Design of the tunable resonator

The schematic of a proposed tunable resonator is exhibited in Fig. 1(a), which is composed of two coupled lines ((Z _1e, Z _1o, θ ₁), (Z _1e, Z _1o, θ ₂)) and one varactor diode (C _t). Due to the symmetry of the designed resonator, the (even-)odd-mode method is adopted to analyze the operating mechanisms. Moreover, the equivalent circuits of the designed tunable resonator under (even-)odd-mode excitation can be simplified as Fig. 1(b) and 1(c), respectively.

Fig. 1.

The schematic of (a) the presented tunable resonator, (b) the even-mode equivalent circuit, (c) the odd-mode equivalent circuit.

As observed from Fig. 1(b) and 1(c), the admittance of Port 1 in the even- and odd-mode equivalent circuits (Y _ine, Y _ino) can be deduced as equation (1) based on transmission line theory, and the transmission zero can be introduced when Y _ine = Y _ino. Moreover, the even- and odd-mode resonant conditions can be expressed as equation (2). By submitting equation (1) into (2), the even-(odd-)mode resonance frequencies (f _e, f _d) are derived as equation (3) based on Taylor expansion. It is concluded from equation (3) that f _e is influenced by the value of varactor diode (C _t), the electrical length (θ, θ = θ ₁ + θ ₂) and the even-mode characteristic impedance (Z _1e) of the coupled line, while f _d is only determined by the electrical length (θ) of a coupled line.

(1a)$$Y_{ine} = Y_{1e}\displaystyle{{\,jY_{1e}\cdot \tan \theta _1 + j\omega _e\cdot C_t/2} \over {Y_{1e}-\omega _e\cdot ( C_t/2) \cdot \tan \theta _1}} + \displaystyle{{Y_{1e}} \over {\,j\tan \theta _2}}$$

(1b)$$Y_{ind} = \displaystyle{{Y_{1o}} \over {\,j\tan \theta _ 1}} + \displaystyle{{Y_{1o}} \over {\,j\tan \theta _2}}$$

(2a)$${\mathop{\rm Im}\nolimits} ( Y_{ine}) = 0$$

(2b)$${\mathop{\rm Im}\nolimits} ( Y_{ind}) = 0$$

(3a)$$f_e\approx \displaystyle{1 \over {2\pi }}\cdot \sqrt {-\displaystyle{{3\left({-\omega_0^3 \sqrt {( Z_{1e}^2 \cdot \theta^2/\omega_0^2 ) + ( 8Z_{1e}\cdot \theta^3/3C_t\cdot \omega_0^3 ) } + Z_{1e}\cdot \omega_0^2 \cdot \theta } \right)} \over {2Z_{1e}\cdot \theta ^3}}} $$

(3b)$$f_d = \displaystyle{{\omega _0} \over {2\theta }}$$

(4)$$f_0 = \displaystyle{1 \over {2\pi }}\cdot \sqrt {-\displaystyle{{3\left({-\omega_0^3 \sqrt {( Z_{1e}^2 \cdot \theta^2/\omega_0^2 ) + ( 8Z_{1e}\cdot \theta^3/3C_t\cdot \omega_0^3 ) } + Z_{1e}\cdot \omega_0^2 \cdot \theta } \right)} \over {2Z_{1e}\cdot \theta ^3}}} $$

To further explain the operating mechanisms of the resonance frequency, the simulated f _e, f _d from Advanced Design Software (ADS) and the calculated f _e, f _d from equation (3) with different values of θ and C _t are plotted in Fig. 2. Accordingly, the calculated resonance frequencies are in good agreements with the simulated resonance frequencies, which verify the correctness of equation (3). Moreover, f _d is much higher than f _e when θ is less than $60^\circ$ @1 GHz (C _t  = 0.63 pF), and f _d is much higher than f _e from 0.5 to 20.0 GHz (θ = 30^o). Thus, there is no coupling between f _e and f _d in these cases, and the operating mechanisms of the designed tunable resonator become simple. Specifically, the resonance frequency (f ₀) is only determined by f _e in these cases, and the expression of f ₀ can be deduced as equation (4). Moreover, it is seen from Fig. 2(b) that f _e decreases with the increased value of C _t, while f _d remains constant with the increased value of C _t. Thus, f ₀ can be adjusted by tuning the value of C _t.

Fig. 2.

The f _e and f _d with different values of (a) θ (C _t  = 0.63 pF), (b) $C_t( \theta = 30^\circ )$.

The frequency tuning range (R _f) is defined as equation (5), while the f _L and f _H mean the lowest and highest resonance frequencies of the designed resonator. It is concluded from equations (4)–(5) that R _f relies on Z _1e, θ and C _t. Then, Fig. 3 plots the calculated f _L, f _H, R _f with different values of θ and Z _1e when the C _t is employed as SMV1405-079 with the tuning range from 0.63 to 2.67 pF (DC bias voltage is from 30 to 0 V). As observed from Fig. 3, larger Z _1e and shorter θ will result in bigger R _f, which provides effective guidance to extend R _f. In this paper, considering the processing technology, the values of θ and Z _1e are selected as $30^\circ$ @1 GHz and 140 Ω to obtain a wide frequency tuning range.

