Singlet: asymmetric iris coupled transverse rectangular ridge resonator

The ridge can be modeled as a length of parallel plate line open circuited at both ends [Reference Bastioli, Marcaccioli and Sorrentino25]. The ridge resonate at the frequency f ₀, when the length ℓ is

(1)$$\ell = {\lambda_0\over 2}$$

where λ₀ is the free space wavelength at f ₀.

Being a half-wave length section of an open-circuited parallel-plate line, the transverse ridge has positive electric field maximum at the one end and a negative maximum at the other end [see Fig. 2(a)]. The TE₁₀ mode having a maximum electric field at the center with diminishing fields at the side walls [see Fig. 2(b)], thus cannot excite the resonant mode of the ridge. The symmetry is broken by introducing asymmetric irises, which then excite the higher-order modes particularly TE ₂₀ mode [see Fig. 2(c)], that has a field configuration aligned to the ridge mode and thus can excite the ridge resonator.

Fig. 2. Cross-sectional views showing electric field distribution for (a) fundamental resonant mode of transverse ridge inside a rectangular waveguide, (b) TE₁₀ mode in a rectangular waveguide, and (c) TE₂₀ mode in a rectangular waveguide.

Thus, in the presence of asymmetric irises, the dominant mode of the rectangular waveguide forms the source to load coupling (J _SL), while the higher-order modes like TE₂₀ couples energy to or from the resonator (J _S1 or J _1L), thus forming a singlet.

The 3 × 3 normalized coupling matrix of the form shown below can be utilized to represent the response of the proposed ridge resonator structures:

(2)$$M = \left[\matrix{ 0 & J_{S1} & J_{SL}\cr J_{S1} & B & J_{L1}\cr J_{SL} & J_{L1} & 0 }\right]$$

The coupling and routing diagram for this singlet structure is shown in Fig. 3.

Fig. 3. Coupling and routing diagram of the proposed transverse rectangular ridge-based singlet.

Different filter responses can be achieved from the proposed singlet structure by modifying the structure.

Transverse rectangular ridge resonator with no irises

Here a rectangular ridge resonator is placed transverse to the waveguide axis [see Fig. 4(a)]. For symmetry reasons explained above this ridge resonator will not be excited by the dominant TE₁₀ mode of the rectangular waveguide and thus behaves as just a capacitive discontinuity. This case does not create any pole or any TZ, i.e. J _S1 = J _1L = 0. Source to load coupling is not zero J _SL ≠ 0, because of the TE₁₀ mode of the waveguide.

Fig. 4. Transverse rectangular ridge resonator with (a) no irises, (b) same side irises, and (c) opposite side irises. Dotted arrows: source to load coupling (J _SL). Solid arrows: input and output to ridge couplings (J _S1, J _1L).

Transverse rectangular ridge resonator with asymmetric irises on the same side

When asymmetric irises are on the same side [see Fig. 4(b)], a TZ below the passband can be realized. As shown in Fig. 5(a), the location of the TZ can easily be varied by varying dimension d of each iris. Some fine adjustment is also achievable by changing distance p of each iris from the ridge [Fig. 5(b)]. In this case, all couplings are non-zero J _S1 ≠ 0, J _1L ≠ 0, and J _SL ≠ 0. Fig. 5(c) shows the source to resonator coupling versus d and p. Fig. 5(d) gives the plot of source to load coupling versus d and p. Note that J _S1 is positive and J _SL is negative for the same side irises.

Fig. 5. Transverse rectangular ridge resonator with asymmetric irises on the same side. (a) S ₂₁ for different values of d, (b) S ₂₁ for different values of p, (c) J _S1 versus variations in d and p, and (d) J _SL versus variations in d and p.

It is worth mentioning that Fig. 5 shows the singlet for which d and p of input and output irises are equal and thus J _S1=J _1L, which is not necessarily required by the structure.

Transverse rectangular ridge resonator with asymmetric irises on the opposite sides

For asymmetric irises located on the opposite sides [see Fig. 4(c)], a TZ above the passband can be realized. Location of the TZ can be changed by varying dimension d and/or p, as shown in Figs 6(a) and 6(b). All coupling coefficients namely J _S1, J _SL, and J _1L are non-zero in this case. As shown in Figs 6(c) and 6(d), both J _S1 and J _SL are positive thus leading to a TZ above the passband. Note that Fig. 6 shows the singlet for which d and p of input and output irises are equal and thus J _S1=J _1L, which is not necessarily required by the structure.

