II. ANALYSIS OF THE SHORT/OPEN-STUB-LOADED RESONATOR

As shown in Fig. 1(a), the proposed resonator is a half-wavelength resonator with a short-stub and an open-stub shunted at the midpoint of the microstrip line, where Z ₁ and Z ₂ represent the characteristic impedances of the microstrip lines, and θ ₁, θ ₂, θ ₃ denote the lengths of half-wavelength resonator, short stub, and open stub, respectively. Since the resonator is symmetrical to the A–A′ plane, it is convenient to analyze it by using the odd- and even-mode methods.

Fig. 1.

(a) Structure of stub-loaded resonator, (b) odd-mode equivalent circuit, (c) even-mode equivalent circuit, (d) λ/4 resonator of even-mode equivalent circuit, (e) λ/2 resonator of even-mode equivalent circuit.

For odd-mode excitation, the symmetrical plane is electric wall, and the equivalent circuit is shown in Fig. 1(b). The input admittance can be expressed as

(1) $$Y_{ino} = \displaystyle{1 \over {jZ_1 \tan \theta _1}}. $$

Under the resonance condition, Y _ino  = 0, then (1) can be deduced as

(2) $$\theta _1 = \displaystyle{{\lambda _o} \over 4} = \displaystyle{c \over {4f_o \sqrt {\varepsilon _e}}} \quad {\rm or}\quad f_o = \displaystyle{c \over {4\theta _1 \sqrt {\varepsilon _e}}}, $$

where c is the speed of the light in free space, and ε _e denotes the effective dielectric constant of the substrate. It is demonstrated that the odd-mode resonant frequencies are not influenced by the stubs.

Under even-mode excitation, the symmetrical plane is magnetic wall, we can get the even-mode equivalent circuit (see Fig. 1(c)), which contains a quarter-wavelength resonator and a half-wavelength resonator. For the quarter-wavelength resonator (as shown in Fig. 1(d)), the input admittance can be obtain as

(3) $$Y_{ine1} = \displaystyle{{jZ_1 \tan \theta _2 + j2Z_2 \tan \theta _1} \over {Z_1 (2Z_2 - Z_1 \tan \theta _1 \tan \theta _2 )}}.$$

Under the resonance condition, then (3) can be deduced as

(4) $$Z_1 \tan \theta _2 = - 2Z_2 \tan \theta _1. $$

The half-wavelength resonator is shown in Fig. 1(e), the input admittance can be expressed as

(5) $$Y_{ine2} = \displaystyle{{Z_1 - 2Z_3 \tan \theta _1 \tan \theta _3} \over {jZ_1 (2Z_3 \tan \theta _3 - Z_1 \tan \theta _1 )}}.$$

Under the resonance condition, then (5) can be deduced as

(6) $$Z_1 = 2Z_3 \tan \theta _1 \tan \theta _3. $$

The resonators, as shown in Figs 1(d) and 1(e), can be considered as quarter-wavelength stepped impedance resonator (SIR) with one end grounded and half-wavelength SIR, respectively, and their resonance condition can be expressed as

(7a) $$\tan \theta _2 = - 2\tan \theta _1, \quad {\rm when}\quad Z_1 = Z_2,$$

(7b) $$1 = 2\tan \theta _1 \tan \theta _3, \quad {\rm when}\quad Z_1 = Z_3.$$

When θ ₂ ≈ θ ₃, we can obtain f _e1 < f _o  < f _e2 from (1)–(3). Thus, the first three resonant frequencies of the proposed stub-loaded resonator could be controlled by tuning θ ₁, θ ₂, and θ ₃. The first resonant frequency is determined by θ ₁ and θ ₂, the second resonator frequency is determined by θ ₁, and the third resonant frequency is determined by θ ₁ and θ ₃.

III. ANALYSIS OF THE TRANSMISSION ZEROS

The tri-bandpass filter structure will produce five transmission zeros. The five transmission zeros are named by TZ₁, TZ₂, TZ₃, TZ₄, and TZ₅ from the left- to the right side. According to equations (3) and (5), if Y = ∞, each of them will produce one transmission zero. The condition can be expressed as

(8a) $$2 = \tan \theta _1 \tan \theta _2, \quad {\rm when}\quad Z_1 = Z_2,$$

(8b) $$1 = 2\tan \theta _1 \tan \theta _3, \quad {\rm when}\quad Z_1 = Z_3.$$

Because the tri-bandpass filter is composed of the two same tri-mode resonators, every passband can be regarded as single-bandpass filter composed of two same single-mode resonators. The topology of the proposed single-bandpass filter is shown in Fig. 2. This topology will produce one transmission zero. So, the tri-bandpass filter will produce three transmission zeros. In summary, the proposed structure will have five transmission zeros at all.

