**2. Structure and Design of The BPF**

The proposed BPF has two independent paths as is shown in Figure 1. Upper path I is composed of a pair of shorted quarter-wavelength resonators in the top layer and one half-wavelength resonator in the bottom layer, with two rectangular slots etched in the middle ground layer to realize slot-coupling. And this path is coupled to two quarter-wavelength short microstrip lines as exciting structure. Lower path II consists of a stub-loaded anti-parallel coupled line, connecting to the feeding ports in parallel with upper path I. Both of these two paths will be analyzed in detail next.

**Figure 1.** Configuration of the proposed bandpass filter (BPF) (**a**) Top layer. (**b**) Middle layer. (**c**) Bottom layer. (**d**) 3D view. (**e**) Front view of the proposed BPF. (**f**) Bottom view of the proposed BPF.

### *2.1. Analysis of Upper Path I*

Figure 2 shows the equivalent circuit of upper path I. The coupling networks consisting of quarterand half-wavelength resonators are marked as red (T1–T5), (T1+T2 and T4+T5 are quarter-wavelength, T2+T3+T4 is half-wavelength), which are excited by two shorted quarter-wavelength resonators marked as blue (T6) from port 1 and 2, respectively. T2 and T4 are two end-to-end coupled lines with even and odd impedance of Z*e*<sup>1</sup> and Z*o*1, electric length of *θ*2. Three Microstrip lines T1, T3 and T5 are connected to coupling networks T2 and T4. T1 and T5 are connected to the shorted lines in T2 and T4 with impedance of Z1 and electric length of *θ*<sup>1</sup> while T3 is used to connect T2 and T4 with impedance of Z2 but two times longer than T1 or T5. The total length of T1 and T2 or T4 and T5 is quarter-wavelength (*θ*<sup>1</sup> + *θ*<sup>2</sup> = *π*/2).

To verify the analysis, we use transfer matrix ABCD which is defined as Equation (1) shows [16]:

$$
\begin{pmatrix} V\_1 \\ I\_1 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} V\_2 \\ -I\_2 \end{pmatrix} \tag{1}
$$

For transmission lines with character impedance *Z*<sup>0</sup> and electrical length *θ* [16]:

$$
\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} \cos \theta & jZ\_0 \sin \theta \\ j\chi\_0 \sin \theta & \cos \theta \end{pmatrix} \tag{2}
$$

Thus the response of the coupling network can be calculated by multiplying the ABCD matrixes as Equation (3).

(**b**)

**Figure 2.** Equivalent circuit and MATLAB simulated results of resonators in upper path I. (**a**) Equivalent circuit of different sections in upper path I. (**b**) MATLAB simulated results of different sections in upper path I.

$$T = T\_1 \times T\_2 \times T\_3 \times T\_4 \times T\_5 \tag{3}$$

where

$$T\_1 = \begin{pmatrix} \cos \theta\_1 & jZ\_1 \sin \theta\_1 \\ jY\_1 \sin \theta\_1 & \cos \theta\_1 \end{pmatrix} = T\_5 \tag{4a}$$

$$T\_3 = \begin{pmatrix} \cos 2\theta\_1 & jZ\_2 \sin 2\theta\_1 \\ jY\_2 \sin 2\theta\_1 & \cos 2\theta\_1 \end{pmatrix} \tag{4b}$$

$$T\_2 = \frac{1}{ab - cf} \begin{pmatrix} a^2 - df & ac - db \\ ac - bf & ac - b^2 \end{pmatrix} \tag{4c}$$

$$T\_4 = \frac{1}{ab - de} \begin{pmatrix} b^2 - ce & db - ac \\ fb - ae & fd - a^2 \end{pmatrix} \tag{4d}$$

Here a to f in Equation (4) represent formulas related to the dimensions of coupling network T2 and T4 [17] as Equation (5).

