Electromagnetism. 1 ENGN6521 / ENGN4521: Embedded Wireless
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1 Electromagnetism 1 ENGN6521 / ENGN4521: Embedded Wireless
2 Radio Spectrum use for Communications 2 ENGN6521 / ENGN4521: Embedded Wireless
3 3 ENGN6521 / ENGN4521: Embedded Wireless
4 Electromagnetism I Gauss s law and Dielectrics 4 ENGN6521 / ENGN4521: Embedded Wireless
5 Topics Surface and line integrals Maxwells equations (DIV and CURL) Charge and current Electric and magnetic field Gauss law for E and B Dielectrics Ohm s law 5 ENGN6521 / ENGN4521: Embedded Wireless
6 The surface integral A E.dA = A E da cos θ (1) 6 ENGN6521 / ENGN4521: Embedded Wireless
7 The line integral C E.dl = C E dl cos θ (2) 7 ENGN6521 / ENGN4521: Embedded Wireless
8 Maxwell s equations in integral form S S E ds = Q ǫ o [Gauss] (3) B ds = 0 [Gauss] (4) E dl = t ˆ S B ds [Faraday] (5) B dl = µ oˆ S j ds+µ o ǫ o t ˆ S E ds [Ampere] (6) 8 ENGN6521 / ENGN4521: Embedded Wireless
9 Maxwell s Equations in differential form 9 ENGN6521 / ENGN4521: Embedded Wireless
10 Gauss Law For the electric field SE ds = Q ǫ o (7) For the magnetic field B ds = 0 S (8) where ǫ o is the permittivity of free-space (Farads per m) 10 ENGN6521 / ENGN4521: Embedded Wireless
11 Charge: Q (positive and negative) The ultimate source of all EM fields Measured in Coulombs Can be postive or negative - note the direction of the electric field Q = V ρ dv (9) 11 ENGN6521 / ENGN4521: Embedded Wireless
12 Charge as a source of the Electric Field: MAXWELL s first equation 12 ENGN6521 / ENGN4521: Embedded Wireless
13 Simple case of a point charge: Coulomb s Law SE ds = Q ǫ o (10) E r 4πr 2 = Q ǫ o (11) E r = Q 4πǫ o r 2 (12) 13 ENGN6521 / ENGN4521: Embedded Wireless
14 Parallel Charged Plates 14 ENGN6521 / ENGN4521: Embedded Wireless
15 Electrostatics: The Electrostatic Potential (Beware - replace this by Faraday s law Consider a situation where there is no changing magnetic field in Faraday s law. Definition: The potential difference between two points x 1 and x 2 is defined by, ˆx 2 ˆ Φ = E.dl = x 1 γ E.dl Since the path γ can be any which connects the points x 1 and x 2 we may conclude that E = Φ. Kirchhoffs voltage law: The sum of the voltage drops in a circuit is zero - BEWARE: to be modified by B t 15 ENGN6521 / ENGN4521: Embedded Wireless
16 Parallel Charged Plates E dl = 0 [Faraday, no changing B] (13) 16 ENGN6521 / ENGN4521: Embedded Wireless
17 Parallel Plate Capacitor Capacitors are reservoirs of charge - a bit like transient batteries C = Q V (Farads) Since V = Ed, C = ǫ 0A d 17 ENGN6521 / ENGN4521: Embedded Wireless
18 Actual capacitors 18 ENGN6521 / ENGN4521: Embedded Wireless
19 The vacuum gap capacitor 19 ENGN6521 / ENGN4521: Embedded Wireless
20 20 ENGN6521 / ENGN4521: Embedded Wireless
21 Moving charges Current The rate of charge crossing a surface is given by (note that charges are positive even if electrons are flowing!), Q t = j ds (14) S Current is measured in Amperes (Coulombs per second) Charge conservation: the charge flows out of a region as a current Q. Why the - sign here? Q t = j ds (15) S Charge conservation leads to Kirchhoff s current rule: currents entering a circuit node sum to zero 21 ENGN6521 / ENGN4521: Embedded Wireless
22 Current Density (current per unit area) 22 ENGN6521 / ENGN4521: Embedded Wireless
23 Kirchhoff s current law 23 ENGN6521 / ENGN4521: Embedded Wireless
24 Dielectrics and Conductors Dielectrics are insulating materials VIZ. do not allow D.C. current to flow through them. Usually we just call them insulators In conductors, charge carriers (electrons) are free to move. e.g. metals. 24 ENGN6521 / ENGN4521: Embedded Wireless
25 Dielectrics 1 Two opposite, cancelling charges separated in space is referred to collectively as a dipole. Electrons and nuclei in dielectrics experience opposing forces in the presence of an imposed electric field. Electrons move opposite to the field and nuclei move in the direction of the field. This produces a distribution of dipoles referred to as polarisation. The charge separation induced by the field acts to reduce the electric field within the dielectric. 25 ENGN6521 / ENGN4521: Embedded Wireless
