ECE 107

Reflected Average Power

(-|Γ|^2|V0|^2)/2Z0

Coaxial C'

(2πε)/ln(b/a) (F/m)

Coaxial G'

(2πσ)/ln(b/a) (S/m)

Coaxial R'

(Rs/2π)(1/a + 1/b) (Ω/m)

cosθt

(εr - sin^2(θi))^(1/2)

Coaxial L'

(μ/2π)ln(b/a) (H/m)

Two-Wire L'

(μ/2π)ln[(D/d) + ((D/d)^2 - 1)^(1/2)] (H/m)

Rs

(πfμc/σc)^(1/2)

Plane wave

In physics of wave propagation, a plane wave is a wave whose wavefronts are infinite parallel planes.

Delivered Average Power

Incident Average Power - Reflected Average Power

Gauss's Law

Law relating the distribution of electric charge to the resulting electric field.

Transverse Wave

Moving wave that consists of oscillations occurring perpendicular to the direction of the propagation of the wave.

Sum of the incident and reflected plane waves

Plane Wave

Short Dipole

Power radiated is delivered to the load

Polarization (waves)

Property applying to transverse waves that specifies the geometrical orientation of the oscillations.

Losslessness in a Transmission Line

R' = G' = 0

Laplace's Equation Definition

Solutions are harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials.

TE10 Mode

The dominant mode in a particular waveguide is the mode having the lowest cutoff frequency. For rectangular waveguides this is the __________.

Faraday's Law - Integral Form

The integral of the electric field bounded by a closed contour is the negative surface integral of a time varying magnetic field with respect to an infinitesimal vector element of the enclosed surface. ∫E*dl = -∫∫(dB/dt)ds

Wave Impedance (η)

The ratio of the transverse components of the electric and magnetic fields (the transverse components being those at right angles to the direction of propagation). η = (μ/ε)^(1/2)

Oblique Incidence: Perpendicular Polarization for Incident Waves

The vector representing the electric field for an incident plane wave at boundary Ei⟂ = yhatE0e^(-jk(xsinθ + zcosθ))

Oblique Incidence: Perpendicular Polarization for Reflected Waves

The vector representing the electric field for an reflected plane wave at boundary Er⟂ = yhatΓ⟂E0e^(-jk(xsinθ - zcosθ))

Differential Form

Useful for solving fields extended over an area because derivatives aren't defined at points r discontinuities.

Integral Form

Useful for solving things where you have discontinuities like wires and points.

Parallel Plate Capacitor Voltage with respect to z

V = V0z/d

Conservative Vector Field

Vector field that is the gradient of some function, known in this context as a scalar potential. Have the property that the line integral is path independent, i.e., the choice of any path does not change the value of the line integral.

Shorted Line

Zin = 0

Input Impedance of Standing Waves on a Lossless Transmission Line

Zin = Z0[(Zl + jZ0tan(βl))/(Z0 + jZltan(βl)]

Boundary Conditions

a condition that is required to be satisfied as all or part of the boundary of a region in which a set of differential equations is to be solved.

Locus

a set of points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.

Faraday's Law - Differential Form

a time-varying magnetic field will always accompany a spatially-varying, non-conservative electric field, and vice versa. ∇ x E = -dB/dt

Standing Wave

a wave in a medium in which each point on the axis of the wave has an associated constant amplitude.

Kelvin-Stokes Theorem

also known as the curl theorem, is a theorem in vector calculus on R^3. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface.

Average Power Density of Wave

amount of time rate of energy transfer per unit volume. abs(Et0)^2/(2*η2)

L'

combine inductance of both conductors per unit length, in H/m

G'

conductance of the insulation medium between the two conductors per unit length.

Characteristic Impedance of Free Space (Z0)

expression of the relationship between the electric field and magnetic field intensities in an EM field propagating through a vacuum. z0 = 120π Ω

Phase Constant (β)

imaginary component of the propagation constant for a plane wave. It represents the change in phase per unit length along the path traveled by the wave at any instant and is equal to real part of the angular wavenumber of the wave. β = 2π/λ

Parallel Polarized Reflection Coefficient

incident light is polarized with its electric field in the same direction to the plane of incidence. Γ|| = (η2cosθt - η1cosθi)/(η2cosθt + η1cosθi)

Perpendicular Polarized Reflection Coefficient

incident light is polarized with its electric field normal to the plane containing the incident, reflected, and refracted rays. This plane is called the plane of incidence. The light is said to be s-polarized, from the German, senkrecht, meaning perpendicular. Γ⟂ = (η2cosθi - η1cosθt)/(η2cosθi + η1cosθt)

