ElectromagEng slides

Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 1 of 13 ELECTROMAGNETIC ENGINEERING EE325 INDEX Ampere's circuital law .....11 Ampere's law ................... 6 angstro m.......................... 2 Avogadro's number ........... 2 B Ampere's circuital law 11 Biot-Savart law ............... 11 Boltzmann's constant ........ 2 capacitance ................... 7, 8 between coaxial cylinders ................................ 7 between concentric spheres .................... 7 between parallel plates .7 between two conductors 7 characteristic impedance ..2 complex c onjugate ............ 1 complex notation .............. 1 conductance ..................... 8 conductivity ..................... 8 semiconductor .............. 8 conservative field law ....... 6 constants .......................... 2 continuity equation ........... 8 coordinate systems .......... 10 coordinate transformations 10 coulomb ........................... 1 Coulomb's law .................. 7 cross product ................... 10 curl................................ ..9 current ............................. 8 current density ................. 7 D flux density ................. 6 del................................ ...8 divergence ........................ 9 dot product ....................... 9 duality of J and D ............. 8 E electric field ................. 5 electric field ..................... 5 electron mass ................... 2 electron volt ..................... 2 electrostatic force ............................ 5 potential ...................... 5 electrostatics .................... 5 elipse ............................... 8 Faraday's law .............. 6, 12 flux density ...................... 6 force electrostatic ................. 5 magnetic ..................... 11 Gauss' law ........................ 6 geometry .......................... 8 grad operator .................... 8 H magnetic field intensity 12 impedance short -circuit ................. 2 induced voltage due to changing magnetic field ........................ 13 due to conductor motion 13 Faraday's law .............. 12 slider problem ............. 13 inductance ....................... 12 J current density .............. 7 joule ................................ 2 Laplacian ......................... 9 Lenz's law ....................... 12 light, speed of .................. 2 line impedance ................. 3 linkage ............................ 12 magnetic energy .............. 12 magnetic field ................. 11 at the center of a circular wire ........................ 11 central axis of a solenoid ............................... 11 due to a finite straight conductor ................ 11 due to an inf inite straight conductor ................ 11 magnetic field intensity ...12 magnetic flux .................. 12 magnetic force ................. 11 magnetization .................. 13 matching transformer inline – reactive load ....3 inline – resistive load ...3 mathematics ..................... 8 Maxwell's equations ......... 6 mutual inductance ........... 12 nabla operator .................. 8 permeability ..................... 2 permittivity ...................... 2 phase constant .................. 2 Planck's constant .............. 2 Poisson's equation ............ 6 potential energy ................ 7 power with phasor notation .....5 reactance .......................... 3 reflection coefficient ......... 2 resistance ......................... 8 Rydberg constant .............. 2 self-inductance ................ 12 series stub ........................ 4 shunt stub ........................ 4 singl e-stub tuning ............. 4 Smith chart ...................... 4 Smith charts ..................... 4 space derivative ............... 8 sphere .............................. 8 standing wave ratio .......... 4 static magnetic field ........ 11 stub length ....................... 4 surface charge density ......6 time average power .......... 5 vector differential equation 8 volume energy density ......7 wave forward -traveling ......... 5 wave equation .................. 2 wavelength ....................... 2 We potential energy ......... 7 we volume energy density 7 X reactance ..................... 3 Zin line impedance ........... 3 Φ electrostatic potential ...................... 5 Γ reflection coefficient ....2 YY magnetic flux ............. 