Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

In design the impedance is given and the width is wanted. Inverting Eq. (7.7.5) by hand is impractical, and Wheeler's inversion formulas give $w / h$ directly to about $1 %$ :

Synthesis (Wheeler)

With

A B = \frac{Z _{0}}{60} \frac{ε _{r} + 1}{2} + \frac{ε _{r} - 1}{ε _{r} + 1} (0.23 + \frac{0.11}{ε _{r}}), = \frac{60 π ^{2}}{Z _{0} ε _{r}},

C = \frac{ε _{r} - 1}{2 ε _{r}} (ln (B - 1) + 0.39 - \frac{0.61}{ε _{r}}), (7.7.6)

\frac{w}{h} = \frac{8 e ^{A}}{e ^{2 A} - 2} if this is < 2, otherwise \frac{w}{h} = \frac{2}{π} [B - 1 - ln (2 B - 1) + C] . (7.7.7)

The same formulas are Sadiku's Eq. (11.72) and Orfanidis's Eqs. (11.3.7)–(11.3.9). Table 7.7.1 lists the $50 Ω$ lines on common substrates, found with Eq. (7.7.7) and checked with Eqs. (7.7.4) and (7.7.5).

Recipe · Designing a microstrip line

Compute $A$ and the first branch of Eq. (7.7.7). If $w / h < 2$ keep it, otherwise use the second branch with $B$ and $C$ .
$w = (w / h) h$ .
Check with the analysis formulas Eqs. (7.7.4) and (7.7.5), and adjust $w$ by a few per cent if the impedance is off.
Compute $λ_{g} = λ_{0} / ε_{eff}$ for the lengths of stubs and transformers.

The tool below starts with the $50 Ω$ line on $1.6 mm$ FR-4 of Table 7.7.1 ( $w = 3.06 mm$ , $ε_{eff} = 3.33$ ). Type a target $Z_{0}$ , press Set w, and compare the field and $ε_{eff}$ of a narrow and a wide strip.

InteractiveLoading simulation…

Table 7.7.1. $50 Ω$ microstrip lines (Wheeler's synthesis, then Hammerstad's analysis).

Substrate ( $ε_{r}$ )	$h$ [mm]	$w$ [mm] ( $w / h$ )	$ε_{eff}$
FR-4 ( $4.4$ )	$1.6$	$3.06$ ( $1.91$ )	$3.33$
FR-4 prepreg ( $4.4$ )	$0.2$	$0.38$ ( $1.91$ )	$3.33$
RO4003C ( $3.55$ )	$0.508$	$1.14$ ( $2.24$ )	$2.78$
PTFE, RT5880 ( $2.2$ )	$0.787$	$2.43$ ( $3.08$ )	$1.87$
alumina ( $9.8$ )	$0.635$	$0.62$ ( $0.98$ )	$6.61$
GaAs, MMIC ( $12.9$ )	$0.1$	$0.074$ ( $0.74$ )	$8.40$

Rule of thumb

A $50 Ω$ line needs $w \approx 2 h$ on FR-4 and $w \approx h$ on alumina. A thin inner layer of a multilayer board therefore allows narrow $50 Ω$ traces, $0.38 mm$ on a $0.2 mm$ prepreg, between densely packed components.

7.7.5 Synthesis: the strip width for a given Z₀