$$a = \frac{1}{2} (\cos \theta\_{2\varepsilon} + \cos \theta\_{2\nu}) \tag{5a}$$

$$b = \frac{1}{2} (\cos \theta\_{2\varepsilon} - \cos \theta\_{2\nu}) \tag{5b}$$

$$\mathbf{c} = j\frac{1}{2}(Z\_{\epsilon1}\cos\theta\_{2\epsilon} + Z\_{\nu1}\cos\theta\_{2\nu})\tag{5c}$$

$$d = j\frac{1}{2}(Z\_{\epsilon1}\cos\theta\_{2\epsilon} - Z\_{\nu1}\cos\theta\_{2\nu})\tag{5d}$$

$$\varepsilon = j \frac{1}{2} (\mathbf{Y}\_{\varepsilon 1} \cos \theta\_{2\varepsilon} + \mathbf{Y}\_{o1} \cos \theta\_{2o}) \tag{5e}$$

$$f = j\frac{1}{2}(\chi\_{\varepsilon1}\cos\theta\_{2\varepsilon} - \chi\_{o1}\cos\theta\_{2o})\tag{5f}$$

*θ*2*<sup>e</sup>* and *θ*2*<sup>o</sup>* are the even and odd electric lengths of slot-coupled resonators while Z*e*1, Z*o*<sup>1</sup> and Y*e*1, Y*o*<sup>1</sup> are the impedances and admittances. All of these parameters can be obtained using quasi-static analysis [18].

It is noteworthy that if the length of the slots *θ*<sup>2</sup> is two-third of quarter-wavelength, discriminating coupling is formed. According to [19], a zero at 3f0 will appear when discriminating coupling occurs, because the voltage distribution is odd on one line and even on the other, leading to null coupling coefficient. Figure 2 shows the simulated result using MATLAB to validate the performance of discriminating. Blue line in Figure 2 demonstrates the simulated result of coupling networks including the connecting microstrip lines (T1-T5) and it is seen that there are two modes and one zero. Thus this line validates that discriminating coupling is effective to introduce a transmission zero at 3f0 in this design. However, the second mode is located at around 2f0 because coupling network T2,T4 and microstrip line section M3 realize a half-wavelength resonator in the bottom layer as is clearly shown in Figure 1. To suppress the harmonic caused by half-wavelength resonator, quarter-wavelength shorted coupled lines (M6) is used as feeding lines to introduce another zero at 2f0. The red line in Figure 2 proves the function of M6.

Finally the black line in Figure 2 gives the MATLAB simulated result of the whole structure including T1-T6. It shows that the second and third harmonics are suppressed by zeros introduced by discriminating coupling and quarter-wavelength resonators, validating the theory and design very well. Also due to multilayer structure, it is easy to introduce source-load coupling using the short pins of M6 as is shown in the top layer of Figure 1. Thus a zero located at lower side-band appears and improves the selectivity. HFSS simulated results will validate the design in the next subsection.

#### *2.2. Analysis of Lower Path Ii*

The lower path II consists of a stub-loaded anti-parallel coupling section connected by microstrip lines, as is shown in Figure 3. Here the impedance and electric length of connecting microstrip lines are Z3 and *θ*3. The anti-parallel coupled line has even and odd impedance of Z*e*<sup>2</sup> and Z*o*<sup>2</sup> and the electric length of *θ*<sup>3</sup> with loaded stubs of Z4 and *θ*4. Formula Equation (6) gives the ABCD matrix of lower path II.

**Figure 3.** Equivalent circuit of anti-parallel coupling network in lower path II.

$$T = \begin{pmatrix} \cos \theta\_3 & jZ\_3 \sin \theta\_3 \\ j\chi\_3 \sin \theta\_3 & \cos \theta\_1 \end{pmatrix} \begin{pmatrix} A\_c & B\_c \\ C\_c & D\_c \end{pmatrix} \begin{pmatrix} \cos \theta\_3 & jZ\_3 \sin \theta\_3 \\ j\chi\_3 \sin \theta\_3 & \cos \theta\_1 \end{pmatrix} \tag{6}$$

A*c*, B*c*, C*c* and D*c* in Equation (6) are the ABCD matrix's elements of the stub-loaded anti-parallel coupled line. The method to calculate these elements is same as the one used for coupling networks T2 or T4 [17] as Equation (7).