12 λ wavelength ................... 2 ρs surface charge density .6 σ conductivity ................. 8 ∇ del............................... 8 ∇× curl ........................... 9 ∇· divergence ................ 9 ∇2 Laplacian ................... 9 COULOMB [C] A unit of electrical charge equal to one amp second, the charge on 6.21×1018 electrons, or one joule per volt. COMPLEX NOTATION ) (ba aejb∠= where b may be in radians or degrees (if noted). COMPLEX CONJUGATES The complex conjugate of a number is simply that number with the sign changed on the imag inary part. This applies to both rectangular and polar notation . When conjugates are multiplied, the result is a scalar. 2 2) )( ( b a jbajba +=− + 2) )( ( A B ABA =°−∠°∠ Other properties of conjugates : *) ** ***()* ( F ED CBA F DE ABC + + = + + jB jBe e+ −=)* ( Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 2 of 13 TRANSMISSION LINES GGL REFLECTION COEFFICIENT [V/V] The reflection co efficient is a value from –1 to +1 which, when multiplied by the wave voltage, determines the amount of voltage reflected at one end of the transmission line. 00 Z ZZ Ze LL j L +−=ρ=Γψ and LL LZ ZΓ−Γ+=11 0 where: LZ is the load impedance LΓ is the load reflection coefficient ρ is the reflection coefficient magnitude ψ is the reflection coefficient phase CLZ=0 is the characteristic impedance THE COMPLEX WAVE EQUATION The complex wave equation is applicable when the excitation is sinusoidal and the circuit is under steady state conditions. )()( 2 22 zV zdzVdβ−= where 2LCπβ=ω=λ is the phase constant The complex wave equation above is a second -order ordinary differential equation commonly found in the analysis of physical systems. The general solution is: zj zjeV eVzVβ+− β−++ =)( where zjeβ− and zjeβ+ represent wave propagation in the +z and –z directions respectively. The same equation applies to current: zj zjeI eI zIβ+− β−++ =)( and 0)(ZeV eVzIzj zj β+− β−++= where 0/ ZLC= is the characteristic impedance of the line. These equations represent the voltage and current phasors . SHORT -CIRCUIT IMPEDANCE [Ω] ()l jZ Zscβ =tan0 where: 0Z is the characteristic imped ance λπ= ω=β2LC is the phase constant l is the length of the line [ m] CONSTANTS Avogadro’s number [molecules/mole] 231002.6 ×=AN Boltzmann’s constant 231038.1−×=k J/K 51062.8−×= eV/K Elementary charge 191060.1−×=q C Electron mass 31 01011.9−×=m kg Permittivity of free space 12 01085.8−×=ε F/m Permeability constant 7 0104−×π=µ H/m Planck’s constant 341063.6−×=h J-s 151014.4−×= cV-s Rydberg constant 678,109=R cm-1 kT @ room temperature 0259.0=kT eV Speed of light 810 998.2 × =c m/s 1 Å (angstrom) 10-8 cm = 10-10 m 1 µm (micron) 10-4 cm 1 nm = 10 Å = 10-7 cm 1 eV = 1.6 × 10-19 J 1 V = 1 J/C 1 N/C = 1 V/m 1 J = 1 N·m = 1 C·V ll WAVELENGTH [m] fvp =λ vp = velocity of propagation ( 2.998×108 m/s for a line in air) f = frequency [ Hz] Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 3 of 13 ¼ -WAVELENGTH INLINE MATCHING TRANSFORMER – resistive load For use with a purely resistive load that does not match the line impedance. The load is matched to the line by inserting a ¼ -wavelength segment having a characteristic impedance ZQ. Z0ZQλ/4 RL L QRZ Z 0= Z0 = characteristic impedance of the transmission line [Ω] λ = wavelength [meters ] RL = resistance of the load [Ω] ZQ = characteristic impedance of the ¼-wave matching segment [ Ω] ¼ -WAVELENGTH INLINE MATCHING TRANSFORMER – reactive load For use with a react ive load. The load is matched to the line by inserting a ¼ -wavelength segment having a characteristic impedance ZQ at a distance l from the load. l is the length of transmission line required to produce the first voltage maximum —closest to the load. I f the load is inductive, the first voltage maximum will be closer than the first voltage minimum, i.e. within ½ wavelength. 