MICROSTRIP LINE DESIGN Design a microstrip line on a 0.5 mm alumina substrate (εr = 9.9, tan δ = 0.001) for a 50 Ω characteristic impedance. Find the length of this line required to produce a phase delay of 270° at 10 GHz, and compute the total loss on this line, assuming copper conductors.

Question

MICROSTRIP LINE DESIGN

Design a microstrip line on a 0.5 mm alumina substrate [latex]\ \left(\epsilon _{r} = 9.9, an \delta = 0.001 ight)[/latex] for a 50 Ω characteristic impedance. Find the length of this line required to produce a phase delay of 270° at 10 GHz, and compute the total loss on this line, assuming copper conductors. Compare the results obtained from the approximate formulas of

(3.195) [latex]\ \epsilon _{e}=\frac{\epsilon _{r}+1}{2}+\frac{\epsilon _{r}-1}{2} \frac{1}{\sqrt{1+12d/W} } [/latex] – (3.199) [latex]\ \alpha _{c} =\frac{R_{s}}{Z_{0}W} Np/m[/latex] with those from a microwave CAD package.

Accepted Answer

First find W/d and [latex]\ Z_{0}=50\Omega [/latex] , and initially guess that W/d < 2. From (3.197)

[latex]\ \frac{W}{d}=\begin{cases}
\frac{8e^{A}}{e^{2A}-2} & ext{for} W/d\lt 2 \ \frac{2}{\pi } \left[B-1-\ln \left(2B-1 ight)+\frac{\epsilon _{r}-1}{2\epsilon _{r}}\left\{\ln \left(B-1 ight)+0.39-\frac{0.61}{\epsilon _{r}} ight\} ight] & ext{for} W/d\gt 2,\end{cases}[/latex]

A = 2.142, W/d = 0.9654.

So the condition that W/d < 2 is satisfied; otherwise we would use the expression for W/d > 2. Then the required line width is W = 0.9654d = 0.483 mm. From (3.195) the effective dielectric constant is [latex]\ \epsilon _{e}= 6.665[/latex]. The line length [latex]\ell[/latex] , for a 270° phase shift is found as

[latex]\ \phi =270°=\beta \ell =\sqrt{\epsilon _{e}}k_{0}\ell , [/latex]

[latex]\ k_{0}=\frac{2\pi f}{c} =209.4m^{-1}, [/latex]

[latex]\ l=\frac{270°\left(\pi /180° ight)}{\sqrt{\epsilon_{e}}k_{0}}=8.72mm [/latex]

Attenuation due to dielectric loss is found from (3.198)

[latex]\ \alpha _{d}=\frac{k_{0}\epsilon _{r}\left(\epsilon _{e}-1 ight) an \delta }{2\sqrt{\epsilon _{e}}\left(\epsilon _{r}-1 ight) } Np/m[/latex]

as [latex]\ \alpha _{d}= 0.255 Np/m =0.022 dB/cm[/latex]. The surface resistivity for copper at 10 GHz is 0.026 , and the attenuation due to conductor loss is, from (3.199) [latex]\ \alpha _{c}= 0.0108 Np/cm = 0.094 dB/cm.[/latex]The total loss on the line is then 0.101 dB.

A commercial microwave CAD package gives the following results: W =0.478 mm, [latex]\ \epsilon _{e}= 6.83, = 8.61 mm,[/latex][latex]\ \alpha _{d}= 0.022 dB/cm,[/latex] and [latex]\ \alpha _{c} = 0.054 dB/cm.[/latex]
The approximate formulas give results that are within a few percent of the CAD data for linewidth, effective dielectric constant, line length, and dielectric attenuation. The greatest discrepancy occurs for the attenuation constant for conductor loss.

Question 3.7: MICROSTRIP LINE DESIGN Design a microstrip line on a 0.5 mm ...

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