(5)$$R_f = \displaystyle{{\,f_H-f_L} \over {0.5\cdot ( {\,f_H{\rm} + f_L} ) }}$$

Fig. 3.

The f _L, f _H and R _f with different values of (a) $Z_{1e}( \theta = 30^\circ )$, (b) θ @1 GHz (Z _1e = 140 Ω).

Operating mechanism of the external quality factor

To analyze the external quality factor (Q _e) of the feeding resonator, the lumped-element equivalent circuit should be utilized. Based on [Reference Hong and Lancaster25], the transmission lines ((Z _1e, θ ₁), (Z _1e, θ ₂)) in the even-mode equivalent circuit (Fig. 4(a)) could be considered as inductances (L ₁, L ₂), and the expressions of L ₁, L ₂ are derived as equation (6). Thus, the lumped-element equivalent circuit of the even-mode resonator can be extracted as Fig. 4(b), which also can be simplified as a traditional L′ − C′ circuit (Fig. 4(c)). Accordingly, the input admittances (Y _ine(b), Y _ine(c)) in Figs 4(b) and 4(c) also can be expressed as equation 7(a)–7(b). To make sure the equivalence of Fig. 4(b) and 4(c), the relationship between Y _ine(b) and Y _ine(c) should satisfy equation 7(c). Following the circuit theory, the external quality factor $( Q_e^{\rm {\prime}} )$ of the traditional L′ − C′ circuit can be derived as equation (8). Thus, the external quality factor (Q _e) in Fig. 4(b) can be derived and shown in equation (9) by submitting equation (7) into (8).

(6a)$$L_1 = \displaystyle{{Z_{1e}\cdot \tan ( \omega \cdot \theta _1/\omega _0) } \over \omega }$$

(6b)$$L_2 = \displaystyle{{Z_{1e}\cdot \tan ( \omega \cdot \theta _2/\omega _0) } \over \omega }$$

(7a)$$Y_{ine( b) } = \displaystyle{1 \over {( 1/j\omega ( C_t/2) ) + j\omega L_1}} + \displaystyle{1 \over {\,j\omega L_2}}$$

(7b)$$Y_{ine( c) } = j\omega \cdot C^{\rm {\prime}} + \displaystyle{1 \over {\,j\omega \cdot L^{\rm {\prime}}}}$$

(7c)$$Y_{ine( b) } = Y_{ine( c) }$$

(8)$$Q_e^{\rm {\prime}} = R\sqrt {\displaystyle{{C^{\rm{\prime}}} \over {L^{\rm{\prime}}}}} $$

(9)$$Q_e = R\sqrt {\displaystyle{{{( L_1 + L_2) }^3\cdot C_t/2} \over {L_2^4 }}} $$

Fig. 4.

(a) The even-mode equivalent circuit, (b) Lumped-element equivalent circuit, (c) Simplified lumped-element equivalent circuit.

The f ₀, Q _e calculated from equations (4), (9) and simulated from Advanced Design Software (ADS) are plotted in Fig. 5. Accordingly, the Q _e will increase with increased C _t, and f ₀ will shift to lower frequency with bigger C _t. It can be observed from Fig. 5(b) that bigger $\theta _2/\theta _1( \theta _1 + \theta _2 = 30^\circ )$ will result in smaller Q _e, and f ₀ remains constant with different values of θ ₂/θ ₁. Consequently, the f ₀ can be adjusted by tuning the value of varactor diode (C _t), and initial Q _e is mainly determined by θ ₂/θ ₁.

Fig. 5.

The external quality factors and resonance frequencies with different values of (a) $C_t\;( \theta _2 = 11^\circ )$, (b) $\theta _2/\theta _1\,( C_t = 2\,{\rm pF}, \;\theta _1 + \theta _2 = 30^\circ )$. (Dash lines: simulated results, Solid lines: calculated results).

Operating mechanism of the coupling structure

The schematic of the presented coupling structure is exhibited in Fig. 6(a), which is composed of transmission lines (Z _m, θ _m) and varactor diodes (C_m). Moreover, the transmission line (Z _m, θ _m) can be considered as the inductance (L _m), and the value of L _m can be derived as equation (10). Thus, the lumped-element equivalent circuit can be extracted and exhibited in Fig. 6(b), which may produce the spurious band. To verify the analysis above, the transmission coefficients of the designed second-order tunable BPF with different values of C _m are plotted in Fig. 7. Accordingly, the bigger C _m will result in a lower spurious band during the out-of-band.

(10)$$L_1 = \displaystyle{{Z_{1e}\cdot \tan ( \omega \cdot \theta _1/\omega _0) } \over \omega }$$

Fig. 6.

(a) The schematic of the designed tunable coupling structure, (b) The lumped-element equivalent circuit of the designed tunable coupling structure.

Fig. 7.