0Z0Zλ/4 ZQl Zin LZ First find the reflection coefficient in order to determine the value of ψ. Then find the length l of the line that will convert the load to a pure resistance, i.e. produces the first voltage maximum. Find this resistance ( Zin) using the line impedance formula. Then determine the impedance ZQ of the ¼ -wavelength segment tha t will match the load to the line. 00 Z ZZ Ze LL j L +−=ρ=Γψ i.e. ψ∠ρ=ΓL (radians) πψλ=βψ=4 2l l jZ Zl jZ ZZ Z LL in β +β +=tantan 00 0 in QZZ Z 0= ΓL is the load reflection coefficient ψ = phase of the reflection coefficient [radia ns] ρ = magnitude of the reflection coefficient [Ω] Z0 = characteristic impedance [Ω] λπ=β /2 λ = vp/f wavelength [m] Zin = impedance (resistive) of the load combined with the l segment [Ω] ZQ = line impedance of the ¼ -wave matching segme nt [Ω] X REACTANCE [Ω] CjXC ω−= Lj XLω= XC = reactance [Ω] XL = reactance [Ω] j = 1− ω = frequency [ radians ] C = capacitance [F] L = inductance [H] Zin LINE IMPEDANCE [Ω] l jZ Zl jZ ZZ Z LL in β +β +=tantan 00 0 l = distance from load [m] j = 1− β = phase constant Z0 = characteristic impedance [Ω] ZL = load impedance [Ω] The line impedance of a ¼ -wavelength line is the inverse of the load impedance. Impedance is a real value when its magnitude is maximum or minimum. ρ−ρ+==11 0 0 maxZSZ Z ρ+ρ−==11 00 minZSZZ Z0 = characteristic impedance [Ω] S = standing wave ratio ρ = magnitude of the reflection coefficient Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 4 of 13 SMITH CHARTS First normalize the load impedance by dividing by the characteristic impedance, and find this point on the chart. An inductive load will be located on the top half of the chart, a capacitive load on the bottom half. Draw a straight line from the center of the char t through the normalized load impedance point to the edge of the chart. Anchor a compass at the center of the chart and draw an arc through the normalized load impedance point. Points along this arc represent the normalized impedance at various points alo ng the transmission line. Clockwise movement along the arc represents movement from the load toward the source with one full revolution representing 1/2 wavelength as marked on the outer circle. The two points where the arc intersects the horizontal axis are the voltage maxima (right) and the voltage minima (left). Points opposite the impedance (180° around the arc) are admittance . The reason admittance is useful is because admittances in parallel are simply added. zj Le zβΓ=Γ2)( z ezjβ∠=β212 ()1()()1zz z−Γ= +Z Z 11 +Γ−Γ= LL LZ 0ZZL=Z z = distance from load [m] j = 1− ρ = magnitude of the reflection coefficient β = phase constant Γ = reflection coefficient Z = normalized impedance [ Ω] SINGLE -STUB TUNING The basic idea is to connect a line stu b in parallel (shunt) or series a distance d from the load so that the imaginary part of the load impedance will be canceled. Shunt -stub: Select d so that the admittance Y looking toward the load from a distance d is of the form Y0 + jB. Then the stub susceptance is chosen as –jB, resulting in a matched condition. Y Open or short lY00d Y0 YL Series -stub: Select d so that the admittance Z looking toward the load from a distance d is of the form Z0 + jX. Then the stub susceptance is chosen as -jX, resulting in a matched conditio n. LZ l0Z Open or short0Zd 0Z FINDING A STUB LENGTH Example: Find the lengths of open and shorted shunt stubs to match an admittance of 1-j0.5. The admittance of an open shunt (zero length) is Y=0; this point is located at the left end of the Smith Chart x-axis. We proceed clockwise around the Smith chart, i.e. away from the end of the stub, to the +j0.5 arc (the value needed to match –j0.5). The difference in the starting point and the end point on the wavelength scale is the length of the stub in wavelengths. The length of a shorted -type stub is found in the same manner but with the starting point at Y=¥¥. rota gerne rdawoT Admittance (short) Admittance (open) Shorted stub of length .324 matches an admittance of 1-j.5λ .46λ.324 .47 .48 .49.43 .44 .45Y 1.0.42.4.41.38 .390.5= 0j .06 .04 0.01.02.03λ.074 0.1.05Open stub of length .074 matches an admittance of 1-j.5λ .070.5 0.5 1.0.1 .08.09.5 1.0.11.12.33.35.36.37 .342.0 .29.3.31.325.0 .26.27.285.17 2.0 2.15.14 .13 .16 .19 .21Y 5.0.2 .23.25.24.22∞=.18 In this example, all values were in units of admittance. If we were interested in finding a stub length for a series stub problem , the units would be in impedance. The problem would be worked in exactly the same way. Of course in impedance, an open shunt (zero length) would have the value Z=¥¥, representing a point at the right end of the x-axis. SWR STANDING WAVE RATIO [V/V] ρ−ρ+= = =11SWR minmax minmax II VV Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 5 of 13 P(z) TIME -AVERAGE POWER ON A LOSSLESS TRANSMISSION LINE [W] Equal to the power delivered to the load. The power delivered to the load is maximized under matched