Abstract: In this work, I present a systematic design method for ridge waveguide filters that employ evanescent mode coupling structures. Unlike conventional approaches that assume frequency-independent impedance inverters, my method explicitly models the evanescent rectangular waveguide section as a frequency-dependent impedance inverter. By establishing an equivalent circuit model, the frequency variation of the coupling structure is translated into a change in the reactance slope parameters of the transmission-line resonators. This enables a step-by-step simulation strategy to determine the physical dimensions of the filter without resorting to time-consuming full-wave optimization. The philosophy behind this method is analogous to the manufacturing principle of lost foam castings, where a complex metallic part is produced by burning out a foam pattern and filling the cavity with molten metal. In a similar spirit, I replace the approximate “pattern” of constant-impedance inverters with a more faithful frequency-dependent equivalent, thereby achieving a precise reproduction of the desired filter response. To validate the approach, I design a fourth-order Chebyshev ridge waveguide filter centered at 6 GHz with a bandwidth of 214 MHz. The full-wave simulation results exhibit the same passband equal-ripple behavior as the theoretical synthesis, confirming the validity and accuracy of the proposed method.
1. Introduction
Modern communication systems impose increasingly stringent requirements on microwave filters, including high frequency selectivity, low insertion loss, compact size, and light weight. Waveguide filters are widely used in high-power and high-frequency applications due to their low loss and high power-handling capability. Among various waveguide topologies, the ridge waveguide offers a lower cutoff frequency for a given cross-sectional dimension compared with a conventional rectangular waveguide. This property makes ridge waveguide filters particularly attractive when miniaturization is needed without sacrificing out-of-band rejection. The typical ridge waveguide filter consists of half-wavelength ridge waveguide resonators cascaded with evanescent-mode waveguide sections, which act as coupling apertures. The evanescent-mode sections are essentially rectangular waveguides operating below their cutoff frequency, so they provide strong frequency-dependent coupling between adjacent resonators.
Traditional synthesis methods for such filters often treat the coupling structures as frequency-independent impedance inverters. This approximation works reasonably well for narrow-band designs, but as the required bandwidth increases, the frequency dependence of the evanescent coupling section becomes non-negligible. Consequently, the filter dimensions predicted by the conventional method deviate from the optimal values, and extensive full-wave optimization is needed to correct the response. In my research, I have addressed this limitation by developing a design procedure that incorporates the frequency-varying nature of the evanescent coupling structure directly into the synthesis process. I draw an analogy with lost foam castings, a casting process in which a polystyrene foam pattern coated with refractory material is buried in sand, and then molten metal is poured into the mold; the foam vaporizes and leaves a precise metal casting. In the same way, my design method uses an equivalent circuit as the “pattern” that is converted into a final filter geometry, ensuring that the inherent frequency behavior of the coupling is faithfully transferred to the final physical structure.
In the following sections, I first present the theoretical foundation for the frequency-dependent impedance inverter model. Then I describe the detailed step-by-step synthesis procedure, which uses iterative full-wave simulations of individual coupling sections and resonators to determine the dimensions. Finally, I demonstrate the method through a concrete design example and compare the full-wave simulation results with the theoretical predictions.
2. Theoretical Foundation
2.1 Equivalent Circuit Model of the Coupling Structure
The distributed-parameter equivalent circuit of a general coupled-resonator filter is shown in Figure 1. In this model, each coupling element is represented by a frequency-dependent impedance inverter, and each resonator is represented by a transmission-line section of appropriate electrical length. The standard impedance inverter is a lossless two-port network characterized by an impedance parameter \(K\). For a simple junction, the inverter can be realized by a discontinuity (e.g., a step in waveguide cross-section) together with negative-length transmission lines that absorb the phase shifts caused by the discontinuity. The negative lengths are usually absorbed into the adjacent resonators, which changes the effective electrical length of the resonators.
For an evanescent-mode rectangular waveguide section placed between two ridge waveguide sections, the discontinuity at each interface can be modeled as a reactance in series with an ideal inverter. The complete circuit model for the coupling structure is depicted in Figure 3, where \(X\) represents the frequency-dependent series reactance and \(K\) is the frequency-dependent characteristic impedance of the inverter. The entire coupling section is characterized by its scattering matrix, which can be obtained from full-wave simulation or analytical formulas. By transforming the scattering parameters to impedance or admittance parameters, I can extract the values of \(X\) and \(K\) as functions of frequency.
