The following theoretical notes are based on sources such as Stratton "Electromagnetic Theory", McGraw Hill Book Company and Von Hippel "Dielectric Materials and Applications", Technology Press of M. I. T. and John Wiley and Sons.
The formulas and discussions are presented here for the convenience of the designer planning to use Dielectric Materials. Only the most practical formulas here have been selected, and an attempt has been made to simplify those presented. On the other hand, the use of many of the formulas requires the ability to manipulate complex numbers. (Emerson & Cuming, Inc. offers charts which give dielectric data on most commonly used electrical materials.)
It is important to note that the formulas presented here apply, with proper interpretations, equally well to:
Complex Dielectric Constant
where K is the real part of the relative permittivity and is the dielectric loss tangent.
Complex Relative Magnetic Permeability
where Km is the real part of the relative permeability and is the magnetic loss tangent.
d = thickness of flat sheet in free space or length of slug along the transmission line, either coaxial line or waveguide.
is free space wavelength.
is incidence angle, the angle between the direction of propagation of signal impinging on the free space slab and the normal to the free space slab; is always zero for the coaxial line; for wave guide in the TE10 mode.
where is the wave guide wavelength and is given by
where a is the broad dimension of the waveguide.
Intrinsic impedance of a material =
Relative impedance magnitude or absolute value of relative impedance =
Attenuation in dB/cm.
The following equation may be used to calculate the attenuation constant for any material in terms of its dielectric and magnetic properties:
Attenuation constant =
where = free-space wavelength in cm
The attenuation values tabulated or computed from the equation must be interpreted as in the following example. Suppose that a slug of lossy material completely fills a section of a coaxial transmission line. Once energy has entered the lossy material, it will be diminished at the given rate as it propagates in the section of line that is filled by the material. The tabulated or calculated rate does not include any effects of energy reflections at the input or output interfaces between the lossy material and any other materials such as air in adjacent sections of the line.
Polarization
Before presenting transmission and reflection formulas, the two principal polarization cases for electromagnetic impingement on a flat sheet must be defined. These are perpendicular and parallel polarization, identified respectively in the formulas by the subscripts and .
The terms refer to the direction of the electric field in a linearly polarized wave. They also refer to an incidence plane which is the plane formed by two lines, one of which is the normal to a flat sheet of material on which the wave is impinging, and the other which is the direction of propagation of the impinging wave. The incidence angle, q previously defined lies in this incidence plane. In a coaxial line the incidence angle is zero; therefore polarization does not apply. In the TE10 waveguide mode, the polarization is always perpendicular.
Interface Voltage Reflection Coefficients
There are two interface voltage reflection coefficients (r) one for each polarization:
Electrical Thickness,
This is the same for both polarizations and is given by
At normal incidence in free space and in a coaxial line, the last three formulas are simplified by the incidence angle, being zero and become:
and
Transmission and Reflection
In terms of the interface reflection coefficient r, and electrical thickness . We can now write the transmission and reflection equations for the flat sheet in free space, the solid slug in a coaxial line, and the rectangular solid in the waveguide.
from which the reflection loss in decibels is given by
and the reflection phase or delay due to energy lingering in the sample prior to reflection is simply R’, the argument of R*.
Metal Backed Reflection Coefficient
If the dielectric material is backed by metal or short circuited, the reflection coefficient becomes:
in which again reflection loss is given by
Open Circuit Reflection Coefficient
If a quarter wave of air is placed between the dielectric material and the short circuit, the material is said to be open circuited, and
and again reflection loss is given by
This device is often used to take full advantage of the reflection loss potential in a lossy dielectric used for a termination.