4.5: Microstrip Design Formulas

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The formulas developed in Section 4.4.2 enable the electrical characteristics to be determined given the material properties and the physical dimensions of a microstrip line. In design, the physical dimensions must be determined given the desired electrical properties. Several people have developed procedures that can be used to synthesize microstrip lines. This subject is considered in much more depth in [4], and here just one approach is reported. The formulas are useful outside the range indicated, but with reduced accuracy. Again, these formulas are the result of curve fits, but starting with physically based equation forms.

4.5.1 High Impedance

For narrow strips, that is, when \(Z_{0} > (44 − 2\varepsilon_{r})\:\Omega\),

\[\label{eq:1}\frac{w}{h}=\left(\frac{\text{exp }H'}{8}-\frac{1}{4\text{ exp }H'}\right)^{-1} \]

where

\[\label{eq:2}H'=\frac{Z_{0}\sqrt{2(\varepsilon_{r}+1)}}{119.9}+\frac{1}{2}\left(\frac{\varepsilon_{r}-1}{\varepsilon_{r}+1}\right)\left(\ln\frac{\pi}{2}+\frac{1}{\varepsilon_{r}}\ln\frac{4}{\pi}\right) \]

For \(Z_{0}>(63-2\varepsilon_{r})\:\Omega\)

\[\label{eq:3}\varepsilon_{e}=\frac{\varepsilon_{r}+1}{2}\left[1+\frac{29.98}{Z_{0}}\left(\frac{2}{\varepsilon_{r}+1}\right)^{1/2}\left(\frac{\varepsilon_{r}-1}{\varepsilon_{r}+1}\right)\left(\ln\frac{\pi}{2}+\frac{1}{\varepsilon_{r}}\ln\frac{4}{\pi}\right)\right]^{2} \]

The formula for \(\varepsilon_{e}\) is accurate to better than \(1\%\) for \(Z_{0} > (44 − 2\varepsilon_{r})\:\Omega\) (i.e \(w/h < 1.3\)) for \(8 < \varepsilon_{r} < 12\). Overall the synthesis of \(w/h\) has an accuracy of better than \(1\%\).

4.5.2 Low Impedance

Strips with low \(Z_{0}\) are relatively wide and the formulas below can be used when \(Z_{0} < (44 − 2\varepsilon_{r})\:\Omega\). The cross-sectional geometry is given by

\[\label{eq:4}\frac{w}{h}=\frac{2}{\pi}[(d_{\varepsilon_{r}}-1)-\ln (2d_{\varepsilon_{r}}-1)]+\frac{(\varepsilon_{r}-1)}{\pi\varepsilon_{r}}\left[\ln (d_{\varepsilon_{r}}-1)+0.293-\frac{0.517}{\varepsilon_{r}}\right] \]

where

\[\label{eq:5}d_{\varepsilon_{r}}=\frac{59.95\pi ^{2}}{Z_{0}\sqrt{\varepsilon_{r}}} \]

For \(Z_{0} < (63 − 2\varepsilon_{r})\:\Omega\)

\[\label{eq:6}\varepsilon_{e}=\frac{\varepsilon_{r}}{0.96+\varepsilon_{r}(0.109-0.004\varepsilon_{r})[\log (10+Z_{0})-1]} \]

The expression for \(\varepsilon_{e}\) is accurate to better than \(1\%\) for \(8 < \varepsilon_{r} < 12\) and \(8 ≤ Z_{0} ≤ (63 − 2\varepsilon_{r})\:\Omega\).

Example \(\PageIndex{1}\): Microstrip Design

Design a microstrip line to have a characteristic impedance of \(75\:\Omega\) at \(10\text{ GHz}\). The microstrip a substrate that is \(500\:\mu\text{m}\) thick with a relative permittivity of \(5.6\).

What is the width of the line?
What is the effective permittivity of the line?

Solution

The high-impedance (or narrow-strip) formula (Equation \(\eqref{eq:1}\)) is to be used for \(Z_{0} > (44 − \varepsilon_{r}) [= (44 − 5.6) = 38.4]\:\Omega\).
With \(\varepsilon_{r} = 5.6\) and \(Z_{0} = 75\:\Omega\), Equation \(\eqref{eq:2}\) yields \(H′ = 2.445\). From Equation \(\eqref{eq:1}\), \(w/h = 0.704\), thus \(w = w/h\times h = 0.704\times 500\:\mu\text{m} = 352\:\mu\text{m}\).
The effective permittivity formula is Equation \(\eqref{eq:3}\), and so \(\varepsilon_{e} = 3.82\).