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Electromagnetism INEL 4152 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayag ü ez, PR

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Cruz-Pol, Electromagnetics UPRM Electricity => Magnetism In 1820 Oersted discovered that a steady current produces a magnetic field while teaching a physics class. In 1820 Oersted discovered that a steady current produces a magnetic field while teaching a physics class.

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Cruz-Pol, Electromagnetics UPRM Would magnetism would produce electricity? Eleven years later, and at the same time, Mike Faraday in London and Joe Henry in New York discovered that a time-varying magnetic field would produce an electric current! Eleven years later, and at the same time, Mike Faraday in London and Joe Henry in New York discovered that a time-varying magnetic field would produce an electric current!

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Cruz-Pol, Electromagnetics UPRM Electromagnetics was born! This is the principle of motors, hydro-electric generators and transformers operation. This is the principle of motors, hydro-electric generators and transformers operation. *Mention some examples of em waves This is what Oersted discovered accidentally:

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Cruz-Pol, Electromagnetics UPRM

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Electromagnetic Spectrum

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Cruz-Pol, Electromagnetics UPRM Some terms E = electric field intensity [V/m] E = electric field intensity [V/m] D = electric field density D = electric field density H = magnetic field intensity, [A/m] H = magnetic field intensity, [A/m] B = magnetic field density, [Teslas] B = magnetic field density, [Teslas]

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Cruz-Pol, Electromagnetics UPRM Maxwell Equations in General Form Differential form Integral Form Gausss Law for E field. Gausss Law for H field. Nonexistence of monopole Faradays Law Amperes Circuit Law

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Cruz-Pol, Electromagnetics UPRM Maxwells Eqs. Also the equation of continuity Also the equation of continuity Maxwell added the term to Amperes Law so that it not only works for static conditions but also for time-varying situations. Maxwell added the term to Amperes Law so that it not only works for static conditions but also for time-varying situations. This added term is called the displacement current density, while J is the conduction current. This added term is called the displacement current density, while J is the conduction current.

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Cruz-Pol, Electromagnetics UPRM Maxwell put them together And added J d, the displacement current And added J d, the displacement current I S2S2 S1S1 L At low frequencies J>>J d, but at radio frequencies both terms are comparable in magnitude.

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Cruz-Pol, Electromagnetics UPRM Moving loop in static field When a conducting loop is moving inside a magnet (static B field, in Teslas), the force on a charge is When a conducting loop is moving inside a magnet (static B field, in Teslas), the force on a charge is Encarta®

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Cruz-Pol, Electromagnetics UPRM Who was NikolaTesla? Find out what inventions he made Find out what inventions he made His relation to Thomas Edison His relation to Thomas Edison Why is he not well know? Why is he not well know?

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Cruz-Pol, Electromagnetics UPRM Special case Consider the case of a lossless medium Consider the case of a lossless medium with no charges, i.e.. with no charges, i.e.. The wave equation can be derived from Maxwell equations as What is the solution for this differential equation? The equation of a wave! The equation of a wave!

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Cruz-Pol, Electromagnetics UPRM Phasors & complex #s Working with harmonic fields is easier, but requires knowledge of phasor, lets review complex numbers and complex numbers and phasors phasors

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Cruz-Pol, Electromagnetics UPRM COMPLEX NUMBERS: Given a complex number z Given a complex number zwhere

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Cruz-Pol, Electromagnetics UPRM Review: Addition, Addition, Subtraction, Subtraction, Multiplication, Multiplication, Division, Division, Square Root, Square Root, Complex Conjugate Complex Conjugate

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Cruz-Pol, Electromagnetics UPRM For a time varying phase Real and imaginary parts are: Real and imaginary parts are:

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Cruz-Pol, Electromagnetics UPRM PHASORS For a sinusoidal current For a sinusoidal current equals the real part of The complex term which results from dropping the time factor is called the phasor current, denoted by (s comes from sinusoidal) The complex term which results from dropping the time factor is called the phasor current, denoted by (s comes from sinusoidal)

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Cruz-Pol, Electromagnetics UPRM To change back to time domain The phasor is multiplied by the time factor, e j t, and taken the real part. The phasor is multiplied by the time factor, e j t, and taken the real part.

