The DBR bandpass filter article helps to understand and analyze the synthesis of a DBR [1]-[5] bandpass filter in microstrip technology. From a DBR resonator it is possible to synthesize filters for which the simultaneous control of the bandwidth and the attenuated bands is possible, this is the strong interest of this filter.

## Résonateur coupe bande

Initially we have a transmission line, an elementary LC cell connected in parallel, then a second transmission line (see figure 1).

**Figure 1. **Cascading transmission line and LC cell

This elementary LC cell makes it possible to achieve a transmission zero at a frequency f_{z}, i.e. to prevent any passage of energy between the input of the first line and the output of the second. At the frequency Fz it brings a zero impedance in parallel between the two transmission lines, it therefore behaves like a parallel short-circuit.

We present the synthesis of elementary “filtering” structures allowing to introduce a transmission zero (cf figure 2)

Consider two transmission lines with characteristic impedance *Z _{c}*=50 Ohm and electrical length

*θ*. An open circuit stub type structure with characteristic impedance and physical and electrical length l

_{c}_{s1}is connected between two lines (Figure 2).

**Figure 2 : **elementary “filtering” structure

The structure is band cut if the stub brings back a short circuit, i.e. if Zin_{s1}= 0. This is done if θs1 is equal to π / 2 (modulo π), that is ls1 = λs1 / 4 at the frequency fs1.

We will then design this bandstop resonator using microstrip technology. We use an FR4 type substrate (h=1 mm, εr= 4.4, tanδ=0.01). You can therefore determine the length l_{s1} necessary in order to position a transmission zero at 1.5 GHz in the case where the line impedance is 45 Ω. the length l_{s1} is 27.2 mm and thanks to the Elliptika calculatorwe can determine the line width for with an FR4 substrate, it is 2.22 mm.

The circuit simulation of this resonator is presented in figure 3, we observe a band stop function at 1.5 GHz with an attenuation of 50 dB in transmission (S12).

**Figure 3**: Circuit simulation of a notch resonator

## Résonateur DBR

The principle of the DBR resonator is based on the association in parallel of two different stop band structures. The schematic diagram is shown in Figure 4. the different stop-band structures can be, for example, open-circuit or closed-circuit stubs. the originality of this filter lies in the independent control of its adjustment parameters, which makes it possible to separately fix the position of the transmission zeros. The first stub of length ls1 and impedance Zs1 provides a transmission zero at low frequency, the second stub of length ls2 and impedance Zs2 achieves a transmission zero at higher frequency. between the two transmission zeros, a pole obtained by constructive recombination achieves a bandwidth.

This principle is summarized by the following equation, it presents the paralleling of the input impedances of the two stubs. Z is zero when Z1 or Z2 is zero. Z becomes infinite when Z1 or Z2 are opposite.

**Figure 4 :** Schematic diagram of the DBR resonator

the impedances brought back by each stub on the main line.

*At f0 the two stubs associated in parallel bring an open circuit to the main line:*

From this expression we express Zs2 as a function of Zs1

The first transmission zero is positioned at f_{s1}=1.5 GHz. One fixes the impedance of this stub Z_{s1}=45Ω . The second transmission zero will be placed at f_{s2}=2.65 GHz and the resonance frequency f_{0} at 2 GHz. We can then determine the impedance Z_{s2} necessary to have the resonance of the DBR resonator at f_{0}, we then find Zs2 = 64 Ω [1]. Then, using the Elliptika calculator, the impedance line widths Zs1, Zs2 and the line lengths Ls1 and Ls2 are determined in order to position the zeros at 1.5 GHz and 2.02 GHz for a central frequency at 2 GHz. we obtain for the stub at low frequency1 Ls1=27.2 mm and Ws1=2.2 mm and for the second stub at high frequency Ls2=15.8 mm and Ws2=1.2 mm

figure 4 presents the ideal simulation of this resonator and figure 5 presents the comparison between the ideal simulation and the circuit simulation. we observe a slight difference between the two results, in fact, the “tee” necessary to connect the two stubs in circuit adds a capacitive effect.

