# 4.4: Microstrip Transmission Lines


Microstrip has conductors embedded in two dielectric mediums and cannot support a pure TEM mode. In most practical cases, the dielectric substrate is electrically thin, that is, $$h ≪\lambda$$. Then the transverse field is dominant and the fields are called quasi-TEM.

## 4.4.1 Microstrip Line in the Quasi-TEM Approximation

In this section relations are developed based on the principle that the phase velocity of an EM wave in an air-only homogeneous transmission with a TEM field line is just $$c$$. As a first step, the potential of the conductor strip is set to $$V_{0}$$ and Laplace’s equation is solved using an EM simulator for the electrostatic potential everywhere in the dielectric. Then the per unit length (p.u.l.) electric charge, $$Q$$, on the conductor is determined. Using this in the following relation gives the line capacitance:

$C=\frac{Q}{V_{0}}\nonumber$

In the next step, the process is repeated with $$\varepsilon_{r} = 1$$ to determine $$C_{\text{air}}$$ (the capacitance of the line without a dielectric).

If the microstrip line is now an air-filled lossless TEM structure,

$\label{eq:1} v_{p,\text{air}}=c=\frac{1}{LC_{\text{air}}}$

and so

$\label{eq:2} L=\frac{1}{c^{2}C_{\text{air}}}$

$$L$$ is not affected by the dielectric properties of the medium. $$L$$ calculated above is the desired p.u.l. inductance of the line with the dielectric as well as in free space. Once $$L$$ and $$C$$ have been found, the characteristic impedance can be found using

$\label{eq:3} Z_{0}=\sqrt{\frac{L}{C}}$

rewritten as

$\label{eq:4} Z_{0}=\frac{1}{c}\frac{1}{\sqrt{CC_{\text{air}}}}$

and the phase velocity is

$\label{eq:5} v_{p}=\frac{1}{\sqrt{LC}}=c\sqrt{\frac{C_{\text{air}}}{C}}$

Now the field is distributed in the inhomogeneous medium and in free space, as shown in Figure $$\PageIndex{1}$$(a). So the effective relative permittivity, εe, of the equivalent homogeneous microstrip line (see Figure $$\PageIndex{1}$$(b)) is defined by

$\label{eq:6}\sqrt{\varepsilon_{e}}=\frac{c}{v_{p}}$

Combining Equations $$\eqref{eq:5}$$ and $$\eqref{eq:6}$$, the effective relative permittivity (usually just the term effective permittivity is used) is obtained:

$\label{eq:7}\varepsilon_{e}=\frac{C}{C_{\text{air}}}$

The effective permittivity can be interpreted as the permittivity of a homogeneous medium that replaces the air and the dielectric regions of the

Figure $$\PageIndex{1}$$: Microstrip line: (a) cross section; and (b) eps .. equivalent structure where the strip is embedded in a dielectric of semi-infinite extent with effective relative permittivity $$\varepsilon_{e}$$.

microstrip, as shown in Figure $$\PageIndex{1}$$. Since some of the field is in the dielectric and some is in air, the effective relative permittivity must satisfy

$\label{eq:8} 1<\varepsilon_{e} <\varepsilon_{r}$

However, the minimum $$\varepsilon_{e}$$ will be greater than $$1$$ as electrical energy will be distributed in air and dielectric. The wavelength on a transmission line, the guide wavelength $$\lambda_{g}$$, is related to the free space wavelength by $$\lambda_{g} = \lambda_{0}/ \sqrt{\varepsilon_{e}}$$.

Example $$\PageIndex{1}$$: Microstrip Calculations

A microstrip line has a characteristic impedance $$Z_{0}$$ of $$50\:\Omega$$ derived from reflection coefficient measurements and an effective permittivity, $$\varepsilon_{e}$$, of $$7$$ derived from measurement of phase velocity. What is the line’s per-unit-length inductance, $$L$$, and capacitance, $$C$$?