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Cruz-Pol, Electromagnetics UPRM Advantages of phasors Time derivative is equivalent to multiplying its phasor by j Time derivative is equivalent to multiplying its phasor by j Time integral is equivalent to dividing by the same term. Time integral is equivalent to dividing by the same term.

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Cruz-Pol, Electromagnetics UPRM Time-Harmonic fields (sines and cosines) The wave equation can be derived from Maxwell equations, indicating that the changes in the fields behave as a wave, called an electromagnetic field. The wave equation can be derived from Maxwell equations, indicating that the changes in the fields behave as a wave, called an electromagnetic field. Since any periodic wave can be represented as a sum of sines and cosines (using Fourier), then we can deal only with harmonic fields to simplify the equations. Since any periodic wave can be represented as a sum of sines and cosines (using Fourier), then we can deal only with harmonic fields to simplify the equations.

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Cruz-Pol, Electromagnetics UPRM Maxwell Equations for Harmonic fields Differential form* Gausss Law for E field. Gausss Law for H field. No monopole Faradays Law Amperes Circuit Law * (substituting and )

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Cruz-Pol, Electromagnetics UPRM A wave Start taking the curl of Faradays law Start taking the curl of Faradays law Then apply the vectorial identity Then apply the vectorial identity And youre left with And youre left with

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Cruz-Pol, Electromagnetics UPRM A Wave Lets look at a special case for simplicity without loosing generality: The electric field has only an x-componentThe electric field has only an x-component The field travels in z directionThe field travels in z direction Then we have

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Cruz-Pol, Electromagnetics UPRM To change back to time domain From phasor From phasor …to time domain …to time domain

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Ejemplo 9.23 In free space, In free space, Find k, J d and H using phasors and maxwells eqs. Find k, J d and H using phasors and maxwells eqs. Cruz-Pol, Electromagnetics UPRM

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Several Cases of Media 1. Free space 2. Lossless dielectric 3. Lossy dielectric 4. Good Conductor Recall: Permittivity e o =8.854 x [ F/m] Permeability m o = 4 x [H/m]

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Cruz-Pol, Electromagnetics UPRM 1. Free space There are no losses, e.g. Lets define The phase of the wave The phase of the wave The angular frequency The angular frequency Phase constant Phase constant The phase velocity of the wave The phase velocity of the wave The period and wavelength The period and wavelength How does it moves? How does it moves?

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Cruz-Pol, Electromagnetics UPRM 3. Lossy Dielectrics (General Case) In general, we had In general, we had From this we obtain From this we obtain So, for a known material and frequency, we can find j So, for a known material and frequency, we can find j

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Cruz-Pol, Electromagnetics UPRM Intrinsic Impedance, Intrinsic Impedance, If we divide E by H, we get units of ohms and the definition of the intrinsic impedance of a medium at a given frequency. If we divide E by H, we get units of ohms and the definition of the intrinsic impedance of a medium at a given frequency. *Not in-phase for a lossy medium

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Cruz-Pol, Electromagnetics UPRM Note… E and H are perpendicular to one another E and H are perpendicular to one another Travel is perpendicular to the direction of propagation Travel is perpendicular to the direction of propagation The amplitude is related to the impedance The amplitude is related to the impedance And so is the phase And so is the phase

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Cruz-Pol, Electromagnetics UPRM Loss Tangent If we divide the conduction current by the displacement current If we divide the conduction current by the displacement current

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Cruz-Pol, Electromagnetics UPRM Relation between tan and c

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Cruz-Pol, Electromagnetics UPRM 2. Lossless dielectric Substituting in the general equations: Substituting in the general equations:

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Cruz-Pol, Electromagnetics UPRM Review: 1. Free Space Substituting in the general equations: Substituting in the general equations:

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Cruz-Pol, Electromagnetics UPRM 4. Good Conductors Substituting in the general equations: Substituting in the general equations: Is water a good conductor???

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Cruz-Pol, Electromagnetics UPRMSummary Any medium Lossless medium ( =0) Low-loss medium ( / <.01) Good conductor ( / >100) Units 0 [Np/m] [rad/m] [ohm] u c u p /f [m/s] [m] **In free space; **In free space; o =8.85 x F/m o =4 x H/m

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Cruz-Pol, Electromagnetics UPRM Skin depth, Skin depth, Is defined as the depth at which the electric amplitude is decreased to 37% Is defined as the depth at which the electric amplitude is decreased to 37% We know that a wave attenuates in a lossy medium until it vanishes, but how deep does it go?

