Example 7.1

A cubical cavity resonator made of copper has a side length of 20 mm. Find the resonance frequency of the fundamental mode and the quality factor.

Solution

The fundamental mode is TE101 . When a = b = d = 20 mm, n = l = 1, and m = 0, we get, from (7.21), the resonance frequency as f 101 = 10.6

GHz. The conductivity of copper is s = 5.8 × 107 S/m. At the resonance frequency the surface resistance is R s = √vm0 /(2s ) = 0.027V. From (7.31) or (7.32) we solve Q = 10,400. In practice Q is lower because R s is higher due to surface roughness and because couplings have some loss.

In principle, the quality factor of other resonance modes may be calcu-

lated the same way as the quality factor of the TE101 mode. Note that the name of a resonance mode also depends on the choice of the coordinate

system. For example, the TE101 mode is called the TM110 mode if the y -axis is chosen to be ‘‘the direction of propagation.’’ At a given frequency, the

higher the order of the resonance mode, the larger the cavity needed. As the size of the cavity increases, the ratio of the volume to the surface area increases, leading to a higher quality factor. In the case of a large cavity, however, the resonance frequencies of different modes are close to each other, and it is difficult to excite only a particular mode.

A cylindrical cavity is a section of a circular waveguide. The lowest order resonance mode is TE111 . At this mode the height of the cylinder is

lg /2 at the fundamental mode TE11 . The resonance mode TE011 of a cylindrical cavity is exceptionally important. This mode has a high quality

factor and no axial surface currents, which facilitates the realization of an adjustable cavity because the moving short does not need to make a good contact with the cylinder walls.

7.1.6 Dielectric Resonators

It is not possible to make high-quality resonators with microstrip techniques because microstrip lines are rather lossy and radiate easily. However, dielectric resonators [4] can easily be used in connection with microstrip circuits.

Dielectric resonators are usually small, cylindrical pills made of ceramic materials such as Ba2Ti9O20 , BaTi4O9 , or (Zr-Sn)TiO4 . Such materials have a good temperature stability, low loss, and high dielectric constant, typically er = 10 to 100. Because of the large dielectric constant, the size of