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

## Theory

### Resonant Cavity:

To provide a resonant circuit at UHF and higher frequencies, an enclosure completely surrounded by conducting walls is required. Such a shielded enclosure confines electromagnetic fields inside and furnishes large areas for current flow. Thus eliminating radiation and high-resistance effects. These enclosures have natural resonant frequencies and a very high Q (Quality factor), and are called cavity resonator. Cavity has both ends short circuited. Designation of TE and TM modes in resonator cavity is not unique because we are free to choose x or y or z as the "direction of propagation"; that is, there is no unique "longitudinal direction." For example, a TE mode with respect to the z-axis could be a TM mode with respect to the y-axis. A three-symbol (mnp) subscript is needed to designate a TM or TE standing wave pattern in a cavity resonator.Like waveguides, different modes having the same resonant frequency are called degenerate modes.TMmnp and TEmnp are always degenerate if none of the mode indices is zero. The mode with lowest resonant frequency for a given cavity size is referred as the dominant mode.

Resonant frequency of the cavity can be defined as:

 ωmnp = (1/√µε)* √[(m*π/a)2 + (n*π/b)2 + (p*π/d)2] fmnp = u/2*√[(m/a)2 + (n/b)2 + (p/d)2]   (Hz)

where integers m,n,p denote the number of half-wave variations in the x, y, and z direction, respectively.

Quality factor, Q of a resonator, is associated with the loss factor of the cavity resonator and is defined as:
Q=2π * (Time average energy stored at a resonant frequency)/(Energy dissipated in one period of this frequency)
The Quality factor, Q of a cavity resonator is extremely high in comparision with that obtainable from lumped L-C resonant circuits. When a sample is inserted in the cavity resonator, the resonant frequency is shifted and the curve broadens, which increases Δf. Hence quality factor decreases. For high values of Q, it can also be find out as:

Q = fr / Δf

where fr is the resonant frequency, and Δf is the bandwidth taken at the 0.707 of the peak frequency.