Fig. 2.

Topology of the proposed single-bandpass filter.

IV. DESIGN OF THE PROPOSED POWER DIVIDER

Based on the analysis above, the proposed resonator has three resonant frequencies and can be utilized to design tri-bandpass power divider. The configuration of the proposed tri-band power divider is shown in Fig. 3. There are two tri-band bandpass filters embedded in the presented circuit structure. A shunt isolation resistor R is located between the two output ports. Owing to the symmetric topology, it can be analyzed by using the even- and odd-mode methods.

Fig. 3.

Configuration of the proposed tri-band power divider.

A) Even-mode

For the even-mode case, the even-mode equivalent circuit can be obtained and given in Fig. 4(a) when magnetic wall is applied to the symmetrical plane. Since the symmetrical plane is open circuit, there is no current flowing through the isolation resistor R; thus the isolation resistor can be considered invalid. Moreover, the characteristic impedance of input port 1 is 100 Ω.

Fig. 4.

Equivalent circuit: (a) even-mode, (b) odd-mode.

As can be seen, the even mode equivalent circuit is similar to that of an open-loop tri-band bandpass filter. It is demonstrated that the proposed power divider has tri-band bandpass filtering response because the bandpass filter is embedded in the power divider.

B) Odd-mode

For the odd-mode case, the odd-mode equivalent circuit is given in Fig. 4(b) when electric wall is applied to the symmetrical plane. Since the symmetrical plane is short circuit, port 1 is shorted and the voltage along the middle of the isolation element is zero. Thus, the impedance of the isolation element is R/2. Under the odd-mode excitation, there are two transmission paths between two ports. Paths I and II are illustrated in Fig. 5. It can be seen that the even- and odd-mode equivalent circuits have the same structure between points A and C. It is convenient for the analysis of even- and odd-mode equivalent circuits.

Fig. 5.

Input impedance of the odd-even equivalent: (a) path I, (b) path II.

For part 1, the coupled line can be considered as a four-port circuit. As shown in Fig. 5(a), two of the ports are open circuit and another is shorted. Then, it becomes a one-port network. The input impedance of part 1 can be deduced as

(9) $$Z_{in\_1} = Z_{11}^{(1)} - \displaystyle{{Z_{14}^{(1)}} \over {Z_{11}^{(1)}}}, $$

where Z ₁₁ ⁽¹⁾ and Z ₁₄ ⁽¹⁾ are the Z-parameters of the parallel coupled line in part 1. Part 1 just can be considered as a load of the transmission line with characteristic Z ₁ and electric length θ ₁. Then the input impedance of this transmission line can be obtained

(10) $$Z_{in\_2} = Z_1 \displaystyle{{Z_{in\_1} + jZ_1 \tan \theta _1} \over {Z_1 + jZ_{in\_1} \tan \theta _1}}. $$

From node P to the open stub, the input impedance Z _{in_2} is

(11) $$Z_{in\_2} = \displaystyle{{Z_2} \over {j\tan \theta _2}}. $$

From node P to the open stub, the input impedance Z _{in_3} is

(12) $$Z_{in\_3} = \displaystyle{{Z_3} \over {j\tan \theta _3}}. $$

Then, the input impedance Z _{in_p} from node P to the left part can be got

(13) $$Z_{in\_p} = Z_{in\_2} //Z_{in\_3} //Z_{in\_4}. $$

Also,

(14) $$Z_{in\_5} = Z_4 \displaystyle{{Z_{in\_p} + jZ_4 \tan \theta _4} \over {Z_4 + jZ_{in\_p} \tan \theta _4}}. $$

For part 2, it also can be considered as a four-port circuit. For simplifying the circuit analysis, the open-circuited microstrip line with length θ ₃ is viewed as a capacitance in the equivalent circuit, where C ₁ = tanθ ₅/2πfZ ₅. As shown in Fig. 5(b), one of the ports is opened and another is loaded with a capacitance. Then, it becomes a two-port network 1 and 2 are refined as the ports of the coupled line. The two-port elements of the impedance matrix are

(15) $$Z^{\prime}_{11} = Z_{11}^{(2)} - \displaystyle{{Z_{14}^{(2)} Z_{14}^{(2)}} \over {Z_{11}^{(2)} + X_1}}, $$

(16) $$Z^{\prime}_{12} = Z^{\prime}_{21} = Z_{13}^{(2)} - \displaystyle{{Z_{12}^{(2)} Z_{14}^{(2)}} \over {Z_{11}^{(2)} + X_1}}, $$

(17) $$Z^{\prime}_{22} = Z_{11}^{(2)} - \displaystyle{{Z_{12}^{(2)} Z_{12}^{(2)}} \over {Z_{11}^{(2)} + X_1}}, $$

where Z ₁₁ ⁽²⁾, Z ₁₂ ⁽²⁾, Z ₁₃ ⁽²⁾, and Z ₁₄ ⁽²⁾are the Z-parameters of the parallel coupled line in part 2. Moreover, Z′₁₁, Z′₂₁, Z′₁₂, and Z′₂₂ are the new Z-parameters of the two-port parallel coupled line, and X ₁ = 1/j2πfC ₁.