**Figure 4.** Ideal simulation of a DBR resonator

**Figure 5.** Comparison ideal simulation and simulation of a DBR resonator

#### Filtre passe bande DBR d'ordre 2

The Global Synthesis [1] makes it possible to produce filters of order N. involving N DBR resonators (N poles in the band, N transmission zeros in the lower attenuation and N transmission zeros in the upper attenuation band). .

Figure 6 shows the schematic of a 2nd order open circuit stub DBR band pass filter. This filter consists of four open-circuit *λ*/4 stubs placed in parallel and interconnected by three *λ*_{0}/4 [2,p .433]. *λ*_{0} is the wavelength in the propagation medium at the center frequency *f*_{0} of the filter. The open-circuit stubs synthesize the resonators of the band-pass filter coupled by the quarter-wave lines *λ*_{0}/4 synthesizing admittance inverters.

**Figure 6.** Symmetrical 2nd order DBR filter

The filter is symmetrical, the slope parameter b is therefore identical for resonators 1 and 2. As a reminder, this slope parameter is a degree of freedom making it possible to obtain achievable impedances according to the technologies used. The technology used here is microstrip, and the impedances are between 25 Ω and 85 Ω. The following equations make it possible to determine the impedances Zs1 and Zs2 for an order 2. the equations are generalized to order N here [1].

We therefore wish to produce a filter of 2nd order centered at 2 GHz and with a relative bandwidth of 9%. This filter will be based on the Tchebyschef approximation with a ripple in the passband *A*_{m} = 0.01 dB. the characteristic impedances Z_{01} Z_{12}and Z_{23} and the lengths L_{01} L_{12} and L_{23} to be used for the inverters are therefore determined.

we find the following impedances Z01=39.5Ω, Z12=29.70 Ω and Z23=39.5Ω, for line lengths L_{01} L_{12} and L_{23} around 22 mm.

For the symmetrical DBR band pass filter, two identical resonators are used. Each DBR resonator is therefore composed of two identical low frequency stubs (Zs1= 45 and Ls1=29 mm) and two identical high frequency stubs (Zs2= 65 and Ls2=17 mm)

In FIG. 7 we observe the result of the ideal simulation compared with the circuit simulation of the DBR band pass filter with a relative bandwidth of 9%. the ideal simulation corresponds well to the set objective of positioning zero and e the central frequency. on the other hand, a difference is observed at the level of the circuit simulation which is due to the capacitive effects of the connection of the stubs and to the electromagnetic couplings between the stubs.

Reference

[1] E. Rius, C. Quendo, A. Manchec, Y. Clavet, C. Person and J.F. Favennec “Design of microstrip dual behavior resonator fiters: a practical guide” Microwave journal, december, 8 2006

[2]. C. Quendo, E. Rius and C. Person, “An Original Topology of Dual-band Filter with Transmission Zero,” *2003 IEEE International Microwave Symposium Digest*, Vol. II, pp. 1093–1095.

[3] A. Manchec, E. Rius, C. Quendo, C. Person, J.F. Favennec, P. Moroni, J.C. Cayrou and J.L. Cazaux, “Ku-band Microstrip Diplexer Based on Dual Behavior Resonator Filter,” *2005 IEEE International Microwave Symposium Session WE1F*.

[4] A. Manchec, C. Quendo, J-F. Favennec, E. Rius, Ch. Person, “Synthesis of Capacitive-Coupled Dual Behavior Resonator (CCDBR) Filters”, IEEE Transactions on Microwave Theory and Techniques, vol. 54, n°6, pp 2346-2355, June 2006

[5] Alexandre Manchec, Cédric Quendo, Eric Rius Christian Person, Jean-François Favennec “Synthesis of Dual Behavior Resonator (DBR) Filters With Integrated Low-Pass Structures for Spurious Responses Suppression” IEEE Microwave and Wireless Components Letters, Vol. 16, N° 1, January 2006