Fig. 6. Transverse rectangular ridge resonator with asymmetric irises on the opposite sides. (a) S ₂₁ for different values of d, (b) S ₂₁ for different values of p, (c) J _S1 versus variations in d and p, and (d) J _SL versus variations in d and p.

It is worth mentioning, that for cases discussed in subsections ‘Transverse rectangular ridge resonator with asymmetric irises on the same side’ and ‘Transverse rectangular ridge resonator with asymmetric irises on the opposite sides’, dimensions d and p, of each input and output irises can be adjusted independent of each other. This means J _S1 and J _1L do not necessarily have to be the same, leading to more flexibility in realization of desired filter responses.

Third-order pseudoelliptic filter with one TZ

A third-order filter with one TZ in the lower stopband is designed. Lower stopband TZ is realized using the singlet with the same side irises [see Fig. 4(b)]. The coupling and routing diagram for this filter is shown in Fig. 7. The coupling matrix is of the form

(3)$$M = \left[\matrix{ 0 & M_{S1}& 0& 0& 0\cr M_{S1}& M_{11}& M_{12}& M_{13}& 0\cr 0& M_{12}& M_{22}& M_{23}& 0\cr 0& M_{13}& M_{23}& M_{33}& M_{3L}\cr 0& 0& 0& M_{3L}& 0 }\right]$$

Fig. 7. Coupling and routing diagram for the third-order filter with one TZ.

The coupling matrix for the required filter response is synthesized using optimization-based technique [Reference Amari, Rosenberg and Bornemann27]. The coupling matrix response is shown in Fig. 8. The synthesized coupling matrix is given below:

(4)$$M = \left[\matrix{ 0& 1.0821& 0& 0& 0\cr 1.0821& -0.0597& 1.0041& -0.2663& 0\cr 0& 1.0041& 0.2670& 1.0041& 0\cr 0& -0.2663& 1.0041& -0.0597& 1.0821\cr 0& 0& 0& 1.0821& 0 }\right]$$

To design the filter from this coupling matrix, an equivalent circuit shown in Fig. 9 has been synthesized. The equivalent circuit is comprised of input and output couplings (K _S1 and K _3L), waveguide sections (WG ₁ and WG ₃ of lengths ℓ₁ and ℓ₃, respectively) and singlet (S2). Using the basic microwave network theory [Reference Pozar28], each block of the equivalent circuit can be represented as ABCD matrix. The resulting ABCD matrix of the complete filter is obtained by multiplying individual matrices and is given as:

(5)$$\eqalign{ABCD &= ABCD_{KS1}\times ABCD_{WG1}\times ABCD_{S2} \cr& \quad \times ABCD_{WG3}\times ABCD_{K3L} }$$

where ABCD matrices for input and output couplings and waveguide sections may be given as below [Reference Pozar28, Reference Hong29]:

$$ABCD_{KS1} = \left[\matrix{ 1& jK_{S1} \cr j/K_{S1}& 0 }\right]$$

$$ABCD_{K3L} = \left[\matrix{ 1& jK_{3L} \cr j/K_{3L}& 0 }\right]$$

$$ABCD_{WG1} = \left[\matrix{ \cos( \beta\ell_1)& j\sin( \beta\ell_1) \cr j\sin( \beta\ell_1)& \cos( \beta\ell_1) }\right]$$

$$ABCD_{WG3} = \left[\matrix{ \cos( \beta\ell_3)& j\sin( \beta\ell_3) \cr j\sin( \beta\ell_3)& \cos( \beta\ell_3) }\right]$$

where β is the phase constant of the TE₁₀ mode in the rectangular waveguide.

Fig. 8. Comparison of coupling matrix, circuit, FEST3D, and HFSS responses for the third-order filter.

Fig. 9. Equivalent circuit for third-order filter with one TZ.

The singlet ABCD matrix is obtained from scattering parameters by using standard conversion tables [Reference Pozar28]. Singlet S-parameters can be obtained from its coupling matrix of (2), using equations from [Reference Amari, Rosenberg and Bornemann27].

To synthesize the equivalent circuit of Fig. 9, optimization-based methodology [Reference Amari, Rosenberg and Bornemann27] is utilized. The design parameters to be obtained through optimization are given as the vector below:

(6)$$x = \left[\matrix{ K_{S1}& \ell_1& J_{S1}& J_{SL}& B& J_{1L}& \ell_3& K_{L3} }\right]$$

The cost function utilized is given in (7), where the S-parameters for each iteration and each frequency point are obtained using (5) and converting the ABCD to S-parameters.

