Question 6.6: DESIGN OF A GAP-COUPLED MICROSTRIP RESONATOR A resonator is ...
DESIGN OF A GAP-COUPLED MICROSTRIP RESONATOR
A resonator is made from an open-circuited 50 Ω microstrip line and is gap coupled to a 50 Ω feedline, as in Figure 6.13a. The resonator has a length of 2.175 cm, an effective dielectric constant of 1.9, and an attenuation of 0.01 dB/cm near its resonance. Find the value of the coupling capacitor required for critical coupling, and the resulting resonant frequency.
The first resonant frequency will occur when the resonator is about ℓ = λ_{g}/2 in length. Ignoring fringing fields, we find that the approximate resonant frequency is
f_{0}=\frac{v_{p}}{\lambda _{g}}=\frac{c}{2\ell \sqrt{ϵ_{e}} }=\frac{3 \times 10^{8}}{2(0.02175)\sqrt{1.9} }=5.00GHz,This result does not include the effect of the coupling capacitor. From (6.35) the unloaded Q of this resonator is
Q_{0}=\frac{\beta }{2\alpha } =\frac{\pi }{\lambda _{g}\alpha } =\frac{\pi }{2\ell \alpha } =\frac{\pi (8.7) dB/N_{p}}{2(0.02175 m)(1 dB/m)}=628From (6.82) the normalized coupling capacitor susceptance is
b_{c}=\sqrt{\frac{\pi }{2Q_{0}} }=\sqrt{\frac{\pi }{2(628)} }=0.05,so the coupling capacitor has a value of
C=\frac{b_{c}}{\omega Z_{0}} =\frac{0.05}{2\pi (5\times 10^{9}(50)}=0.032 pF,which should provide critical coupling of the resonator to the 50 Ω feedline. Now that C is determined, the exact resonant frequency can be found by solving the transcendental equation of (6.78). Because we know from the graphical solution of Figure 6.17 that the actual resonant frequency is slightly lower than the unloaded resonant frequency of 5.0 GHz, it is an easy matter to calculate (6.78) for several frequencies in this vicinity, which leads to a value of about 4.918 GHz. This is about 1.6% lower than the unloaded resonant frequency. Figure 6.18 shows a Smith chart plot of the input impedance of the gap-coupled resonator for coupling capacitor values that lead to under coupled, critically coupled, and over-coupled resonators.
\tan \beta \ell +b_{c}=0,
