Electromagnetic Theory
Cartesian Coordinate System
x y z axes, unit vectors, position vector.
Cylindrical Coordinate System
rho phi z, transformation to Cartesian.
Spherical Coordinate System
r theta phi, transformation to Cartesian.
Gradient
Del V, direction of maximum rate of increase, normal to surface.
Divergence
Del dot A, source or sink of vector field, flux density.
Laplacian Operator
Del squared, second order differential operator.
Divergence Theorem
Volume to surface integral conversion, Gauss theorem.
Helmholtz Theorem
Decomposition into irrotational and solenoidal parts.
Coulomb Law
Force between point charges, superposition principle.
Electric Field Intensity
E = F/q, field from point charges and distributions.
Electric Field of Line Charge
Infinite line charge, E = rho_L/(2*pi*epsilon*r).
Electric Field of Surface Charge
Infinite sheet, E = rho_S/(2*epsilon), uniform field.
Electric Flux Density
D = epsilon*E, displacement vector, flux concept.
Gauss Law
Total flux through closed surface equals enclosed charge.
Gauss Law Applications
Sphere, cylinder, infinite plane using symmetry.
Electric Potential
V = -integral E dot dL, reference at infinity.
Potential and Field Relationship
E = -grad V, equipotential surfaces.
Electric Dipole
Dipole moment p = Qd, potential and field of dipole.
Boundary Conditions Electrostatics
Tangential E and normal D conditions at interface.
Capacitance
C = Q/V, parallel plate, coaxial, spherical capacitors.
Energy Stored in Electric Field
W = (1/2)epsilon*E² per unit volume, total energy.
Poisson and Laplace Equations
Del²V = -rho/epsilon and Del²V = 0, uniqueness.
Biot-Savart Law
dH = IdL cross aR / (4*pi*R²), magnetic field from current.
Magnetic Field of Infinite Wire
H = I/(2*pi*r), concentric circles.
Magnetic Field of Circular Loop
Field on axis, magnetic moment m = NIA.
Ampere Circuital Law
Closed line integral of H = enclosed current.
Ampere Law Applications
Infinite wire, solenoid, toroid, coaxial cable.
Magnetic Flux Density
B = mu*H, magnetic flux phi = integral B dot dS.
Magnetic Force on Current
F = IL cross B, force between parallel conductors.
Lorentz Force
F = q(E + v cross B), charged particle in EM fields.
Inductance
Self inductance L = N*phi/I, mutual inductance M.
Inductance Calculations
Solenoid, toroid, coaxial cable inductance formulas.
Magnetic Energy
W = (1/2)L*I² = (1/2)mu*H² per unit volume.
Faraday Law of Induction
emf = -d(phi)/dt, time-varying magnetic flux.
Lenz Law
Induced current opposes change in flux, conservation of energy.
Displacement Current
Jd = dD/dt, Maxwell correction to Ampere law.
Maxwell Equations Integral Form
All four equations in integral form, physical meaning.
Maxwell Equations Differential Form
Point form, curl and divergence equations.
Boundary Conditions General
Tangential and normal conditions from Maxwell equations.
Wave Equation Derivation
From Maxwell equations, second order wave equation.
Uniform Plane Wave
E and H perpendicular to propagation, TEM mode.
Phase Velocity and Wavelength
v = omega/beta = 1/sqrt(mu*epsilon), lambda = 2*pi/beta.
Intrinsic Impedance
eta = E/H = sqrt(mu/epsilon), 377 ohms for free space.
Poynting Vector and Power Flow
S = E cross H, average power, Poynting theorem.
Wave Propagation in Lossy Media
Attenuation constant alpha, skin depth.
Skin Effect
Current concentration at surface, skin depth delta.
Linear Polarization
E-field in single plane, horizontal and vertical.
Circular Polarization
Equal amplitude, 90 degree phase, RHCP and LHCP.
Reflection at Normal Incidence
Reflection and transmission coefficients, standing waves.
Reflection at Oblique Incidence
Snell law, Fresnel equations, Brewster angle.
Transmission Line Parameters
R L G C per unit length, distributed model.
Transmission Line Equations
Telegrapher equations, voltage and current waves.
Characteristic Impedance
Z0 = sqrt((R+jwL)/(G+jwC)), lossless Z0 = sqrt(L/C).
Reflection Coefficient
Gamma = (ZL-Z0)/(ZL+Z0), magnitude and phase.
VSWR
Voltage standing wave ratio, relation to reflection coefficient.
Input Impedance of TL
Zin = Z0*(ZL+jZ0*tan(beta*l))/(Z0+jZL*tan(beta*l)).
Quarter Wave Transformer
Lambda/4 matching, Zin = Z0²/ZL, impedance transformation.
Smith Chart Basics
Normalized impedance, resistance and reactance circles.
Single Stub Matching
Shunt stub, series stub, position and length.
Antenna Fundamentals
Radiation mechanism, near and far field regions.
Radiation Pattern
Main lobe, side lobes, back lobe, beamwidth.
Antenna Parameters
Directivity, gain, efficiency, effective aperture.
Hertzian Dipole
Infinitesimal dipole, radiation resistance, pattern.
Half Wave Dipole
Lambda/2 antenna, 73 ohm impedance, pattern.
Quarter Wave Monopole
Ground plane antenna, 36.5 ohm, vertical polarization.
Friis Transmission Equation
Power received, path loss, link budget calculation.
Polarization
Dielectric polarization P, electric susceptibility.
Double Stub Matching
Two stub tuner, forbidden region, design.
Parabolic Reflector Antenna
Dish antenna, focal point feed, high gain.
Vector Addition and Subtraction
Parallelogram law, component-wise operations.
Dot Product
Scalar product, projection, work done calculation.
Cross Product
Vector product, area, torque, direction by right hand rule.
Scalar Triple Product
Volume of parallelepiped, [A B C] determinant form.
Magnetic Materials
Diamagnetic, paramagnetic, ferromagnetic, permeability.
Boundary Conditions Magnetostatics
Tangential H and normal B conditions at interface.
Maxwell Equations in Phasor Form
Frequency domain, complex permittivity.
Elliptical Polarization
General case, axial ratio, polarization ellipse.
Total Internal Reflection
Critical angle, evanescent wave, fiber optics.
Lossless Transmission Line
Alpha=0, standing wave patterns, special cases.
Smith Chart Applications
Impedance matching, stub matching using Smith chart.
Antenna Arrays
Linear array, pattern multiplication, steering.
Yagi-Uda Antenna
Driven element, reflector, directors, directional.
Stokes Theorem
Surface to line integral conversion, curl relation.
Curl
Del cross A, circulation per unit area, rotational measure.