Design of Two-Layer Spiral Filters Using Electromagnetic CAD Tools (SONNET Software)

Coupling Coefficient

Microwave band-pass filter design commonly integrates resonator coupling coefficients, allowing for the precise synthesis of narrow-band filters. This accurate technique extends to the optimization synthesis of wideband filters, utilizing initial structure parameters. Notably, intelligent filter optimization techniques often leverage the physical properties of resonator filters, as recognized by Cohn (1957) and Hong and Lancaster (2009).

The asymmetrical slopes of baseband filters result from the frequency dispersion of coupling coefficients. Simultaneously, attenuation poles become null primarily due to the impact of resonator coupling coefficients. The peculiar dependence of coupling coefficients on the distance between certain resonators is elucidated by an energy-based approach to coupling coefficients, as elucidated by Belyaev et al. in 1996. Presently, a standardized definition for a resonator coupling coefficient is unavailable. In the context of wideband filters characterized by strong coupling, the distinctive nature of the resonator coupling coefficient is further revealed.

• Resonant Band-Pass Networks

The first person who introduced the coupling coefficients into the microwave filter theory was M.Aisal (Hong and Lancaster, 1999). The researcher started his work from a bandpass network of alternate and/or parallel resonant (dissipative) circuits set to the same resonance frequency.

The coupling coefficient he defined for this network between adjacent resonant circuits is as follows:

             ,                                                              (3)

Where,

 C_s = Capacitance of the Resistor Inductor Capacitor (RLC) resonant circuit, and C_p = Capacitance of R.L.C resonant circuit (parallel). The Network Transfer Immitance expression is a ratio of the frequency ω, proportional to the product of coupling coefficients of every bordering resonant-circuits. In contrast, the denominator comprises () with the maximum exponent n, the number of resonant circuits. Each coefficient in the polynomials is an exclusive function of the resonant circuits' coupling coefficients and the Q-factors. The algebraic expression can be given as a Chebyshev polynomial at some values of the coupling coefficients of the circuits.  The simultaneous equations for those values have been expressed and derived by Hong and Lancaster in 1999 for the value of n up to or equal to four. So, the coupling coefficient k is defined by Cohn, 1957 for an unlimited value range.

The ladder-chain of RLC parallel resonant circuits, including capacitive and mutual inductive coupling band-pass network, was considered. Physical accomplishment/production of such a network is much more practical. It has also been expressed that the transfer immittance fraction of the supplementary network is not a polynomial(s) of;

( )

The 2 band-pass networks are more or less equivalent to a narrow pass-band filter case, i.e., when ‘|k|’ << ‘1’. Their transfer function coincides in the region of ω₀ if |k| = |k (ω)|, where

                                  (4)

                                          (5)

                   (6)

The C_1., L_1., C_2., L_2.= Inductances, Capacitances of 2 parallel coupled resonant circuits, and C_m, L_m= Coupling capacitance and mutual inductances. k_L and k_C are inductive and capacitive constants for coupling coefficients of 2 parallel resonant circuits.

Thus, the coupling coefficient k(ω) for 2 parallel RLC resonant circuits is the sum/addition of two frequency-dependent expressions. The first expression is said to be related to capacitive coupling, whereas the second expression represents inductive coupling. The mathematical sum in Eq. (4) may disappear at a specific frequency, given as ω_p, in which the transfer function (of band-pass network) has an attenuation pole. The phenomenon indicates that inductive coupling and capacitive capacitive coupling remunerate each other. With a resonant frequency of ω_{0, both expressions' absolute value(s)} in the mathematical addition (4) are limited to unity.

In research conducted by Bovbysh and Tyunev (2010), it was reported that the physical realization of narrow-band filters having no attenuation poles at finite frequencies should provide the real values for resonant-frequency (ω₀₎) resonator-factors (Q), and k_i;i+1 which are the coupling-coefficients (b/w resonators). The expression for the coupling coefficient for such a scenario can be given as follows;

                                       (7)

Where ω_e and ω₀ = Frequency (s) of coupled-oscillations of resonators, ω₀ = Resonant-frequency (s), that is, for all coupling-reactances. In other words, the coupling coefficient in a narrow passband case is defined as a modelled resonant-frequency constant. However, this simplified approach may not always prove to be successful.