(7)$$\eqalignb{\Phi &= \sum_{i = 1}^{n}w_{\,pi}\left\vert S_{11}\left(\,f_{\,pi}\right)\right\vert ^2 + \sum_{\,j = 1}^{m}w_{zj}\left\vert S_{21}\left(\,f_{zj}\right)\right\vert ^2 \cr &\quad + \left(\left\vert S_{11}\left(\,f_{1}\right)\right\vert -S_{11req} \right)^2 + \left(\left\vert S_{11}\left(\,f_{2}\right)\right\vert -S_{11req} \right)^2 }$$

where S _11req = 10^−RL/20. f _pi and f _zj are the required poles and TZs frequencies, respectively. w _pi and w _zj are the weights for poles and zeros, respectively, which may be adjusted to give more importance to certain terms in the cost function. w _pi = 1 and w _zj = 100 have been used for this third-order filter design. RL is the required return loss in the passband of the filter. f ₁ and f ₂ are the frequency points at the edges of the passband. m is the number of TZs and n is the number of poles. For this filter, f ₁ = 10.1 GHz, f ₂ = 10.4 GHz, RL = 20 dB, n = 3, and m = 1.

The synthesized equivalent circuit parameters are shown in Table 1.

Table 1. Synthesized equivalent circuit parameter values for the designed third-order filter.

The frequency response of this synthesized equivalent circuit has been plotted in Fig. 8, and it compares well to the coupling matrix response.

Using the synthesized equivalent circuit [see Table 1], the physical dimensions of the filter can be determined. Using K _S1 and K _3L, the iris width of the input and output irises, respectively, can be determined using the well-known inductive iris-based filter design procedure [Reference Cameron, Mansour and Kudsia30]. The physical lengths for the two waveguide sections are already synthesized and are given in Table 1. For the singlet S2, full-wave electromagnetic simulations have been used to match the S-parameter response of the synthesized singlet coupling matrix, to the actual structure of Fig. 4(b). The comparison of circuit response and the simulation response is shown in Fig. 10, which shows the two responses are almost similar. The reference planes at the input and output of the simulated singlet structure are adjusted, to achieve phase response almost similar to that of the singlet coupling matrix phase response.

Fig. 10. Comparison of circuit and simulation responses of the singlet S2.

The resulting dimensions are then utilized to draw the structures in FEST3D and HFSS. Some optimizations are essential in these three-dimenstional electromagnetic simulation tools, since the circuit model does not cater for all the higher-order mode effects. Optimizations are first carried out in FEST3D and then finalized in HFSS. The resulting responses are shown in Fig. 8, which indicate that FEST3D and HFSS responses are a good match to the coupling matrix and circuit responses.

The designed third-order filter is manufactured using CNC milling of aluminum blocks. Photograph of manufactured prototype is shown in Fig. 11.

Fig. 11. Manufactured prototype of the third-order filter.

The measurements are carried out on a vector network analyzer. The measured results along with simulated response are shown in Fig. 12. These results indicate a good agreement of the measured S-parameters to the simulated response. Detailed view of the passband insertion loss is shown in the inset of Fig. 12, which indicates the measured insertion loss of 0.28 dB at the center frequency of 10.25 GHz, instead of the simulated value of 0.1 dB. Measured 3 dB bandwidth is 483.8 MHz in contrast to the simulated bandwidth of 489 MHz.

Fig. 12. Comparison of measured and HFSS responses of the third-order filter.

Fig. 13 shows the comparison of measured and simulated spurious responses. It can be seen that the upper stopband extends up to 14.65 GHz while no spurious response is observed in the lower stopband.

Fig. 13. Comparison of measured and FEST3D broadband responses of the third-order filter.

Fifth-order pseudoelliptic filter with two TZs

A fifth-order pseudoelliptic filter with two TZs above the passband is designed by using two singlets and three half-wave waveguide sections. The singlets utilized make use of the opposite side irises as shown in Fig. 4(c). The coupling and routing diagram for this filter is shown in Fig. 14.

Fig. 14. Coupling and routing diagram for the fifth-order filter with two TZs.