conditions, i.e. ρ = 0, otherwise part of the power is reflected back to the source. To calculate power, it may be simpler to find the input impedance and use P = I2R or P = IV. ()2 02 12)(P ρ− =+ ZV z [] { }1P()()()*2zVzIz=Re V+ = the voltage of the forward -traveling wave [V] Z0 = characteristic impedance [Ω] ρ = magnitude of the reflection coefficient Re = "the real part" POWER USING PHASOR NOTATION [W] *21VI S= S = power [W] V = volts [V] I* = complex co njugate of current [A] V+ FORWARD -TRAVELING WAVE ( )( )lj Llj S inin e eZ ZVZVβ− β+ Γ+ +=20 1 V+ = the voltage of the forward - traveling wave [V] V0 = source voltage [V] Zin = input impedance [Ω] ZS = source impedance [ Ω] β = phase constant l = length of the line [m] ΓL = load reflection coefficient ELECTROSTATICS F ELECTROSTATIC FORCE 3 1 21 2 21 012) ( 41 rrrrF −− πε= QQ 9 010941×= πε F12 = the force exerted by charge Q1 on Q2. [N] r1 = vector from the origin to Q1 r2 = vector from the origin to Q2. When finding the force on one charge due to multiple charges, the result can be found by summing the effects of each charge separately or by converting the multiple c harges to a single equivalent charge and solving as a 2 -charge problem. E ELECTRIC FIELD ∑= ′−′− πε=n kkk k pQ 13 041 rrrrE ()ldrdl′ ′−′ρ πε=2 0ˆ 41 rrR E () ∫ ′ ′−′ρ πε= ldrl 2 0ˆ 41 rrR E Electric field from a potential: Φ−∇=E refer to the NABLA notes on page 8. *NOTE: The l symbols could be replaced by a symbol for area or volume. See Working With … on page 9. Ep = electric field at point p due to a charge Q or charge density ρ [V/m] dE = an increment of electric field [ V/m] Q = electric charge [ C] ε0 = permittivity of free space 8.85 × 10-12 F/m ρl = charge density; charge per unit length* [ C/m] dl' = a small segment of line l* Rˆ = unit vector pointing from r' to r , i.e. in the direction of r - r'. r' = vector location of the source charge in relation to the origin r = vector location of the point at which the value of Ep is observed ∇ = Del, Grad, or Nabla operator FF ELECTROSTATIC POTENTIAL [V] ∑=′− πε=Φn k kkQ 1 041 rr rr′−′ρ πε=Φlddl 041 ldl′′−ρ πε=Φ∫rr041 Potential due to an electric field: ∫−=Φb aabdlE· To evaluate voltage at all points. ()∫∞−=Φr d r lE· *NOTE: The l symbols could be replaced b y a symbol for area or volume. See Working With … on page 9. Φ = the potential [ V] dΦ = an increment of potential [V] Φab = the potential difference between points a and b [V] E = electric field dl' = a small segment of line l* dl = the differential vector displacement along the path from a to b ε0 = permittivity of free space 8.85 × 10-12 F/m Q = electric charge [ C] ρl = charge density along a line* [ C/m] rk' = vector location of source charge Qk r' = vector location of the source charge in relation to the origin r = vector location of electrostatic potential Φ in relation to the origin Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 6 of 13 MAXWELL'S EQUATIONS Maxwell's equations govern the principles of guiding and propagation of electromagnetic energy and provide the foundat ions of all electromagnetic phenomena and their applications. t∂∇×∂BE=- Faraday's Law ∇⋅ρD= Gauss' Law t∂∇×+∂DH=J Ampere's Law* 0 ∇⋅B= no name law, where: E = electric field [V/m] B = magnetic field [T] t = time [s] D = electric flux density [C/m2] ρ = volume charge density [C/m3] H = magnetic field intensity [A/m] J = current density [A/m2] *Maxwell added the t∂ ∂D term to Ampere's Law. POISSON'S EQUATION 02 ερ−=Φ∇ rrs SURFACE CHARGE DENSITY [C/m2] n sE0ε=ρ En·ˆ=nE ε0 = permittivity of free space 8.85 × 10- 12 F/m En = electric field normal to the surface [ V/m] D FLUX DENSITY [C/m2] or ELECTRIC DISPLACEMENT PER UNIT AREA 24ˆ rQ π≡rD E Dε= Q = electric charge [ C] ε = dielectric constant rεε=ε0 E = electric field [ V/m] GAUSS'S LAW The net flux passing through a surface enclosing a charge is equal to the charge. Careful, what this first integral really means is the surface area multiplied by the perpendicular electric field. There may not be any integration involved. encSQd=ε∫sE·0 ∫∫ =ρ=VencSQ dv dsD· ε0 = permittivity of free space 8.85 × 10-12 F/m E = electric field [ V/m] D = electric flux density vector [ C/m2] ds = a small increment of surface S ρ = volume charge density [ C/m3] dv = a small increment of volume V Qenc = total electric charge enclosed by the Gaussian surface [S] The differential version of