Microwaves

Microwaves in the Electromagnetic Spectrum (300 MHz - 300 GHz)

ELF Extremely Low Frequency 3-30 HzSLF Super Low Frequency 30-300 HzULF Ultra Low Frequency 300 Hz - 3 kHzVLF Very Low Frequency 3 kHz - 30 kHzLF Low Frequency 30 kHz - 300 kHzMF Medium Frequency 300 kHz - 3 MHzHF High Frequency 3 MHz - 30 MHzVHF Very High Frequency 30 MHz - 300 MHzUHF 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 GHzIR Infared 3000 GHz - 416,000 GHz

Microwave Properties

High bandwidth - The microwave frequency range (300 MHz - 300GHz) is 999 times that of the entire frequency range below it.

Effect of the ionosphere - When lower frequency waves are directedupward into the atmosphere, they experience significantreflection due to the ionosphere. The lower frequency waveswhich pass through the ionosphere suffer distortion.Microwaves pass through the ionosphere with little effect andare therefore utilized in satellite communications and spacetransmissions.

Line-of-sight transmission/reception - The microwave receiveantenna must be within the line-of-sight of the transmit antenna.Long distance communication on earth requires that microwaverelay stations be used.

Electromagnetic noise characteristics - The electromagnetic noiselevel in nature over the 1-10 GHz frequency range is small.This allows for the detection of very low signal levels usingsensitive receivers.

Antenna gain and directivity - The gain of an antenna is directlyproportional to its electrical size. The beamwidth of an antennais inversely proportional to the electrical size of its maximumdimension. Shorter wavelengths at microwave frequenciesallow for smaller antennas. At higher frequencies (visible light-lasers), the beamwidth gets very small and pointing accuracy ofthe detector becomes a problem.

Target reflection of electromagnetic waves (radar cross section) - Ingeneral, electrically large conducting radar targets reflect moreenergy (shape is also a factor - stealth design). Thus, the higherfrequencies of microwaves are preferred for radar systems. Atmillimeter waves, the wavelength becomes comparable to thesize of raindrops which results in attenuation of the incidentwaves.

Absorption at resonant frequencies - various materials absorbmicrowave energy (dissipated in the form of heat) at specificresonant frequencies.

Applications of Microwaves

Wireless communicationsPersonal Communications Systems (PCS)

(pagers, cell phones, etc.)Global Positioning Satellite (GPS) SystemsWireless Local Area Computer Networks (WLANS)Direct Broadcast Satellite (DBS) TelevisionTelephone Microwave/Satellite Links, etc.

Remote sensingRadar (active remote sensing - radiate and receive)

Military applications (target tracking)Weather radarGround Penetrating Radar (GPR)Agricultural applications

Radiometry (passive remote sensing - receive inherentemissions)Radio astronomy

Industrial and home applications Cooking, drying, heatingMicrowave spectroscopy - molecular properties of materials

can be determined by passing microwaves through asample of the material and measuring the absorptionspectrum.

Analysis Techniques in Microwave Theory

In general, circuit theory is not applicable to microwave problems.Circuit theory is derived from Maxwells equations based on certainassumptions about the fields within the circuit elements. Specifically, thecircuit elements must be small relative to wavelength for circuit equationsto be valid. In this sense, microwave components must be modeled bydistributed elements, not lumped elements. For this reason, we must usefield theory solutions (Maxwells equations) for microwave applications.

Maxwells Equations

Maxwells Equations (instantaneous, symmetric form)

(Faradays law)

(Amperes law)

(Gauss law - electric fields)

(Gauss law - magnetic fields)

E, H, D, B, J, M - instantaneous vectors [E =E (x,y,z,t), etc.]m, - instantaneous scalars [ = (x,y,z,t), etc.]

E - electric field intensity (V/m) H - magnetic field intensity (A/m)D - electric flux density (C/m ) B - magnetic flux density (Wb/m )2 2

J - electric current density (A/m ) M - magnetic current density (V/m )2 2

m - electric charge density (C/m ) - magnetic charge density (Wb/m )3 3

The quantities of magnetic current density M and magnetic charge densitym are nonphysical and included in the symmetric forms of Maxwells

equation for mathematical convenience. These magnetic sources may beused to simplify the mathematics of particular problems involving actualelectric currents and charges.

