Determination of the Physical Dimensions

The final stage in the design process is to find the dimensions of the physical filter, so it can be manufactured. It will be carried out with the support of the previously synthesized distributed model and a full-wave EM simulator. The EM simulator should take into account all the effects of the real waveguide structure shown in Figure 2.1, and at the same time, it must be efficient enough to perform the required simulations in a reasonable time. The full-wave EM solver FEST3D [23] has been chosen in this case. The details of the design procedure are given below.

2.4.1 Iris Dimensions

The first step is to find the dimensions of all the irises in the filter structure. In this case, a T-network has been chosen as the equivalent circuit of an iris with a certain thickness (see Figure 2.12). For the general case, which considers an iris with a different waveguide in each side (like in the input/output irises), the T-network equivalent circuit is the one shown in Figure 2.13 [24].

W L

t

Figure 2.12 Geometry of a rectangular thick iris.

Figure 2.13 Equivalent circuit of a K-inverter with a T-network for modeling a thick asymmetric iris.

The values of all the elements of the equivalent circuit in Figure 2.13 can be obtained from the S-parameters of the iris (obtained with an EM solver) as:

DETERMINATION OF THE PHYSICAL DIMENSIONS 25 jXs1 = (1 + S11)(1 − S22) + S 2 21− 2S21 (1 − S11)(1 − S22) − S221 (2.19a) jXs2 = (1 − S11)(1 + S22) + S212 − 2S21 (1 − S11)(1 − S22) − S221 (2.19b) jXp = 2S21 (1 − S11)(1 − S22) − S212 (2.19c) φ1 = − arctan  _Σ Π−  − arctan ∆ Π+  (2.19d) φ2 = − arctan  _Σ Π−  + arctan ∆ Π+  (2.19e) where Σ = Xs1+ Xs2+ 2Xp (2.20a) ∆ = Xs1− Xs2 (2.20b) Π+ = 1 + Xs1Xs2+ Xp(Xs1+ Xs2) (2.20c) Π₋ = 1 − Xs1Xs2− Xp(Xs1+ Xs2) (2.20d)

which reduces to the well-known equations given in [25] and [22] when symmetrical irises are considered.

The inverter valueK produced by the iris can be obtained as

K = s     1 + Γe−jφ1 1 − Γe−jφ1     (2.21a) Γ = j∆ − Π+ jΣ − Π− (2.21b) Using the previous equations, and with the distributed model in hand, the iris dimensions are derived through the following iterative process:

1. Choose the initial dimensions for the iris. They could be, for example, a certain fraction of the width of the adjacent waveguides.

2. Employing an EM solver, perform a simulation of a circuit formed by the iris and the adjacent waveguides with their lengths equal to zero. The simulation is made over a single frequency point, which is the center frequency of the filter, and the S-parameters of the iris are obtained.

3. Calculate the equivalentK of the iris using (2.19) and (2.21).

4. Change the iris dimensions (up or down depending on the comparison of the current and the desired value ofK) and go back to step 2. It is possible to vary

only one iris dimension, usually the length L (see Figure 2.12), or more than one dimension at the same time. Repeat the process until theK value of the iris reaches the correspondingK value in the distributed model.

As it can be seen in Figure 2.13, the equivalent circuit of aK-inverter is composed of a T-network and two adjacent transmission lines. However, the equivalent circuit of an iris is just a T-network without the transmission lines, which means that when the designed irises are introduced in the filter structure, the two transmission lines calculated with (2.19d) and (2.19e) must be added at both sides of the iris. Since for this particular case the electrical lengths obtained are always negative, this leads to a length reduction of the adjacent waveguides (i.e. the circular cavities and the input/output waveguides).

All the irises in the filter structure are designed separately, repeating the iterative algorithm explained above as many times as the number of present irises. Note that the cruciform iris can be interpreted as two orthogonal irises, so they can be designed as two independent irises. In practice, in order to increase the accuracy, these irises are first designed as independent irises, and then each of them is re- designed considering the presence of the other one, thus taking into account the small variation introduced in the coupling value.

When the inverter valueK is negative, as it is the case of the M14coupling of a

4-pole dual-mode filter with real transmission zeros, the iris is designed as if it were positive, and then the oblique screws of the adjacent cavities will be placed in such a position that there is a difference of 90 degrees between them (see Figure 2.1). Proceeding in that way, the vertical modes inside the two cavities are forced to have opposite sign, thus obtaining the so called virtual negative coupling [22].

2.4.2 Dimensions of Cavities and Screws

The initial length of the cavities is calculated using equation (2.4), and subtracting the corresponding phase values obtained according to (2.19) in both sides of the cavity.

At this point, a problem with the dual cavity length arises. The dual cavity is modeled by two lines in the equivalent distributed model. These lines are equal in electrical length, but once the real irises are introduced, they produce differentφ values according to (2.19). Therefore, there are severalφ’s to be embedded (one at each end of the cavity for the vertical polarization, and only one for the horizon- tal polarization) in the cavity length. As a result, the same physical cavity has to show different electrical length for each orthogonal mode. This problem is solved by means of the vertical and horizontal tuning screws (see Figure 2.1). Changing the penetration of the vertical/horizontal screws, the resonant frequency of the corre- sponding vertical/horizontal mode is shifted. In fact, after increasing the penetration depth of the tuning screw, the corresponding resonant frequency is decreased.

The adjustment of the screws and the cavities is done stage by stage (i.e. one cavity at a time), employing the distributed model. It is carried out by comparing the

DETERMINATION OF THE PHYSICAL DIMENSIONS 27

responses of the circuits of figures 2.14 and 2.15, which correspond to one stage of the distributed model and one stage of the EM model, respectively.

Figure 2.14 One stage (representing one cavity) of the distributed model.

Figure 2.15 One stage (i.e. one cavity) of the EM model.

Figure 2.16 shows a typical response of the circuits of Figures 2.14 and 2.15. ParameterS21represents the path between the input/output iris (vertical mode) and

the horizontal arm of the cruciform iris (vertical mode), whileS31refers to the path

between the input/output iris (vertical mode) and the vertical arm of the cruciform iris (horizontal mode).

The goal here is to match the EM response of each cavity with the response of the corresponding stage of the distributed model. A full-wave EM simulator (FEST3D) is going to be used in order to obtain the response of the structure shown in Fig- ure 2.15. The procedure to design each filter cavity is described below.

1. Insertion of the vertical tuning screw. The penetration of the vertical tuning