The approximate expression was given by Cohn (1957) as;

                         (8)

Matthaei et al. (1964) found the Expression (8) for a bandpass network of coupled non-dissipative microwave resonators. Yet another definition of the coupling coefficient was used therein:

                                  (9)

Where,

i + 1 is an equivalent network of a microwave filter

b_i stands for the susceptance slope parameter for the resonator

The trait admittance of the inverter between parallel resonators i is given by

Furthermore, it should be noted that at resonant frequency ω_0, all resonators are bound to have 0 susceptance. Equation (10) can be used to determine the slope parameter.

.                                      (10)

The coupled oscillation frequencies ω₀ and ω_e of 2 similar resonators are mainly responsible for determining the significance of the coupling coefficients at a resonant frequency, as shown by approximate formulas (7) and (8). Therefore, the coupling coefficient of two identical resonators was expressed by Atia and Williams (1972) as:

                           (11)

For a weak coupling case, formula (7) and formula (11) are expected to match. A sole mutual-inductive coupling capitulates the |k| value, coinciding with (5) and (6) in sole capacitive coupling. For formula (11) to be applicable, i.e., equation (6) must suit the given below summarizing regulation for coupling coefficients (Atia and Williams, 1972).

                          (12)

Considering the theory of relativity, formula (12) can be seen as very similar to the velocity-addition equation. The approximate formula for a weak coupling case matches the formula (12) (Ruiz et al., 2008).

                                                 (13)

Values for k, k_L and k_C can be either negative or positive, as assumed by Formulas (11) and (12). The coupling-coefficient sign has a meaningful/considerable value compared to another coupling, such as in simulations and modelling.

As for the case of two coupled series and parallel resonant circuits, the frequencies of coupled oscillations ω± are related to the coupling coefficient (3) as follows:

.                          (14)

Solving (14) gives the coupling coefficient

                            (15)

This formula agrees with the approximate expression (7). Being equivalent to (3), it may serve as a definition of k.

Using (11) and (15), one can compare two different coupling coefficients, k and ĸ. They are related as

.                                                (16)

One can see that |ĸ| is greater than |k|. It should be noted that this procedure has never been mentioned in the literature before.

In a bandpass network obtained from a lowpass prototype filter using the bandpass frequency transformation

,                                    (17)

Equation (18) allows us to calculate the frequencies of coupled mode oscillations for resonant circuits.

,            (18)

where Ω is the frequency of the lowpass prototype filter with normalized parameters g_i.

For a bandpass network with alternate series and parallel resonant circuits, using (15) and (18), it is possible to compute the coupling coefficient values.

.                                  (19)

It should be noted that the results obtained from the approximate formula k_i,i+1 match with those found from the exact method ĸ_i,i+1.

• Coupling Coefficients and Filter Optimization

Various optimization techniques are used in the software packages for microwave modelling.  The smart/intelligent methods for microwave optimization have been proven to be much more useful and effective than ordinary optimization techniques because intelligent techniques use preliminary design information. One such technique was presented by (Belyaev et al. 2006) for band-pass microwave filters. Smart and intelligent optimization techniques aim to produce the required pass-band and not vary the design parameters much. These parameters are kept constant to be used for further optimizations, for example, miniaturization. An objection feature of the intelligence optimization technique is the deflection vector D with n + 1 components.

For n+1 components, n represents the number of resonators in the filter. The first three components are defined as

                                      (20)

                              (21)

,                        (22)

where ω_c is the current centre frequency, ∆ω₀ is the current bandwidth, and R_i is the ith maximum reflected power numbered in the frequency axis direction [5, 6]. Definitions of other components depend on n. In the case of a four-resonator filter,

                                   (23)

,                               (24)

Understanding how to build a proper conjugate correction operation for each deflection component is possible by following certain rules applicable to the intelligence optimization method.