Gauss's law is: ρ= ∇D· or ( )ρ=εE· div 0 GAUSS'S LAW – an example problem Find the intensity of the electric field at distance r from a straight conducto r having a voltage V. Consider a cylindrical surface of length l and radius r enclosing a portion of the conductor. The electric field passes through the curved surface of the cylinder but not the ends. Gauss's law says that the electric flux passing through this curved surface is equal to the charge enclosed. VlCl QdlrE d l l enc rS=ρ==φ ε=ε∫ ∫π2 00 0·sE so VCdrE l r=φ ε∫π2 00 and rVCEl r 02πε= Er = electric field at distance r from the conductor [ V/m] l = length [ m] r dff = a small increment of the cylindrical surface S [m2] ρl = charge density per unit length [ C/m] Cl = capacitance per unit length [ F/m] V = voltage on the line [V] CONSERVATIVE FIELD LAW 0=×∇E 0 ·=∫SdlE E = electric field [ V/m] ds = a small increment of length Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 7 of 13 COULOMB'S LAW ρ= ∇D· ∫∫ ρ=V Sdv dsD· D = electric flux density vector [ C/m2] ρ = volume charge density [ C/m3] ds = a small increment of surface S We POTENTIAL ENERGY [J] The energy required to bring charge q from infinity to a distance R from charge Q. RQqq We πε=Φ=4 ∫ ∫=Φρ=V Vedv dv W ED·21 21 Φ = the potential between q and Q [V] q,Q = electric charges [ C] ε = permittivity of the material R = distance [ m] ρ = volume charge density [ C/m3] E = electric field [ V/m] D = electric flux density vector [C/m2] we VOLUME ENERGY DENSITY [J/m3] for the Electrostatic Field 2 21·21E weε= =ED Φ = the potential between q and Q [V] ε = permittivity of the material R = distance [ m] E = electric field [ V/m] D = electric flux density vector [C/m2] CAPACITANCE C CAPACITANCE [F] Φ=QC VCl lρ = Q = total electric charge [C] Φ = the potential between q and Q [V] Cl = capacitance per unit length [ F/m] ρl = charge density per unit length [ C/m] V = voltage on the line [V] C CAPACITANCE BETWEEN TWO PARALLEL SOLID CYLINDRICAL CONDUC TORS This also applies to a single conductor above ground, where the height above ground is d/2. ()adC/lnπε= , where da? or 1cosh2Cd a−πε = C = capacitance [F/m] ε = permittivity of the material d = separation (center-to- center) [ m] a = conductor radius [ m] C CAPACITANCE BETWEEN PARALLEL PLATES ACdε= C = capacitance [ F] ε = permittivity of the material d = separation of the plates [ m] A = area of one p late [m2] C CAPACITANCE BETWEEN COAXIAL CYLINDERS ()2 ln/Cbaπε= C = capacitance [ F/m] ε = permittivity of the material b = radius of the outer cylinder [m] a = radius of the inner cylinder [m] C CAPACITANCE OF CONCENTRIC SPHERES 4abCbaπε=− C = capacitance [ F/m] ε = permittivity of the material b = radius of the outer sphere [ m] a = radius of the inner sphere [m] J CURRENT DENSITY The amount of current flowing perpendicularly through a unit area [ A/m2] E Jσ= ∫=Sd I sJ· In semiconductor material: decqnv J= σ = conductivity of the material [S/m] E = electric field [ V/m] I = current [ A] ds = a small increment of surface S nc = the number of conduction band electrons qe = electron charge -1.602×10-19 C vd = a small increment of surface S Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 8 of 13 CONTINUITY EQUATION 0 · =∂ρ∂+∇tJ J = current de nsity [ A/m2] E Jσ= ρ = volume charge density [C/m3] DUALITY RELATIONSHIP of J and D RESISTANCE, CAPACITANCE, CURRENT, CONDUCTIVITY Where current enters and leaves a condu cting medium via two perfect conductors (electrodes) we have: εσ=εσ= σ= =∫∫∫Qd d d IS S SsD sE sJ · · · J = current density [ A/m2] E Jσ= E = electric field [ V/m] D = electric flux density vector [ C/m2] E Dε= As a result of this, we have the following relation, useful in finding the resistance between two conductors: σε=RC R = resistance [ Ω] C = capacitance [ F] ε = permittivity of the material σ = conductivity of the material [S/m] G CONDUCTANCE [Ω−1] ΔΦ==I RG1 ∫∫− +σ = lEsE ddS ·· R = resistance [ Ω] I = current [ A] ΔΦ = voltage potential [V] σ = conductivity of the material [S/m] ss SEMICONDUCTOR CONDUCTIVITY [Ω−1] deNqµ≈σ σ = conductivity of the material [S/m]G = conductance [ Ω−1] q = electron charge -1.602×10-19 C µe = electron mobility [m2/(V-s)] Nd = concentration of donors, and thereby the electron concentration in the transition region [m-3] MATHEMATICS WORKING WITH LINES, SURFACES, AND VOLUMES ρl(r') means "the charge density along line l as a