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**Electromagnetism INEL 4152**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Electromagnetism INEL 4152 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayagüez, PR Electrical Engineering, UPRM

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**Electricity => Magnetism**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Electricity => Magnetism In 1820 Oersted discovered that a steady current produces a magnetic field while teaching a physics class. Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Would magnetism would produce electricity?**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics 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! Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Electromagnetics was born!**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Electromagnetics was born! This is the principle of motors, hydro-electric generators and transformers operation. This is what Oersted discovered accidentally: *Mention some examples of em waves Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Electromagnetic Spectrum**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Electromagnetic Spectrum Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Maxwell Equations in General Form**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Maxwell Equations in General Form Differential form Integral Form Gauss’s Law for E field. Gauss’s Law for H field. Nonexistence of monopole Faraday’s Law Ampere’s Circuit Law Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Maxwell’s Eqs. Also the equation of continuity Maxwell added the term to Ampere’s 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. Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Maxwell put them together**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Maxwell put them together And added Jd, the displacement current I S1 L S2 At low frequencies J>>Jd, but at radio frequencies both terms are comparable in magnitude. Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Moving loop in static field**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics 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 Cruz-Pol, Electromagnetics UPRM Encarta® Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Special case Consider the case of a lossless medium 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! Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

COMPLEX NUMBERS: Given a complex number z where Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**For a time varying phase**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics For a time varying phase Real and imaginary parts are: Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

PHASORS 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) Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**To change back to time domain**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics To change back to time domain The phasor is multiplied by the time factor, ejwt, and taken the real part. Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Time-Harmonic fields (sines and cosines)**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics 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. 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. Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Maxwell Equations for Harmonic fields**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Maxwell Equations for Harmonic fields Differential form* Gauss’s Law for E field. Gauss’s Law for H field. No monopole Faraday’s Law Ampere’s Circuit Law * (substituting and ) Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

A wave Start taking the curl of Faraday’s law Then apply the vectorial identity And you’re left with Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**To change back to time domain**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics To change back to time domain From phasor …to time domain Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Cruz-Pol, Electromagnetics UPRM**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Several Cases of Media Free space Lossless dielectric Lossy dielectric Good Conductor Recall: Permittivity eo=8.854 x 10-12[ F/m] Permeability mo= 4p x 10-7 [H/m] Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**3. Lossy Dielectrics (General Case)**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics 3. Lossy Dielectrics (General Case) In general, we had From this we obtain So , for a known material and frequency, we can find g=a+jb Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Intrinsic Impedance, h 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 Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Relation between tanq and ec**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Relation between tanq and ec Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Summary Any medium Lossless medium (s=0) Low-loss medium (e”/e’<.01) Good conductor (e”/e’>100) Units a [Np/m] b [rad/m] h [ohm] uc l w/b 2p/b=up/f [m/s] [m] **In free space; eo =8.85 x F/m mo=4p x 10-7 H/m Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Short Cut … You can use Maxwell’s 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). Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**EM waves don’t need a medium to propagate**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics EM waves don’t need a medium to propagate Sound waves need a medium like air or water to propagate EM wave don’t. 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. Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Exercises: Wave Propagation in Lossless materials**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Exercises: Wave Propagation in Lossless materials A wave in a nonmagnetic material is given by Find: direction of wave propagation, wavelength in the material phase velocity Relative permittivity of material Electric field phasor Answer: +y, up= 2x108 m/s, 1.26m, 2.25, Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Poynting Vector Derivation**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Poynting Vector Derivation Start with E dot Ampere’s Apply vector identity And end up with: Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Poynting Vector Derivation…**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Poynting Vector Derivation… Substitute Faraday in 1rst term Rearrange Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Poynting Vector Derivation…**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Poynting Vector Derivation… Taking the integral wrt volume 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. Total power across surface of volume Rate of change of stored energy in E or H Ohmic losses due to conduction current Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Power: Poynting Vector**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Power: Poynting Vector 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. Represents the instantaneous power density vector associated to the electromagnetic wave. Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Total Power in W The total power through a surface S is Note that the units now are in Watts 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) Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Exercises: Power 1. At microwave frequencies, the power density considered safe for human exposure is 1 mW/cm2. 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 2. A 5GHz wave traveling In a nonmagnetic medium with er=9 is characterized by Determine the direction of wave travel and the average power density carried by the wave Answer: Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Polarization: Why do we care?? 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 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 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. Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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, Ex Horizontal, Ey x y z x y x y Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Several Cases Linear polarization: d=dy-dx =0o or ±180on Circular polarization: dy-dx =±90o & Eox=Eoy Elliptical polarization: dy-dx=±90o & Eox≠Eoy, or d=≠0o or ≠180on even if Eox=Eoy Unpolarized- natural radiation Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Linear polarization Front View d =0 @z=0 in time domain x y Ey Ex Back View: y x Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Circular polarization**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Circular polarization Both components have same amplitude Eox=Eoy, d =d y-d x= -90o = Right circular polarized (RCP) d =+90o = LCP Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Elliptical polarization**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Elliptical polarization X and Y components have different amplitudes Eox≠Eoy, and d =±90o or d ≠±90o and Eox=Eoy Or d ≠0,180o, Or any other phase difference, for example d =56o Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

Cell phone & brain Computer model for Cell phone Radiation inside the Human Brain SAR Specific Absorption Rate [W/Kg] FCC limit 1.6W/kg, ~.2mW/cm2 for 30mins Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Cruz-Pol, Electromagnetics UPRM**

Human absorption * 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. MHz is where the human body absorbs RF energy most efficiently Cruz-Pol, Electromagnetics UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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 12-18 GHz GHz, Weather studies K 18-27 GHz GHz Water vapor content Ka 27-40 GHz GHz Cloud, rain V 40-75 GHz 59-64 GHz Intra-building comm. W GHz 76-81 GH, GHz Rain, tornadoes millimeter GHz Tornado chasers Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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. The power reaches the inside as soon as the oven in turned on! Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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

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**Dr. S. Cruz-Pol, INEL 4152-Electromagnetics**

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 |Eo|= 10V/m. [At 1kHz, sea water has er=81, s=4]. At what depth the amplitude of E has decreased to 1% its initial value at z=0 (sea surface)? Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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**Exercise: Lossy media propagation**

Dr. S. Cruz-Pol, INEL 4152-Electromagnetics Exercise: Lossy media propagation For each of the following determine if the material is low-loss dielectric, good conductor, etc. Glass with mr=1, er=5 and s=10-12 S/m at 10 GHZ Animal tissue with mr=1, er=12 and s=0.3 S/m at 100 MHZ Wood with mr=1, er=3 and s=10-4 S/m at 1 kHZ Answer: low-loss, a= 8.4x10-11 Np/m, b= 468 r/m, l= 1.34 cm, up=1.34x108, hc=168 W general, a= 9.75, b=12, l=52 cm, up=0.5x108 m/s, hc=39.5+j31.7 W Good conductor, a= 6.3x10-4, b= 6.3x10-4, l= 10km, up=0.1x108, hc=6.28(1+j) W Cruz-Pol, Electromagnetics UPRM Electrical Engineering, UPRM

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