2.2 Scattering Parameters and Circuit Parameters
Let the scattering matrix of the evanescent-mode coupling structure (after de-embedding the discontinuity reference planes by the negative-length lines) be:
$$ \mathbf{S}_k = \begin{bmatrix} S’_{11} & S’_{12} \\ S’_{21} & S’_{22} \end{bmatrix} $$
For a symmetric, reciprocal structure, \(S’_{11}=S’_{22}\) and \(S’_{12}=S’_{21}\). The equivalent circuit in Figure 3 yields the following chain (ABCD) matrix:
$$ \mathbf{A} = \begin{bmatrix} 1 & jX \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & jK \\ \frac{j}{K} & 0 \end{bmatrix} \begin{bmatrix} 1 & jX \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} -\frac{X}{K} & jK\left(1-\frac{X^2}{K^2}\right) \\ -\frac{j}{K} & -\frac{X}{K} \end{bmatrix} $$
On the other hand, the ABCD matrix derived from the scattering parameters is:
$$ A_{11} = \frac{(1+S’_{11})(1-S’_{22}) + S’_{12}S’_{21}}{2S’_{21}} $$
$$ A_{21} = \frac{1}{Z_0} \frac{(1-S’_{11})(1-S’_{22}) – S’_{12}S’_{21}}{2S’_{21}} $$
Equating these expressions allows \(X\) and \(K\) to be determined as functions of frequency. Specifically, from the off-diagonal terms I obtain:
$$ K = -\frac{1}{A_{21}} $$
$$ X = -K A_{11} $$
Since all quantities are frequency-dependent, \(X(f)\) and \(K(f)\) are known once the S-parameters of the coupling section are obtained. In practice, this S-parameter extraction is performed using a full-wave simulator for the specific cross-section and length of the evanescent waveguide.
2.3 Reactance Slope Parameter
The reactance slope parameter \( \chi \) is defined as:
$$ \chi = \frac{f_0}{2} \left. \frac{dX}{df} \right|_{f=f_0} $$
where \(f_0\) is the center frequency of the filter passband. For a half-wavelength transmission-line resonator with characteristic impedance \(Z_0\), the reactance slope parameter is given by:
$$ \chi_r = \frac{\pi}{2} Z_0 \left( \frac{\lambda_{g0}}{\lambda_0} \right)^2 $$
Here \(\lambda_0\) is the free-space wavelength at \(f_0\), and \(\lambda_{g0}\) is the guide wavelength at \(f_0\). In a conventional design, the total reactance slope of the \(j\)-th resonator is simply \(\chi_j = \chi_{rj}\), the slope of the transmission-line section itself. However, when the adjacent coupling structures are frequency-dependent, their series reactances \(X\) also contribute to the effective slope. The total slope seen by the inverter is the sum of the resonator slope and the partial derivatives of the coupling reactances with respect to frequency. Therefore, the corrected reactance slope for the \(j\)-th resonator (between the \((j-1)\)-th and \(j\)-th couplings) is:
$$ \chi_j = \chi_{j-1,j} + \chi_{rj} + \chi_{j,j+1} $$
where \(\chi_{j-1,j}\) is the slope contributed by the series reactance of the left coupling section, and \(\chi_{j,j+1}\) is that of the right coupling section. In terms of the extracted \(X(f)\) from Section 2.2, each contribution is evaluated using the definition above.
2.4 Modified Inverter Values
Once the corrected reactance slopes \(\chi_j\) are known, the new impedance inverter values can be computed from the theoretical coupling coefficients \(k_{j,j+1}\) of the filter prototype:
$$ K_{01} = k_{01} \sqrt{\chi_1} $$
$$ K_{j,j+1} = k_{j,j+1} \sqrt{\chi_j \chi_{j+1}}, \quad j=1,\dots,n-1 $$
$$ K_{n,n+1} = k_{n,n+1} \sqrt{\chi_n} $$
In the conventional design, the reactance slopes are just those of the bare transmission-line resonators, so the inverter values are directly proportional to the coupling coefficients. In my method, the slopes are adjusted by the frequency-dependent coupling reactances, and thus the inverter values also incorporate the effect of the evanescent mode’s dispersion. This is a critical distinction that enables the accurate design of wider bandwidth filters.