According to the conversion between the transmission matrix ABCD and the impedance matrix (Z), the transmission matrix elements for the part 2 can be obtained. The input impedance of path I Z _{in_p1} is then calculated as follows:

(18) $$Z_{in\_p1} = \displaystyle{{AZ_{in\_5} + B} \over {CZ_{in\_5} + D}}.$$

Z _{in_p2} (input impedance of path II) can be obtained by using the similar method as that of Z _{in_p1}. When Z _{in_p1} and Z _{in_p2} is obtained, the input impedance Z _in can be calculated by using parallel of the two input impedances

(19) $$Z_{in} = Z_{in\_p1} \vert \vert Z_{in\_p2}. $$

The equivalent circuit of the structure from A to B is shown in Fig. 6. Then, the input impedance of the transmission line with characteristic Z ₆ and electric length θ ₆ is

(20) $$Z_{in\_6} = Z_6 \displaystyle{{Z_{in} + jZ_6 \tan \theta _6} \over {Z_6 + jZ_{in} \tan \theta _6}}. $$

Fig. 6.

Equivalent circuit of the structure from A to B.

At port 2 (as shown in Fig. 6), it can be considered that the resistor of R/2 is parallel with Z _{in_6}. After that, the input impedance Z _{in_p1} is calculated by using

(21) $$Z_{in\_out} = Z_{in\_6} //\displaystyle{R \over 2} = 50\;\Omega. $$

As a result, it is convenient to select R to get a good output match and isolation between the two output ports.

V. SIMULATION AND MEASUREMENT

To verify and demonstrate the proposed tri-band power divider circuit, a tri-band power divider has been designed and implemented. A RF-35 substrate with a relative permittivity of 3.5, thickness of 0.508 mm, and loss tangent of 0.0018 is used. After simulation and optimization using HFSS, the dimensions of the fabricated tri-band filtering-response power divider are chosen as follows: L ₁ = 37.5 mm, L ₂ = 4.5 mm, L ₃ = 11 mm, L ₄ = 16.25 mm, L ₅ = 34.45 mm, W ₁ = W ₂ = 0.6 mm, W ₃ = 0.6 mm, S ₁ = S ₃ = 0.15 mm, S ₂ = 0.3 mm, S ₄ = 0.6 mm, S ₅ = 1 mm. The photograph of the fabricated tri-band filtering-response power divider is shown in Fig. 7.

Fig. 7.

Photograph of the fabricated tri-band power divider.

An Agilent 8757D network analyzer is used for measurements. The measured and simulated results are shown in Fig. 8. The measured 3-dB fractional bandwidths for the three passbands centered at 1.61, 2.48, and 3.54 GHz are found to be 5.6, 10, and 5%, respectively. The measured minimum insertion losses for the three passbands are 4.98, 4.34, 4.97 dB, and the return losses within the three passbands are 12.9, 19.7, and 17.9 dB, respectively. Moreover, more than 20 dB rejection levels within the lower stopbands, two middle stopbands, and upper stopbands are observed, featuring high out-of-band rejection level. In addition, five transmission zeros are located at 1.54, 2.2, 2.98, 3.64, and 4.25 GHz, respectively. These transmission zeros can improve the passband frequency selectivity and the out-of-band rejection level.

Fig. 8.

Simulated and measured results: (a) input return loss and insertion loss, (b) output return loss, (c) isolation, (d) phase difference and magnitude difference (pink frequency bands are the −10 dB frequency ranges).

Figure 8(b) presents the simulated and measured output return loss and the isolation between two output ports. It can be seen that the measured output return loss of the three passbands are more than 20, 16, and 15 dB, respectively. In addition, the measured isolation of the three passbands are greater than 22, 20, and 17 dB, respectively, as shown in Fig. 8(c). Meanwhile, the measured phase difference and magnitude difference between the output ports are less than 2° and 0.2 dB over the three passbands, respectively. The measured results show good agreement with the simulated ones. Slight deviation is observed, which could be attributed to fabrication tolerance in the implementation. Table 1 shows some reported performance of triple-band bandpass filters in recent years for comparison.

Table 1.

Comparison with some filtering-response power divider structures.