Solution

The key equations are $$Z_{0} =\sqrt{L/C}, \varepsilon_{e} = C/C_{\text{air}}$$, and in air $$v_{p} = 1/ \sqrt{LC_{\text{air}}} = c$$. Also assume that $$\mu_{r} = 1$$ which is the default if not specified otherwise and also that $$L$$ does not change if only the dielectric is changed. Thus

$C_{\text{air}}=\frac{C}{\varepsilon_{e}}\quad\text{and then}\quad L=\frac{\varepsilon_{e}}{c^{2}C}\quad\text{so that}\quad Z_{0}=\sqrt{\frac{L}{C}}=\frac{\sqrt{\varepsilon_{e}}}{cC},\quad\text{that is}\quad C=\frac{\sqrt{\varepsilon_{e}}}{cZ_{0}}\nonumber$

So $$C = \sqrt{7}/(2.998\cdot 10^{8}\times 50) = 1.765\cdot 10^{−10} = 176.5\text{ pF/m}\text{ and }L = Z_{0}^{2}C = 44.13\:\mu\text{H/m}$$.

## 4.4.2 Effective Permittivity and Characteristic Impedance

This section presents formulas for the effective permittivity and characteristic impedance of a microstrip line. These formulas are fits to the results of detailed EM simulations. Also, the form of the equations is based on good physical understanding. First, assume that the thickness, $$t$$, is zero. This is not a bad approximation, as $$t ≪ w, h$$ for most microwave circuits.

Hammerstad and others provide well-accepted formulas for calculating the effective permittivity and characteristic impedance of microstrip lines [1–3]. Given $$\varepsilon_{r},\: w,$$ and $$h$$, the effective relative permittivity is

$\label{eq:9}\varepsilon_{e}=\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r}-1}{2}\left(1+\frac{10h}{w}\right)^{-a\cdot b}$

where

$\label{eq:10} a(u)|_{u=w/h}=1+\frac{1}{49}\ln\left[\frac{u^{4}+\left\{ u/52\right\}^{2}}{u^{4}+0.432}\right] +\frac{1}{18.7}\ln\left[ 1+\left(\frac{u}{18.1}\right)^{3}\right]$

and

$\label{eq:11} (\varepsilon_{r} )=0.564\left[\frac{\varepsilon_{r}-0.9}{\varepsilon_{r}+3}\right]^{0.053}$

Take some time to interpret Equation $$\eqref{eq:9}$$, the formula for effective relative permittivity. If $$\varepsilon_{r} = 1$$, then $$\varepsilon_{e} = (1 + 1)/2+0 = 1$$, as expected. If $$\varepsilon_{r}$$ is not that of air, then $$\varepsilon_{e}$$ will be between 1 and εr, dependent on the geometry of the line, or more specifically, the ratio $$w/h$$. For a very wide line, $$w/h ≫ 1,\: \varepsilon_{e} = (\varepsilon_{r} + 1)/2+(\varepsilon_{r} − 1)/2 = \varepsilon_{r}$$, corresponding to the EM energy being confined to the dielectric. For a thin line $$w/h ≪ 1, \varepsilon_{e} = (\varepsilon_{r} + 1)/2$$, the average of the dielectric and air permittivities.

Note

Mostly the term “effective permittivity” is used to mean effective relative permittivity (check the magnitude).

The characteristic impedance is given by

$\label{eq:12} Z_{0}=\frac{Z_{01}}{\sqrt{\varepsilon_{e}}}$

where the characteristic impedance of the microstrip line in free space is

$\label{eq:13} Z_{01} =Z_{0}|_{(\varepsilon_{r}=1)}=60\ln\left[\frac{F_{1}h}{w}+\sqrt{1+\left(\frac{2h}{w}\right)^{2}}\right]$

and

$\label{eq:14} F_{1}=6+(2\pi -6)\text{exp}\left\{ -(30.666h/w)^{0.7528}\right\}$

The accuracy of Equation $$\eqref{eq:9}$$ is better than $$0.2\%$$ for $$0.01 ≤ w/h ≤ 100$$ and $$1 ≤ \varepsilon_{r} ≤ 128$$. Also, the accuracy of Equation $$\eqref{eq:13}$$ is better than $$0.1\%$$ for $$w/h < 1000$$. Note that $$Z_{0}$$ has a maximum value when w is small and a minimum value when w is large.

