Chapter 11. The uniform plane wave

Radiation from a current filament

Spherical waves from a source of finite dimension Transmitted waves from a finite sized source behave like spherical waves.

TV antenna TV signal transmitting antenna TV signals are transmitted as EM wave with horizontal polarization.

Polarization of EM waves Source current Source current

Reflection from ionosphere

Reflection from ground and troposphere EM waves undergo reflections and diffractions while propagating toward receivers.

Concept of plane waves EM waves transmitted from a finite sized source spread spherically in the space. With great distances from the source, EM waves behave as a plane wave locally.

E & H in source free region In source free region, E and H can be obtained easily. With J and ρ equal to zero, a source free wave equation is obtained.

If the source current away from the field point has the direction of x-axis, the electric field can have only x component. If the extent of the source is infinite along x and y directions, variations of the field along those directions become zero. If the propagating direction is in +z-axis, the Helmholtz equation becomes a simple expression. The first term represent the wave propagating toward +z direction, while the second stands for the wave propagating to the opposite direction. Considering only the wave propagating in the positive direction, wave impedance of free space

Plane wave propagating general direction If the propagating direction is other than x, y or z direction, the phase term in the exponential function can be obtained by the inner product of the k-vector and the position vector. the k-vector can be compared to the normal vector of equi-phase plane.

Magnetic field of a plane wave The magnetic field can be obtained.

Wave propagation in dielectrics If the wave is propagating into a medium other than free space, the molecules in that medium vibrate under the action of electric field. Meanwhile, the energy of the wave is dissipated and the medium become heated. The dissipation can be modeled by a complex permittivity. 3. The ratio of the real part of εr to the imaginary part is called the loss tangent of that media.

Propagation constant in a lossy dielectric From the equations on the left, it can be seen that the phase of displacement current leads that of conduction current by 90 degree. That is, electric field propagates first, then charges move under the action of that electric field. Helmholtz equation in a lossy medium becomes,

To obtain the approximate expression of α and β, we consider the two extreme cases of Good dielectric (ε’>>ε’’) Good conductor (ε’<<ε’’)

Plane wave in good dielectrics Case 1)

Plane wave in good conductors Case 2) δ is called as the skin depth of the medium at the given frequency. If the electric field penetrate into a lossy medium as much as one skin depth, its strength decreases by 1/e. That is its strength becomes 36.7% of the original value.

Example Find the skin depth of sea water at the frequency of 1MHz. In sea water σ = 4 [S/m], εr’=81. good conductor

Example Calculate the resistance of a round copper of 1mm radius and 1km length at DC and 1MHz.

Poynting’s theorem and wave power among the right side terms in the above equation, the first term represents ohmic loss, the second term the rate of change in the stored electric and magnetic energy.

12.5 Wave polarization a changing direction of electric field observed at a position. Among the components of an electric field vector, only two of them is independent. The Helmholtz equation has three components (x, y, z). But the divergenceless condition imposes that ∇∙E = 0, which is a constraint among the vector components of E-field. That is, all the component of Ex, Ey, Ez are not independent. From the above condition, only two components among the Ex, Ey, Ez are independent. That is, two kinds of independent polarizations comprise an arbitrary E-field.

Polarization Example of a electric polarization of a wave propagating in +z direction. Linear polarization Circular polarization

Polarization diversity antenna