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Free tool · Coplanar structures
Coplanar waveguide and asymmetric stripline impedance calculator
Solve a grounded coplanar waveguide's impedance and slot, or an asymmetric stripline's, and see what the backside plane costs against the unbacked closed form.
The short answer
A coplanar waveguide puts its return path on the same layer as the trace, as two ground rails either side of a slot, so its impedance is set by the centre width and the slot rather than the dielectric thickness. A 0.3 mm line with 0.2 mm slots on 0.2 mm FR-4 sits near 80 ohms unbacked, and a backside plane pulls that down a few percent. An asymmetric stripline is the opposite case: its two planes sit at different distances, so any offset lowers the impedance below the symmetric value.
Calculator
Coplanar waveguide and asymmetric stripline calculator
Choose the structure, enter the cross-section, and the tool returns the impedance, the effective permittivity and the delay. For a coplanar line it shows the unbacked conformal closed form next to a full 2-D solve of the grounded section, including the backside plane, the copper thickness and the soldermask. For an asymmetric stripline it solves the two half-spacings and compares the answer with the symmetric case the same trace would have.
A laminate fills Dk and Df together.
Structure: grounded coplanar waveguide or asymmetric stripline. Open the button for its units, why it matters and a typical value.
Cross-section
Laminate dielectric constant (Dk). Open the button for its units, why it matters and a typical value.
Loss tangent (Df). Open the button for its units, why it matters and a typical value.
Substrate thickness from the trace to the backside plane. Open the button for its units, why it matters and a typical value.
Slot gap to the coplanar ground. Open the button for its units, why it matters and a typical value.
Centre conductor width. Open the button for its units, why it matters and a typical value.
Copper thickness. Open the button for its units, why it matters and a typical value.
Impedance target to solve the gap or width for. Open the button for its units, why it matters and a typical value.Ω
Frequency the answer is reported at. Open the button for its units, why it matters and a typical value.GHz
Soldermask thickness over the copper. Open the button for its units, why it matters and a typical value.
Soldermask dielectric constant. Open the button for its units, why it matters and a typical value.
Coplanar waveguide
Closed form first, then the grounded section from a 2-D solve.
Unbacked coplanar impedance 72.13 ohms.
- Unbacked Z0
- 72.13Ω
- Effective Dk
- 3.300
- Delay
- 6.059ps/mm
- Centre/slot ratio
- 0.429
- Grounded Z0
- —Ω
- Delta
- —
Conformal closed form, no backside plane.
Half the field is in air.
w/(w + 2s).
From the 2-D solve.
What the backside plane costs.
Model notes
- The substrate is thinner than the coplanar span, so the backside plane is part of the cross-section: this unbacked answer is an upper bound, not the grounded impedance.
What this does not cover
Neither model includes the coplanar ground's own width, the via fence that ties it to the backside plane, or the launch from a connector. Those are 3-D effects; the extraction on our SI/PI engagement is where they are handled.
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Want the derivation? Jump to the math behind the calculator.
The math
The formulas, with the working shown
Each equation is rendered with KaTeX, which also emits a MathML twin for screen readers and answer engines, explained in plain language, and copyable as LaTeX from the button in its corner. The variable list says what every symbol means; the model-limit note says where the formula stops being trustworthy.
The coplanar modulus
Every coplanar formula is written in one dimensionless shape factor: how much of the total span is conductor.
- w
- centre conductor width
- s
- slot gap to the coplanar ground
- k_0
- centre-to-span ratio
The centre conductor is w wide and the two slots add 2s to the span. Their ratio is what the conformal map cares about, not the absolute size, which is why a coplanar line scales and keeps its impedance while the aspect stays the same.
The complementary modulus is the square root of one minus k squared, and the ratio K(k)/K(k prime) is the shape factor the impedance is built from.
Model limits
The fit covers roughly 0.05 < k < 0.95. Outside it the centre conductor is a hairline or the slots are negligible, and the tool warns instead of returning a confident number.
Effective permittivity with a finite substrate
Half the field is in air and half in laminate, so the effective permittivity starts at half the Dk plus one half and moves toward the laminate value as the substrate thins.
- \varepsilon_r
- laminate relative permittivity
- k_1
- modulus with the substrate thickness folded in
- K
- complete elliptic integral of the first kind
The first ratio is the air half. The second folds in the substrate thickness through k1, the same modulus with the hyperbolic sine of the geometry over the substrate height.