Now consider the special case where $$w$$ is vanishingly small. Then $$\varepsilon_{e}$$ has its minimum value:

$\label{eq:15}\varepsilon_{e}=\frac{1}{2}\left(\varepsilon_{r}+1\right)$

This leads to an approximate (and convenient) form of Equation $$\eqref{eq:9}$$:

$\label{eq:16}\varepsilon_{e}=\frac{(\varepsilon_{r}+1)}{2}+\frac{(\varepsilon_{r}-1)}{2}\frac{1}{\sqrt{1+12h/w}}$

This approximation has its greatest error for low and high εr and narrow lines, $$w/h ≪ 1$$, where the maximum error is $$1\%$$. Again, Equation $$\eqref{eq:12}$$ is used to calculate the characteristic impedance. The more exact analysis, represented by Equation $$\eqref{eq:9}$$, was used to develop Table $$\PageIndex{1}$$, which can be used in the design of microstrip.

Example $$\PageIndex{2}$$: Microstrip Characteristic Impedance Calculation

The strip of a microstrip line has a width of $$600\:\mu\text{m}$$ and is fabricated on a lossless substrate that is $$635\:\mu\text{m}$$ thick and has a relative permittivity of $$4.1$$.

1. What is the effective relative permittivity?
2. What is the characteristic impedance?
3. What is the propagation constant at $$5\text{ GHz}$$ ignoring any losses?

Solution

Use the formulas for effective permittivity, characteristic impedance, and attenuation constant from Section 4.4.2 with $$w = 600\:\mu\text{m};\: h = 635\:\mu\text{m};\: \varepsilon_{r} = 4.1;\text{ w/h} = 600/635 = 0.945$$.

Figure $$\PageIndex{2}$$

1. $\varepsilon_{e}=\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r}-1}{2}\left( 1+\frac{10h}{w}\right)^{-a\cdot b}\nonumber$
From equations $$\eqref{eq:10}$$ and $$\eqref{eq:11}$$,
\begin{aligned} a&=1+\frac{1}{49}\ln\left[\frac{(w/h)^{4}+\left\{ w/(52h)\right\} ^{2}}{(w/h)^{4}+0.432}\right]+\frac{1}{18.7}\ln\left[1+\left(\frac{w}{18.1h}\right)^{3}\right] =0.991 \\ b&=0.564 \left[\frac{\varepsilon_{r}-0.9}{\varepsilon_{r}+3}\right]^{0.053}=0.541 \end{aligned} \nonumber
From Equation $$\eqref{eq:9}$$, $$\varepsilon_{e}=2.967$$
2. In free space,
$Z_{0}|_{\text{air}}=60\ln\left[\frac{F_{1}\cdot h}{w}+\sqrt{1+\left(\frac{2h}{w}\right)^{2}}\right]\nonumber$
where $$F_{1}=6 + (2π − 6)\text{ exp}\left\{ − (30.666h/\omega )^{ 0.7528}\right\} ,\quad Z_{0} = Z_{0}|_{\text{air}} /\sqrt{\varepsilon_{e}}$$
$Z_{0}|_{\text{air}}=129.7\:\Omega\quad\text{and}\quad Z_{0}=Z_{0}|_{\text{air}} /\sqrt{\varepsilon_{e}}=75.4\:\Omega\nonumber$
3. $$f=5\text{ GHz},\omega =2\pi f,\:\gamma =\jmath\omega\sqrt{\mu_{0}\varepsilon_{0}\varepsilon_{e}}=\jmath 180.5/\text{ m}$$.