function of r'." This might be a value in C/m or it could be a function. Similarly, ρs(r') would be the charge density of a surface and ρv(r') is the charge density of a volume. For example, a disk of r adius a having a uniform charge density of ρ C/m2, would have a total charge of ρπa2, but to find its influence on points along the central axis we might consider incremental rings of the charged surface as ρs(r') dr'= ρs2πr' dr'. If dl' refers to an incre mental distance along a circular contour C, the expression is r'dff, where r' is the radius and dff is the incremental angle. GEOMETRY SPHERE Area 24r A π= Volume 3 34r V π= ELLIPSE Area AB Aπ= Circumference 222 2b aL+π≈ Ñ Ñ NABLA, DEL OR GRAD OPERATOR [+ m-1] Compare the ∇ operation to taking the ti me derivative. Where ∂/∂t means to take the derivative with respect to time and introduces a s-1 component to the units of the result, the ∇ operation means to take the derivative with respect to distance (in 3 dimensions) and introduces a m-1 component t o the units of the result. ∇ terms may be called space derivatives and an equation which contains the ∇ operator may be called a vector differential equation . In other words ∇A is how fast A changes as you move through space. in rectangular coordinates: ˆˆ ˆAAAxyzxyz∂∂∂∇=++∂∂∂A in cylindrical coordinates: 1ˆ ˆ ˆAAArzrrz∂∂∂∇=+φ+∂∂φ∂A in spherical coordinates: 11ˆˆ ˆ sinAAArrrr∂∂∂∇=+θ+φ∂∂θθ∂φA Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 9 of 13 ÑÑ2 THE LAPLACIAN [+ m-2] in rectangular coordinates: 0 ˆ ˆ ˆ2 2 2 2=∇+∇+∇=∇ z y xA A A z y xA 022 22 22 2=∂∂+∂∂+∂∂≡∇z y x in spherical and cylindrical coordinates: ( ) ()()A AA A A curl curl div grad·2 − =×∇×∇−∇∇≡∇ for example, the Laplacian of electro - static potential: 022 22 22 2=∂Φ∂+∂Φ∂+∂Φ∂=Φ∇z y x ÑÑ· DIVERGENCE [+ m-1] The del operator followed by the dot product operator is read as "the divergence of" and is an operation performed on a vector. In rectangular coordinates, ∇⋅ means the sum of the partial derivatives of the magnitudes in the x, y, and z directions with respect to the x, y, and z variables. The result is a scalar, and a factor of m-1 is contributed to the units of the result. For example, in this form of Gauss' law, where D is a density per unit area, ∇⋅D becomes a density per unit volume. divy x zD D D xyz∂∂ ∂=∇⋅=++=ρ∂∂∂DD D = electric fl ux density vector D = εE [C/m2] ρ = source charge density [C/m3] In the electrostatic context, the divergence of D is the total outward flux per unit volume due to a source charge. The divergence of vector D is: in rectangular coordinates: zD yD xD z y x ∂∂+∂∂+∂∂=Ddiv in cylindrical coordinates: ()zD D rrDrrz r ∂∂+φ∂∂+∂∂=φ 1 1divD in spherical coordinates: ( ) ( ) φ∂∂ θ+θ∂θ∂ θ+∂∂=φ θD rD r rDr rr sin1 sin sin1 1div2 2D ÑÑ× CURL [+ m-1] The circulation around an enclosed area. The curl of vector B is in rectang ular coordinates: curl ˆˆ ˆyy xx zzBB BB BBxyzyzzxxy=∇×= ∂∂  ∂∂ ∂∂ −+−+− ∂∂∂∂∂∂  BB in cylindrical coordinates: ()curl 11ˆ ˆ ˆzrzrrB B BBBBrzrzzrrrφ φ=∇×= ∂ ∂ ∂∂∂∂ −+φ−+−   ∂φ∂∂∂∂∂φ   BB in spherical coordinates: ( ) () ()sin 1ˆ curlsin 111ˆˆ sinrrB Brr rB rB BB rrrrφθ φ θ∂θ ∂=∇×=−+ θ∂θ∂φ  ∂ ∂ ∂∂θ−+φ− θ∂φ∂∂∂θ  BB The divergence of a curl is always zero: ( )0 · =×∇∇H DOT PRODUCT [= units2] The dot product is a scalar value. ( )( ) zz yy xx z y x z y xBA BA BA B B B A A A + + = + + + + = z y x z y x BA ˆ ˆ ˆ•ˆ ˆ ˆ • ABcos • ψ =BA BA 0ˆ•ˆ =yx , 1ˆ•ˆ =xx ( )y z y xB B B B = ++= y z y x yB ˆ•ˆ ˆ ˆ ˆ• ψB A A•B Projection of B along â: ( )aaB ˆˆ• B ψ â âψB The dot product of 90° vectors is zero. The dot product is commutative and distributive : ABBA • •= ( ) CABA CBA • • • + =+ Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 10 of 13 CROSS PRODUCT ( )( ) ( )( )( )xy yx zx xz yz zyz y x z y x BA BA BA BA BA BAB B B A A A − + − + − =++×++=× z y xz y x z y x BA ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ABsin ˆ ψ =× BAnBA where nˆ is the unit vector normal to both A and B (thumb of right -hand rule). BA AB ×−=× zyx =× z xy −=× 0=×xx φ×=zr φ×=−rz The cross product is distributive: ( ) CABA CB A ×+×=+× Also, we have: ( )( ) ( ) ××=⋅−⋅ABCACBABC n ψA×B A B COORDINATE SYSTEMS Cartesian or Rectangular Coordinates: ˆˆ ˆ (,,)xyzxxyyzz =++ r ˆx is a unit vector 2 2 2z y x ++=r Spherical Coordinates: ),,( φθrP r is distance from center θ is angle from vertical φ is the CCW angle from the x-axis ˆr, ˆθ, and ˆφ are unit