One of the most vexing problems facing designers of lumped element filters and other lumped element circuits is the problem of accounting for internal leakages. The capacitors and coils protrude above the surface and are not located uniformly with respect to any axis of the network. Accordingly, there are potentially radiative coupling effects between the capacitors and mutual coupling (also radiative in nature) between the inductors. Such additional couplings cause multipath propagation from input to output and thus spurious passbands occur. To compensate for these problems in filter applications, it is frequently necessary to separate the various resonant portions of the network with isolating barriers. These barriers cannot fully extend across the filter because it is necessary to make connections from one section to the next. The barriers, therefore, have intrinsic inter-section leakage characteristics which cause additional unexpected and undesired spurious passbands, not predicted with conventional simulation techniques. Because these circuits are three-dimensional and irregular in shape, performing electromagnetic simulation is particularly difficult.
However, treatment of the enclosure as a partially-loaded section of waveguide, with the separating barriers considered as coupling irises, enables very good first order prediction for the frequency-domain position and amplitude for the undesirable spurious responses. Consider the rectangular filter enclosure used as an example in Fig. 1 below. The lumped element circuitry is mounted on a printed circuit substrate. The substrate has a dielectric constant of about 2.5, and a thickness of 0.031". The filter resonant sections are separated by copper foil barriers, with a small space left under each barrier to enable the printed circuit substrate connection along the full length of the filter. The spaces between the foil sections can be treated as rectangular waveguide TE101 mode resonators, with resonant frequencies perturbed by the presence of the printed card, the dielectric of the capacitors and the conductivity of the metallic coils. Coupling between the equivalent resonators is determined primarily by the capacitive iris openings represented by the space between the bottom of the foil barriers and the printed substrate. Perturbation theory is essentially applicable when the volume of perturbing material is less than 5% of the total open volume. We have that case in most lumped element circuit applications. The interresonator coupling coefficients can be measured or directly computed using E-M simulation or simply from electric polarizability data applicable to capacitive arises [Ref. 1]. Assuming the cover to be soldered firmly onto the filter thus forming a completely conductive waveguide wall, knowledge of the resonant frequency for each section of waveguide and the inter-resonator coupling coefficients enables computation of the equivalent waveguide bandpass filter which has "sneaked" into the lumped-element picture.
In the case represented by Fig. 1, the lumped element bandpass is centered at 160 MHz, bandwidth of 14 MHz, while the perturbed waveguide bandpass is centered at about 3 GHz. Thus, for the example, we have a first spurious resonance at 3 GHz, with a bandwidth of about 300 MHz. The bandwidth results from the width of the opening at the bottom of the foil barriers, which is large enough (all the way across the filter housing) to provide large inter-resonator coupling coefficients. Fig. 2 below is a similar example structure, but with lumped filter circuitry providing a passband of 2 to 6 GHz, with first waveguide spurious resonance of above 34 GHz. Fig. 2 shows the value of constraining the open interior portions of the lumped element filter to the smallest possible dimensions, when maximum enhancement of spurious-free stopband is the intention.
Reference 1: Matthaei, Young, Jones. "Microwave Filters, Impedance Matching Networks...", The Filter Bible, Fig. 5.10-3.