Figure $$\PageIndex{3}$$: Dependence of the q factor of a microstrip line at $$1\text{ GHz}$$ for various permittivities and aspect ($$\text{w/h}$$) ratios. (Data obtained from EM field simulations using Sonnet.)

## 4.4.3 Filling Factor

Defining a filling factor, $$q$$, provides useful insight into the distribution of energy in an inhomogeneous transmission line. The effective microstrip permittivity is

$\label{eq:17}\varepsilon_{e}=1+q(\varepsilon_{r}-1)$

where for a microstrip line $$q$$ has the bounds $$\frac{1}{2} ≤ q ≤ 1$$ and is almost independent of $$\varepsilon_{r}$$. A $$q$$ of $$1$$ indicates that all of the fields are in the dielectric region. The dependence of the $$q$$ of a microstrip line at $$1\text{ GHz}$$ for various permittivities and aspect ($$\text{w/h}$$) ratios is shown in Figure $$\PageIndex{3}$$. Fitting yields:

$\label{eq:18} q=\frac{1}{2}\left(1+\frac{1}{\sqrt{1+12h/w}}\right)$

$$Z_{0}$$ $$\varepsilon_{r} =4\text{ (SiO}_{2}\text{, FR4)}$$ $$\varepsilon_{r}=10\:\text{(Alumina)}$$ $$\varepsilon_{r}=11.9\:\text{(Si)}$$
($$\Omega$$) $$u$$ $$\varepsilon_{e}$$ $$u$$ $$\varepsilon_{e}$$ $$u$$ $$\varepsilon_{e}$$
$$140$$ $$0.171$$ $$2.718$$ $$0.028$$ $$5.914$$ $$0.017$$ $$6.907$$
$$139$$ $$0.176$$ $$2.720$$ $$0.029$$ $$5.917$$ $$0.018$$ $$6.910$$
$$138$$ $$0.181$$ $$2.722$$ $$0.030$$ $$5.919$$ $$0.019$$ $$6.914$$
$$137$$ $$0.185$$ $$2.723$$ $$0.031$$ $$5.922$$ $$0.020$$ $$6.919$$
$$136$$ $$0.190$$ $$2.725$$ $$0.032$$ $$5.924$$ $$0.021$$ $$6.923$$
$$135$$ $$0.195$$ $$2.727$$ $$0.033$$ $$5.927$$ $$0.022$$ $$6.925$$
$$134$$ $$0.201$$ $$2.729$$ $$0.035$$ $$5.931$$ $$0.022$$ $$6.927$$
$$133$$ $$0.206$$ $$2.731$$ $$0.036$$ $$5.933$$ $$0.023$$ $$6.930$$
$$132$$ $$0.212$$ $$2.733$$ $$0.037$$ $$5.936$$ $$0.024$$ $$6.934$$
$$131$$ $$0.217$$ $$2.734$$ $$0.038$$ $$5.939$$ $$0.025$$ $$6.937$$
$$130$$ $$0.223$$ $$2.736$$ $$0.040$$ $$5.942$$ $$0.026$$ $$6.941$$
$$129$$ $$0.229$$ $$2.738$$ $$0.043$$ $$5.949$$ $$0.028$$ $$6.948$$
$$128$$ $$0.235$$ $$2.740$$ $$0.044$$ $$5.951$$ $$0.029$$ $$6.951$$