vectores and are functions of position —their orientation depends on where they are located. Cylindrical Coordinates: ),,( zrφ C r is distance from the vertical ( z) axis φ is the CCW angle from the x-axis z is the vertical distance from origin COORDINATE TRANSFORMATIONS Rectangular to Cylindrical: To obtain: ˆ ˆ ˆ (,,) rz rzrAAzA φφ=+φ+A 2 2y x Ar+= ˆˆˆ cossin rxy=φ+φ xy1tan−=φ ˆ ˆˆsincosxy φ=−φ+φ zz= ˆˆzz= Cylindric al to Rectangular: To obtain: ˆˆ ˆ (,,)xyzxxyyzz =++ r φ =cosrx ˆ ˆˆ coscos xr=φ−φφ φ =sinry ˆˆˆsincosryφ=φ+φ zz= ˆˆzz= Rectangular to Spherical: To obtain: ˆˆ ˆ (,,) rrrAAAθφθφ=+θ+φA 2 2 2z y x Ar++= ˆˆˆ ˆ sincossinsincos rxyz=θφ+θφ+θ 2 2 21cos z y xz ++=θ− ˆˆˆ ˆ coscoscossinsinxyzθ=θφ+θφ−θ xy1tan−=φ ˆ ˆˆsincosxy φ=−φ+φ Spherical to Rectangular: To obtain: ˆˆ ˆ (,,)xyzxxyyzz =++ r φ θ = cos sinrx ˆˆ ˆˆ sincoscoscossin xr=θφ−θθφ−φφ φ θ = sin sinry ˆˆ ˆˆ sinsincossincos yr=θφ+θθφ+φφ θ =cosrz ˆ ˆˆ cossin zr=θ−θθ Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 11 of 13 THE STATIC MAGNETIC FIELD F F12 MAGNETIC FORCE [N/m] due to a conductor If the current in the two wires travels in opposite directions, the force will be attractive. dII πµ=2ˆ 210 12xF F12 = the force exerted by conductor 1 carrying current I on conductor 2. [N/m] µ0 = permeability constant 4 π×10-7 [H/m] I = current [A] d = distance b etween conductors [m] BP BIOT -SAVART LAW Determines the B field vector at any point P identified by the position vector r, due to a differential current element I dl' located at vector r'. 20 4ˆ' RdIdP π×µ=RlB ( ) ∫−−× πµ=CPd I 30 '' ' 4 rrrrlB ''ˆ rrrrR−−= BP = magnetic field vector [T] µ0 = permeability constant 4π×10-7 [H/m] I dl' = current element [ A] Rˆ = unit vector pointing from the current element to point P R = distance between the current element and point P [m] B AMPERE'S CI RCUITAL LAW Ampere's law is a consequence of the Biot-Savart law and serves the same purpose as Gauss's law . Ampere's law states that the line integral of B around any closed contour is equal to µ0 times the total net current I passing through the surface S enclosed by the contour C. This law is useful in solving magnetostatic problems having some degree of symmetry. Id dS C 00· · µ=µ=∫∫ sJ lB B = magnetic field vector, equal to B times the appropriate unit vector [ T] µ0 = permeability constant 4 π×10-7 [H/m] dl = an increment of the line which is the perimeter of contour C [m] J = current density [ A/m2] E Jσ= ds = an increment of surface [ m2] B MAGNETIC FIELD [T or A/m ] due to an infinite straight conductor May also be applied to the magnetic field close to a conductor of finite length. 0 ˆ 2PI rµ=φπB BP = magnetic field vector [ T] µ0 = permeability constant 4 π×10-7 [H/m] I = current [ A] r = perpendicular distance from the conductor [ m] B MAGNETIC FIELD [T] due to a finite straight conductor at a point perpendicular to the midpoint 0 22ˆ 2PIa rraµ=φ π+B a r I BP = magnetic field vector [ T] µ0 = permeability constant 4π×10-7 [H/m] I = current [ A] a = half the length of the conductor [ m] r = perpendicular distance from the conductor [ m] B MAGNETIC FIELD [T] at the center of a circular wire of N turns aNIBctr 2ˆ0µ=z B = magnetic field [ T] µ0 = permeability const. 4 π×10-7 [H/m] N = number of turns of the coil I = current [ A] a = radius [ m] B MAGNETIC FIELD [T] along the central axis of a solenoid ()() ()() ()    −+−− +++ µ=2 2 2 20 2/2/ 2/2/ 2ˆ lz alz lz alz lNIzB z and at the center of the coil: lNIBctr0ˆµ≈z B = magnetic field [ T] µ0 = permeability constant 4π×10-7 [H/m] N = number of turn s I = current [ A] l = length of the solenoid [ m] z = distance from center of the coil [ m] a = coil radius [ m] Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 12 of 13 H MAGNETIC FIELD INTENSITY [A/m] The magnetic field intensity vector is directly analogous to the electric f lux density vector D in electrostatics in that both D and H are medium - independent and are directly related to their sources. MBH −µ≡ 0 t∂∂+=×∇DJ H H = magnetic field [ A/m] B = magnetic field vector [ T] µ0 = permeability const. 