The configuration of the SSACR is shown in Figure 1. It consists of five layers: two metal layers and three substrate layers, among which Layers 1 and 5 are metal plates, and Layers 2, 3, and 4 utilize the same type of microstrip substrate. Layers 2 and 4 are hollowed out to form a rectangular air cavity which is surrounded by an array of metallic via holes. Within the upper and lower metallic surfaces of Layer 3, two rectangular patterns with the same size of that in Layers 2 and 4 are etched, respectively, which is also surrounded by an array of metallic via holes. In addition, within the upper surface of Layer 3, a metallic line acts as the transition between the microstrip line and the SSACR. Simultaneously, it also acts as the probe for exciting EM energy within the SSACR. Finally, all five layers are integrated tightly with screws to form the proposed SSACR. This specific configuration makes the SSACR effectively retain EM energy so that high Q can be obtained for the operation.

To facilitate practical application of the proposed SSACR, it is necessary to reveal its field distribution. Figure 2A shows a rectangular substrate-suspended waveguide. In essence, the SSACR can be considered as a rectangular substrate-suspended waveguide resonator. However, it is not easy to directly analyze the model in Figure 2A. Instead, we use an equivalent model, i.e., a rectangular waveguide filled with a uniform medium with the equivalent permittivity ε_eff to simplify direct analysis of the SSACR.

In the following, the effectiveness of such approach will be verified. The verification process includes the following several steps. First, a rectangular substrate-suspended waveguide is simulated to obtain the phase at the cut-off frequency. Secondly, the ε_eff of the rectangular waveguide filled with the uniform medium is calculated. For a resonator based on the rectangular waveguide filled with a uniform medium, the fundamental resonance frequency is calculated. If the fundamental resonance frequency of this rectangular waveguide resonator filled with a uniform medium is close to that of the SSACR, the equivalence between these two resonators is verified.

At first, the propagation constant β of the rectangular substrate-suspended waveguide is written as

where φ is the phase, and l is the length. In addition, the propagation constant of the rectangular waveguide filled with a uniform medium is

where μ₀ and ε₀ are the magnetic permeability and the dielectric constant in a vacuum, respectively. Then, ε_eff can be obtained from (1) and (2). For example, that of the TE₁₀ mode is obtained as

Furthermore, the resonance condition of the rectangular waveguide resonator filled with uniform medium is

When n = 1, the electromagnetic field in the cavity is TE₁₀₁ mode, and its resonance frequency can be obtained by (2), (3) and (4), i.e,

For demonstration, a substrate with relative permittivity of 3.66 and thickness of 0.508 mm is used in this paper. Based on that, an example of the SSACR is set up for EM analysis. The structural parameters are chosen as the following: a = 16.0 mm, b = 1.524 mm and l = 16.0 mm, respectively. The fundamental resonance frequency of the SSACR is 11.525 GHz. According to the verification process, first, a rectangular substrate-suspended waveguide is simulated to obtain the phase at the cut-off frequency. The simulation shows that the phase φ = 0.0418 rad at the cut-off frequency f _c = 8.157 GHz. Then, according to (3), ε_eff = 1.32. The resonant frequency of the rectangular waveguide resonator filled with uniform medium is calculated as f = 11.535 GHz, using (5). Meanwhile, its field distribution is shown in Figure 2B, which is similar to that of the SSACR. Obviously, the simulated resonant frequency of the SSACR is very close to that of the rectangular waveguide resonator filled with a uniform medium. Therefore, the SSACR can be considered equivalent to a rectangular waveguide resonator filled with a uniform medium. The verification steps above can also be used to determine the structural dimensions of the SSACR at a particular frequency, which will be applied in the design of the FSN based on the SSACR.