$$127$$ $$0.241$$ $$2.742$$ $$0.046$$ $$5.955$$ $$0.030$$ $$6.954$$
$$126$$ $$0.248$$ $$2.744$$ $$0.048$$ $$5.958$$ $$0.031$$ $$6.957$$
$$125$$ $$0.254$$ $$2.746$$ $$0.050$$ $$5.962$$ $$0.033$$ $$6.963$$
$$124$$ $$0.261$$ $$2.748$$ $$0.052$$ $$5.966$$ $$0.034$$ $$6.966$$
$$123$$ $$0.268$$ $$2.750$$ $$0.054$$ $$5.970$$ $$0.035$$ $$6.969$$
$$122$$ $$0.275$$ $$2.752$$ $$0.056$$ $$5.973$$ $$0.038$$ $$6.977$$
$$121$$ $$0.283$$ $$2.755$$ $$0.058$$ $$5.977$$ $$0.039$$ $$6.980$$
$$120$$ $$0.290$$ $$2.757$$ $$0.061$$ $$5.982$$ $$0.041$$ $$6.985$$
$$119$$ $$0.298$$ $$2.759$$ $$0.063$$ $$5.985$$ $$0.043$$ $$6.990$$
$$118$$ $$0.306$$ $$2.761$$ $$0.066$$ $$5.990$$ $$0.045$$ $$6.995$$
$$117$$ $$0.314$$ $$2.763$$ $$0.068$$ $$5.993$$ $$0.047$$ $$6.999$$
$$116$$ $$0.323$$ $$2.766$$ $$0.071$$ $$5.998$$ $$0.049$$ $$7.004$$
$$115$$ $$0.331$$ $$2.768$$ $$0.074$$ $$6.003$$ $$0.051$$ $$7.008$$
$$114$$ $$0.340$$ $$2.771$$ $$0.077$$ $$6.007$$ $$0.053$$ $$7.013$$
$$113$$ $$0.349$$ $$2.773$$ $$0.080$$ $$6.012$$ $$0.055$$ $$7.017$$
$$112$$ $$0.359$$ $$2.776$$ $$0.083$$ $$6.016$$ $$0.057$$ $$7.022$$
$$111$$ $$0.368$$ $$2.778$$ $$0.086$$ $$6.021$$ $$0.060$$ $$7.028$$
$$110$$ $$0.378$$ $$2.781$$ $$0.089$$ $$6.025$$ $$0.062$$ $$7.032$$
$$109$$ $$0.389$$ $$2.783$$ $$0.093$$ $$6.031$$ $$0.065$$ $$7.038$$
$$108$$ $$0.399$$ $$2.786$$ $$0.097$$ $$6.036$$ $$0.068$$ $$7.044$$
$$107$$ $$0.410$$ $$2.789$$ $$0.100$$ $$6.040$$ $$0.071$$ $$7.050$$
$$106$$ $$0.421$$ $$2.791$$ $$0.104$$ $$6.046$$ $$0.074$$ $$7.055$$
$$105$$ $$0.432$$ $$2.794$$ $$0.109$$ $$6.052$$ $$0.077$$ $$7.061$$
$$104$$ $$0.444$$ $$2.797$$ $$0.113$$ $$6.057$$ $$0.080$$ $$7.066$$
$$103$$ $$0.456$$ $$2.800$$ $$0.117$$ $$6.062$$ $$0.084$$ $$7.073$$
$$102$$ $$0.468$$ $$2.803$$ $$0.122$$ $$6.069$$ $$0.087$$ $$7.079$$
$$101$$ $$0.481$$ $$2.806$$ $$0.127$$ $$6.075$$ $$0.091$$ $$7.085$$
$$100$$ $$0.494$$ $$2.809$$ $$0.132$$ $$6.081$$ $$0.095$$ $$7.092$$
$$99$$ $$0.507$$ $$2.812$$ $$0.137$$ $$6.087$$ $$0.099$$ $$7.099$$
$$98$$ $$0.521$$ $$2.815$$ $$0.143$$ $$6.094$$ $$0.103$$ $$7.105$$
$$97$$ $$0.535$$ $$2.819$$ $$0.148$$ $$6.100$$ $$0.108$$ $$7.113$$
