# Measurement of the dielectric constant and loss tangent of materials in microwave frequency band using a rectangular waveguide cavity

## Theory Continued...

### Cavity Perturbation Technique

As we know, that a piece of sample material affects the resonant frequency (fr) and quality fator (Q) of the cavity. From these parameters, the complex permittivity (εr) or permeability (µr) of material can be calculated at a single frequency. There are many different types of cavities and methods. For calculation of these parameters, most widely used method is Cavity Perturbation Technique. This method uses a rectangular waveguide with iris coupled end plates, operating in TE10n mode. For a dielectric measurement the sample should be placed in a maximum electric field and for a magnetic measurement, in a maximum magnetic field. The figure above shows a rectangular cavity resonator inserted with a sample. If the sample is inserted through a hole in the middle of the waveguide length, then an odd number of half wavelengths (n=2k+1) will bring the maximum electric field to the sample location, so that the dielectric properties of sample can be measured.

The cavity perturbation method requires a very small sample such that the fields in the cavity are only slightly disturbed to shift the measured resonant frequency (fr) and cavity quality factor, Q. In this experiment, a rectangular waveguide for TE107 mode (7 half wavelength long at the designated frequency) with a hole drilled exactly at the middle of the waveguide length and the two iris-coupled end plates is considered. The dimension of the iris hole is typically b/2.2, where b is the narrow dimension of the waveguide cross section. When the resonator is loaded with a sample, it shifts the resonant frequency (fr) and the resonance curve broadens, which results in a lower quality factor (Q). The figure below shows the comparision of resonant frequency fr of three different samples with the empty cavity. After calculating the resonant frequency and quality factor for various samples, values of ε'r and ε"r can be obtained as:
ε'r = [Vc*(fc - fs)/(2*Vs*fs)] + 1

ε"r = (Vc / 4*Vs)*[(1/Qs) - (1/Qc)]

Vc is the volume of the empty cavity
Vs is the volume of the sample inserted
fc is the resonant frequency of the empty cavity
fs is the resonant frequency of the sample inserted
Qc is the quality factor of the empty cavity
Qs is the quality factor of the sample inserted

Here we designed the resonator of follwing volume
Vc=32.516 cm3, Volume of the empty cavity
Vs=0.046 cm3, Volume of the cavity where sample is to be inserted