The well-known Leeson model is written as

where L is the feedback loop insertion loss, F is the additional noise figure of the amplifier, K is the Boltzman constant, T is the ambient temperature, P_c is the oscillator output power, ω_c is the flicker noise corner frequency, ω_m is the deviation frequency, and τ_d is the feedback loop group delay, respectively, [11]. According to (6), both the insertion loss and group delay of the FSN have a great impact on the phase noise of an oscillator, between which compromise has to be made. Compared with a single resonator, a bandpass filter comprising more than one resonance might form a more flexible and complicated response to meet the requirements of an oscillator. The SSACR in the previous subsection could be used to construct a bandpass filter and then act as the FSN in microwave oscillator.

It should be noted that there is some difference between the performance of a conventional filter and that of the FSN used in oscillator. The latter aims to reduce low phase noise for an oscillator, and thus there is a specific requirement for its magnitude and phase responses simultaneously. In practice, the design procedure of a conventional bandpass filter can be applied, and then at least initial values can be obtained for starting. Through full-wave simulation, the performance of the FSN can be finally optimized to meet the requirements of microwave oscillator.

For example, we design a second-order Chebyshev bandpass FSN by coupling two SSACRs, as shown in Figure 3. In order to prevent electromagnetic leakage due to the gap between the substrates and the metal covers, two screw holes are arranged between two SSACRs. To apply the proposed FSN to the following X-band oscillator, the center frequency of the FSN is set at 11.0 GHz, and the bandwidth is 35 MHz. The physical mechanism of the FSN in Figure 3 can be described by the coupling topology in Figure 4. According to the design procedure of a conventional bandpass filter described in [12], the element parameters of the low-pass prototype are calculated as: g₀ = 1.0000, g₁ = 0.9635, g₂ = 0.6581, and g₃ = 1.4642, respectively. Then, the coupling coefficient M and the external Q_e factor can be expressed as:

where FBW is the fractional bandwidth. For this case, Q_e1 = 302.8, Q_e2 = 302.8, M_{S, 1} = 0.0032, M_{1, 2} = 0.0040, and M_{2, L}= 0.0032, respectively.

The model to extract actual external Q_e-factor is shown in Figure 5. For narrowband operation, the dimensions of a single SSACR can be determined by (5). For this example, siw_x = siw_y = 16.2 mm. In order to avoid the influence of the screw holes on the electromagnetic field in the SSACR, some of the structural parameters in Figure 5 are: B_d = 1.0 mm, rad₁ = 0.4 mm, rad₂ = 2.1 mm, rad_d = 0.2 mm, and rad_d₂ = 0.5 mm, respectively. For the sake of the external 50-ohm microstrip line being well matched to the SSACR, some structural parameters are IN_g = 2.0, feel_s₁ = 0.5 mm, and feel_s₂ = 1.0 mm, respectively.

Then, the external Q_e-factor can be extracted by using the following formula [12].

where τ_s11(ω₀) is the group delay of S₁₁ at the resonant frequency ω₀. In Figure 6, the relationship between some structural parameters and the external Q_e-factor is depicted. Then, the desired value can be located on the curve so that the corresponding structural parameter can be determined. For this example, IN_l = 4.6 mm, and IN_w = 0.2 mm, respectively.

To extract the coupling coefficient k, the following formula is used.

where f _p1 and f _p2 are the two resonant frequencies under weak coupling. In Figure 7, the relationship between the coupling gap feel_l and the coupling coefficient is depicted. For this example, when feel_l = 3.4 mm, k = 0.0030, which is very close to the desired value of 0.0032.

Following the design procedure for a conventional bandpass filter, we can obtain a set of initial values for the proposed FSN in Figure 3. Then, according to oscillator requirement, the performance of the FSN is further optimized. Finally, IN_l = 5.2 mm, IN_w = 0.2 mm, and feel_l = 3.2 mm, respectively.

For this case, the simulated results are shown in Figure 8. Clearly, the second-order Chebyshev bandpass response is formed. Figure 8A depicts the magnitude response, and the minimum insertion loss at the center frequency is about 2.63 dB. Specially, high group delay is formed within a narrow range around the center frequency, as shown in Figure 8B. For example, the ones at 11.06, 11.08 and 11.10 GHz are 9.96, 8.37, and 9.98 ns, respectively. Such specific response is beneficial to frequency stability and phase noise reduction in the following oscillator design.