$$96$$ $$0.550$$ $$2.822$$ $$0.154$$ $$6.106$$ $$0.112$$ $$7.120$$
$$95$$ $$0.565$$ $$2.825$$ $$0.160$$ $$6.113$$ $$0.117$$ $$7.127$$
$$94$$ $$0.580$$ $$2.829$$ $$0.167$$ $$6.121$$ $$0.122$$ $$7.135$$
$$93$$ $$0.596$$ $$2.832$$ $$0.173$$ $$6.127$$ $$0.128$$ $$7.144$$
$$92$$ $$0.612$$ $$2.836$$ $$0.180$$ $$6.134$$ $$0.133$$ $$7.151$$
$$91$$ $$0.629$$ $$2.839$$ $$0.187$$ $$6.142$$ $$0.139$$ $$7.159$$
$$90$$ $$0.646$$ $$2.843$$ $$0.195$$ $$6.150$$ $$0.145$$ $$7.168$$
$$89$$ $$0.664$$ $$2.847$$ $$0.202$$ $$6.157$$ $$0.151$$ $$7.176$$
$$87$$ $$0.701$$ $$2.855$$ $$0.219$$ $$6.173$$ $$0.164$$ $$7.193$$
$$86$$ $$0.721$$ $$2.859$$ $$0.228$$ $$6.182$$ $$0.171$$ $$7.203$$
$$85$$ $$0.740$$ $$2.863$$ $$0.237$$ $$6.190$$ $$0.179$$ $$7.213$$
$$84$$ $$0.761$$ $$2.867$$ $$0.246$$ $$6.198$$ $$0.187$$ $$7.223$$
$$83$$ $$0.782$$ $$2.872$$ $$0.256$$ $$6.208$$ $$0.195$$ $$7.233$$
$$82$$ $$0.804$$ $$2.876$$ $$0.266$$ $$6.216$$ $$0.203$$ $$7.242$$
$$81$$ $$0.826$$ $$2.881$$ $$0.277$$ $$6.226$$ $$0.212$$ $$7.253$$
$$80$$ $$0.849$$ $$2.885$$ $$0.288$$ $$6.235$$ $$0.221$$ $$7.263$$
$$79$$ $$0.873$$ $$2.890$$ $$0.299$$ $$6.245$$ $$0.230$$ $$7.274$$
$$78$$ $$0.898$$ $$2.895$$ $$0.311$$ $$6.255$$ $$0.240$$ $$7.285$$
$$77$$ $$0.923$$ $$2.900$$ $$0.324$$ $$6.265$$ $$0.251$$ $$7.297$$
$$76$$ $$0.949$$ $$2.905$$ $$0.337$$ $$6.276$$ $$0.262$$ $$7.309$$
$$75$$ $$0.976$$ $$2.910$$ $$0.350$$ $$6.286$$ $$0.273$$ $$7.321$$
$$74$$ $$1.003$$ $$2.915$$ $$0.364$$ $$6.297$$ $$0.285$$ $$7.333$$
$$73$$ $$1.032$$ $$2.921$$ $$0.379$$ $$6.309$$ $$0.297$$ $$7.345$$
$$72$$ $$1.062$$ $$2.926$$ $$0.394$$ $$6.320$$ $$0.310$$ $$7.359$$
$$71$$ $$1.092$$ $$2.932$$ $$0.410$$ $$6.332$$ $$0.323$$ $$7.371$$
$$70$$ $$1.123$$ $$2.937$$ $$0.426$$ $$6.34$$ $$0.338$$ $$7.386$$
$$69$$ $$1.156$$ $$2.943$$ $$0.444$$ $$6.357$$ $$0.352$$ $$7.399$$
$$68$$ $$1.190$$ $$2.949$$ $$0.462$$ $$6.369$$ $$0.368$$ $$7.414$$
$$67$$ $$1.224$$ $$2.955$$ $$0.480$$ $$6.382$$ $$0.384$$ $$7.429$$
$$66$$ $$1.260$$ $$2.961$$ $$0.500$$ $$6.396$$ $$0.400$$ $$7.444$$
