Simple modelling of DGS to design 1D‐PBG low‐pass filters

Abstract

After solving Eq. (7), we obtain 1 ϭ 0.433. This gives us Z 1 ϭ 1 Z 0 ϭ 21.65⍀ with Z 0 ϭ 50⍀. From Eqs. (4) and (6), we get Z 2 ϭ 7.21⍀. Upon the substitution of the above values into Eq. (3), we obtain the scattering parameter S 21 ( z) as follows:which has the same form as Eq. (1) except for a multiplication constant. EXPERIMENTAL RESULTSTo construct a microwave differentiator, we employ microstrips to emulate transmission lines. Figure 2 shows the physical layout of the microstrips, which is built on a Duroid substrate with thickness 31 mil (0.78 mm) and relative dielectric constant r ϭ 2.5. To implement the transmission lines having characteristic impedances 21.65⍀ and 7.21⍀, we use a parallel configuration, that is, the equivalent finite microstrips are placed symmetrically on both sides of the 50⍀ line. The ideal propagation delay time of each finite line is 25 picoseconds, which corresponds to a maximum operating frequency of 10 GHz [8]. To account for the discontinuity effect at each junction, we adjust the physical length of each finite line [9]. Therefore, the final propagation delay time of each finite line may not be exactly 25 picoseconds. The ground termination of shunted finite lines is implemented by multiple via-holes along the edge. Figure 3 shows theoretical values as well as experimental results of the differentiator for the frequencies extending from DC to 6 GHz. Note that 6 GHz represents full-band normalized frequency of 0.6. Apparently, the measured transmission coefficient S 21 ( f ) is in good agreement with H( f ), where f is the frequency. The magnitude responses of both H( f ) and S 21 ( f ) increase linearly as the frequencies increase. The magnitude of S 21 ( f ) is 0.94 when the signal frequency is 6 GHz. As stated previously, H( f ) in Eq. (1) deviates from the characteristic of an ideal differentiator significantly for the frequencies greater than 0.6 of full-band normalized frequency. Therefore, the frequency responses of the differentiator in that portion are omitted. Figure 4 shows both simulated and measured reflection coefficients S 11 ( f ) of the circuit in Figure 2. We use the Ansoft HFSS8.0 software to simulate the reflection coefficient of the differentiator while the measured result is attained by using the HP8720D network analyzer. CONCLUSIONA microwave differentiator was implemented by using microstrip transmission lines. In particular, the Z-domain formulations of scattering characteristics of nonuniform transmission lines facilitated the implementation of a discrete-domain differentiator in microwave circuits. It is plausible that many other circuits developed in DSP studies will also be implemented by using nonuniform transmission lines for microwave applications. Figure 4 Simulated and measured reflection coefficients S 11 ( f ) of the circuit in Fig. 2 SIMPLE MODELLING OF DGS TO DESIGN 1D-PBG LOW-PASS FILTERS

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