As the substrate grows thick, k1 approaches k0, the second ratio tends to one, and the permittivity becomes exactly (Dk + 1)/2 — the sanity check an engineer applies first, and the one this page's test suite pins.
Model limits
Permeability is taken as that of vacuum, which is exact for copper and FR-4. The formula assumes the backside of the substrate is air or a plane far enough away to ignore.
Unbacked coplanar impedance
Free-space impedance scaled by the elliptic shape factor and the effective permittivity.
- Z_0
- characteristic impedance, ohm
- k_0'
- complementary modulus, sqrt(1 - k^2)
- \varepsilon_{\text{eff}}
- effective permittivity of the cross-section
The 30 pi is the free-space impedance of a line divided by the geometry factor; the elliptic ratio carries the shape and the square root of the effective permittivity carries the dielectric loading.
Because the ratio falls as the centre conductor widens, the impedance falls with width exactly as a design table says. Because the slot appears in the same ratio, widening the slot raises the impedance — the second knob a coplanar layout has that a microstrip does not.
Model limits
This is the unbacked value. A backside plane adds capacitance and lowers the answer; the 2-D solve on the tool quantifies it, and for an ordinary 0.2 mm substrate the difference is a few percent.
Asymmetric stripline from two symmetric halves
Each side is half of a symmetric stripline, so the result is the harmonic mean and any offset lowers the impedance.
- Z_u
- upper half-spacing as a symmetric stripline
- Z_l
- lower half-spacing as a symmetric stripline
Each half is solved as a symmetric stripline whose plane spacing is twice the distance from the trace to that plane, which is the capacitance that half contributes.
Two capacitances in parallel invert to the harmonic mean. When the halves are equal it collapses to the symmetric closed form exactly, which is the identity the model's test carries.
Model limits
The approximation holds while the trace is roughly centred. Past about 60/40 the field is no longer two independent halves, so the tool warns and points at the 2-D solve.
Structures
Why a coplanar line behaves differently from a microstrip
A microstrip's return current runs in the plane underneath it, so everything about the line depends on the dielectric thickness. A coplanar waveguide carries its return on the same layer, in two rails either side of the slot, so most of the field closes across the slot and never reaches the backside plane. The impedance is then set by the centre width and the slot, and only weakly by the substrate.
That is why a coplanar line can be 50 ohms on almost any layer without changing the stackup, and why it is the usual choice for an outer layer that has to carry a controlled-impedance trace among unrelated copper. The price is a via fence: the rails have to be tied to the backside plane often enough that the return current never has to find its own way.
- Impedance is set by the centre width and the slot, not by the dielectric thickness.
- The backside plane lowers the impedance by a few percent and raises the effective Dk.
- A sparse via fence is a different line from a dense one, and neither closed form knows it.
- Soldermask sits in the slot as well as over the trace, so it pulls harder than on a microstrip.
Inner layers
The stripline that is not centred
A symmetric stripline has the trace exactly between two planes, and its closed form assumes that. Real builds break the assumption constantly: the prepreg above a signal layer is a different style from the core below it, so the trace ends up closer to one plane than the other.
The tool treats each side as half of a symmetric stripline and adds the two capacitances, which reduces to the harmonic mean of the two impedances. Because a harmonic mean is always at or below the arithmetic one, any offset lowers the impedance, and the error grows as the trace approaches a plane. That is the number a copied symmetric table gets wrong.
Past about a 60/40 split the two-half approximation loses accuracy and a 2-D solve is the honest check. Our SI/PI verification service work includes that extraction on the real stackup.
Limits
What neither model contains
The coplanar closed form is the unbacked case on a substrate of finite thickness. It has no backside plane, so it is an upper bound on the grounded impedance; the 2-D solve on the same page is what includes the plane, the copper thickness and the mask. If the two disagree by more than a few percent, the backside plane is part of the problem and the stackup should say so.
Neither model includes the ground rails' own width, the via fence, or the launch from a connector or package. Those are three-dimensional effects that change the answer at the frequencies where coplanar lines are usually chosen.
The high-speed PCB layout service and DFM/DFA layout service pages describe the routing rules and the CAM checks that follow from the geometry this tool produces.
FAQ
Common questions
When the layer has to carry a controlled-impedance trace while the dielectric thickness is fixed by something else, or when the trace must sit among unrelated copper. The rails give a return path on the same layer, so the slot rather than the substrate sets the impedance. The cost is routing room for the rails and the via fence that ties them to ground.
Want this checked on your board?
Send the stackup and interface list. We will check the coplanar geometry against the real layout before the quote is fixed.