$$65$$ $$1.298$$ $$2.968$$ $$0.520$$ $$6.410$$ $$0.418$$ $$7.460$$
$$64$$ $$1.336$$ $$2.974$$ $$0.541$$ $$6.424$$ $$0.436$$ $$7.476$$
$$63$$ $$1.376$$ $$2.980$$ $$0.563$$ $$6.439$$ $$0.455$$ $$7.492$$
$$62$$ $$1.417$$ $$2.987$$ $$0.586$$ $$6.454$$ $$0.475$$ $$7.509$$
$$61$$ $$1.460$$ $$2.994$$ $$0.610$$ $$6.470$$ $$0.496$$ $$7.527$$
$$60$$ $$1.504$$ $$3.001$$ $$0.635$$ $$6.486$$ $$0.518$$ $$7.545$$
$$59$$ $$1.551$$ $$3.008$$ $$0.661$$ $$6.502$$ $$0.541$$ $$7.564$$
$$58$$ $$1.598$$ $$3.015$$ $$0.688$$ $$6.519$$ $$0.564$$ $$7.583$$
$$57$$ $$1.648$$ $$3.022$$ $$0.717$$ $$6.538$$ $$0.589$$ $$7.603$$
$$56$$ $$1.700$$ $$3.030$$ $$0.746$$ $$6.556$$ $$0.616$$ $$7.624$$
$$55$$ $$1.753$$ $$3.037$$ $$0.777$$ $$6.575$$ $$0.643$$ $$7.645$$
$$54$$ $$1.809$$ $$3.045$$ $$0.809$$ $$6.594$$ $$0.672$$ $$7.667$$
$$53$$ $$1.867$$ $$3.053$$ $$0.843$$ $$6.614$$ $$0.702$$ $$7.690$$
$$52$$ $$1.927$$ $$3.061$$ $$0.878$$ $$6.635$$ $$0.733$$ $$7.713$$
$$51$$ $$1.991$$ $$3.069$$ $$0.915$$ $$6.657$$ $$0.766$$ $$7.738$$
$$50$$ $$2.056$$ $$3.077$$ $$0.954$$ $$6.679$$ $$0.800$$ $$7.763$$
$$49$$ $$2.125$$ $$3.086$$ $$0.995$$ $$6.702$$ $$0.837$$ $$7.790$$
$$48$$ $$2.197$$ $$3.094$$ $$1.037$$ $$6.726$$ $$0.875$$ $$7.817$$
$$47$$ $$2.272$$ $$3.103$$ $$1.081$$ $$6.750$$ $$0.914$$ $$7.845$$
$$46$$ $$2.350$$ $$3.112$$ $$1.128$$ $$6.775$$ $$0.956$$ $$7.874$$
$$45$$ $$2.432$$ $$3.121$$ $$1.177$$ $$6.801$$ $$1.000$$ $$7.904$$
$$44$$ $$2.518$$ $$3.131$$ $$1.229$$ $$6.828$$ $$1.047$$ $$7.936$$
$$43$$ $$2.609$$ $$3.140$$ $$1.283$$ $$6.856$$ $$1.096$$ $$7.968$$
$$42$$ $$2.703$$ $$3.150$$ $$1.340$$ $$6.884$$ $$1.147$$ $$8.002$$
$$41$$ $$2.803$$ $$3.160$$ $$1.400$$ $$6.913$$ $$1.201$$ $$8.036$$
$$40$$ $$2.908$$ $$3.171$$ $$1.464$$ $$6.944$$ $$1.259$$ $$8.072$$
$$39$$ $$3.019$$ $$3.181$$ $$1.531$$ $$6.974$$ $$1.319$$ $$8.108$$
$$38$$ $$3.136$$ $$3.192$$ $$1.602$$ $$7.006$$ $$1.384$$ $$8.147$$
$$37$$ $$3.259$$ $$3.203$$ $$1.677$$ $$7.039$$ $$1.452$$ $$8.186$$
$$36$$ $$3.390$$ $$3.214$$ $$1.757$$ $$7.073$$ $$1.524$$ $$8.226$$
$$35$$ $$3.528$$ $$3.226$$ $$1.841$$ $$7.108$$ $$1.600$$ $$8.268$$
