**4. Planar and 3D realizations of the constant-BW tunable filter based on the EVCM synthesis**

The synthesis method can be used to extract the EVCM according to the prescribed filter specifications. This section will present the planar and 3D tunable filters to realize the extracted EVCMs practically.

#### **4.1 Planar realization example of the constant-BW tunable filter**

The resonator of the planar filter example is shown in **Figure 16**, and the frequency range design method is given in **Figure 17**. The resonant frequency is tuned by changing variable capacitor *Ct*, and the tuning range can be predefined by *l*1. The capacitor loading position *lt* does not noticeably affect the tuning frequency.

**Figure 16.**

*Transmission line model of the synchronously tunable filter resonator.*

**Figure 17.** *Tunable resonant frequency vs. Ct, with different l1 or lt for (a) lt = 11 mm (b) l1= 48.6 mm.*

#### *Tunable Filter DOI: http://dx.doi.org/10.5772/intechopen.104391*

**Figures 18** and **19**, respectively, present the electrical coupling (EC) and magnetic coupling (MC) configurations between two planar resonators. Their corresponding coupling coefficient curves extracted from the configurations are shown in **Figures 20** and **21**. As can be seen, both the slope and position of coupling coefficient curves can be independently controlled with these two coupling configurations. Thus, the coupling elements in the EVCM can be physically realized, and both EC and MC are available.

Two feeding structures, i.e. short-end and open-end feeding structures, can be employed to feed this type of filter, as shown in **Figure 22a** and **b**. **Figures 23** and **24**, respectively, present the extracted external quality factor *Qe* curves for these two configurations. It can be seen that both the slope and position of *Qe* curves are approximately defined by adjusting their physical dimensions independently.

With all filter parts mentioned earlier, two tunable filters using EC and MC as the mainline coupling path to implement the constant BW are demonstrated. Two filters

**Figure 18.** *EC configuration between resonators.*

**Figure 19.** *MC configuration between resonators.*

#### **Figure 20.**

*Coupling coefficient extracted from the EC configuration (Figure 18) with different lt or lc when sc = 0.3 mm for (a) lc = 6.8 mm (b) lt = 15 mm.*

#### **Figure 21.**

*Coupling coefficient extracted from the MC configuration (Figure 19) with different lt or ls for (a) ls = 5 mm (b) lt = 8 mm.*

**Figure 22.** *(a) S-EF configuration. (b) O-EF configuration.*

*External quality factor Qe extracted from S-EF configuration (Figure 22a)with different wt or st for (a) st = 0.2 mm (b) wt = 0.4 mm.*

are the four-pole cascade quartet topology, and their prototypes are the 4th-degree General Chebyshev polynomials with 20-RL and 130-MHz ABW. The transmission zeros of two filters are prescribed at [10j, 2j, 2j, 10j]. The tuning range of the EC filter is predefined from 1.2 GHz to 1.6 GHz, and the EC filter is from 1.05 GHz to 1.45 GHz. According to the presented synthesis method, two EVCMs can be extracted, which are

*Tunable Filter DOI: http://dx.doi.org/10.5772/intechopen.104391*

#### **Figure 24.**

*External quality factor Qe extracted from O-EF configuration (Figure 22b) with different wt or st for (a) st = 0.22 mm (b) wt = 2.5 mm.*

$$\begin{cases} [m]\_{4-pole} = \begin{cases} m\_{12} = 0.091646 + m\_{kk} \cdot 0.04582 \\ m\_{23} = 0.080922 + m\_{kk} \cdot 0.04046 \\\\ m\_{14} = -0.0184 - m\_{kk} \cdot 0.0092 \end{cases} \end{cases} \tag{22}$$

$$Q\_{e4-pole} = 9.30255 - m\_{kk} \cdot 4.5322, m\_{sl} = 0.00016$$

for EC filter and

$$\begin{cases} \left[m\right]\_{4-pole} = \begin{cases} m\_{12} = 0.081612 + m\_{kk} \cdot 0.040806\\ m\_{23} = 0.072063 + m\_{kk} \cdot 0.036031\\ m\_{14} = -0.01638 - m\_{kk} \cdot 0.00819 \end{cases} \end{cases} \tag{23}$$

$$Q\_{e4-pole} = 10.39153 - m\_{kk} \cdot 5.0897, m\_{sl} = 0.00019$$

for MC filter. Besides, all filter examples are designed on the Rogers RT/duroid 5880 (*h* = 0.787 mm, *ε<sup>r</sup>* = 2.2, tan*δ* = 0.0009) and the variable capacitors are SMV1234 (*Ct* = 1.3-9.6 pF, *Rs* = 0.8 Ω).