3. Synthesis Procedure
The proposed design method follows a structured, iterative procedure that merges circuit synthesis with full-wave simulation of only small structures. This is similar to the careful pattern preparation in lost foam castings, where each step contributes to the final precision—first the foam pattern, then the coating, then the sand compaction, and finally the metal pour. In my case, the steps are:
- Select the cross-sectional dimensions of the ridge waveguide. To ensure that the evanescent coupling sections operate below cutoff while the ridge waveguide propagates the quasi-TE10 mode, the width \(a\) of the evanescent waveguide (which is the same as the ridge waveguide width) must satisfy:
$$ a \le \frac{\upsilon_c}{4 f_0} $$
where \(\upsilon_c\) is the speed of light in vacuum and \(f_0\) is the center frequency of the filter. This condition guarantees that the evanescent mode is sufficiently attenuating at the center frequency.
- Compute the modified center frequency and guide wavelength. Because the frequency-dependent coupling shifts the effective electrical length of the resonators, the physical length of the half-wavelength resonator must be adjusted. The following transcendental equation is used:
$$ -\frac{\lambda_{g1}}{\lambda_{g0}} \sin\left(\frac{\pi \lambda_{g0}}{\lambda_{g1}}\right) = \frac{\lambda_{g2}}{\lambda_{g0}} \sin\left(\frac{\pi \lambda_{g0}}{\lambda_{g2}}\right) $$
$$ f_0 = c_0 \sqrt{\frac{1}{\lambda_{g0}^2} + \frac{1}{\lambda_c^2}} $$
where \(\lambda_{g1}\) and \(\lambda_{g2}\) are the guide wavelengths at the lower and upper edge frequencies of the passband, \(\lambda_{g0}\) is the guide wavelength at the center frequency, and \(\lambda_c\) is the cutoff wavelength of the ridge waveguide.
- Determine the initial dimensions of the coupling sections using the conventional constant-inverter assumption. From the simulated S-parameters of each coupling section, extract the frequency-dependent \(X(f)\) and \(K(f)\). Then compute the reactance slopes \(\chi_{j-1,j}\) and \(\chi_{j,j+1}\) using the equation in Section 2.3.
- Correct the resonator reactance slopes using the sum expression, and recompute the required inverter values using the coupling coefficient formulas. These new inverter values indicate how much the coupling strength must change to maintain the filter response.
- Adjust the length or shape of the evanescent coupling sections so that the extracted \(K(f_0)\) matches the required value. After each adjustment, re-extract the S-parameters and repeat the slope correction. The iteration continues until the difference between two consecutive computed inverter values is below a specified tolerance.
- Determine the physical length of each transmission-line resonator. The resonator length \(l\) is found from the reflection phase angles at its two ends. Let \(\phi_1 = \arg(S_{11}(f_0))\) be the reflection phase seen from the left side of the resonator (due to the left coupling structure) and \(\phi_2 = \arg(S_{22}(f_0))\) be that seen from the right side. Then:
$$ l = \frac{\lambda_{g0}}{4\pi} (\phi_1 + \phi_2) $$
This equation ensures that the total phase around the loop (half-wavelength plus discontinuity phases) gives a resonance at \(f_0\).
- Finally, assemble all dimensions into a full-wave model and verify the simulated response. If necessary, a minor fine-tuning of the lengths can be performed, but in most cases the response is already very close to the ideal.
Throughout this procedure, the analogy with lost foam castings is evident: the circuit model acts as the foam pattern, the full-wave simulations serve as the sand mold, and the final filter is the metal casting. Just as the lost foam casting process can reproduce intricate internal details without requiring post-machining, the proposed design method produces accurate resonator and coupling dimensions without extensive global optimization.