The filter can be represented by a 7 × 7 coupling matrix of the form shown below:

(8)$$M = \left[\matrix{ 0& M_{S1}& 0& 0& 0& 0& 0\cr M_{S1}& M_{11}& M_{12}& M_{13}& 0& 0& 0\cr 0& M_{12}& M_{22}& M_{23}& 0& 0& 0\cr 0& M_{13}& M_{23}& M_{33}& M_{34}& M_{35}& 0\cr 0& 0& 0& M_{34}& M_{44}& M_{45}& 0\cr 0& 0& 0& M_{35}& M_{45}& M_{55}& M_{5L}\cr 0& 0& 0& 0& 0& M_{5L}& 0 }\right]$$

This coupling matrix is synthesized using the same method as explained in Section ‘Third-order pseudoelliptic filter with one TZ’, and is given below: M =

(9)$$\scriptsize\left[\matrix{0& 1.0135& 0& 0& 0& 0& 0\cr 1.0135& 0.0325& 0.8606& 0.0938& 0& 0& 0\cr 0& 0.8606& -0.0947& 0.6307& 0& 0& 0\cr 0& 0.0938& 0.6307& 0.0867& 0.5937& 0.2690& 0\cr 0& 0& 0& 0.5937& -0.3495& 0.8228& 0\cr 0& 0& 0& 0.2690& 0.8228& 0.0325& 1.0135\cr 0& 0& 0& 0& 0& 1.0135& 0 }\right]$$

The frequency response of the coupling matrix of (9) is shown in Fig. 15. To design this filter, the equivalent circuit of Fig. 16 is utilized. The design procedure makes use of optimization-based strategy and is similar to the one described in Section ‘Third-order pseudoelliptic filter with one TZ’. Each block of the equivalent circuit is represented by its ABCD matrix. The synthesized parameters of the equivalent circuit of Fig. 16 are shown in Table 2.

Fig. 15. Comparison of coupling matrix, circuit, FEST3D, and HFSS responses for the fifth-order filter.

Fig. 16. Equivalent circuit for the fifth-order filter with two TZs.

Table 2. Synthesized equivalent circuit parameter values for the designed fifth-order filter.

For each singlet S2 and S4, the frequency response of the circuit [see Table 2] is matched using HFSS. The S-parameter response comparison between circuit and full-wave simulation for each singlet is shown in Fig. 17. The simulated response matches the circuit response for each singlet reasonably well.

Fig. 17. Comparison of circuit and simulation responses of the singlets.

Using the same procedure as in Section ‘Third-order pseudoelliptic filter with one TZ’, the resulting dimensions are realized in FEST3D and HFSS for the fifth-order filter. The results of simulations are shown in Fig. 15, which shows good consistency with the coupling matrix and circuit responses. The filter is manufactured and is shown in Fig. 18.

Fig. 18. Manufactured prototype of the fifth-order filter.

The measured results for this fifth-order waveguide filter are shown in Fig. 19, which also compares the measured S-parameter response with the HFSS simulated response. The measured S ₂₁ match well with the simulated response. The measured S ₁₁ response indicates that the return loss is deteriorated partially in the passband, which is attributed to machining tolerances.

Fig. 19. Comparison of measured and HFSS responses of the fifth-order filter.

To understand the cause for this deterioration in return loss, tolerance analysis is carried out using FEST3D. For each iteration, random error with a Gaussian distribution is added to all filter dimensions, except for the dimensions a and b of the rectangular waveguide. In FEST3D, 50 iterations are performed with the standard deviation of 12.5 μm. The results are shown in Fig. 20. It is observed that the return loss is more sensitive to manufacturing errors while the locations of TZs are less influenced by these errors.

Fig. 20. Tolerance analysis of the fifth-order filter.

The inset of Fig. 19 shows the close-up view of the in-band insertion loss. Measured insertion loss is 0.475 dB at 10.175 GHz, instead of the simulated value of 0.24 dB. Measured insertion loss is 0.862 dB at the center frequency of 10.25 GHz, instead of the simulated value of 0.25 dB. The increase in insertion loss at the center frequency is because of the deteriorated return loss at this frequency, which can be estimated using the S-parameters unitary condition for a lossless network [Reference Pozar28] that implies |S ₁₁|² + |S ₂₁|² = 1. Thus the measured dissipation loss at 10.25 GHz is 0.45 dB. Using this measured dissipation loss, unloaded quality factor of a waveguide section is estimated as $Q_u^{WG} = 3680$ and that of a ridge resonator as $Q_u^{ridge} = 1363$. Measured 3 dB bandwidth is 343.4 MHz in contrast to the simulated bandwidth of 353.3 MHz.

The comparison of measured and simulated spurious responses is shown in Fig. 21, which indicates that the upper stopband extends up till 14.6 GHz while no spurious response is observed in the lower stopband.

Fig. 21. Comparison of measured and FEST3D broadband responses of the fifth-order filter.