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Cruz-Pol, Electromagnetics UPRM Short Cut … You can use Maxwells or use You can use Maxwells or use where k is the direction of propagation of the wave, i.e., the direction in which the EM wave is traveling (a unitary vector).

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Cruz-Pol, Electromagnetics UPRM Waves Static charges > static electric field, E Static charges > static electric field, E Steady current > static magnetic field, H Steady current > static magnetic field, H Static magnet > static magnetic field, H Static magnet > static magnetic field, H Time-varying current > time varying E(t) & H(t) that are interdependent > electromagnetic wave Time-varying current > time varying E(t) & H(t) that are interdependent > electromagnetic wave Time-varying magnet > time varying E(t) & H(t) that are interdependent > electromagnetic wave Time-varying magnet > time varying E(t) & H(t) that are interdependent > electromagnetic wave

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Cruz-Pol, Electromagnetics UPRM EM waves dont need a medium to propagate Sound waves need a medium like air or water to propagate Sound waves need a medium like air or water to propagate EM wave dont. They can travel in free space in the complete absence of matter. EM wave dont. They can travel in free space in the complete absence of matter. Look at a wind wave; the energy moves, the plants stay at the same place. Look at a wind wave; the energy moves, the plants stay at the same place.

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Cruz-Pol, Electromagnetics UPRM Exercises: Wave Propagation in Lossless materials A wave in a nonmagnetic material is given by A wave in a nonmagnetic material is given by Find: (a) direction of wave propagation, (b) wavelength in the material (c) phase velocity (d) Relative permittivity of material (e) Electric field phasor 2.25, Answer: +y, u p = 2 x 10 8 m/s, 1.26m, 2.25,

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Cruz-Pol, Electromagnetics UPRM Power in a wave A wave carries power and transmits it wherever it goes A wave carries power and transmits it wherever it goes See Applet by Daniel Roth at The power density per area carried by a wave is given by the Poynting vector.

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Cruz-Pol, Electromagnetics UPRM Poynting Vector Derivation Start with E dot Amperes Start with E dot Amperes Apply vector identity Apply vector identity And end up with: And end up with:

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Cruz-Pol, Electromagnetics UPRM Poynting Vector Derivation… Substitute Faraday in 1 rst term Substitute Faraday in 1 rst term Rearrange

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Cruz-Pol, Electromagnetics UPRM Poynting Vector Derivation… Taking the integral wrt volume Taking the integral wrt volume Applying Theorem of Divergence Applying Theorem of Divergence Which means that the total power coming out of a volume is either due to the electric or magnetic field energy variations or is lost in ohmic losses. Which means that the total power coming out of a volume is either due to the electric or magnetic field energy variations or is lost in ohmic losses. Total power across surface of volume Rate of change of stored energy in E or H Ohmic losses due to conduction current

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Cruz-Pol, Electromagnetics UPRM Power: Poynting Vector Waves carry energy and information Waves carry energy and information Poynting says that the net power flowing out of a given volume is = to the decrease in time in energy stored minus the conduction losses. Poynting says that the net power flowing out of a given volume is = to the decrease in time in energy stored minus the conduction losses. Represents the instantaneous power density vector associated to the electromagnetic wave.

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Cruz-Pol, Electromagnetics UPRM Time Average Power The Poynting vector averaged in time is The Poynting vector averaged in time is For the general case wave: For the general case wave:

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Cruz-Pol, Electromagnetics UPRM Total Power in W The total power through a surface S is Note that the units now are in Watts Note that the units now are in Watts Note that power nomenclature, P is not cursive. Note that power nomenclature, P is not cursive. Note that the dot product indicates that the surface area needs to be perpendicular to the Poynting vector so that all the power will go thru. (give example of receiver antenna) Note that the dot product indicates that the surface area needs to be perpendicular to the Poynting vector so that all the power will go thru. (give example of receiver antenna)

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Cruz-Pol, Electromagnetics UPRM Exercises: Power 1. At microwave frequencies, the power density considered safe for human exposure is 1 mW/cm 2. A radar radiates a wave with an electric field amplitude E that decays with distance as |E(R)|=3000/R [V/m], where R is the distance in meters. What is the radius of the unsafe region? Answer: m Answer: m 2. A 5GHz wave traveling In a nonmagnetic medium with r =9 is characterized by Determine the direction of wave travel and the average power density carried by the wave Answer: Answer:

