LS Note 179: The Effect of Small Ellipsoidal Material on the Resonant Frequency of a Cavity

LS Note 179

The Effect of Small Ellipsoidal Material on the Resonant Frequency of a Cavity

T. Khoe

We assume that the medium inside the cavity has no losses with farads/meter and henrys/meter. Excited at resonance, the fields inside the cavity can be written in the form

The electric and magnetic fields are 90° out of phase. In other words, if E_o is real H_o is imaginary and vice versa. Insertion of a small piece of material with and will change the field values and the resonant frequency

Note that will be a complex quantity if the inserted material is lossy. Substitution of equations (1) in Maxwell's equations

give

Noting that the fields E_o, D_o, H_o and B_o satisfy the same Maxwell's equations we obtain:

Multiplication of equation (2a) by H_o and equation (2b) by E_o and addition give

Using the vector relation

we can rewrite the L.H.S. of equation (3) in the form

Comparison of equations (3) and (4) gives

In practice >> so that in the expression ( + ) (E_o · D₁ - H_o · B₁) we can neglect with respect to and obtain

Outside the ellipsoid we have

₁

and equation (5) reduces to

j(E_o · D_o - H_o · B_o) = - div (H_o x E₁ + E_o x H₁).

We integrate this equation over the volume bounded by the cavity wall and the surface of the ellipsoid

where V = volume of the cavity

and V = volume of the ellipsoid.

Using the divergence theorem, the R.H.S. of equation (6) can be written as a surface integral

where s + s is the surface bounding the volume V-V. Since the cavity is assumed to be a good conductor, E_o and E₁ will be practically perpendicular to the cavity surface and the contribution of the cavity wall to the surface integral can be neglected. In this case we find

where s = surface of the ellipsoid. Note that ds is in the direction of the now outward normal to the ellipsoid surface. Inside the ellipsoid we have

₁

where P = polarization or electric dipole moment per unit volume, and

Substitution in equation (5) gives

₁

Integrating over the volume of the ellipsoid and using the divergence theorem we obtain

Comparison of equations (7) and (8) and some manipulation gives

or, since H_o and B_o are imaginary if E_o and D_o are real, we can write

where U = cavity stored energy and E_o, P, H_o and M are now all real quantities. For the small region in and around the ellipsoid, E_o and H_o are practically uniform. For an ellipsoid of semi-axis a, b and c with a field parallel to the a-axis, P and M are given by (see M. Mason and W. Weaver, Dover Publications, § 36).

where _r = relative permittivity, _r = relative permeability, and

Axial Symmetrical Ellipsoid

For axial symmetry about the a-axis we have b = c, and this reduces the integral to an elementary one,

Performing the integration we find for a oblate spheroid (a < b)

and for a prolate spheroid (a > b)

For both cases the spheroid reduces to a sphere for e 0 and L = 1/3. For the limit as a 0 (e ), the oblate spheroid becomes a circular disk of radius b and L = 1. On the other hand, as b 0 the prolate spheroid becomes a very thin rod of length 2a and L = 0.