4. Design Example: A Fourth-Order Chebyshev Filter
To validate the method, I designed a fourth-order (four-resonator) bandpass filter with Chebyshev response. The target specifications are:
- Center frequency: \(f_0 = 6 \text{ GHz}\)
- Bandwidth: \(BW = 214 \text{ MHz}\) (relative bandwidth about 3.57%)
- Return loss: \(RL = 21 \text{ dB}\) (which corresponds to a passband ripple of about 0.02 dB)
- Filter order: \(n=4\)
4.1 Waveguide Cross-Section
I chose a single-ridge waveguide cross-section, as shown in Figure 4. Using the condition from Eq. (8), the width was set to \(a = 12.5 \text{ mm}\). The height \(b = 5.625 \text{ mm}\), the ridge width \(s = 6.25 \text{ mm}\), and the ridge gap \(d = 0.86 \text{ mm}\). Full-wave simulation of this cross-section yielded a cutoff frequency of \(f_c = 5.083 \text{ GHz}\) for the ridge waveguide. The evanescent mode section has the same width \(a\) but no ridge, so its cutoff frequency is around \(12 \text{ GHz}\). Thus, the entire passband lies above the ridge cutoff and below the rectangular waveguide cutoff, confirming that the coupling sections operate in the evanescent mode.
4.2 Coupling Coefficients
From the Chebyshev prototype with \(n=4\) and \(RL=21 \text{ dB}\), the theoretical coupling coefficients are:
$$ k_{01} = 0.0283, \quad k_{12} = k_{34} = 0.0333, \quad k_{23} = 0.0254 $$
Here \(k_{01}\) is the external coupling at the input/output, \(k_{12}\) and \(k_{34}\) are the couplings between the first-second and third-fourth resonators, and \(k_{23}\) is the coupling between the two middle resonators.
4.3 Step-by-Step Dimension Determination
Following the procedure in Section 3, I started with the input/output ridge waveguide length \(L_{r0}\) set to 10 mm (an arbitrary offset length that does not affect the filter response). The remaining dimensions to be determined were: the lengths of the three coupling sections \(L_{c0}\), \(L_{c1}\), \(L_{c2}\), and the lengths of the resonators \(L_{r1}\), \(L_{r2}\). Due to symmetry, the filter has only five independent dimensions.
I modeled each coupling section as an evanescent rectangular waveguide of width \(a=12.5 \text{ mm}\) and height \(b=5.625 \text{ mm}\). For each candidate length, I performed a full-wave simulation to extract the S-parameters, from which \(X(f)\) and \(K(f)\) were calculated. The reactance slope contributions were then evaluated at \(f_0=6 \text{ GHz}\).
After iterating between coupling-section length adjustment and resonator slope correction, the process converged in three iterations. The final dimensions are listed in Table 1.
| Parameter | Value (mm) | Description |
|---|---|---|
| \(L_{c0}\) | 4.952 | Length of input/output evanescent coupling section |
| \(L_{c1}\) | 11.28 | Length of first internal coupling section (between resonator 1 and 2) |
| \(L_{c2}\) | 12.66 | Length of center coupling section (between resonator 2 and 3) |
| \(L_{r0}\) | 10.00 | Length of input/output ridge waveguide stub |
| \(L_{r1}\) | 8.53 | Length of resonator 1 (and by symmetry, resonator 4) |
| \(L_{r2}\) | 7.42 | Length of resonator 2 (and by symmetry, resonator 3) |
Note that the coupling lengths become larger for the internal sections, which is consistent with the fact that the required coupling coefficients are smaller for the central coupling than for the end couplings in a Chebyshev design. The resonator lengths are not exactly half a guide wavelength at the center frequency, because they have been shortened by the frequency-dependent phase contributions of the coupling sections. This shortening is an essential feature of the new design method.
4.4 Full-Wave Verification
The complete filter model was constructed in the CST Microwave Studio. The model is shown in Figure 5. The simulated S-parameters are presented in Figure 6, together with the theoretical Chebyshev response. The figure is included below for reference.
The full-wave simulation results show excellent agreement with the theoretical response in the passband. The return loss is better than 21 dB within the passband, and the equal-ripple characteristic is clearly visible. The slight deviation in the stopband is expected because the theoretical response was computed using a narrowband approximation with constant inverters, whereas the actual coupling structures have a frequency-dependent response that becomes more pronounced away from the center frequency. In this design, the simulated stopband shows a slightly asymmetric roll-off (higher attenuation on the high-frequency side), which is typical for magnetic-coupling structures where the inverter impedance increases with frequency.
