit vector in the direction o f the vect or electric field. The arbitrary phase shift ö allows us to use the cosine function to represent any sinusoidal time var iation relative to time t = 0. According to Euler’s identity, we may write the equ ation above as ~~~~~~~ Complex vector (phasor vector) [magnitude/phase/direction] We m ay write all v ector quantities in the instantaneous M axwell’s equations in terms of pha sors according to the relationship above . The derivatives with respect to time in the instantaneous equations yield jù terms in the phasor e quations.E E Maxwell’s Equat ions ( phas or form) E, H, D, B, J, M - phasor vect ors m ñ, ñ - phasor sca lars Relation of instantaneous quantities to phasor quantities ... E(x,y,z,t) = Re{E(x,y,z)e}, etc.jùt Complex Permittivity and Permeability In order to account for dielec tric an d magnetic losses in media where time-harmonic electromagnetic fields exi st, we may define a comp lex permittivity and permeabi lity. In the case of dielectrics, we may combine the condu ctivity losses with the dielectric losses according to Maxwell’s equations. The conduction current density J in a g iven medi um is defi ned by where ó is the conduct ivity of t he m edium i n S/m (É/m). We may write a single equation whi ch includes dielectric and condu ctor losses by incorporating the complex permittivity and the condu ction current equation into Am pere’s law. ~~~~~~ ~ ~~~~~~~~~ Displacement co nductor +dielectric current losses The ratio of the overall condu ctor and dielectric losses to the displacement current is defi ned as the loss tange nt [tan ä] since it is related to the tangent of the complex number m ultiplying the electri c field phasor. Material Classifications A given medium is characterize d by its th ree constitutive parameters defined as (ì,å,ó). We may classify medi a according to the char acteristics of the con stitutive paramet ers. Homogeneous - the constitutive parameters of the medium are not functions of position (otherwise - inhomoge neous). Linear - the constitutive parameters of the medium are not functions of the magnitude of the applied field (otherwise - nonl inear). Isotropic - the constitutive parameters of the medium are not functions of t he direction of the app lied fi eld (otherwise - anisotropic). Electromagne tic Field Boundar y Condi tions Knowledge of how the components of an electromagnetic field behave at the interface between two different media is important in the solution of many p roblems in mi crowave eng ineering. A simple interface between two media is sh own below. The vector n is defined as the unit normal to the interface pointing into region 2. The general boundary con ditions are: Note that the individual comp onents of the vect or fields [n @ {} defines the normal components of t he v ector while n × {} d efines t he tangential compone nts of the vector] are discontinuou s at the interface by an amount equal to the respect ive surf ace current or charge on the boundary. In mo st applications, we do not encou nter all of the surf ace sources. These general boundary conditions can be specialized to problems involving specific comb inations of mat erials. Interface Between Two Lossless Dielectric Materials If the two media are lossless dielectrics (perfect insulators defined by 1 2 12 1 2ìO = ìO = åO = åO = ó = ó = 0), then no surface charge or current will occu r naturally. The boundary con ditions then becom e Thus , the nor mal compone nts of electric and m agnetic flux a nd t he tangential comp onents of electric and magn etic field are conti nuous across a lossless dielectric interface. Perfect Electric Conductor (PEC) 1If reg ion 1 is assumed to be a perfect electric co nductor (ó 64) while region 2 is a dielectric, no electromagnetic field can penetrate into region 111 (E = H = 0) . Electric surface currents and charges are found on the PEC (no magn etic char ge or current) which gives Note that the tangential electric field is always zero on the surface of a PEC. The tangential magnetic field on a PEC i s equal to the surface current while the norma l electric flux is equal to the surf ace char ge. Perfect Magnetic Conductor (PMC) If region 1 is assumed to be a perfect magnetic conductor (its m1equivalent magn etic cond uctivity ó 64) while region 2 is a dielectric, no 11electromagnetic field can penetrate into region 1 (E = H = 0). Magnetic surface currents and charges are found on the PMC (no electric charge or current) which gives Note that the tangential magn etic field is always zero on the surf ace of a PMC. The tangential electric field on a PEC i s equal to the negative of the surface magnetic current while the normal electric flux is equal to the surface magn etic char ge. Electrom agnetic Waves Maxwell’s equations show that the electric field and magnetic field m in a sou rce-free (J = M = 0, ñ = ñ= 0), homogeneous, linear, isotropic medi um satisfy wave equ ations (Helmholtz equat ions). The source-free Maxwell’s equations in phasor form a re Note that taking the di vergence of (1) and (2) yields (3) and (4) since L@L×F = 0 for any vector F. Thus, in a source-free region, (3) and (4) are not necessary. Taking the curl of (1) and inserting (2) yields while taking the curl of (2) and