26 Dielectrics 2 26 ENGN6521 / ENGN4521: Embedded Wireless
27 Dielectrics 3: Polarisation Polarisation P is the dipole moment per unit volume induced by an imposed, external electric field At the edge of a polarised dielectric there is a charge density left over by the displacement of the dipoles given by σ p = P.ˆn where ˆn is the unit vector normal to the surface. σ p belongs to the material. It is not a free charge. 27 ENGN6521 / ENGN4521: Embedded Wireless
28 Dielectrics 4: Relative dielectric constant For a slither of dielectric, the surface charge is related to the E-field within the dielectric that produces it by σ p = P ˆn = ǫ 0 (ǫ r 1)E ˆn ǫ r at D.C. is a positive dimensionless number and ǫ r > 1 for dielectrics. E.g. ǫ r = 2 for teflon, ǫ r = 1 for air or vacuum. When the electric field oscillates, ǫ r is a complex function of frequency -lossy The ratio of the imaginary to real components of ǫ r is termed the loss tangent of the dielectric: tan[δ(ω)] = Im(ǫ r(ω)) Re(ǫ r (ω)) (16) Even though dielectrics are insulators at DC they can be lossy conductors at radiofrequency. 28 ENGN6521 / ENGN4521: Embedded Wireless
29 Exercise 1: Compute the capacitance of a parallel plate air-gap capacitor (E 0) A = Q ǫ o = σ A ǫ o => E = σ ǫ o (17) V = E d => C = Q V = ǫ oa d Q σα = E = capacitor plates (18) d E > area = A enclosed by dots electric fieldlines 29 ENGN6521 / ENGN4521: Embedded Wireless
30 Exercise 2: Compute the capacitance for a gap of dielectric constant ǫ r Use the definition of ǫ r and comnpute the E field inside the dielectric E = σ cap σ p ǫ o = σ p ǫ o (ǫ r 1) (19) (σ cap σ p )(ǫ r 1) = σ p => σ p = σ cap(ǫ r 1) ǫ r (20) V = E d = σ p d ǫ o (ǫ r 1) = σ capd ǫ o ǫ r (21) Q cap = σ cap A => C = Q cap V = ǫ rǫ o A d (22) 30 ENGN6521 / ENGN4521: Embedded Wireless
31 Conductors and Ohm s law Ohm s law: The current density in a conductor is proportional to the electric field within the conductor. j = σe (23) Conductors are completely specified by the conductivity σ. For copper, σ = mhos/meter. Ohm s law is assumed to be an accurate result for metals at all radiofrequencies :) I.E. σ is always a real number and independent of frequency. 31 ENGN6521 / ENGN4521: Embedded Wireless
32 Resistance Resistance is obtained by converting j to I and E to V for a short length of metal. I A = σv l (24) V = l σa I From which we define resistance R = l σa We also define resistivity, η from σ η = 1 σ (25) (26) (27) 32 ENGN6521 / ENGN4521: Embedded Wireless
33 Conductors vs Dielectrics for dielectrics: σ p = ǫ 0 (ǫ r 1)E ˆn If σ p oscillates as a function of time then, jωσ p = jωǫ 0 (ǫ r 1)E ˆn since the dipoles are reversing sign at rate ω, we may write, j P ˆn = jωσ p, where j P is the polarisation current. Dielectrics obey Ohm s law with conductivity defined by σ = jωǫ 0 (ǫ r 1) The difference between conductors and insulators is simply that σ is resistive and frequency independent for a conductor, but is mainly reactive and frequency dependent for insulators. As we already know, ǫ r is a complex function of frequency and the real and imaginary frequency functions of ǫ r can be computed from each other (Kramers-Kronig). 33 ENGN6521 / ENGN4521: Embedded Wireless
34 Charge as a source of the Electric Field: MAXWELL s first equation the field can get distorted with complex charge distributions makes no difference to the integral The coulomb force... F = QE (28) 34 ENGN6521 / ENGN4521: Embedded Wireless
35 Electromagnetism II Moving Charges. Faraday s law and Ampere s law. Ferrites 35 ENGN6521 / ENGN4521: Embedded Wireless
36 ELECTROMAGNETISM SUMMARY RF shielding Magnetostatics Ampere s law and Faraday s law Inductance Maxwell s equations 36 ENGN6521 / ENGN4521: Embedded Wireless
37 The Floating Conducting Object Ohm s law for metals and Gauss s law for the electric field give rise to the concept of electromagnetic shielding 37 ENGN6521 / ENGN4521: Embedded Wireless
38 The Earthed Conducting Object 38 ENGN6521 / ENGN4521: Embedded Wireless
39 Images A charge near a conducting surface attracts charge of the opposite sign to the nearest point on the surface. These surface charges arrange themselves so that the tangntial component of electric field is zero on the surface. One may compute the electric field outside the conductor by assuming (mathematically) that there is an image charge within the conductor. 39 ENGN6521 / ENGN4521: Embedded Wireless