The flux and field quantities are related by the constitutive relations:

D = E B = H

where is the permittivity (F/m) and is the permeability (H/m) of themedium in which the fields are located. The permittivity and permeabilityof a given medium may be defined in terms of the free space (vacuum)

o ovalues [ = 8.85410 F/m, = 410 H/m] and unitless relative!12 !7

r rvalues ( , ) such that r o r o = =

BME

D JH

D

B

The instantaneous Maxwells equations are valid given any type oftime-dependence for the electromagnetic fields. Most applications inmicrowave engineering involve fields which have a sinusoidal (harmonic)time-dependence. This harmonic time-dependence allows us to simplifyMaxwells equations by writing them in terms of phasors just like we usein circuit analysis.

For time-harmonic fields, we may separate the dependence on timeand space. The real-valued instantaneous electric field E (x,y,z,t) may bewritten as

~~~~~~~~~~ Real vector [magnitude/direction]

Ewhere a is a unit vector in the direction of the vector electric field. Thearbitrary phase shift allows us to use the cosine function to represent anysinusoidal time variation relative to time t = 0. According to Eulersidentity, we may write the equation above as

~~~~~~~ Complex vector (phasor vector) [magnitude/phase/direction]

We may write all vector quantities in the instantaneous Maxwellsequations in terms of phasors according to the relationship above. Thederivatives with respect to time in the instantaneous equations yield jterms in the phasor equations.

E

E

Maxwells Equations (phasor form)

E, H, D, B, J, M - phasor vectors m, - phasor scalars

Relation of instantaneous quantities to phasor quantities ...

E (x,y,z,t) = Re{E(x,y,z)e }, etc.jt

Complex Permittivity and Permeability

In order to account for dielectric and magnetic losses in media wheretime-harmonic electromagnetic fields exist, we may define a complexpermittivity and permeability.

In the case of dielectrics, we may combine the conductivity losses with thedielectric losses according to Maxwells equations. The conduction currentdensity J in a given medium is defined by