inserting (1) yields where k = ù%ì&å& is defi ned as the waven umber of the medium. Using the vector identity in (5) and (6) gives However, the divergence terms in (7) and (8) are zero in the sou rce-free region. This gives the wave equations (Helmholtz equations) for the electric and m agnetic field. Wave equations (Helmholtz Eq uations) The w ave equations for the E and H show that ener gy w ill propagate aw ay from a time-varying electromagnetic source in the form of electromagnetic waves. Plane W aves Plane waves are the most commonly encountered wave type in electromagnetic appli cations and are the easi est to define mat hemat ically. Plane wave - the electric and m agnetic field of a plane wave lie in the plane which is perpendicular to the direction o f wave propagation (the d irection of E × H is the direction o f wave propagation). Uniform Plane w ave - the electri c and magnetic field s of a uniform plane wave are uniform in the plane which is perpendicular to the d irection of propag ation (the mag nitude of E and H vary only in the direction of wave propagation). Examp le (Uniform pl ane w ave) The uniform pl ane wave for this examp le has only a z-compone nt of electric field and a n x-component of magnetic field which are both functions of on ly y. The vector Laplacia n operator (L) which appears in2 the wave equations for E and H may be exp anded i n rectangular coordinates as Linear, homogeneous, 2 order D.E.’sndGiven the vector Laplacia n definition, the wave equations for E and H reduce to where the partial derivatives hav e been r eplaced by pure der ivatives gi ven that the field compone nts are functions of only one variable. Note that the right hand side of the equations above is the zero vector. Thus , by equating the vector comp onents on both sides of the equ ation, we may write scalar zxequations for E and H. The general solutions to these D.E.’s are 12 1 2where E, E, H, and H are constants. The instantaneous forms of the wave field components are zE The direction of propagation for the plane wave may be determined by investigating the points of const ant phase on the w aves. y1 1Given the +a traveling wave of o ur examp le, the con stants E and H must be zer o so that Plane wave parameters pThe velocity of propagation ( v) of the plane wave is found by differentiating the position of the point of constant phase with respect to position. p ooIn free space, v = 1/ %ì&å& = c (speed of light = 3 × 10 m/s).8 p rrIn media with ì > 1 and/or å > 1, v < c.xH The radian frequency of the plane wave is defined by pWith the w ave t raveling at a vel ocity of v, it takes one period (T) for the wave to travel one wavelength (ë). The w aven umber defi nition in terms of ë shows that the al l waves se e a phase chang e of 2 ð radians per wavelength. Plane waves have the characteristic that the ratio of the electric field to m agnetic field at any poi nt is a constant which is related to the constitutive parameters of the medium. This property can be illustrated by using Maxwell’s equations with our examp le plane w ave. ooIn free space, the w ave i mpedan ce is %ì&/&å& . 120 ð = 37 7 Ù. Plane Waves in Lossy Media A plane wave loses energy as it propagat es through a lossy medium. A medi um is defi ned as a lossy medium if it is characterized by any or all of the fol lowing loss mechani sms: conduction losses Y(ó > 0) dielectric losses Y(åO > 0) magn etic losses Y(ìO > 0) Dielectric and m agnetic losses are typically small and can be neglected for most materials. H owever, condu ction l osses can be significant for commo nly encou ntered mat erials. If we include conduction losses in a h omogeneo us, isotropic, linear medium while assuming that th e dielec tric an d magnetic lo sses are negligible (µ and 0 are real), Maxwell’s equations be come ~~~ conduction losses Following the same techn iques used i n the lossless problem, we find that E and H satisfy wave equ ations (5) and (6) which include a c omplex propag ation constant ã (as opposed t o a real waven umber in the lossless case). ã - propagation constant á - attenuation constant â - phase constant Note that the propagation constant reduces to ã = jk (â = k) when ó = 0. The sol utions for the at tenuation and phase constants in terms of ì, å and ó are Given the same +y-directed un iform plane wave assumed in the lossless example, the differential equations governing the plane wave field components in the lossy medium are which have general solutions of the form ~ ~~~~~~~ ~~~~~~~~~~ !y directed +y directed wave wave The instantaneous fields of the plane wave in a lossy medium are Since t he p articular solution con tains o nly a + y traveling wave, the 11constants E and H must be zero. Note that the phase constant â defines the phase associated with the plane wave propagating in a lossy medium. The resulting equations for the wave par amet ers must be adj usted accor dingly (replace k with â). The wave impedance in the lossy medium is complex as shown using Maxwell’s equations.E H Alternatively, the con duction losses may be i ncluded in the com plex permittivity as defined by the loss tangent. Thus , using the effective complex permittivity term in brackets above , the condu ctor and di electric losses may be included without explicitly writing a conduct ion cur rent term. Plane Waves in Good Conductors For good con ductors (ó >> ùå), the propagation c onstant may be approximated by The inverse of the at tenuation constant for good conductors is defi ned as sthe skin depth ä. The skin depth defines the distance over which a plane traveling in a goo d condu ctor wave decays by an amount of e = 0. 