40 Proximity to Floating (coupled) and Earthed Conductors (decoupled) 40 ENGN6521 / ENGN4521: Embedded Wireless
41 Magnetostatics: The static magnetic field Gauss s law for the magnetic field: B da = 0 A (29) There is no static sink or source of the magnetic field. Also generally true. However current is the source of a static magnetic field. Using Ampere s law (with no chaning E-field) B.dl = µ 0 I (30) γ The line integral of B around a closed circuit γ bounding a surface A is equal to the current flowing through A. Ampere s law 41 ENGN6521 / ENGN4521: Embedded Wireless
42 Important Correction to Ampere s Law A time varying Electric field is also a source of the time varying magnetic field, γ B.dl = ( µ 0 A ) E j+ǫ 0 t.da (31) 42 ENGN6521 / ENGN4521: Embedded Wireless
43 Why this Correction to Ampere s Law? I Ampere s law without E contradicts charge conservation γ B.dl = ( µ 0 A ) E j+ǫ 0 t.da (32) Consider the new Ampere s law on a close surface area, A. 43 ENGN6521 / ENGN4521: Embedded Wireless
44 Why this Correction to Ampere s Law? II If we shrink the closed contour γ on the left hand side to zero then we obtain ( ) E 0 = µ 0 j+ǫ 0.dA (33) t A However the two terms on the right hand side are E.dA = q A ǫ 0 (34) and A j.da = q t - Q.E.D. In fact it simply confirms how capacitors work!! (35) 44 ENGN6521 / ENGN4521: Embedded Wireless
45 Faraday s Law The law of electromagnetic induction or Lenz s / Faraday s law A time varying magnetic field is also the source of electric field, ˆ B E.dl =.da (36) t γ A 45 ENGN6521 / ENGN4521: Embedded Wireless
46 Inductance I circuite dl = A B t da = Φ B t Note that Kirchhoff s voltage law still works. I B field inductor V 46 ENGN6521 / ENGN4521: Embedded Wireless
47 Inductors II Magnetic field of a solenoid Consider Ampere s law for a solenoid with a static current I t = 0. If the solenoid is long and axially uniform then the magnetic field is directed along the axis. By choosing a curve along the axis of the solenoid, Ampere s law gives, ˆ d 0 dlb = µ o NI => B = µ oni d => B = µ o N l I (37) where µ o = 4π 10 7 Henriess/meter is the permeability of free-space, N is the number of turns enclosed by the contour and N l is the number of turns per unit length. 47 ENGN6521 / ENGN4521: Embedded Wireless
48 Inductance III Inductance of a coil 48 ENGN6521 / ENGN4521: Embedded Wireless
49 Inductance IV: Inductance of a coil A simple formula defining the inductance of a coil can be found by noting that if there are N turns on a coil then the total flux linking the coil is Φ = N B A From Faraday, the electromotive force (E.M.F.) at frequency ω generated by the changing flux is given by V = jω N B A Next we use the formula for magnetic field B of an infinite solenoid as an approximation for a coil (a solenoid of finite length) V = jω N (µ o N l I) A = jω ( µ on 2 A )I (38) d From which we define the inductance, L, V = jωli, L = µ on 2 A d (39) More accurate formula (lab2)... L(µH) = 0.394r2 N 2 9r +10d (40) 49 ENGN6521 / ENGN4521: Embedded Wireless
50 Ferromagnetic Materials Whereas a dielectric reduce E field with ǫ r 1, magnetic materials known as ferrites that can amplify an internal magnetic field. In Ferrites, magnetic domains align with the imposed field (see the figure below). If the above solenoid were wound on a ferromagnetic core, then the formulae for the B field and the inductance L would become B = µ r µ o N l I and L = µ rµ o N 2 A d 50 ENGN6521 / ENGN4521: Embedded Wireless
51 Inductors and Ferrites 51 ENGN6521 / ENGN4521: Embedded Wireless
52 A comparison between ferrites and dielectrics A dielectric decreases the imposed E-field whereas a ferromagnetic material increases the imposed B-field. ǫ r > 1 increases C because it decreases the voltage for a given charge on the capacitor electrodes. µ r > 1 increases L because it increases the magnetic flux for a given current, thus increasing the E.M.F. 52 ENGN6521 / ENGN4521: Embedded Wireless
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