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Cruz-Pol, Electromagnetics UPRM TEM wave Transverse ElectroMagnetic = plane wave There are no fields parallel to the direction of propagation, only perpendicular (=transverse). If have an electric field E x (z) …then must have a corresponding magnetic field H x (z) The direction of propagation is a E x a H = a k z x y z x

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Cruz-Pol, Electromagnetics UPRM PE 10.7 In free space, H=0.2 cos ( t- x) z A/m. Find the total power passing through a square plate of side 10cm on plane x+z=1 square plate of side 10cm on plane x+z=1 square plate at z=3 square plate at z=3 Answer; P tot = 53mW HzHz EyEy x Answer; P tot = 0mW!

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Cruz-Pol, Electromagnetics UPRM Polarization: Why do we care?? Antenna Antenna applications – Antenna can only TX or RX a polarization it is designed to support. Straight wires, square waveguides, and similar rectangular systems support linear waves (polarized in one direction) Round waveguides, helical or flat spiral antennas produce circular or elliptical waves. Remote Remote Sensing and Radar Applications – Many targets will reflect or absorb EM waves differently for different polarizations. Using multiple polarizations can give more information and improve results. Absorption Absorption applications – Human body, for instance, will absorb waves with E oriented from head to toe better than side-to-side, esp. in grounded cases. Also, the frequency at which maximum absorption occurs is different for these two polarizations. This has ramifications in safety guidelines and studies.

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Cruz-Pol, Electromagnetics UPRM x x y y z z Polarization of a wave IEEE Definition: The trace of the tip of the E-field vector as a function of time seen from behind. Simple cases Vertical, E x Vertical, E x Horizontal, E y Horizontal, E y x x y y kl.de/software/java/polarisation/

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Cruz-Pol, Electromagnetics UPRM Polarization In general, plane wave has 2 components; in x & y In general, plane wave has 2 components; in x & y And y-component might be out of phase wrt to x- component, is the phase difference between x and y. And y-component might be out of phase wrt to x- component, is the phase difference between x and y. x y EyEy E x y x Front View

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Cruz-Pol, Electromagnetics UPRM Several Cases Linear polarization: y - x =0 o or ±180 o n Linear polarization: y - x =0 o or ±180 o n Circular polarization: y - x =±90 o & E ox =E oy Circular polarization: y - x =±90 o & E ox =E oy Elliptical polarization: y - x =±90 o & E oxE oy, or 0 o or 180 o n even if E ox =E oy Elliptical polarization: y - x =±90 o & E oxE oy, or 0 o or 180 o n even if E ox =E oy Unpolarized- natural radiation Unpolarized- natural radiation

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Cruz-Pol, Electromagnetics UPRM Linear polarization =0 in time in time domain x y EyEy E x Front View y x Back View:

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Cruz-Pol, Electromagnetics UPRM Circular polarization Both components have same amplitude E ox =E oy, Both components have same amplitude E ox =E oy, = y - x = -90 o = Right circular polarized (RCP) = y - x = -90 o = Right circular polarized (RCP) =+90 o = LCP =+90 o = LCP

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Cruz-Pol, Electromagnetics UPRM Elliptical polarization X and Y components have different amplitudes E oxE oy, and = ±90 o or ±90 o and E ox =E oy X and Y components have different amplitudes E oxE oy, and = ±90 o or ±90 o and E ox =E oy Or 0,180 o, Or 0,180 o, Or any other phase difference, for example =56 o Or any other phase difference, for example =56 o

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Cruz-Pol, Electromagnetics UPRM Polarization example Polarizing glasses Unpolarized radiation enters Nothing comes out this time. All light comes out

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Cruz-Pol, Electromagnetics UPRM Example Determine the polarization state of a plane wave with electric field: Determine the polarization state of a plane wave with electric field:a. b.c.d. =105, Elliptic =0, linear a 30 o c.+180, LP a 45 o d.-90, RHCP