368.-1 The w ave i mpedan ce w ithin a g ood conductor is Poynting’s Theorem Poynting’s theorem is the basic conservation law for electromagnetic energy. It defines the balance of complex po wer given sources of electromagnetic energy, energy storage and di ssipation. The direction and density of el ectromagnetic power flow at a po int is defined by the Poynting vector. The instantaneous form o f the Poynting vector S is S = E × H The corresponding phasor form o f the Poynting vector S is Given a vo lume V encl osed b y the surf ace S which contains electric and magn etic sources J and M, Poynting’s theorem for the volume may be written as Complex po wer delivered by the sou rces Complex po wer flow out of the volume V ~~~~~~~~~ ~~ ~~~~~~ ~~~~~~~ Conduction Dielectric M agnetic losses losses losses ~~~~~~~~~~~~ ~ ~~~~~~~~~~~ Stored electric energy Stored magnetic energy Poynting’s theorem st ates that the complex po wer produ ced by t he sou rces is equal to the power transmitted out of the volume plus that dissipated in the form of heat (through condu ctor, dielectric and magnetic losses) plus the 2ù times t he net reactive st ored energy. Plane Wave Reflection/Transmission at a Planar Inte rface Norma l Incidence Incident wave fi elds Transmitted wave fields Reflected wave fields Bound ary Condi tions (Reflection coefficient) (Transmission coefficient) Surface Impedance of a Good Condu ctor If region 1 is air and region 2 is a good conductor, th e wave is attenuated rapidly as it penetrates the conductor. The electric field within the goo d condu ctor is given by where the propagation constant in the condu ctor may be written in terms of sthe skin depth ä as The con duction current density within the con ductor is given by We ma y determine the total current per uni t width (y-direction) by integrating the current density over all z. Given this value, we may assume that this total current is spread uniformly from the surface of the condu ctor sto a depth ä. The transmission coefficient for the air-good conductor example is 21Since *ç* << *ç*, the transmi ssion coeff icient may be w ritten as 2s 2Using the equations for ã and T in the equation for J gives Our approximation for the curr ent density within conductor beco mes The power dissipated within the condu ctor is given by We may express the dissipated power as swhere R is defined as the surf ace resistance of t he con ductor and is given by Oblique Incidence (assume lossless dielectrics) Parallel Polarization i è - angle of i ncidence r è - angle of reflection t è - angle of transmission Application o f the bou ndary condi tions at the interface yields Perpendicular Polarization Reciprocity Theorem The reciproci ty theorem is a useful mathemat ical tool used to recast certain electromagnetic problems into different forms. Assume that a 11volume V encl osed b y the surf ace S contains two set s of sourc es (J, M) 22and (J, M). The reciproci ty theorem r elates the respon se at one source due to the second source to the respons e at the second source due to the first source. Starting with Maxwell’s equations defining the field responses to the two sets of sourc es we may use vect or identities to relate these r espon ses. If we subtract the two divergence equations, we find We may integrate both sides of t he equ ation above over the volume V and apply the divergence theorem to find or Sour ce-Free Regi on 11 22If S is a sou rce fr ee region (J = M = J = M = 0 ), the integral reduces to Enclosed PEC Structure If S is a an encl osed P EC structure, then Uniquen ess Theorem Given a vo lume V enclosed by a surface S which is completely filled with lossy medi a, the fields withing S are uniquely determined by the sources within S and the tangential comp onents of E or H on S. When solving boundary value problems, if we find a solution to Maxwell’s equations which satisfies the a ppropriate bou ndary condi tions, the uniqueness t heorem ensur es that the sol ution is unique. Image Theory Given a current in the pr esence of a per fect conducting ground plane, [perfect electric conductor (PEC), perfect magnetic conductor (PMC) ] we may use image theory to formulate the total fields without ever having to determ ine the surface currents induced on the ground plane. Image theory is based o n the electric or magn etic field boundary condition on the surf ace of the per fect conductor (the tangential electric field is zero on the surf ace of a PEC, the tan gential magnetic field is ze ro on the surface of a PMC). Using image theory, the ground plane can be replaced by the equivalent image current located an equal distance below the ground plane. The original current and its image are now r adiating in a hom ogeneous medium of infinite extent and we ma y use the correspond ing hom ogeneous medium equations. Example (vertical electric current) Currents over a PEC Currents over a PMC