---
jupyter:
  description: 'Microstrip impedance with planex2d and NGSolve: the cross section of a trace on FR-4 over a ground plane,
    two electrostatic solves for the capacitance per length with and without the substrate, the characteristic
    impedance and the effective permittivity against Hammerstad and Jensen, and the trace width for 50 ohms.'
---

# Microstrip impedance

How wide must a trace on a circuit board be for a characteristic impedance of 50 Ω? A microstrip is a trace on top of
a substrate, with a ground plane below it. Part of its field runs through the substrate and part through the air
above, so a signal on it travels as if the whole line were embedded in an effective permittivity
$\varepsilon_{\mathrm{eff}}$ between 1 and the substrate's $\varepsilon_r$. Its impedance depends on the ratio of the
trace's width $W$ to the substrate's thickness $h$, and for a thin trace, the closed-form formulas of Hammerstad and
Jensen give $Z_0$ and $\varepsilon_{\mathrm{eff}}$.

The example computes both from the cross section itself. Two electrostatic solves give the capacitance per length of
the line with its substrate and of the same line in air, and from them $Z_0$ and $\varepsilon_{\mathrm{eff}}$ follow
without any fitted formula. They must agree with Hammerstad and Jensen, and a root search over the width then finds
the 50 Ω trace.

![The electric potential around a 50 ohm trace on FR-4, over the ground plane](../_static/gallery/microstrip-impedance.png){.result}

```python
import math

import ngsolve as ngs
from scipy.optimize import brentq
import planex as px
```

```python nbsphinx="hidden"
import matplotlib.pyplot as plt
from docs_figures import Chart, Row
```

## The cross section

The example is a trace on FR-4, a glass-epoxy laminate, here with $\varepsilon_r = 4.4$ and $h = 1.6\,\text{mm}$. The trace is
infinitely thin, as in Hammerstad and Jensen's formulas. The section lies in a box 160 times as wide as the substrate
is thick and half as high, whose walls are far enough away to change the result only in the fifth digit, and the box's
bottom is the ground plane. `section` glues the substrate, the air above it and the trace into one shape. The trace is
a line on the interface, which splits the interface's edge, so the trace has an edge of its own, named `trace`. The
field is singular at the trace's two edges, so its mesh size is 0.01 mm (Figure 1):

```python
h, er = 1.6, 4.4  # the substrate's thickness in mm and its relative permittivity


def section(W, box=160 * h):
    """the cross section of a trace W wide on the substrate, in a box whose bottom is the ground plane"""
    substrate = px.QuickSketch((-box / 2, 0)).Line(box, 0, name="ground").Line(0, h, name="shield").Line(-box, 0)
    substrate = substrate.Close(name="shield").Face()
    substrate.name = "substrate"
    air = px.QuickSketch((-box / 2, h)).Line(box, 0).Line(0, box / 2 - h, name="shield").Line(-box, 0, name="shield")
    air = air.Close(name="shield").Face()
    air.name = "air"
    trace = px.Line((-W / 2, h), (W / 2, h))
    trace.name = "trace"
    trace.maxh = 0.01
    return px.Glue([substrate, air, trace])


board = section(3)
print("faces", [face.name for face in board.faces], "| boundaries", sorted({e.name for e in board.edges if e.name}))
```

```python nbsphinx="hide-input"
view = px.QuickSketch((-6, 0)).Rectangle(12, 4).Face()
trace = px.Line((-1.5, h), (1.5, h))
Row((px.Glue([face * view for face in board.faces] + [trace]), "Around the trace", {"highlight": trace.edges}),
    caption="The substrate and the air around a trace 3 wide, the trace in cyan")
```

## Capacitance per length

The trace is held at 1 V, the ground plane and the box at 0 V. NGSolve solves
$\nabla \cdot (\varepsilon_r \nabla\varphi) = 0$ for the potential $\varphi$, and the energy of the field gives the
capacitance per length, $C = \varepsilon_0 \int \varepsilon_r\,|\nabla\varphi|^2\,dA$. The same section with air in
place of the substrate gives $C_{\mathrm{air}}$. The line's effective permittivity is the ratio of the two,
$\varepsilon_{\mathrm{eff}} = C/C_{\mathrm{air}}$, and its characteristic impedance is
$Z_0 = 1/(c\sqrt{C\,C_{\mathrm{air}}})$, with the speed of light $c$. A cross section has no length, so $C$ comes out
per meter whatever the unit of the drawing:

```python
def line(W, order=5):
    """the characteristic impedance, the effective permittivity and the potential of a trace W wide"""
    mesh = ngs.Mesh(px.OCCGeometry(section(W)).GenerateMesh(maxh=8 * h))
    mesh.Curve(order)
    fes = ngs.H1(mesh, order=order, dirichlet="ground|shield|trace")
    u, v = fes.TnT()
    C, potentials = [], []
    for eps_r in (er, 1):
        eps = mesh.MaterialCF({"substrate": eps_r}, default=1)
        a = ngs.BilinearForm(eps * ngs.grad(u) * ngs.grad(v) * ngs.dx).Assemble()
        phi = ngs.GridFunction(fes)
        phi.Set(1, definedon=mesh.Boundaries("trace"))
        rest = phi.vec.CreateVector()
        rest.data = -a.mat * phi.vec
        phi.vec.data += a.mat.Inverse(fes.FreeDofs(), inverse="sparsecholesky") * rest
        C.append(8.8541878128e-12 * ngs.Integrate(eps * ngs.grad(phi) * ngs.grad(phi), mesh))
        potentials.append(phi)
    return 1 / (299792458 * math.sqrt(C[0] * C[1])), C[0] / C[1], potentials[0], mesh


Z0, eff, phi, mesh = line(3)
print("W 3 mm | Z0", Z0, "ohm | effective permittivity", eff)
```