4 π×10-7 [H/m] M = magnetization [ A/m] J = current density [ A/m2] E Jσ= D = electric flux density vector [C/m2] Y, LY, L (,lambda) MAGNETIC FLUX, LINKAGE Flux linkage Λ is the a bility of a closed circuit to store magnetic energy . It depends, in part, on the physical layout of the conductors. It is the total magnetic field due to circuit #1 passing through the area enclosed by the conductors of circuit #2. The text seemed to de scribe Ψ as the flux due to one turn and Λ as the flux due to all of the turns of the coil, but was not consistent so be careful. ∫=Ψ22 1 12·SdsB 121 12Ψ=ΛN ∫=ΛSd N sB· Ψ12 = the magnetic flux passing through coil 2 that is produ ced by a current in coil 1 [ Wb] Λ = total flux linkage [ Wb] B = magnetic field vector [ T] N = number of turns of the coil ds = an increment of surface [ m2] LENZ'S LAW Induced voltage causes current to flow in the direction that produce s a magnetic flux which opposes the flux that induced the voltage in the first place. This law is useful in checking or determining the sign or polarity of a result. L INDUCTANCE [H] Inductance is the ability of a conductor configu ration to "link magnetic flux", i.e. store magnetic energy. Two methods of calculating inductance are given below. ILΛ= 22 IWLm= Λ = flux linkage [ Wb] I = current [ A] Wm = energy stored in a magnetic field [J] L11 SELF -INDUCTANCE [H] When a current in coil 1 induces a current in coil 2, the induced current in coil 2 induces a current back in coil 1. Thi s is self -inductance. 1112 11111 11 ININL Ψ=Λ= N = number of turns of the coil Λ11 = the total flux linked by a single turn of coil 1 [ Wb] I1 = current in coil 1 [ A] Ψ11 = the magnetic flux produced by a single turn of coil 1 and linked by a single turn of coil 1 [ Wb] L12 MUTUAL INDUCTANCE [H] The mutual inductance between two coils. 1121 2 112 2 12 INN INLΨ=Λ= Neumann formula: ∫∫− πµ=1 2 '· 42 1 2 1 0 12CCdd NNLrrll N = number of turns of the coil Λ = flux linkage [ Wb] I = current [ A] Ψ = magne tic flux [ Wb] r = vector to the point of observation r' = vector to source Wm MAGNETIC ENERGY [J] Energy stored in a magnetic field [Joules]. ∫µ=VmdvB W '21 2 0 Wm = energy stored in a magnetic field [ J] µ0 = permeability constant 4π×10-7 [H/m] B = magnetic field [ T] FARADAY'S LAW When the magnetic flux enclosed by a loop of wire changes with time, a current is produced in the loop. The variation of the magnet ic flux can result from a time-varying magnetic field, a coil in motion, or both. t∂∂−=×∇BE ∇×E = the curl of the electric field B = magnetic field vector [ T] Another way of expressing Faraday's law is that a changing magnetic field induces an electric field. ··indCSdVdddt==−∫∫ElBsÑ where S is the surface enclosed by contour C. (see also Induced Voltage below) Tom Penick [email protected] www.teicontrol s.com/notes 10/22/2000 Page 13 of 13 Vind INDUCED VOLTAGE The voltage induced in a coil due to a changing magnetic field is equal to the number of turns in the coil times the rate at which the magnetic field is changing (could be a change in field strength or coil area normal to the field). dtdN VindΨ −= ∫=Cindd V lE· N = number of turns of t he coil Ψ = the magnetic flux produced by a single turn of the coil [ Wb] Vind INDUCED VOLTAGE DUE TO MOTION When conductors move in the presence of magnetic fields, an induced voltage is produced even if the magnetic fields do not vary in time. For the voltage produced due to both a changing magnetic field and a conductor in motion: () ·· indSCVdd t∂ =−+×∂∫∫BsvBlÑ B = magnetic field vector [ T] v = velocity vector of the conductor [ m/s] ds = increment of the surfa ce normal to the magnetic field vector [ m2] dl = incremental length of conductor [m] INDUCED VOLTAGE – SLIDER PROBLEM A frictionless conducting bar moves to the right at velocity v produces a current I. RI 0B vd h An expandi ng magnetic field area having a static magnetic field directed into the page produces a CCW current. 0 indVBhv= 0ˆ magBIh=Fx 2dEIRv= Vind = induced voltage [ V] B0 = static magnetic field [ T] h = distance between the conductor rails [T] v = velocity of the conductor [ m/s] Fmag = magnetic force opposing slider [N] ˆx = unit vector in the direction against conductor movement [ m/s] I = current [ A] E = energy produced [ J or W/s ] R = circuit resistance [ Ω] d = distance the conductor moves [m] M MAGNETIZATION [A/m] The induced magnetic dipole moment per unit volume. ee maNq 422BM −= or 0µχ=BMm where ee m maNq 4022µ−=χ N = number of turns of the coil qe = electron charge - 1.602×10-19 C a = orbit radius of an electron [ m] B = magnetic field vector [ T] µ0 = permeability constant 4 π×10-7 [H/m] me = who knows? χm = magnetic susceptibility