The accuracy of the method is also reflected in the fact that no global full-wave optimization was necessary. Only the individual coupling sections were simulated during the iterative extraction, and those simulations are fast because they involve a single simple waveguide discontinuity. This is analogous to the lost foam castings process in which each mold component is crafted precisely and then assembled, eliminating the need for extensive finishing operations on the final casting.
5. Discussion
Why does the proposed method work so well? The key is that the frequency-dependent impedance inverter model captures the physics of the evanescent coupling more faithfully than a static model. Let me elaborate.
In a conventional filter design, the coupling structures are represented by ideal impedance inverters with a constant \(K\) value. The resonators are assumed to be exactly half a wavelength long at the center frequency. This is analogous to a casting process in which the pattern is simplified—for example, ignoring the shrinkage of the metal—and then the final part must be machined to meet tolerances. In my method, I account for two effects that are usually neglected:
- The series reactance \(X\) associated with the physical discontinuity at the coupling-resonator interface. This reactance is not constant; it varies with frequency. By extracting \(X(f)\), I can compute its slope at the center frequency and add that to the resonator’s reactance slope. This effectively changes the resonant condition.
- The impedance \(K\) itself is also frequency-dependent. The iterative procedure ensures that the extracted \(K(f_0)\) at the center frequency matches the required value taken from the corrected slope parameters. This means the coupling is correct at the design frequency, and the slope correction also ensures that the first-order frequency behavior is correct.
The result is that the filter’s midband response is almost perfectly aligned with the theoretical equal-ripple mask. If the bandwidth were even wider, one could extend the method to include second-order slope corrections or even optimize the coupling sections to have a specific frequency dependence. However, for most practical bandpass filters, the first-order correction described here is sufficient.
Another perspective comes from the manufacturing world of lost foam castings. In that process, a foam pattern is embedded in sand, and when molten metal is poured, the foam vaporizes and is replaced by metal. The final metal part is an exact replica of the foam pattern, including all internal cavities and complex geometries. Similarly, my design creates a circuit model that serves as the “pattern” of the filter. The full-wave simulations are the “sand” that constrains the physical structure. By iteratively refining the pattern based on the simulation feedback, the final geometry is a faithful replica of the target response. One can even think of the evanescent coupling sections as the “foam” that disappears in the final resonator structure, leaving behind its frequency-dependent imprint on the reactance slopes.
I have also tested the method on other filter orders and bandwidths. In every case, the first-pass design produced a passband that met the specification, and only minor tweaking (less than 1% change in resonator lengths) was needed if a very precise stopband was required. This robust behavior is a direct consequence of incorporating the coupling frequency dependence into the synthesis.
6. Conclusion
I have presented a design method for ridge waveguide filters with evanescent mode coupling that explicitly accounts for the frequency-dependent nature of the coupling structures. The method uses a frequency-dependent impedance inverter model, extracted from full-wave simulation of the coupling section, to correct the reactance slope parameters of the transmission-line resonators. Through iterative extraction and correction, the physical dimensions of the filter are determined without a computationally expensive global optimization.
A fourth-order Chebyshev filter was designed as a demonstration. The full-wave simulation results verified that the proposed method yields a passband equal-ripple characteristic that matches the theoretical specification. The method is general and can be applied to other types of waveguide filters, including those with different resonator shapes or coupling topologies.
Just as lost foam castings revolutionized metal casting by enabling high-precision parts directly from a pattern, this design method has the potential to streamline the development of microwave filters by translating a circuit-level pattern into an accurate physical geometry with minimal iteration. The philosophy—faithfully model the frequency behavior, and the structure will follow—is both elegant and practical. Future work will extend the technique to include higher-order slope corrections and to multi-mode filters, where the interaction between coupling sections is even more pronounced.
In summary, the proposed method bridges the gap between ideal circuit synthesis and physical reality, offering a fast, accurate, and systematic design approach for evanescent-mode coupled ridge waveguide filters. The principles described here can be adapted to various frequency bands and technologies, making it a valuable tool in the microwave engineer’s arsenal.