Question 8.10: CAPACITIVELY COUPLED SHUNT RESONATOR BANDPASS FILTER DESIGN ...
CAPACITIVELY COUPLED SHUNT RESONATOR BANDPASS FILTER DESIGN
Design a third-order bandpass filter with a 0.5 dB equal-ripple response using capacitively coupled short-circuited shunt stub resonators. The center frequency is 2.5 GHz, and the bandwidth is 10%. The impedance is 50 Ω. What is the resulting attenuation at 3.0 GHz?
We first calculate the attenuation at 3.0 GHz. Using (8.71) to convert 3.0 GHz to normalized low-pass form gives
\omega \leftarrow \frac{1}{\Delta } (\frac{\omega }{\omega _{0}} – \frac{\omega _{0}}{\omega } ) =\frac{1}{0.1}(\frac{3.0}{2.5}-\frac{2.5}{3.0}) =3.667.
Then, to use Figure 8.27a, the value on the horizontal axis is
\left|\frac{\omega }{\omega _{c}} \right|-1 = \left|-3.667\right| -1=2.667,from which we find the attenuation as 35 dB. Next we calculate the admittance inverter constants and coupling capacitor values using (8.136) and (8.137):
Z_{0}J_{01}=\sqrt{\frac{\pi \Delta }{4g_{1}} },
Z_{0}J_{n,n+1}=\frac{\pi \Delta }{4\sqrt{g_{n}g_{n+1}} },
Z_{0}J_{N,N+1}=\sqrt{\frac{\pi \Delta }{4g_{N}g_{N+1}} }
C_{01}=\frac{J_{01}}{\omega _{0}\sqrt{1-(Z_{0}J_{01})^{2}}} ,
C_{n,n+1}=\frac{J_{n,n+1}}{\omega _{0}} ,
C_{N,N+1}=\frac{J_{N,N+1}}{\omega _{0}\sqrt{1-(Z_{0}j_{N,N+1})^{2}} }|
n |
g_{n} |
Z_{0}J_{n-1,n} |
C_{n-1,n}(pF) |
|
|
1 |
1.5963 |
Z_{0}j_{01} = 0.2218 |
C_{01} = 0.2896 |
|
|
2 |
1.0967 |
Z_{0}j_{12} = 0.0594 |
C_{12} = 0.0756 |
|
|
3 |
1.5963 |
Z_{0}j_{23} = 0.0594 |
C_{23} = 0.0756 |
|
|
4 |
1 |
Z_{0}j_{34} = 0.2218 |
C_{34} = 0.2896 |
|
Then we use (8.138), (8.140), and (8.141) to find the required resonator lengths:
C_{n}^{′}=C_{n}+\Delta C_{n}=C_{n}-C_{n-1,n}-C_{n,n+1},
\Delta \ell =\frac{Z_{0}\omega _{0}C}{\beta } =(\frac{Z_{0}\omega _{0}C}{2\pi } )\lambda ,
\ell _{n}=\frac{\lambda }{4}+(\frac{Z_{0}\omega _{0}\Delta C_{n}}{2\pi } )\lambda ,
|
n |
ΔC_{n}(pF) |
Δ\ell_{n}(λ) |
\ell(deg) |
|
1 |
-0.3652 |
-0.04565 |
73.6 |
|
2 |
-0.1512 |
-0.0189 |
83.2 |
|
3 |
-0.3652 |
-0.04565 |
73.6 |
Note that the resonator lengths are slightly less than 90° (λ / 4 ) . The calculated amplitude response of this design is shown in Figure 8.54. The stopband rolloff at high frequencies is less than at lower frequencies, and the attenuation at 3 GHz is seen to be about 30 dB, while our calculated value for a canonical lumped-element bandpass filter was 35 dB.