$$34$$ $$3.675$$ $$3.237$$ $$1.931$$ $$7.143$$ $$1.682$$ $$8.311$$
$$33$$ $$3.831$$ $$3.250$$ $$2.027$$ $$7.180$$ $$1.769$$ $$8.355$$
$$32$$ $$3.997$$ $$3.262$$ $$2.129$$ $$7.218$$ $$1.862$$ $$8.402$$
$$31$$ $$4.174$$ $$3.275$$ $$2.238$$ $$7.258$$ $$1.961$$ $$8.449$$
$$30$$ $$4.364$$ $$3.288$$ $$2.355$$ $$7.298$$ $$2.067$$ $$8.498$$
$$29$$ $$4.567$$ $$3.301$$ $$2.480$$ $$7.340$$ $$2.181$$ $$8.549$$
$$28$$ $$4.875$$ $$3.315$$ $$2.615$$ $$7.384$$ $$2.304$$ $$8.601$$
$$27$$ $$5.020$$ $$3.329$$ $$2.760$$ $$7.428$$ $$2.436$$ $$8.655$$
$$26$$ $$5.273$$ $$3.344$$ $$2.917$$ $$7.475$$ $$2.579$$ $$8.712$$
$$25$$ $$5.547$$ $$3.359$$ $$3.087$$ $$7.523$$ $$2.734$$ $$8.770$$
$$24$$ $$5.845$$ $$3.374$$ $$3.272$$ $$7.573$$ $$2.902$$ $$8.831$$
$$23$$ $$6.169$$ $$3.390$$ $$3.474$$ $$7.625$$ $$3.086$$ $$8.894$$
$$22$$ $$6.523$$ $$3.407$$ $$3.694$$ $$7.679$$ $$3.287$$ $$8.960$$
$$21$$ $$6.912$$ $$3.424$$ $$3.936$$ $$7.734$$ $$3.508$$ $$9.028$$
$$20$$ $$7.341$$ $$3.441$$ $$4.203$$ $$7.793$$ $$3.752$$ $$9.100$$
$$19$$ $$7.815$$ $$3.459$$ $$4.499$$ $$7.854$$ $$4.022$$ $$9.174$$
$$18$$ $$8.344$$ $$3.478$$ $$4.829$$ $$7.917$$ $$4.323$$ $$9.252$$
$$17$$ $$8.936$$ $$3.497$$ $$5.199$$ $$7.983$$ $$4.661$$ $$9.334$$
$$16$$ $$9.603$$ $$3.517$$ $$5.616$$ $$8.053$$ $$5.043$$ $$9.419$$
$$15$$ $$10.361$$ $$3.538$$ $$6.090$$ $$8.126$$ $$5.476$$ $$9.509$$
$$14$$ $$11.229$$ $$3.559$$ $$6.633$$ $$8.202$$ $$5.972$$ $$9.604$$
$$13$$ $$12.233$$ $$3.581$$ $$7.262$$ $$8.282$$ $$6.547$$ $$9.704$$
$$12$$ $$13.407$$ $$3.604$$ $$7.997$$ $$8.367$$ $$7.219$$ $$9.809$$
$$11$$ $$14.798$$ $$3.628$$ $$8.868$$ $$8.456$$ $$8.016$$ $$9.920$$
$$10$$ $$16.471$$ $$3.652$$ $$9.916$$ $$8.550$$ $$8.975$$ $$10.038$$

Table $$\PageIndex{1}$$: Microstrip line normalized width $$u (= w/h)$$ and effective permittivity, $$\varepsilon_{e}$$, for specified characteristic impedance $$Z_{0}$$. Data derived from the analysis in Section 4.4.2.

This page titled 4.4: Microstrip Transmission Lines is shared under a CC BY-NC license and was authored, remixed, and/or curated by Michael Steer.