The transmission coefficients of the designed second-order tunable BPF with different values of C _m.

As observed from Fig. 6(b), the coupling coefficients are determined by the C_m and L _m, while L _m is influenced by the characteristic impedance (Z _m) and electrical length (θ _m) of the transmission line (Z _m, θ _m). Then, the coupling coefficients with different values of C _m, Z _m and θ _m are illustrated in Fig. 8. As observed from Fig. 8(a), the increased Z _m and θ _m will result in smaller M _ij. It can be concluded from Fig. 8(b) that M _ij will become bigger when the value of capacitance (C _m) increases. Thus, the initial value of M _ij is determined by Z _m, θ _m, and the value of M _ij can be further adjusted by tuning the value of C _m.

Design of the fully tunable BPFs

By employing the designed tunable resonators and coupling structures, the schematic diagram of the Nth-order fully tunable BPF is presented and exhibited in Fig. 9. To realize Nth-order Chebyshev filtering response, the values of coupling coefficient (M _ij) and external quality factor (Q _e) should satisfy equation (11), and the detailed values of g _i−1, g _i, g _i+1 are given in [Reference Hong and Lancaster25].

(11)$$M_{ij} = \displaystyle{{FBW} \over {\sqrt {g_i, \;g_{i + 1}} }}{\kern 1pt} , \;\quad Q_e = \displaystyle{{g_{i-1}\cdot g_i} \over {FBW}}\quad ( i\in 1, \;2, \;\ldots , \;N) $$

Fig. 8.

The coupling coefficients with different parameters of (a) θ _m, Z _m (C _m = 0.5  pF), (b) $C_m( \theta _m = 9.2^\circ @1\,{\rm GHz}, \;\;Z_m = 148\,\Omega )$.

Fig. 9.

The schematic of the presented Nth-order fully tunable BPF.

Based on the analysis above, the detailed design procedure for Nth-order fully tunable BPF is summarized as following:

1. (1) Given the required center frequency (f ₀), passband bandwidth (BW), frequency tuning range (R_f), frequency selectively and harmonic suppression of the Nth-order tunable BPF.

2. (2) Confirm the order of tunable BPF, and calculate the values of Q_e, M_ij from equation (11).

3. (3) Realize the resonance at f ₀ by determining the relationship between Z _1e, θ and C_t, based on equation (4).

4. (4) Optimize the Q_e of a designed resonator by adjusting the electrical length ratio of coupled lines (θ ₂/θ ₁).

5. (5) Obtain the required R_f by selecting appropriate C_t, Z _1e, θ based on Fig. 3.

6. (6) Realize the initial M_ij by tuning the values of Z_m and θ_m, then adjust the M _ij by tuning the value of C _m.

7. (7) Construct the initial layout of Nth-order fully tunable BPF by employing the designed tunable resonators and coupling structures, then simulate and optimize the tunable BPF based on the operating mechanisms.

To demonstrate the generality of the presented Nth-order tunable BPF structure, the second- and third-order tunable BPFs have been designed and simulated. Moreover, the detailed layout and corresponding transmission coefficients with different values of C _t and C _m are plotted in Figs 10 and 11. Accordingly, the center frequency in the designed Nth-order tunable BPF can be adjusted by tuning the value of C _t, and the passband bandwidth in the designed Nth-order tunable BPF could be adjusted by changing the value of C _m, which have good agreements with the analysis above.

Fig. 10.

(a) The layout of the designed second-order tunable BPF $( l_1 = 14.7\,{\rm mm}, \;l_2 = 6.4\,{\rm mm}, \;l_3 = 8.7\,{\rm mm}, \;w_0 $ $= 2.2\,{\rm mm}, \; w_1 = 0.45\,{\rm mm}, \; s_1 = 0.5\, {\rm mm},$ $w_3 = 0.2\,{\rm mm}, \;R_b = 100\,{\rm k}, \;C_s = 6.8\,{\rm pF}, \;L_s $ $= 0.6\,{\rm nH}, \;R_s = 0.07, \;C_d = 100\,{\rm pF})$, (b) The transmission coefficients of the designed second-order tunable BPF with different values of C _t, (c) The transmission coefficients of the designed second-order tunable BPF with different values of C _m.

Fig. 11.

(a) The layout of the designed third-order tunable BPF $( l_1 = 15.1\,{\rm mm}, \;l_2 = 7.7\,{\rm mm}, \;l_3 = 5.2\,{\rm mm}, \;w_0 $ $ = 2.2\,{\rm mm}, \;w_1 = 0.45\,{\rm mm}, \; s_1 = 0.5\,{\rm mm},$ $w_3 = 0.2\,{\rm mm}, \;R_b = 100\,{\rm k}, \;C_s = 6.8\,{\rm pF},$ $L_s = 0.4\,{\rm nH}, \;R_s = 0.07, \;C_d = 100\,{\rm pF})$, (b) The transmission coefficients of the designed third-order tunable BPF with different values of C _t, (c) The transmission coefficients of designed third-order tunable BPF with different values of C _m.