Perturbation Method

I

t is known that the electromagnetic field problems can be expressed in integral integration form or in differential equation form. While the differential equation form gives an exact solution of the electromagnetic problem, on the contrast, the integral equation form is especially useful for obtaining approximate solution. Owing to the above characteristics, the integral equation can be employed in perturbation method for approximate solutions of propagation constant of the specified waveguide systems.

To “perturb” means to change the status of a system slightly. The perturbation method is useful to calculate the changes of a physical quantity due to a small change in the system.

There are two circumstances involved in the analysis of a perturbation system. First is the

“unperturbed” one, which is with known solution of certain parameters. Then is the

“perturbed” one, which is perturbed slightly from the “unperturbed” one in the system.

Considering two types of perturbations in a cylindrical waveguide, or a waveguide has identical cross section. One type is on the perturbation of the width of the waveguide wall, and the other type is on the addition of a dielectric material inside the waveguide. The following perturbation formulae for the change in propagation constant in the loss-free waveguide are derived in the Harrington’s textbook [38], however we re-write, use the results, and modify the results to show the physical insight of the two kinds of perturbations mentioned above for beam steering. In the first case the perturbation is on the constituent material inside the waveguide, the change of phase constant from unperturbed value β₀to β due to perturbation is given as:

And, in the case of a waveguide wall perturbation, the change of phase constant from unperturbed value β₀to β due to perturbation is given as:

In the case of shallow, smooth deformation of the waveguide wall, e and h can be approximated by e0 and h0, respectively. So, equations (6) and (7) can be approximated by

An example demonstrates waveguide perturbation is shown in figure 2. It shows a slitted waveguide beam steering antenna. The idea of this beam steering mechanism is to change the longitudinal field propagation constant by changing the waveguide width, that is, by

deforming the waveguide sidewall longitudinally. Concerning the dominant TE10 mode, the operating region where the transverse fields are perturbed is with weak electric field strength but strong magnetic field strength. In this case equation (8b) can be reduced to

( )

Change of the propagation constant is approximately proportional to the magnetic field energy stored in the deformed region of the waveguide transverse plane, where magnetic field is of the strongest strength. The slitted waveguide antenna steers its main beam by changing its width of waveguide wall. Assuming that the perturbation is shallow and slight, and the magnetic field strength can similarly as above assumed to be approximately constant over the region of waveguide deformation. Then equation (9) can be further reduced to

_S

(

^{e h hte ah}

)

^{z ds⎟⎟}_⎠^⋅∆b

Where a is the height of the waveguide, ∆ represents the deformed width of the waveguide, b and ht is the magnetic field strength in the deformed waveguide region. Note that the change in propagation constant depends nearly linearly on the deformation width in the waveguide.

This technique has been realized and demonstrated successfully during WWII on the air-borne beam steering radar developed for the aircraft B-29 bomber [2].

The other case of inserting a thin dielectric slab in the waveguide with moderately low relative dielectric constant as shown in figure 3(a), that is we have slight dielectric perturbation. Then equation (8a) becomes

( )

It is obvious that maximum change in propagation constant exists when the dielectric slab is inserted in the area where the electric field is of maximum magnitude. This is certainly the case that we put the dielectric slab around the central region of the waveguide, for TE10 mode field under consideration. In this case, the antenna radiation main beam may experience the largest angle shifting. If the dielectric slab is thin enough, it is reasonable to assume that the electric field is almost concentrated and being almost constant within the volume of dielectric slab, then equation (19) can be further approximated to

( )

Where a is the height of the waveguide, and t is the thickness of the dielectric slab and et is the electric field strength in the dielectric slab. It shows that the change of the propagation constant can be approximately proportional to the value of

(

^∆ε^⋅t

)

, as long as the perturbation is shallow and smooth, that is, assuming product term

(

^∆ε^⋅t

)

is small. That means a smaller angle scan in the antenna maim-beam results when we put a thinner dielectric slab inside the waveguide. Similarly, when we put the dielectric slab closer to the waveguide sidewall, where the electric field is relatively weak in comparison with the other position, smaller angle scan in the antenna maim-beam is derived. By using the variational principles for electromagnetic resonators and waveguide, the variation of propagation constants (between exact and approximate results) vs. frequency for dielectric slab of various widths in a rectangular waveguide could be found in [39].

在文檔中 波束可掃描之波導漏波天線設計－理論分析與天線製作量測 (頁 16-19)