this article makes it possible to understand the operation of a bandpass stub filter and to analyze the synthesis of an open circuit or short circuit stub filter [1, p.595-608] in microstrip technology.

## Short-circuited stub bandpass filter

Figure 1 shows a “line” diagram of a 4th order shorted stub band pass filter. This filter consists of four short-circuited *λ*_{0} / 4 stubs placed in parallel and interconnected by three *λ*_{0} / 4 lines [1, p.595-599]. *λ*_{0} is the wavelength in the propagation medium at the central frequency *f*_{0} of the filter. The shorted stubs synthesize the band-pass filter resonators coupled by quarter-wave lines synthesizing admittance inverters. On the diagram *Y*_{A} and *Y*_{B} represent the admittances of the inlet and outlet of the filter., *Y*_{1}, *Y*_{2}, *Y*_{3} and *Y*_{4} are the characteristic admittances of the four short-circuited stubs (resonators) and *Y*_{12}, *Y*_{23} and *Y*_{34} are the characteristic admittances of the three interconnection lines (admittance inverters).

**Figure 1.** Line diagram of a 4th order short-circuited stub band-pass filter.

Then, you must specify the relative bandwidth *w*(*w* = (BP) / *f*_{0}) and the input (*Y*_{A}) and output (*Y*_{B}) admittances of the filter. It is then possible to calculate the characteristic admittances of short-circuited stubs and interconnection lines using equations (1) to (10) **[1, p.597]** :

where *d* is a dimensionless number (0< *d* ≤ 1) used in particular to adjust the level of admittances inside the filter so that stubs and interconnection lines are feasible.

The characteristic admittances of the four short-circuited stubs are given by equations (7) to (9):

The characteristic admittances of the interconnection lines are given by equation (10):

__Note__: Equations (1) to (10) are valid for a cut-off pulse of the prototype low-pass filter w_{c} = 1 rad / s.

*Example of Synthesis of a 4th order filter with f _{0} = 1.5 GHz and BP = 750 MHz *

To design a 4th order bandpass filter, we chose to use the Tchebyscheff approximation with a ripple in the passband *A*_{m} = 0.01 dB. the coefficients *g*_{k} of the prototype low-pass filter are given in Table 1 for Am = 0.01 dB. According to this table, for a defined order *n* = 4 we have : *g*_{0} = 1, *g*_{1} = 0.7128, *g*_{2} = 1.2003, *g*_{3} = 1.3212, *g*_{4} = 0.6476, *g*_{5} = 1.1007.

**Table 1** : Coefficients values *g*_{k} of a normalized Tchebyscheff low-pass filter with *A*_{m} = 0.01 dB, *g*_{0} = 1 and *ω*_{c}’ = 1 rad/s. According to **[1, p. 100]**.

Knowing that the filter will be supplied by lines of characteristic 50 Ω impedance (*Z*_{A} = *Z*_{B} = 50 Ω → *Y*_{A} = *Y*_{B} = 0.02 S), We can determine with the help of the preceding equation (1-10) the values of the characteristic impedances *Z*_{1}, *Z*_{2}, *Z*_{3}, *Z*_{4}, *Z*_{12}, *Z*_{23} and *Z*_{34 }for values other than *d* : 0.3, 0.6, 0.9 and 1.0.

- For
*d*= 0.3 :*Z*_{1}=*Z*_{4}= 35.8 Ω,*Z*_{2}=*Z*_{3}= 106.2 Ω,*Z*_{12}=*Z*_{34}= 83.8 Ω,*Z*_{23}= 147.2 Ω. - For
*d*= 0.6 :*Z*_{1}=*Z*_{4}= 42.4 Ω,*Z*_{2}=*Z*_{3}= 47.8 Ω,*Z*_{12}=*Z*_{34}= 59.2 Ω,*Z*_{23}= 73.6 Ω. - For
*d*= 0.9 :*Z*_{1}=*Z*_{4}= 50.0 Ω,*Z*_{2}=*Z*_{3}= 30.1 Ω,*Z*_{12}=*Z*_{34}= 48.4 Ω,*Z*_{23}= 49.1 Ω. - For
*d*= 1.0 :*Z*_{1}=*Z*_{4}= 52.8 Ω,*Z*_{2}=*Z*_{3}= 26.7 Ω,*Z*_{12}=*Z*_{34}= 45.9 Ω,*Z*_{23}= 44.2 Ω.

After analyzing the value of *d* which makes it possible to obtain characteristic impedances *Z*_{12}, *Z*_{23} et *Z*_{34} proches de 50 Ω est 0,9. Indeed these values are easily achievable in microstrip technology.

We decided to make this filter using microstrip technology on a substrate (*ε*_{r} = 3.8, tan (*δ*) = 0.007) of thickness *h* = 0.711 mm (copper metallization of thickness *t* = 17 μm). the engraving precision of the metal tracks requires that the width *W* of the metal tracks be greater than or equal to *W*_{min}= 0.15 mm.

Table 1 shows the widths of the characteristic impedance microstrip lines calculated using the Elliptika calculator necessary for the construction of the filter with *d* = 0.9.

**Table 2.** Values of the widths of the characteristic impedance lines *Z*_{1} = *Z*_{4} , *Z*_{2} = *Z*_{3} , *Z*_{12} = *Z*_{34} = , *Z*_{23}, necessary for the construction of the filter for *d* = 0.9.

Using simulation software such as keysight’s ADS, we perform a circuit simulation between 0 and 3 GHz of the 4th order filter with *d* = 0.9 in the “ideal lines” and “microstrip lines” versions up to at 10 GHz. Figure 2 shows the frequency responses of the moduli of the reflection (Sii) and transmission (Sij, with i ≠ j) parameters of the 4th order filter with *d* = 0.9 in the “ideal lines” (S11 and S21, black curves) and “microstrip lines” (S33 and S43, blue curves) versions obtained during a circuit simulation.