$$A\_{\varepsilon} = \frac{1}{G} \left[ (\varepsilon + aZ\_s)(a + \varepsilon Z\_s) - (d + bZ\_s)(b + fZ\_s) \right] \tag{7a}$$

$$B\_{\varepsilon} = \frac{1}{G} \left[ (\mathfrak{c} + a\mathbf{Z}\_{\mathfrak{s}})(\mathfrak{c} + a\mathbf{Z}\_{\mathfrak{s}}) - (\mathfrak{d} + b\mathbf{Z}\_{\mathfrak{s}})(\mathfrak{d} + b\mathbf{Z}\_{\mathfrak{s}}) \right] \tag{7b}$$

$$\mathcal{L}\_{\mathbb{C}} = \frac{1}{G} [(a + eZ\_{\mathbb{S}})(a + eZ\_{\mathbb{S}}) - (b + fZ\_{\mathbb{S}})(b + fZ\_{\mathbb{S}})] \tag{7c}$$

$$D\_{\varepsilon} = \frac{1}{G} [ (a + eZ\_{\mathfrak{s}})(c + aZ\_{\mathfrak{s}}) - (b + fZ\_{\mathfrak{s}})(d + bZ\_{\mathfrak{s}}) ] \tag{7d}$$

where

$$G = (d + bZ\_s)(a + eZ\_s) - (b + fZ\_s)(c + aZ\_s) \tag{8}$$

and

$$\mathbf{Z}\_{\\$} = -j\mathbf{Z}\_{\mathbf{4}} \cot \theta\_{\mathbf{4}} \tag{9}$$

Similarly, a to f in Equation (7) and Equation (8) are same as Equation (5) but the even and odd parameters are replaced by *θ*5*e*, *θ*5*o*, Z*e*2, Z*o*2, Y*e*<sup>2</sup> and Y*o*2. Formula Equation (9) is the input impedance of the open stubs.

MATLAB can also be used to calculate the response of combined upper and lower path. However, the formulas will be very complicated and the simulated result will not be clear enough to demonstrate the function of lower path II. Thus full-wave simulation software Ansys HFSS 19 is used. Blue line in Figure 4 shows the HFSS simulated results of upper path I while the red line gives the ones of lower path II. As is shown in Figure 4, the upper path I has bandpass response with the second and third harmonics suppressed by TZ2 and TZ3, and another transmission zero TZ1 located at lower sideband is introduced by inductive source-load coupling as is mentioned in the last subsection. It can also be seen that the response of lower path II has same amplitude as upper path I in some frequencies but at the same time they are out of phase. So in these frequencies, transmission zeros will appear because of signal cancellation. In this design, three extra transmission zeros TZ4, TZ5 and TZ6 are introduced due to signal cancellation. TZ5 located at upper sideband is used to improve the selectivity while the lower and upper stopband is improved by TZ4 and TZ6. However, TZ2 and TZ6 are both at around 2f0. So overall, there are five TZs used to improve selectivity and out-of-band suppression performance.

**Figure 4.** HFSS Simulated results of lower path (red line) and upper path (blue line). (**a**) Amplitude of S21 (dB). (**b**) Phase of S21 (deg)

#### **3. Simulated and Measured Results**

As shown in Figure 5, the proposed BPF is designed on two RO4350B substrate with relative permittivity of 3.66 and thickness of 0.508 mm. A 0.1 mm RO4450B prepreg is used as paste layer and so the total height of the PCB is 1.116 mm (0.508 + 0.1 + 0.508). The final parameters are: *<sup>W</sup>*<sup>1</sup> = 0.3, *<sup>W</sup>*<sup>2</sup> = 0.5, *<sup>W</sup>*<sup>3</sup> = 0.6, *<sup>W</sup>*<sup>4</sup> = 0.8, *<sup>W</sup>*<sup>5</sup> = 0.4, *<sup>L</sup>*<sup>1</sup> = 18.2, *<sup>L</sup>*<sup>2</sup> = 13.8, *<sup>L</sup>*<sup>3</sup> = 5.7, *<sup>L</sup>*<sup>4</sup> = 12.7, *<sup>L</sup>*<sup>5</sup> = 3, *<sup>L</sup>*<sup>6</sup> = 1, *<sup>D</sup>* = 2, *<sup>S</sup>*<sup>1</sup> = *<sup>S</sup>*<sup>2</sup> = 0.2, *<sup>R</sup>* = 0.15, *<sup>W</sup>*<sup>50</sup> = 1.1 (unit: mm). To validate the design, a prototype is fabricated and measured using Keysigth ENA network analyser E5671C. The photograph of the proposed bandpass filter is shown in Figure 1.

