Question 3.8: FREQUENCY DEPENDENCE OF EFFECTIVE DIELECTRIC CONSTANT Use th...
FREQUENCY DEPENDENCE OF EFFECTIVE
DIELECTRIC CONSTANT
Use the approximate formula of (3.200) to plot the change in effective dielectric constant over frequency for a 25 αmicrostrip line on a substrate having a relative permittivity of 10.0 and a thickness of 0.65 mm. Compare the approximate data with results from a CAD model for frequencies up to 20 Ω Compare the calculated phase delay at 10 GHz through a 1.093 cm length of line when using ϵ_{e}(0) versus ϵ_{e}(10 GHz).
The required linewidth for a 25 Ω impedance is w = 2.00 mm. The effective dielectric constant for this line at low frequencies can be found from (3.195) to be ϵ_{e}(0) = 7.53. A short computer program was used to calculate the effective dielectric constant as a function of frequency using (3.200), and the result is shown in Figure 3.27. Comparison with a commercial microwave CAD package shows that the approximate model is reasonably accurate up to about 10 GHz but gives an overestimate at higher frequencies.
ϵ_{e}=\frac{ϵ_{r}+1}{2}+\frac{ϵ_{r}-1}{2}\frac{1}{\sqrt{1+12d/W} }ϵ_{e}(f)=ϵ_{r} – \frac{ϵ_{r}-ϵ_{e}(0)}{1+G_{f}} ,
Using an effective dielectric constant of ϵ_{e}(0) = 7.53, we find the phase delay through a 1.093 cm length of line to be \phi _{0} = \sqrt{ϵ_{e}(0)} k_{0}ℓ = 360°. The effective dielectric constant at 10 GHz is 8.120 (CAD), with a corresponding phase delay of \phi _{10} = \sqrt{ϵ_{e}(10 GHz)} k_{0}ℓ = 374°—an error of about 14°.
