**2. Element variable coupling matrix (EVCM)**

The coupling matrix is the most commonly used technique to synthesize the fixed filter. One can prescribe or optimize the filter function in terms of the fixed filter specification, and then the coupling matrix can be directly extracted from the filter function. After a few iterations of mathematical manipulation/optimization, the coupling matrix can physically correspond to the filter structures. This is the synthesis process for the fixed filter. However, for the tunable filter, the frequency of the resonator is variable. When the filter is tuning (tunable frequency or tunable BW), there are numerous filter prototypes (filter matrices) corresponding to the same filter response. As illustrated in **Figure 2**, one can extract three different coupling matrices according to the same filter response using different tuning frequencies (*fd*) and different tuning bandwidths (BWs). This means three coupling matrices can be corresponding to the same filter response. Obviously, the conventional coupling matrix is not appropriate to synthesize the tunable filter, and therefore the element variable coupling matrix (EVCM) is introduced.

The EVCM is derived from the conventional coupling matrix and can be treated as the tunable version of the coupling matrix. First of all, the conventional coupling matrix can be extracted using the frequency-fixed filter synthesis (e.g. using the matrix manipulation method [5] or optimization process [38]) based on the prescribed frequency response. For the lossless NN coupling matrix with the termination impedance *R*1, the response is calculated based on the current loop model and voltage node model as:

**Figure 2.** *Same response of the three different classical coupling matrices.*

$$\begin{cases} \mathbf{S}\_{21} = \frac{2\mathbf{R}\_1}{\boldsymbol{w}\_0 \boldsymbol{L} \times \boldsymbol{F} \mathbf{B} \mathbf{W}\_d} [\boldsymbol{A}]^{-1} {}\_{N,1} = \frac{2}{\boldsymbol{w}\_0 \boldsymbol{C} \mathbf{R}\_1 \times \boldsymbol{F} \mathbf{B} \mathbf{W}\_d} [\boldsymbol{A}]^{-1} {}\_{N,1} \\\\ \mathbf{S}\_{11} = \mathbf{1} - \frac{2\mathbf{R}\_1}{\boldsymbol{w}\_0 \boldsymbol{L} \times \boldsymbol{F} \mathbf{B} \mathbf{W}\_d} [\boldsymbol{A}]^{-1} {}\_{1,1} = \mathbf{1} - \frac{2}{\boldsymbol{w}\_0 \boldsymbol{C} \mathbf{R}\_1 \times \boldsymbol{F} \mathbf{B} \mathbf{W}\_d} [\boldsymbol{A}]^{-1} {}\_{1,1} \end{cases} \tag{1}$$

where

$$[A] = \begin{bmatrix} 1/(\mathcal{Q}\_{\varepsilon} \times \text{FBW}\_{d}) + \Lambda\_{d} + j\mathcal{M}\_{11} & j\mathcal{M}\_{12} & \cdots & j\mathcal{M}\_{1n} \\ & j\mathcal{M}\_{21} & & \Lambda\_{d} + j\mathcal{M}\_{22} & \cdots & j\mathcal{M}\_{2n} \\ & \vdots & & \vdots & & \vdots \\ & & j\mathcal{M}\_{n1} & & j\mathcal{M}\_{n2} & \cdots & 1/(\mathcal{Q}\_{\varepsilon} \times \text{FBW}\_{d}) + \Lambda\_{d} + j\mathcal{M}\_{kk} \\ & & & & & (2) \end{bmatrix} \tag{2}$$

$$\begin{cases} \frac{1}{Q\_c} = \frac{R\_1}{w\_0 L} = \frac{1}{R\_1 w\_0 C} \\\\ \Lambda\_d = j \left(\frac{f}{f\_d} - \frac{f\_d}{f}\right) / FBW\_d \\\\ M\_{kk} = \left(\frac{f\_d}{f\_{kk}} - \frac{f\_{kk}}{f\_d}\right) / FBW\_d \end{cases} \tag{3}$$