**Figures 25** and **26** present the EC and MC filter examples where the cascade quartet configurations are used, thus yielding the prescribed tunable responses with constant BW. The measurement results of the two design examples are shown in **Figures 27** and **28**. The BW of the two filters is approximately 133 MHz and the tuning ranges are over 27%. Good agreement between simulations and measurements is

**Figure 25.** *EC filter example.*

**Figure 26.** *MC filter example.*

**Figure 27.** *Measurement results of the EC filter example (Figure 25).*

achieved, and the design objective is fully implemented. It is noted that the MC filter has a more stable BW because of the more flexible choosing range of coupling and *Qe* curves with the wider adjusting ranges.

### **4.2 3D realization example of the constant-BW tunable filter**

The ceramic monoblock waveguide filter featured a low-cost and competitive Qu/size ratio that attracted a great deal of attention for 5G applications [39]. The 3D realization example of the constant-BW tunable filter presented in this section will be based on the ceramic monoblock waveguide structure and the high-Q mechanically tunable mechanism.

**Figure 29** present the tunable ceramic monoblock waveguide resonator. It generally contains the fixed block and the movable cylinder. The low-dielectric (*ε<sup>r</sup>* = 19.5)

**Figure 28.** *Measurement results of the MC filter example (Figure 26).*

fixed block is coated with the silver layer except for the big center hole (*ht*21, *ht*22, *Rt*2). The hole on the bottom (*ht*1, *Rt*1) is metalized. The movable cylinder (*ε<sup>r</sup>* = 90) is not coated with the silver layer and can be dragged up and down in the non-metalized hole. The Teflon screw (M0.8) is employed to control the movable cylinder via a driving mechanism. The non-metalized hole with a lid constructs an air cavity to accommodate the movable cylinder. As the ceramic cylinder moves in the air cavity, the resonator's resonant frequency is tuned. **Figure 30** presents the tuning frequency ranges and the *Qu* performance by changing *ht*1. As expected, the frequency is tuned by changing the positions of the movable cylinder, and the *Qu* is always kept on a high level. The frequency tuning range can be predefined by *ht*1.

The coupling configuration between two resonators is shown in **Figure 31**. The extracted coupling coefficient curves with different key dimensions are presented in **Figure 32**. It is seen that the coupling coefficient curve is controlled by *hm*<sup>1</sup> and *Rm*1. *hm*<sup>1</sup> and *Rm*<sup>1</sup> have more influence in the coupling curve's high-frequency area and low-frequency area, respectively.

The feeding configuration for the tunable ceramic waveguide resonator is given in **Figure 33**, where the resonator is fed by a coplanar waveguide (CPW) line on the bottom of the fixed block. The external *Qe* can be evaluated by the single-end group delay of the fed resonator structure using *Qe* = 2π*f*<sup>0</sup>*τ*11(*f*0)/4[20]. **Figure 34** shows the

**Figure 30.**

*Resonant behavior of the tunable resonator with different ht1. (a) Resonant frequency tuning ranges and (b) unloaded Qu factor performance.*

**Figure 31.** *Coupling configuration between two resonators.*

**Figure 32.** *Extracted coupling coefficient curves with different (a) hm1 and (b) Rm1.*

**Figure 33.** *Feeding structures excited using stripline.*

**Figure 34.** *Group delay responses of the feeding structures with different (a) l0 and (b) w0.*

extracted group delay. The observation implies that the variation tendency of the *Qe* is controlled by *l*<sup>0</sup> and the magnitude is controlled by *w*<sup>0</sup> independently.

With the tunable resonator, coupling structure, and feeding configuration discussed earlier, the ceramic waveguide tunable filter can be constructed. For the demonstration, the six-degree Chebyshev low-pass prototype with 17-dB return loss and two TZs (�1.19511j) is employed to form the two-section cascade-triplet topology. The EVCM is extracted for the 5.8–6.2 GHz constant-400-MHz bandwidth filter as:


EM design according to the extracted EVCM is carried out, and the optimized filter structure with the manufactured design sample is given in **Figure 35**. The feeding circuit is on Rogers RT/duroid 5880.

**Figure 35.** *Six-pole quasi-elliptic ceramic waveguide filter with constant absolute bandwidth. (a) Filter structure and (b) filter photograph.*

*Tunable Filter DOI: http://dx.doi.org/10.5772/intechopen.104391*

**Figure 36.** *Measurement results of the manufactured design sample.*

**Figure 36** presents the measured responses of the tunable ceramic waveguide filter. The measured passband moves from 5.76 GHz to 6.22 GHz, but the 3-dB bandwidth maintains 429 7 MHz. The rectangular factor of two skirts is better than 12 dB/20 MHz because of two symmetric transmission zeros. The fitted *Qu* of the tunable filter is kept from 570 to 720.
