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Microstrip Mitered Bend Calculator — Optimum Mitre (Chamfer)

Optimum mitre for a 90° microstrip bend from the Douville and James relation: the percentage of the corner diagonal to cut away, the cut depth along the diagonal and the chamfer length along each outer edge, ready to draw in a PCB layout.

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Formula

M=52+65 e−1.35 W/h %,d=W2,x=M d,a=x2M = 52 + 65\,e^{-1.35\,W/h}\ \%,\quad d = W\sqrt{2},\quad x = M\,d,\quad a = x\sqrt{2}

Reference: Douville & James, IEEE Trans. Microwave Theory Tech., MTT-26(3), pp. 175–181 (1978)

W— Strip width (mm)
h— Substrate height (mm)
M— Optimum mitre, as a fraction of the diagonal (%)
d— Diagonal of the unmitred corner, outer to inner (mm)
x— Depth cut away along the diagonal, from the outer corner (mm)
a— Chamfer length along each outer edge, from the old corner (mm)

How It Works

A right-angle corner in a microstrip line adds metal at the outside of the bend. The extra area is excess capacitance to ground, which reflects part of the signal, more so as the frequency rises. Cutting off the outer corner, a mitre, removes that capacitance; cut too much and the bend turns inductive instead. The optimum lies between, and Pozar's Microwave Engineering (4th ed., section 4.6) describes mitring as the usual compensation when there is no room for a swept bend.

The Douville and James optimum

Douville and James measured symmetric microstrip bends over a range of widths and substrates and fitted the mitre that gives the lowest reflection (IEEE Transactions on Microwave Theory and Techniques, 1978):

M=52+65 e−1.35 W/h %M = 52 + 65\,e^{-1.35\,W/h}\ \%
MM is the percentage of the corner diagonal cut away, measured from the outer corner. The diagonal of the unmitred corner, from the outer to the inner corner, is d=W2d = W\sqrt{2}, so the cut lies x=M dx = M\,d in from the outer corner, at right angles to the diagonal. Narrow lines need more mitre than wide ones: 98.38% at W/h=0.25W/h = 0.25, 68.85% at W/h=1W/h = 1, falling toward 52% for wide lines.

Drawing it

The cut removes a right isosceles triangle from the outside of the corner. Along each outer edge it starts x2x\sqrt{2} back from where the old corner was, so in a layout tool draw the plain corner and chamfer it with a 45° line whose ends sit that distance from the old corner. What remains between the cut and the inner corner, d−xd - x, is the narrowest copper the signal passes.

Validity

The relation is an empirical fit; it does not depend on εr\varepsilon_r or on frequency. Douville and James give it for W/h≥0.25W/h \geq 0.25 and εr≤25\varepsilon_r \leq 25, from measurements spanning W/hW/h from 0.25 to 2.75 and εr\varepsilon_r from 2.5 to 25. The calculator flags inputs outside either bound. Below W/hW/h of about 0.225 the extrapolated mitre would exceed 100%, cutting past the inner corner and severing the line, and the calculator refuses it. The relation is for 90° bends between lines of equal width.

Worked Example

Problem: A 50 Ω microstrip line, 1.10 mm wide on a 0.508 mm RO4003C substrate (εr = 3.55), turns through 90°. How should the corner be mitred?

Step 1 - Width-to-height ratio: W/h = 1.10 / 0.508 = 2.165

Step 2 - Optimum mitre (Douville and James): M = 52 + 65 × e^(−1.35 × 2.165) = 55.49%

Step 3 - Corner diagonal: d = 1.10 × √2 = 1.556 mm

Step 4 - Cut depth along the diagonal: x = 0.5549 × 1.556 = 0.863 mm

Step 5 - Chamfer along each outer edge: 0.863 × √2 = 1.221 mm from the old corner

Step 6 - Copper left at the inner corner: d − x = 1.556 − 0.863 = 0.692 mm

W/h and εr are inside the measured range, so no warning appears. A 100 Ω line on the same board, 0.254 mm wide (W/h = 0.5), needs a deeper mitre: M = 85.10%, x = 0.306 mm, leaving only 0.054 mm of copper at the inner corner, so check that against the fabricator's minimum width.

Practical Tips

  • ✓Find the line width for your target impedance with the microstrip calculator first, then mitre with that width.
  • ✓Above W/h of 3 the optimum barely moves (53.13% at 3, tending to 52%), so wide lines can share one mitre.
  • ✓Where the board has room, a swept bend with a radius of at least three line widths avoids the corner discontinuity altogether.
  • ✓The mitre corrects the reflection, not the bend's electrical length; include the bend in the line length of a phase-critical path.

Common Mistakes

  • ✗Measuring the mitre from the inner corner. M is the fraction of the corner diagonal cut away from the outer corner.
  • ✗Using the same 45° chamfer for every line. The optimum depends on W/h: a narrow line needs nearly the whole corner cut, a wide line little more than half.
  • ✗Mitring a bend between lines of different widths with this relation. It was fitted to symmetric bends of a single width.
  • ✗Ignoring the copper left at the inner corner on narrow lines. At high mitre percentages it can fall below the fabricator's minimum feature.

Frequently Asked Questions

A right-angle microstrip corner with its outer corner cut off at 45°. Removing that copper cancels the excess capacitance of the square corner, which would otherwise reflect part of the signal at high frequencies.
Douville and James found M = 52 + 65 × exp(−1.35 W/h) percent of the corner diagonal, cut from the outer corner. For a line as wide as its substrate is thick that is 68.85%; it rises to 98% for narrow lines and falls toward 52% for wide ones.
No. The relation depends only on the ratio of strip width to substrate height. It was measured on substrates with relative permittivity from 2.5 to 25, and outside that range the calculator warns that it is extrapolating.
A curved, swept bend with a radius of at least three line widths has almost no discontinuity, but it takes more board area. A mitred corner fits where a tight right angle is needed and, at the optimum, keeps the reflection low.
Draw the square corner, then chamfer the outside with a 45° line whose ends sit the chamfer length, x√2, back from the old corner along both outer edges. The calculator gives that length directly.

Methodology & References

References

  • Experimental Study of Symmetric Microstrip Bends and Their Compensation — R. J. P. Douville and D. S. James, IEEE Transactions on Microwave Theory and Techniques, vol. MTT-26, no. 3, pp. 175–181, March 1978 — the optimum-mitre relation
  • Microwave Engineering, 4th ed. — David M. Pozar (2012), §4.6, pp. 209–210 — compensated microstrip discontinuities: mitred bends and why they work

The mitre is Douville and James's relation evaluated directly: 68.85% at W/h = 1, 98.38% at 0.25, falling to 52% for wide strips.

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