**Figure 2.** Frequency responses of the moduli of the reflection (Sii) and transmission (Sij, with i ≠ j) parameters of the 4th order filter with *d* = 0.9 in the “ideal lines” (S11 and S21, black curves) and “microstrip lines” (S33 and S43, blue curves) versions obtained during a circuit simulation.

In figure 2, we observe three harmonic responses of the band-pass filter centered on *f* = *f*_{0} = 1.5 GHz (harmonic 1), *f* = 3 * *f*_{0} = 4.5 GHz (harmonic 3) and *f* = 5 * *f*_{0}= 7.5 GHz ( harmonic 5). we also notice that the filter in “microstrip lines” has insertion losses unlike the filter in “ideal lines”.

### Open-circuited stub bandpass filter

Figure 3 shows a line diagram of a 4th order open circuit stub band pass filter. This filter consists of four open-circuit *λ*_{0} / 2 stubs, placed in parallel and interconnected by *λ*_{0} / 4 lines [1, p.605-608]. *λ*_{0} is the wavelength in the propagation medium at the central frequency *f*_{0} of the filter. Open circuit stubs synthesize the band pass filter resonators coupled by quarter wave lines synthesizing admittance inverters. On the diagram *Y*_{A} and *Y*_{B} represent the admittances of the inlet and outlet of the filter., *Y*_{1}, *Y*_{2}, *Y*_{3} and *Y*_{4} are the characteristic admittances of the four open-circuit stubs (resonators) and *Y*_{12}, *Y*_{23} and *Y*_{34} are the characteristic admittances of the three interconnection lines (admittance inverters).

**Figure 3** Line diagram of a 4th order open circuit stub bandpass filter.

In order to obtain an open-circuit stub band-pass filter still operating at 1.5 GHz, we replace the short-circuited *λ*_{0} / 4 stubs of the 4th order filter with *d* = 0.9 presented above, by open circuit stubs *λ*_{0}/ 2.

Figure 4 shows the frequency responses of the moduli of the reflection (Sii) and transmission (Sij, with i ≠ j) parameters of the open circuit stub filter in the “ideal line” versions (S11 and S21, black curves) and “microstrip lines” (S33 and S43, red dotted curves) obtained during a circuit simulation.

In FIG. 4, we observe the fundamental response (harmonic 1 centered on *f* = *f*_{0} = 1.5 GHz) of the band-pass filter. It is important to underline the presence of two zeros located at*f* = *f*_{0} / 2 = 0.75 GHz and *f*= 3*f*_{0} / 2 = 2.25 GHz on the frequency response of the modulus of the transmission parameter of the filter.

**Figure 4.** Frequency responses of the reflection (Sii) and transmission (Sij, with i ≠ j) parameter moduli of the open circuit stub filter in the “ideal lines” (S11 and S21, black curves) and “microstrip lines” versions (S33 and S43, red dotted curves) obtained during a circuit simulation.

#### Comparison of the responses of open-circuited and short-circuited stub bandpass filters.

Figure 5 shows the frequency responses of the moduli of the reflection (Sii) and transmission (Sij, with i ≠ j) parameters of the “microstrip” versions of the short-circuited stub filters (S11 and S21, blue curves) and open-circuit stubs (S33 and S43, red dotted curves) obtained during a circuit simulation.

By observing Figure 5, it can be seen that the passband of the open-circuit stub filter is lower than that of the short-circuit stub filter.

**Figure 5.** Frequency responses of the reflection (Sii) and transmission (Sij, with i ≠ j) parameter moduli of the open circuit stub filter in the “ideal lines” (S11 and S21, black curves) and “microstrip lines” versions (S33 and S43, red dotted curves) obtained during a circuit simulation.

The origin of the transmission zeros observed at frequencies *f*_{0} / 2 and 3*f*_{0} / 2 on the frequency response of the transmission parameter of the open-circuit stub filter.

The transmission zeros of the open-circuit and short-circuited stub filters appear when the impedance returned by the stubs to the admittance inverters is equal to 0 (short-circuit).

The input impedances of the open-circuit (*Z*_{OC}) and short-circuit (*Z*_{SC}) stubs are given by equations (11) and (12).

In FIG. 4, we observe the fundamental response (harmonic 1 centered on *f* = *f*_{0} = 1.5 GHz) of the band-pass filter. It is important to underline the presence of two zeros located at*f* = *f*_{0} / 2 = 0.75 GHz and *f*= 3*f*_{0} / 2 = 2.25 GHz on the frequency response of the modulus of the transmission parameter of the filter.

where *Z*_{c} and *l* are respectively the characteristic impedance and the length of the stub and *λ*_{g} is the guided wavelength.

By exploiting equations (11) and (12), we find that:

where*k* is a positive or zero integer.

We deduce that:

The first transmission zero induced by the open-circuit stubs is obtained at the frequency *f*_{0} / 2 = 0.75 GHz for which *l* = *λ*_{g} / 4.

The second transmission zero induced by the open-circuit stubs is obtained at the frequency 3 * *f*_{0} / 2 = 2.25 GHz for which *l* = 3 * *λ*_{g} / 4.

The first transmission zero induced by the short-circuited stubs is obtained at the frequency 2 * *f*_{0} = 3.0 GHz for which *l*= *λ*_{g} / 2.

**[1]** G.L. Matthaei, L. Young et E.M.T. Jones, *Microwave filters, impedance-matching networks, and coupling structures*, Artech House, 1980.