*Λ<sup>d</sup>* and *Mij* denote normalized frequency, and element of the coupling coefficient between resonator *i* and *j*. *Mkk* is the self-coupling coefficient or immittance, representing the corresponding resonator's frequency shift. *fkk* is the resonant frequency of each resonator (or resonator *k*). As implied by (1–3), the passband response is calculated from a coupling matrix when the CF (scaling frequency *fd*) and fractional bandwidth (*FBW*) are given. However, for the tunable filter, both BW (or*FBW*) and CF are variable. Thus, it is difficult to form a tunable passband by a coupling matrix and then realize the filter accordingly (as demonstrated in **Figure 2**). The tuning range and behavior are also important since the passband is tunable, but the conventional matrix cannot work. Therefore, the coupling matrix is reformulated as:

$$\begin{cases} \mathbf{S}\_{21} = \frac{2}{Q\_{\epsilon}} [\boldsymbol{B}]^{-1} \boldsymbol{1}\_{N,1} \\\\ \mathbf{S}\_{11} = \mathbf{1} - \frac{2}{Q\_{\epsilon}} [\boldsymbol{B}]^{-1} \boldsymbol{1}\_{1,1} \end{cases} \tag{4}$$

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

$$\begin{aligned} [B] = \begin{bmatrix} 1 \\ \overline{Q\_{\epsilon} + \lambda\_{d} + jm\_{11}} & jm\_{12} & \cdots & jm\_{1n} \\ jm\_{21} & \lambda\_{d} + jm\_{22} & \cdots & jm\_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ jm\_{n1} & jm\_{n2} & \cdots & \overline{Q\_{\epsilon} + \lambda\_{d} + jm\_{kk}} \end{bmatrix} = ([Q\_{\epsilon}] + \lambda\_{d}[W] + j[m\_{\Delta}]) \end{aligned} \tag{5}$$

$$\begin{cases} \lambda\_d = j \left( \frac{f}{f\_d} - \frac{f\_d}{f} \right) \\\\ m\_{kk} = \left( \frac{f\_d}{f\_{kk}} - \frac{f\_{kk}}{f\_d} \right) \end{cases} \tag{6}$$

$$[\mathbf{Q}\_{\epsilon}] = \begin{bmatrix} \mathbf{1} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} \\ \vdots & \vdots & \vdots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \frac{\mathbf{1}}{\mathbf{Q}\_{\epsilon}} \end{bmatrix} \tag{7}$$

Here, the denormalized matrix [*m*△] has an interesting feature. For demonstration, a dual-mode resonator filter (or two-order transversal filter) is taken as an example (as shown in **Figure 3**). The matrix [*m*△] of this filter can be extracted according to the aforementioned techniques as:

$$
\begin{bmatrix} m\_{\Delta} \end{bmatrix} = \begin{bmatrix} m\_{\text{et}} & m\_{\text{co}} \\ m\_{\text{oet}} & m\_{\text{oo}} \end{bmatrix} = \begin{bmatrix} \times & \mathbf{0} \\ \mathbf{0} & \mathbf{y} \end{bmatrix} \tag{8}
$$

For simplification, the external Q factors Qexe/exo for two resonant modes are roughly approximate to *Qe* as:

$$Q\_{\omega} = \frac{(Q\_{\text{exc}} + Q\_{\text{exc}})}{2} \tag{9}$$

**Figure 3.** *Architecture of the dual-mode resonator tunable filter.*

It can be derived that

$$\begin{cases} \mathbf{x} = f\_d / f\_e - f\_e / f\_d\\ \mathbf{y} = f\_d / f\_o - f\_o / f\_d\\ \mathbf{Q}\_{ex\epsilon/exo} = \frac{f\_{ex\epsilon/exo}}{\Delta f\_{\pm 90}} \approx \frac{\mathbf{1}}{\mathbf{F} \mathbf{B} \mathbf{W}\_{ex\epsilon/exo}} = \frac{f\_{ex\epsilon/exo}}{\mathbf{B} \mathbf{W}\_{ex\epsilon/exo}} \end{cases} \tag{10}$$

Now, it can be found that all elements of matrix [*m*△] are directly related to the architecture. As long as the parameters of the architecture are defined with the given tuning ranges and the scaling frequency *fd*, the responses of the tunable filter (e.g. tuning ranges of CF, BW, return loss (RL)) can be completely determined. The second-order filter example is given for demonstration, as shown in **Figure 4**, where the matrix is denoted as *x*, *y*, and *Qe*. **Figure 5** presents the responses of the matrix when the *x* is varied. It is observed that only the even-mode resonant frequency is tuned, implying even-mode resonant frequency is independently controlled by *x*. Similarly, **Figures 6** and **7** show that the odd-mode frequency and RL independently varied by changing *y* and *Qe,* which suggests that the odd-mode resonant frequency and the RL are independently controlled by *y* and *Qe*.