where is the conductivity of the medium in S/m (/m). We may write asingle equation which includes dielectric and conductor losses byincorporating the complex permittivity and the conduction current equationinto Amperes law.

~~~~~~ ~~~~~~~~~~ Displacement conductor +dielectric

current losses

The ratio of the overall conductor and dielectric losses to the displacementcurrent is defined as the loss tangent [tan ] since it is related to the tangentof the complex number multiplying the electric field phasor.

Material Classifications

A given medium is characterized by its three constitutive parametersdefined as (,,). We may classify media according to the characteristicsof the constitutive parameters.

Homogeneous - the constitutive parameters of the medium are notfunctions of position (otherwise - inhomogeneous).

Linear - the constitutive parameters of the medium are not functionsof the magnitude of the applied field (otherwise - nonlinear).

Isotropic - the constitutive parameters of the medium are notfunctions of the direction of the applied field (otherwise -anisotropic).

Electromagnetic Field Boundary Conditions

Knowledge of how the components of an electromagnetic fieldbehave at the interface between two different media is important in thesolution of many problems in microwave engineering. A simple interfacebetween two media is shown below. The vector n is defined as the unitnormal to the interface pointing into region 2.

The general boundary conditions are:

Note that the individual components of the vector fields [n @ {} defines thenormal components of the vector while n {} defines the tangentialcomponents of the vector] are discontinuous at the interface by an amountequal to the respective surface current or charge on the boundary. In mostapplications, we do not encounter all of the surface sources. These generalboundary conditions can be specialized to problems involving specificcombinations of materials.

Interface Between Two Lossless Dielectric Materials

If the two media are lossless dielectrics (perfect insulators defined by1 2 1 2 1 2 O = O = O = O = = = 0), then no surface charge or current will

occur naturally. The boundary conditions then become

Thus, the normal components of electric and magnetic flux and thetangential components of electric and magnetic field are continuous acrossa lossless dielectric interface.

Perfect Electric Conductor (PEC)

1If region 1 is assumed to be a perfect electric conductor ( 64) whileregion 2 is a dielectric, no electromagnetic field can penetrate into region

1 11 (E = H = 0). Electric surface currents and charges are found on the PEC(no magnetic charge or current) which gives

Note that the tangential electric field is always zero on the surface of aPEC. The tangential magnetic field on a PEC is equal to the surface currentwhile the normal electric flux is equal to the surface charge.

Perfect Magnetic Conductor (PMC)

If region 1 is assumed to be a perfect magnetic conductor (itsm1equivalent magnetic conductivity 64) while region 2 is a dielectric, no

1 1electromagnetic field can penetrate into region 1 (E = H = 0). Magneticsurface currents and charges are found on the PMC (no electric charge orcurrent) which gives

Note that the tangential magnetic field is always zero on the surface of aPMC. The tangential electric field on a PEC is equal to the negative of thesurface magnetic current while the normal electric flux is equal to thesurface magnetic charge.

Electromagnetic Waves

Maxwells equations show that the electric field and magnetic fieldm in a source-free (J = M = 0, = = 0), homogeneous, linear, isotropic

medium satisfy wave equations (Helmholtz equations). The source-freeMaxwells equations in phasor form are

Note that taking the divergence of (1) and (2) yields (3) and (4) sinceL @ L F = 0 for any vector F. Thus, in a source-free region, (3) and (4) arenot necessary. Taking the curl of (1) and inserting (2) yields

while taking the curl of (2) and inserting (1) yields

where k = %&& is defined as the wavenumber of the medium. Using thevector identity

in (5) and (6) gives

However, the divergence terms in (7) and (8) are zero in the source-freeregion. This gives the wave equations (Helmholtz equations) for theelectric and magnetic field.

Wave equations(Helmholtz Equations)

The wave equations for the E and H show that energy will propagate awayfrom a time-varying electromagnetic source in the form of electromagneticwaves.

Plane Waves

Plane waves are the most commonly encountered wave type inelectromagnetic applications and are the easiest to define mathematically.

Plane wave - the electric and magnetic field of a plane wave lie in theplane which is perpendicular to the direction of wavepropagation (the direction of E H is the direction of wavepropagation).

Uniform Plane wave - the electric and magnetic fields of a uniformplane wave are uniform in the plane which is perpendicular tothe direction of propagation (the magnitude of E and H varyonly in the direction of wave propagation).

Example (Uniform plane wave)

The uniform plane wave for this example has only a z-component ofelectric field and an x-component of magnetic field which are bothfunctions of only y. The vector Laplacian operator (L ) which appears in2

the wave equations for E and H may be expanded in rectangularcoordinates as

Linear, homogeneous,2 order D.E.snd

Given the vector Laplacian definition, the wave equations for E and Hreduce to

where the partial derivatives have been replaced by pure derivatives giventhat the field components are functions of only one variable. Note that theright hand side of the equations above is the zero vector. Thus, by equatingthe vector components on both sides of the equation, we may write scalar

z xequations for E and H .

The general solutions to these D.E.s are

1 2 1 2where E , E , H , and H are constants. The instantaneous forms of thewave field components are

zE

The direction of propagation for the plane wave may be determined byinvestigating the points of constant phase on the waves.

y 1 1Given the +a traveling wave of our example, the constants E and H mustbe zero so that

Plane wave parameters

pThe velocity of propagation (v ) of the plane wave is found bydifferentiating the position of the point of constant phase with respect toposition.

p o oIn free space, v = 1/%&& = c (speed of light = 3 10 m/s).8

pr rIn media with > 1 and/or > 1, v < c.

xH

The radian frequency of the plane wave is defined by

pWith the wave traveling at a velocity of v , it takes one period (T ) for thewave to travel one wavelength ().

The wavenumber definition in terms of shows that the all waves see aphase change of 2 radians per wavelength.

Plane waves have the characteristic that the ratio of the electric fieldto magnetic field at any point is a constant which is related to theconstitutive parameters of the medium. This property can be illustrated byusing Maxwells equations with our example plane wave.

o oIn free space, the wave impedance is %&/&& . 120 = 377 .

Plane Waves in Lossy Media

A plane wave loses energy as it propagates through a lossy medium.A medium is defined as a lossy medium if it is characterized by any or allof the following loss mechanisms:

conduction losses Y ( > 0)

dielectric losses Y (O > 0)

magnetic losses Y (O > 0)

Dielectric and magnetic losses are typically small and can be neglected formost materials. However, conduction losses can be significant forcommonly encountered materials.

If we include conduction losses in a homogeneous, isotropic, linearmedium while assuming that the dielectric and magnetic losses arenegligible ( and 0 are real), Maxwells equations become

~~~ conduction losses

Following the same techniques used in the lossless problem, we find thatE and H satisfy wave equations (5) and (6) which include a complexpropagation constant (as opposed to a real wavenumber in the losslesscase).

- propagation constant

- attenuation constant

- phase constant

Note that the propagation constant reduces to = jk ( = k) when = 0.The solutions for the attenuation and phase constants in terms of , and are

Given the same +y-directed uniform plane wave assumed in thelossless example, the differential equations governing the plane wave fieldcomponents 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 the particular solution contains only a +y traveling wave, the1 1constants E and H must be zero.

Note that the phase constant defines the phase associated with theplane wave propagating in a lossy medium. The resulting equations for thewave parameters must be adjusted accordingly (replace k with ).