**Figure 5.** laminate layer definition of the fabricated PCB.

Figure 6 illustrates the simulated and measured responses. The measured 3 dB bandwidth is from 2.39 GHz to 2.59 GHz (8.1%) centered at 2.49 GHz with return loss better than −12 dB within the passband. Due to mechanical fabrication error and permittivity difference between simulation and production, the center frequency is a little higher for the measurement results. Also the insertion loss is 3.1 dB, larger than simulated one due to 0.6 dB SMA connector loss. The positions of 5 TZs are in agreement with the simulated ones with a little shift. The depicted discrepancies could be from the fabrication tolerance in the etching process. Due to these 5 TZs, the lower sideband selectivity of the proposed BPF is calculated by 3 dB and 20 dB amplitude response with respect to its frequency point:

$$|\text{Selecttivity}\_{lower\,\,sideband} = |\frac{3-20}{f\_3 - f\_{20}}| = \frac{17}{2.39 - 2.3} = 188.8\,\text{dB/GHz}\tag{10}$$

and the upper sideband selectivity:

$$|Selocity\_{upper\, sideband}| = |\frac{3-20}{f\_3 - f\_{20}}| = \frac{17}{2.65 - 2.59} = 288.3\,\text{dB/GHz}\tag{11}$$

so that the shape factor is BW20/BW3 = 1.75 (BW represents bandwidth). And −15 dB suppression is from 2.64 to 8.84 GHz. The overall size is 0.2*λ<sup>g</sup>* × 0.22*λ<sup>g</sup>* (*λ<sup>g</sup>* is the wavelength at center frequency). Table 1 tabulates the performance comparisons with some previous works. Here CF represents the center frequency of the filter, and FBW is the fractional bandwidth which is calculated by passband (3 dB bandwidth) divide center frequency BW3*dB*/*f*0, and N is the number of TZs, and metal layers are

the number of metal used to form the filter. TZ@sideband is the frequency of sideband transmission zeros divide center frequency f*sideband TZ*/*f*0.

**Figure 6.** Simulated and measured results of the proposed BPF (solid lines: measured results; dash lines: simulated results) (**a**) Simulated and measured amplitude of S parameters (dB). (**b**) Measured group delay (ns). (**c**) Simulated and measured phase of S parameters (deg).



#### **4. Conclusions**

A multilayer bandpass filter with high selectivity and wide stopband is proposed in this letter. TZs introduced by anti-parallel coupling network as well as discriminating coulpling between quarterand half-wavelength resonators improve the selectivity and out-of-band suppression of the BPF. To verify the design, a prototype BPF is fabricated centered at 2.49 GHz. Both simulated and measured results manifest the performance with good selectivity and extended stopband. With these features, The proposed BPF is attractive in modern wireless system.

**Author Contributions:** Conceptualization, J.C. and H.C.; methodology, J.C.; software, H.C.; validation, J.C., H.C.and R.Z.; formal analysis, J.C.; investigation, H.C.; resources, R.Z.; data curation, J.C.; Writing—Original draft preparation, H.C.; Writing—Review and editing, R.Z.; funding acquisition, J.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been supported by National Natural Science Foundation of China 62001232, 61971224 and Jiangsu Provincial Natural Science Foundation under Grants Nos. BK20180457.

**Conflicts of Interest:** The authors declare no conflict of interest.