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Cruz-Pol, Electromagnetics UPRM Cell phone & brain Computer model for Cell phone Radiation inside the Human Brain Computer model for Cell phone Radiation inside the Human Brain SAR Specific Absorption Rate [W/Kg] FCC limit 1.6W/kg, ~.2mW/cm 2 for 30mins SAR Specific Absorption Rate [W/Kg] FCC limit 1.6W/kg, ~.2mW/cm 2 for 30mins Phone/Samsung/Impression+%28SGH-a877%29/ Phone/Samsung/Impression+%28SGH-a877%29/

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Human absorption MHz is where the human body absorbs RF energy most efficiently MHz is where the human body absorbs RF energy most efficiently nts/bulletins/oet56/oet56e4.pdf nts/bulletins/oet56/oet56e4.pdf nts/bulletins/oet56/oet56e4.pdf nts/bulletins/oet56/oet56e4.pdf * The FCC limit in the US for public exposure from cellular telephones at the ear level is a SAR level of 1.6 watts per kilogram (1.6 W/kg) as averaged over one gram of tissue. * The FCC limit in the US for public exposure from cellular telephones at the ear level is a SAR level of 1.6 watts per kilogram (1.6 W/kg) as averaged over one gram of tissue. **The ICNIRP limit in Europe for public exposure from cellular telephones at the ear level is a SAR level of 2.0 watts per kilogram (2.0 W/kg) as averaged over ten grams of tissue. **The ICNIRP limit in Europe for public exposure from cellular telephones at the ear level is a SAR level of 2.0 watts per kilogram (2.0 W/kg) as averaged over ten grams of tissue. Cruz-Pol, Electromagnetics UPRM

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Radar bands Band Name Nominal Freq Range Specific Bands Application HF, VHF, UHF 3-30 MHz0, MHz, MHz MHz , MHz TV, Radio, L 1-2 GHz (15-30 cm) GHz Clear air, soil moist S 2-4 GHz (8-15 cm) GHz > Weather observations Cellular phones C 4-8 GHz (4-8 cm) GHz TV stations, short range Weather X 8-12 GHz (2.5–4 cm) GHz Cloud, light rain, airplane weather. Police radar. Ku GHz GHz, Weather studies K GHz GHz Water vapor content Ka GHz GHz Cloud, rain V GHz GHz Intra-building comm. W GHz GH, GHz Rain, tornadoes millimeter GHz Tornado chasers

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Cruz-Pol, Electromagnetics UPRM Microwave Oven Most food is lossy media at microwave frequencies, therefore EM power is lost in the food as heat. Find depth of penetration if meat which at 2.45 GHz has the complex permittivity given. Find depth of penetration if meat which at 2.45 GHz has the complex permittivity given. The power reaches the inside as soon as the oven in turned on!

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Cruz-Pol, Electromagnetics UPRM Decibel Scale In many applications need comparison of two powers, a power ratio, e.g. reflected power, attenuated power, gain,… In many applications need comparison of two powers, a power ratio, e.g. reflected power, attenuated power, gain,… The decibel (dB) scale is logarithmic The decibel (dB) scale is logarithmic Note that for voltages, fields, and electric currents, the log is multiplied by 20 instead of 10. Note that for voltages, fields, and electric currents, the log is multiplied by 20 instead of 10.

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Cruz-Pol, Electromagnetics UPRM Attenuation rate, A Represents the rate of decrease of the magnitude of P ave (z) as a function of propagation distance Represents the rate of decrease of the magnitude of P ave (z) as a function of propagation distance

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Cruz-Pol, Electromagnetics UPRM Submarine antenna A submarine at a depth of 200m uses a wire antenna to receive signal transmissions at 1kHz. Determine the power density incident upon the submarine antenna due to the EM wave with |E o |= 10V/m. [At 1kHz, sea water has r =81, =4]. At what depth the amplitude of E has decreased to 1% its initial value at z=0 (sea surface)?

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Cruz-Pol, Electromagnetics UPRM Exercise: Lossy media propagation For each of the following determine if the material is low-loss dielectric, good conductor, etc. (a) (a) Glass with r =1, r =5 and = S/m at 10 GHZ (b) (b) Animal tissue with r =1, r =12 and =0.3 S/m at 100 MHZ (c) (c) Wood with r =1, r =3 and =10 -4 S/m at 1 kHZ Answer: (a) (a) low-loss, x Np/m r/m cm u p x c (b) (b) general, cm u p x m/s c j31.7 (c) (c) Good conductor, x x km u p x c j

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