These rules operate in terms of such physical quantities as resonant frequencies of all resonators, external Q factor of input and output resonators, and coupling coefficients for all pairs of adjacent resonators. Components of the deflection vector D are divided into even and odd relative to the centre frequency or centre reflection maximum(s). The odd components are D₁ and D₄. Even components are D₂, D₃, D₅.

Each physical quantity is matched with one of the best suitable structure parameters, e.g., the coupling coefficient |k| is matched with spacing S between resonators.

The correction operation for an odd component involves correction operations for frequency-related structure parameters. The correction operation for an even component involves correction operations for coupling-related structural parameters. For example, the correction operation for D₅ > 0 comes down to increasing the coupling coefficient |k₁₂| and simultaneously decreasing the coupling coefficient |k₂₃| to assure that the value of the product is not changed, i.e. |k12|²|k₂₃| (Belyaev et al., 1996).

All correction operations associated with deflection vector apparatus/components are quasi-orthogonal, which specifies that every correction operation, with the elimination of conjugate-deflection-component, produces other deflection components having an absolute value lower than the absolute value of the initial deflection component.

Computation and examination of strip and microstrip filters and analysis and synthesis have been successfully performed over the last ten years using intelligence optimisation techniques (Belyaev et al., 2007). The intelligence optimisation method has recently become widely used for bandpass filters with dual-mode resonators.  Instead of using one oscillation ode per resonator, the pass band formation in these filters incorporates two vibration modes. It is recommended to treat each dual mode filter resonator as a pair of a couple of single mode resonators for the rules formulated for single mode filter to be applicable. Therefore, the two resonant frequencies of the dual-mode resonator must match the frequencies of coupled vibration in such an imaginary setting. A simulation or experiment can be performed to obtain an answer to how to control the coupling coefficient and resonant frequencies of coupled imaginary resonators.

Another area of application of coupling coefficients for filter optimization is the coupling matrix (Li et al., 2008) and the extended coupling matrix, which are even. Coupling coefficients kij between jth and ith resonators are the off-diagonal elements of these matrices. Cross-coupling between non-adjoining resonators can, therefore, be achieved. It is assumed that the coupling coefficients in these matrices are not dependent on frequency. Furthermore, it is possible to calculate the frequency response of an equivalent network using the extended coupling matrix and the coupling matrix. Space mapping optimisation of coupled resonator microwave filters is also based on these common models (Amari et al., 2006).

A similarity transformation for the extended coupling matrix can be completed as we move to a linear combination from resonator voltages. The overall frequency response is unaffected by these transformations, so it would not be wrong to say that all these transformations are alike. One of these transformed matrices is known as the transversal matrix, which results in all off-diagonal elements being zero. On the other hand, the diagonal elements of the coupling matrix are taken as the Eigenvalues. Additionally, it should be noted that the Eigenmodes are not attached. Amari and Bekheit (2007) thoroughly discuss this concept's similar transformations and physical understanding.

Conclusion

This report details the design process of spiral microstrip filters, specifically focusing on Chebyshev and quasi-elliptic filters. The design considerations encompassed external and internal couplings, elucidated through diagrams, with calculations to determine the optimal structure. The introduction of microstrip patch capacitors between resonators was found to enhance coupling, with a consequential increase in external coupling. Notably, patch capacitors allowed for omitting via holes in the filter structure.

The quasi-quarter-wavelength resonator filters demonstrated a notable advantage by avoiding resonance at twice the main resonance frequency. Moreover, it was observed that maintaining equality in the resonant frequencies of resonators could result in an asymmetric response in a filter with nonadjacent coupling. To address this, non-identical resonators were employed while preserving symmetry, establishing two transmission zeroes on either side of the filter response.

Two spiral filters were designed, simulated, and fabricated—a Chebyshev filter and a quasi-elliptic filter. The measured results are closely aligned with the simulated outcomes. The fabricated Chebyshev filter exhibited a 20% bandwidth, while the quasi-elliptic filter had a 12% bandwidth around 1 GHz. This underscores the efficacy of the quasi-quarter-wavelength spiral resonators method in practical microwave filter applications, yielding highly satisfactory results.