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# Stub bandpass filter

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 λ0 / 4 stubs placed in parallel and interconnected by three λ0 / 4 lines [1, p.595-599]. λ0 is the wavelength in the propagation medium at the central frequency f0 of the filter. The shorted stubs synthesize the band-pass filter resonators coupled by quarter-wave lines synthesizing admittance inverters. On the diagram YA and YB represent the admittances of the inlet and outlet of the filter., Y1, Y2, Y3 and Y4 are the characteristic admittances of the four short-circuited stubs (resonators) and Y12, Y23 and Y34 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) / f0) and the input (YA) and output (YB) 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 wc = 1 rad / s.

Example of Synthesis of a 4th order filter with f0 = 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 Am = 0.01 dB. the coefficients gk 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 : g0 = 1, g1 = 0.7128, g2 = 1.2003, g3 = 1.3212, g4 = 0.6476, g5 = 1.1007.

Table 1 : Coefficients values gk of a normalized Tchebyscheff low-pass filter with Am = 0.01 dB, g0 = 1 and ωc’ = 1 rad/s. According to [1, p. 100]. Knowing that the filter will be supplied by lines of characteristic 50 Ω impedance (ZA = ZB = 50 Ω → YA = YB = 0.02 S), We can determine with the help of the preceding equation (1-10) the values of the characteristic impedances Z1, Z2, Z3, Z4, Z12, Z23 and Z34 for values other than d : 0.3, 0.6, 0.9 and 1.0.

• For d = 0.3 : Z1 = Z4 = 35.8 Ω, Z2 = Z3 = 106.2 Ω, Z12 = Z34 = 83.8 Ω, Z23 = 147.2 Ω.
• For d = 0.6 : Z1 = Z4 = 42.4 Ω, Z2 = Z3 = 47.8 Ω, Z12 = Z34 = 59.2 Ω, Z23 = 73.6 Ω.
• For d = 0.9 : Z1 = Z4 = 50.0 Ω, Z2 = Z3 = 30.1 Ω, Z12 = Z34 = 48.4 Ω, Z23 = 49.1 Ω.
• For d = 1.0 : Z1 = Z4 = 52.8 Ω, Z2 = Z3 = 26.7 Ω, Z12 = Z34 = 45.9 Ω, Z23 = 44.2 Ω.

After analyzing the value of d which makes it possible to obtain characteristic impedances Z12, Z23 et Z34 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 (εr = 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 Wmin= 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 Z1 = Z4 , Z2 = Z3 , Z12 = Z34 = , Z23, 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 = f0 = 1.5 GHz (harmonic 1), f = 3 * f0 = 4.5 GHz (harmonic 3) and f = 5 * f0= 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 λ0 / 2 stubs, placed in parallel and interconnected by λ0 / 4 lines [1, p.605-608]. λ0 is the wavelength in the propagation medium at the central frequency f0 of the filter. Open circuit stubs synthesize the band pass filter resonators coupled by quarter wave lines synthesizing admittance inverters. On the diagram YA and YB represent the admittances of the inlet and outlet of the filter., Y1, Y2, Y3 and Y4 are the characteristic admittances of the four open-circuit stubs (resonators) and Y12, Y23 and Y34 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 λ0 / 4 stubs of the 4th order filter with d = 0.9 presented above, by open circuit stubs λ0/ 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 = f0 = 1.5 GHz) of the band-pass filter. It is important to underline the presence of two zeros located atf = f0 / 2 = 0.75 GHz and f= 3f0 / 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 f0 / 2 and 3f0 / 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 (ZOC) and short-circuit (ZSC) stubs are given by equations (11) and (12).

In FIG. 4, we observe the fundamental response (harmonic 1 centered on f = f0 = 1.5 GHz) of the band-pass filter. It is important to underline the presence of two zeros located atf = f0 / 2 = 0.75 GHz and f= 3f0 / 2 = 2.25 GHz on the frequency response of the modulus of the transmission parameter of the filter. where Zc and l are respectively the characteristic impedance and the length of the stub and λg is the guided wavelength.

By exploiting equations (11) and (12), we find that: wherek is a positive or zero integer.

We deduce that:

The first transmission zero induced by the open-circuit stubs is obtained at the frequency f0 / 2 = 0.75 GHz for which l = λg / 4.

The second transmission zero induced by the open-circuit stubs is obtained at the frequency 3 * f0 / 2 = 2.25 GHz for which l = 3 * λg / 4.

The first transmission zero induced by the short-circuited stubs is obtained at the frequency 2 * f0 = 3.0 GHz for which l= λg / 2.

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