Most of the potential falls across the substrate under the trace, and the lines of equal potential crowd at the
trace's edges, where the field is singular:

```python nbsphinx="hide-input"
fig, ax = plt.subplots(figsize=(9, 3.2))
px.PlotField(phi, mesh, px.Unify(board.faces) * view, ax=ax, min=0, max=1, contours=11)
ax.set_title("The potential around a trace 3 wide, the trace at 1 V and the ground plane at 0 V")
plt.show()
```

## Against Hammerstad and Jensen

Hammerstad and Jensen fitted closed forms for the impedance and the effective permittivity of an infinitely thin
microstrip. With $u = W/h$, the impedance in air is
$Z_{01} = \eta_0/(2\pi)\,\ln\bigl(f(u)/u + \sqrt{1 + (2/u)^2}\bigr)$, with
$f(u) = 6 + (2\pi - 6)\,e^{-(30.666/u)^{0.7528}}$ and the impedance of free space $\eta_0$. The effective permittivity
is $\varepsilon_{\mathrm{eff}} = \tfrac{\varepsilon_r + 1}{2} + \tfrac{\varepsilon_r - 1}{2}\,(1 + 10/u)^{-ab}$, with
$a(u)$ and $b(\varepsilon_r)$ as in `hammerstad`, and $Z_0 = Z_{01}/\sqrt{\varepsilon_{\mathrm{eff}}}$. Both are
compared for traces from a third of the substrate's thickness to three times it:

```python
def hammerstad(W):
    """Hammerstad and Jensen's characteristic impedance and effective permittivity of a trace W wide"""
    u = W / h
    a = 1 + math.log((u**4 + (u / 52) ** 2) / (u**4 + 0.432)) / 49 + math.log(1 + (u / 18.1) ** 3) / 18.7
    b = 0.564 * ((er - 0.9) / (er + 3)) ** 0.053
    eff = (er + 1) / 2 + (er - 1) / 2 * (1 + 10 / u) ** (-a * b)
    f = 6 + (2 * math.pi - 6) * math.exp(-((30.666 / u) ** 0.7528))
    z01 = 376.730313668 / (2 * math.pi) * math.log(f / u + math.sqrt(1 + (2 / u) ** 2))
    return z01 / math.sqrt(eff), eff


widths = [0.5, 1, 2, 3, 4, 5]
for W in widths:
    Z0, eff = line(W)[:2]
    z, e = hammerstad(W)
    print("W", W, "mm | Z0", round(Z0, 3), "formula", round(z, 3), "| effective permittivity", round(eff, 5),
          "formula", round(e, 5))
```

The two agree within about 0.1 %, which is within the 0.2 % that Hammerstad and Jensen's formula is said to hold for
the effective permittivity.

## A 50 Ω trace

SciPy's `brentq` finds the width at which the computed impedance is 50 Ω, between traces of 1 and 5 mm, and the same
for the formula:

```python
W50 = brentq(lambda W: line(W)[0] - 50, 1, 5, xtol=1e-5)
W50_formula = brentq(lambda W: hammerstad(W)[0] - 50, 1, 5, xtol=1e-9)
print("50 ohm: W", round(W50, 4), "mm, W/h", round(W50 / h, 4), "| formula: W", round(W50_formula, 4), "mm")
```

The 50 Ω trace on this board is 3.06 mm wide, 1.9 times the substrate's thickness, and the formula's width differs
from it by 0.05 %. The rule of thumb that a 50 Ω trace on FR-4 is twice as wide as the substrate is thick
holds for $\varepsilon_r$ of about 4. At the 4.4 taken here, the trace is a little narrower.

See [Gluing faces](../core/basics/faces-shapes-compounds.md#Gluing:-faces-that-share-edges) for `px.Glue` and
[Meshing a face](../integrations/netgen/meshing.md) for meshes with named regions and boundaries.

```python nbsphinx="hidden"
import sys
sys.path.insert(0, "..")
from gallery import save_figure
Z0, eff, phi, mesh = line(W50)
fig, ax = plt.subplots(figsize=(6, 4.2))
px.PlotField(phi, mesh, px.Unify(section(W50).faces) * px.QuickSketch((-6, 0)).Rectangle(12, 8).Face(), ax=ax, min=0,
             max=1, contours=11)
for (x0, x1), y in (((-W50 / 2, W50 / 2), h), ((-6, 6), 0)):  # the trace and the ground plane, in copper
    ax.plot([x0, x1], [y, y], color="#b87333", lw=5, solid_capstyle="butt", zorder=5)
save_figure(fig, "microstrip-impedance", bare=True, panels=[0], bars={1: "potential (V)"})
plt.close(fig)
```