X-couplings in filters.
How do they work?

X-couplings  or non adjacent couplings  are couplings between resonators, which are not placed just next to each other in the signal main-path. The reason to include x-couplings in a filter is most often that they create notches  or transmission zeroes  in the filter characteristic. With x-couplings it is therefore possible to improve stop-band performance without increasing the order of the filter.

To explain why x-couplings do generate notches the most common approach is to use lumped element representations of resonators and couplings. Though correct - this approach does not explain what really happens in the filter. To get a hunch about this one has to look at the electromagnetic fields in the filter  so lets do that.

We will take a look at the folded waveguide filter shown in figure 1.

The filter is of  5th order, with a x-coupling between resonator 2 and 4. The arrows show entry and exit ports of the filter. The resonators are coupled by irises going from top to bottom in the filter. Full-height irises are known to be inductive (positive) by nature and hence all couplings in figure 1 are inductive. The corresponding coupling diagram is shown in figure 2. Normally the filter topology in figure 2 generates a notch above the pass-band, but for the waveguide filter in figure 1 the notch will appear below the pass-band.

The reason for this is that resonator 3 is a full-wave resonator, in opposition to all the other resonators, which are half-wave resonators. The extra half wavelength in resonator 3 changes the phase by 180 degrees. The 180 degree shift in resonator 3 has a similar effect as changing the sign of the x-couplings. This means that an inductive x-coupling will behave like a capacitive x-coupling  and produce a notch above the pass-band (and vice-versa).

In order to explain exactly how this works we have to look at the magnetic field patterns in the cavities.

Since only resonators 2, 3 and 4 are involved we will limit ourselves only to look at these three cavities.
The key to explain what goes on, is the fundamentally different magnetic field patterns that exist in the cavities - below and above the center frequency. These patterns are sketched in figure 3 and figure 4. # Above fc (figure 3)

At frequencies above the filter center frequency the field patterns are as shown in Figure 3. The x-coupled fields are here in phase with the original fields in cavities 2 and 4, which means that in the upper stop band  signal energy takes a shortcut from input to output via the x-coupling aperture. A x-coupling as above therefore has the effect of decreasing the stop band attenuation above the pass band. This is also seen in practice.

# Below fc (figure 4)

At frequencies below the filter center frequency, the field patterns are as shown in Figure 4. The blue horizontal H-field line is the x-coupled field originating from cavity 2.
A similar field is introduced in cavity 2 originating from cavity 4.
In the pass-band, signal strengths in the cavities are equal and the two x-coupled fields will phase each other out, since they are equal in magnitude but opposite in direction (phase).
Below the pass-band the two x-coupled field components are not equal in magnitude, since the field in cavity 2 is much stronger than the field in cavity 4 (the field in  cavity 4 is heavily attenuated by passing the detuned resonators 2 & 3). The cross coupled field in cavity 4 originating from cavity 2 is now much stronger than the opposite x-coupled field, which means that excess x-coupled field exists in cavity 4. Since this field is in anti-phase with the existing (weak) field in cavity 4  signal extinction is possible and a notch will appear.

If the x-coupling aperture is opened up the x-coupled field will be stronger and extinction of a stronger existing field is then possible, i.e. the notch will move closer to the pass band.

This means that at frequencies below the notch frequency the signal on the output of the filter is predominantly x-coupled field. Since the x-coupled signal below the pass-band is in opposite phase of the main-line signal, one should expect that the phase of the signal changes 180 deg at the notch frequency. This is also observed in practice.

The explained features are summarized in the figure below: Having realized what the x-coupling does from the above example, other configurations can be explained by similar observations.