If (D/d)^2 >> 1 in a Two Wire Line

ln[(D/d) + sqrt((D/d)^2 - 1) = ln(2D/d)

Voltage Standing Wave Ratio

measure of impedance matching of loads to the characteristic impedance of a transmission line or waveguide. Usually thought in terms of the maximum and minimum AC voltages along the transmission line. (1 + |Γ|)/(1 - |Γ|)

Propagation Constant (γ)

measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. γ = α + jβ

Reflection Coefficient

parameter that describes how much of an electromagnetic wave is reflected by an impedance discontinuity in the transmission medium.

Circular Polarization

polarization state in which, at each point, the electric field of the wave has a constant magnitude but its direction rotates with time at a steady rate in a plane perpendicular to the direction of the wave.

Snell's Law

relates the indices of refraction n of the two media to the directions of propagation in terms of the angles to the normal.

Ampere's Law

relates the integrated magnetic field around a closed loop to the electric current passing through the loop.

Poynting Vector

represents the directional energy flux density (the rate of energy transfer per unit area) of an electromagnetic field.

C'

the capacitance of the two conductors per unit length, in F/m

Refraction

the fact of phenomenon of light, radio waves, etc., being deflected in passing obliquely through the interface between one medium and another or though a medium of varying density.

Wavefront

the locus of points characterized by propagation of position of the same phase: a propagation of a line in 1D, a curve in 2D or a surface for a wave in 3D

Magnetic Field (B)

the magnetic effect of electric currents and magnetic materials.

Phase

the position of a point in time on a waveform cycle.

Index of Refraction

the speed of light in a vacuum divided by the speed of light in the medium.

Cosine

the trigonometric function that is equal to the ratio of the side adjacent to an acute angle to the hypotenuse.

Electric Displacement Field (D)

vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials while its sources are the free charges only.

Critical Angle (θc)

when light is incident upon a medium of lesser index of refraction, the ray is bent away from the normal, so the exit angle is greater than the incident angle. The exit angle will then approach 90° for some __________, and for incident angles greater than the critical angle there will be total internal reflection. The __________ can be calculated from Snell's law by setting the refraction angle equal to 90°. For any angle of incidence less than the __________, part of the incident light will be transmitted and part will be reflected.

Incident Average Power

|V0|^2/2Z0

Propagation Constant - General Case

γ = sqrt[(R' + jωL')(G' + jωC')]

Parallel Plate C'

εw/h (F/m)

Angle of Transmittance

θt = sin^-1(sinθi/(εr)^(1/2))

Parallel_Plate L'

μh/w (H/m)

Two-Wire C'

πε/ln[(D/d) + ((D/d)^2 - 1)^(1/2)] (F/m)

Two-Wire G'

πσ/ln[(D/d) + ((D/d)^2 - 1)^(1/2)] (S/m)

Parallel Plate G'

σw/h (S/m)

Ampere's Law - Differential Form

∇ x H = dD/dt + J

Faraday's Law - Phasor Form

∇ x ~E = -jωμ~H

Ampere's Law - Phasor Form

∇ x ~H = jωε~E + ~J

Ampere's Law - Integral Form

∫H*dl = -∫∫(dD/dt + J)ds

Parallel-Plate R'

2Rs/w (Ω/m)

Two-Wire R'

2Rs/πd (Ω/m)

Faraday's Law

Basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF) - a phenomenon called electromagnetic induction.

Capacitance of Parallel Plates

C = Q/V = Q/Ed = Qε/σd = QAε/Qd = Aε/d

R'

Combined resistance of both conductors per unit length in (Ω/m)

Parallel Transmission Coefficient (τ||)

Describes the amplitude, intensity, or total power of a transmitted parallel wave relative to an incident wave. (1 + Γ||)(cosθi/cosθt)

Perpendicular Transmission Coefficient (τ⟂)

Describes the amplitude, intensity, or total power of a transmitted perpendicular wave relative to an incident wave. 1 + Γ⟂

E=-∇V

E is the gradient of the electric potential V, identical to the classical gravitational field.

λ = c/f

Given an incident wave of a certain frequency, you can determine ___.

Z0 = 60/sqrt(εr)ln(b/a)

Given the inner and outer radii of a lossless coaxial line, you can determine ___.

Conductivity Current (in terms of Current Density)

Ic = ∫∫Jc*area

Laplace's Equation Use Case

If no charges are present between a parallel plate capacitor.