With this interesting feature, a one-to-one correspondence between the tunable filter responses and this matrix can be established. As a result, the matrix can uniquely represent tunable filter responses. **Figures 8**–**10** presents the typical responses by purposely varying the matrix elements [*m*△]. As shown in **Figure 8**, the matrix elements *x, y,* and *Qe* are changed from 0.058, 0.034, and 18.765 to 0.028, 0.044, and 27.777, thus forming a 0.8 GHz to 1.2 GHz CF-tunable filter response with a constant 130 MHz BW. Hence, this tunable filter can be synthesized by such a coupling matrix

**Figure 4.** *Theoretical response of a fixed filter example.*

**Figure 5.** *Theoretical curves for varying* x*.*

**Figure 6.** *Theoretical curves for varying* y*.*

**Figure 7.** *Theoretical curves for varying* Qe*.*

#### **Figure 8.**

*Theoretical curves for tuning the CF with 130-MHz 3-dB BW and 20-dB RL.*

with these variable elements. Similarly, 130 MHz to 300 MHz BW-tunable filter responses with the constant 1 GHz CF (as shown in **Figure 9**) and RL-tunable filter responses with 1 GHz CF and130 MHz BW (as shown in **Figure 10**) are formed by specifying the elements of the matrix. So, the element variable coupling matrix (EVCM) can synthesize these tunable filters.

The physical structure is presented to realize the EVCM, as shown in **Figure 11**. The given circuit is controlled by loaded varactor diodes, and each control element can be used to tune the entry of the EVCM separately. For example, the variable capacitor *Co* and *Ce*, respectively, control the odd-mode and even-mode resonant frequencies, and adjusting *Cm* will tune the external quality factor *Qe*. Therefore, this structure is a fully tunable filter. Since the multimode structures are challenging to be coupled with equal energy (*Qexo* = *Qexe*) for every resonant mode, the added circuit configuration

**Figure 9.** *Theoretical curves for tuning the BW with 1 GHz CF and 20-dB RL.*

**Figure 10.** *Theoretical curves for the RL reconfiguration with 1 GHz CF and 130 MHz BW3db.*

**Figure 11.** *Transmission line model and physical circuit.*

with *Ce2* is employed. Properly adjusting *Ce2* will enforce *Qexo* = *Qexe* and fully implement the EVCM.

**Figures 12**–**14** presents the measurement results of the physical circuit corresponding to the calculated results of EVCM. As can be seen, the in-band responses of the filter are all very close to the given EVCM. The CF frequency of the filter can be tuned from 0.8 GHz to 1.2 GHz, and a 130-MHz 3-dB BW is kept nearly

**Figure 12.** *Measurement curves for tuning the CF with constant BW.*

**Figure 13.** *Measurement curves for tuning the BW with constant CF.*

constant, as implied in **Figure 12**. These responses closely agree with the EVCM results as shown in **Figure 8**. The BW varying from 50 to 400 MHz is measured as shown in **Figure 13** when the circuit is fully tuned, which corresponds to the EVCM results as

**Figure 14.** *Measurement curves for tuning the RL with constant CF and BW.*

shown in **Figure 9**. **Figure 14** shows the RL reconfiguration results of the filter. As shown, the passband is fixed at 1 GHz with 130-MHz BW, while the RL is reconfigured as prescribed by the EVCM (**Figure 10**) and the rejection is reshaped. The agreement between the experimental circuit and EVCM is obtained. Note that the practical circuit generates the transmission zeros to enforce *Qexo* = *Qexe,* which causes the major disagreement in the stopband. These three transmission zeros are owing to the added circuit configuration, which introduces an out-band resonant mode that forms one more coupling path from source to load and generates transmission zeros there.