The wave impedance in the lossy medium is complex as shown usingMaxwells equations.

E

H

Alternatively, the conduction losses may be included in the complexpermittivity as defined by the loss tangent.

Thus, using the effective complex permittivity term in brackets above, theconductor and dielectric losses may be included without explicitly writinga conduction current term.

Plane Waves in Good Conductors

For good conductors ( >> ), the propagation constant may beapproximated by

The inverse of the attenuation constant for good conductors is defined assthe skin depth . The skin depth defines the distance over which a plane

traveling in a good conductor wave decays by an amount of e = 0.368.-1

The wave impedance within a good conductor is

Poyntings Theorem

Poyntings theorem is the basic conservation law for electromagneticenergy. It defines the balance of complex power given sources ofelectromagnetic energy, energy storage and dissipation. The direction anddensity of electromagnetic power flow at a point is defined by the Poyntingvector. The instantaneous form of the Poynting vector S is

S = E H

The corresponding phasor form of the Poynting vector S is

Given a volume V enclosed by the surface S which contains electricand magnetic sources J and M, Poyntings theorem for the volume may bewritten as

Complex power delivered by the sources

Complex power flow out ofthe volume V

~~~~~~~~~ ~~~~~~~~ ~~~~~~~ Conduction Dielectric Magnetic

losses losses losses

~~~~~~~~~~~~ ~~~~~~~~~~~~ Stored electric energy Stored magnetic energy

Poyntings theorem states that the complex power produced by the sourcesis equal to the power transmitted out of the volume plus that dissipated inthe form of heat (through conductor, dielectric and magnetic losses) plusthe 2 times the net reactive stored energy.

Plane Wave Reflection/Transmission at a Planar Interface

Normal Incidence

Incident wave fields

Transmitted wave fields

Reflected wave fields

Boundary Conditions

(Reflection coefficient)

(Transmission coefficient)

Surface Impedance of a Good Conductor

If region 1 is air and region 2 is a good conductor, the wave isattenuated rapidly as it penetrates the conductor. The electric field withinthe good conductor is given by

where the propagation constant in the conductor may be written in terms ofsthe skin depth as

The conduction current density within the conductor is given by

We may determine the total current per unit width (y-direction) byintegrating the current density over all z. Given this value, we may assumethat this total current is spread uniformly from the surface of the conductor

sto a depth .

The transmission coefficient for the air-good conductor example is

2 1Since * *

We may express the dissipated power as

swhere R is defined as the surface resistance of the conductor and is givenby

Oblique Incidence (assume lossless dielectrics)

Parallel Polarization

i - angle of incidence

r - angle of reflection

t - angle of transmission

Application of the boundary conditions at the interface yields

Perpendicular Polarization

Reciprocity Theorem

The reciprocity theorem is a useful mathematical tool used to recastcertain electromagnetic problems into different forms. Assume that a

1 1volume V enclosed by the surface S contains two sets of sources (J , M )2 2and (J , M ). The reciprocity theorem relates the response at one source

due to the second source to the response at the second source due to thefirst source.

Starting with Maxwells equations defining the field responses to the twosets of sources

we may use vector identities to relate these responses.

If we subtract the two divergence equations, we find

We may integrate both sides of the equation above over the volume V andapply the divergence theorem to find

or

Source-Free Region

1 1 2 2If S is a source free region (J = M = J = M = 0), the integralreduces to

Enclosed PEC Structure

If S is a an enclosed PEC structure, then

Uniqueness Theorem

Given a volume V enclosed by a surface S which is completely filledwith lossy media, the fields withing S are uniquely determined by thesources within S and the tangential components of E or H on S.

When solving boundary value problems, if we find a solution to Maxwellsequations which satisfies the appropriate boundary conditions, theuniqueness theorem ensures that the solution is unique.

Image Theory

Given a current in the presence of a perfect conducting ground plane,[perfect electric conductor (PEC), perfect magnetic conductor (PMC)] wemay use image theory to formulate the total fields without ever having todetermine the surface currents induced on the ground plane. Image theoryis based on the electric or magnetic field boundary condition on the surfaceof the perfect conductor (the tangential electric field is zero on the surfaceof a PEC, the tangential magnetic field is zero on the surface of a PMC).Using image theory, the ground plane can be replaced by the equivalentimage current located an equal distance below the ground plane. Theoriginal current and its image are now radiating in a homogeneous mediumof infinite extent and we may use the corresponding homogeneous mediumequations.

Example (vertical electric current)

Currents over a PEC

Currents over a PMC

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