1. What Is a Microstrip Line?
A microstrip line is a copper trace of width w and thickness t on a dielectric substrate of height h, with a ground plane on the other face. It is the most common transmission line on printed circuit boards because every component can be soldered to it from the top.
The field lies partly in the substrate and partly in the air above it. Because the two media have different wave speeds, the line cannot carry a pure TEM wave; it carries a quasi-TEM mode that behaves like TEM at low frequency and slowly concentrates in the substrate as the frequency rises. Its two key parameters are the characteristic impedance Z₀ and the effective permittivity εeff.
2. Z₀ and εeff from the FEM Capacitances
For a quasi-TEM line both parameters follow from two capacitances per unit length: C with the substrate in place and Cair with the substrate replaced by air. Step 1 finds each by solving Laplace's equation ∇·(ε∇V) = 0 on a graded triangular mesh of the cross-section, refined at the strip edge where the field is singular:
FEM capacitance method
C = capacitance per unit length with the substrate · Cair = the same line in air · c = speed of light
The capacitance comes from the stored energy, C = ε₀·VᵀKV for a strip at 1 V, where K is the assembled stiffness matrix. Eighteen widths spaced logarithmically from w/h = 0.04 to 9 form the line library, and a cubic spline in log w interpolates between them.
3. The Closed-Form Reference: Hammerstad–Jensen
The library is checked against the Hammerstad–Jensen equations, accurate to about 0.2% for zero thickness. With u = w/h:
Impedance in air
f₁(u) = 6 + (2π − 6)·exp(−(30.666/u)0.7528) · η₀ ≈ 376.7 Ω
Effective permittivity
a(u) and b(εr) are Hammerstad–Jensen's fitting functions; Z₀ = Z₀,air/√εeff
The copper thickness widens the strip electrically; Hammerstad–Jensen add a width correction Δu that depends on t/h. The FEM meshes the copper directly, so the two agree to within a few tenths of a percent for typical boards, as the validation table in step 1 shows.
4. Synthesis, Guided Wavelength and Length
To find the width for a target impedance, the calculator inverts the spline by bisection on log w. The wavelength on the line and the length for an electrical length θ are:
Guided wavelength and physical length
The phase delay is ℓ√εeff/c
5. Line Types: Pairs, GCPW, Stripline, Embedded Lines and Solder Mask
The same FEM solver handles three line types. An edge-coupled differential pair carries two modes. In the odd mode the strips sit at +V and −V, the symmetry plane between them is an electric wall, and much of the field crosses the gap through air. In the even mode both strips sit at +V and the symmetry plane is a magnetic wall. The calculator solves each mode with the matching boundary on the half structure:
Coupled-line impedances
Cmode = capacitance per unit length of one strip in that mode
The odd mode has more of its field in air, so εeff,odd < εeff,even: on microstrip the two modes travel at different speeds, which is why loosely coupled pairs are easier to route. As the gap grows, Zodd and Zeven both approach the single-line Z₀. The closed-form reference for microstrip pairs is Hammerstad and Jensen's coupled-line model, accurate to about 1% for zero thickness; section 10 compares the two.
Grounded coplanar waveguide adds ground copper on the top layer, a gap s either side of the centre strip, with vias stitching it to the bottom ground. The FEM holds those top grounds at 0 V. The closed-form reference is the conformal-mapping result for conductor-backed CPW with zero thickness, where a = w/2, b = w/2 + s:
Conductor-backed CPW (conformal mapping)
K = complete elliptic integral of the first kind, evaluated by the arithmetic–geometric mean · k′ = √(1 − k²)
With thin copper the FEM agrees with this to about 0.3%. Real copper thickness adds capacitance across narrow gaps and lowers Z₀ noticeably once t is more than a few percent of s, which the FEM captures and the closed form does not.
Stripline runs inside the board between two ground planes, a height h below the trace and h₂ above it. The field is entirely in dielectric, so the wave is pure TEM: εeff = εr, there is no dispersion and the line is far better shielded than microstrip. For a centred strip of zero thickness between grounds b apart, Cohn's conformal mapping is exact, and it extends to edge-coupled pairs:
Stripline (Cohn, exact for t = 0)
Coupled pair: Zeven,odd = (30π/√εr)·K(k′e,o)/K(ke,o)
These exact results are what make stripline a strong test of the solver: they check the FEM, the box boundaries and the odd- and even-mode walls without any approximation in the reference. The FEM also handles an off-centre trace, copper thickness and a different εr above the trace (prepreg against core), which the formula does not.
Embedded microstrip is a surface trace buried under a dielectric cover of thickness c, typically an inner layer near the surface or a trace under a thick coating. Air still sits above the cover, so εeff lies between the microstrip value and εr, rising as the cover thickens. The closed-form reference blends the surface value toward εr as 1 − e−1.55c/h, an IPC-style approximation; the FEM meshes the cover directly.
Solder mask is modelled as a conformal dielectric layer of thickness tm and permittivity εr,m over the substrate, the copper and its side walls. It replaces some air in the fringing field, so it raises εeff and lowers Z₀; a typical 20 µm LPI mask with εr ≈ 3.6 lowers a 50 Ω microstrip by about 1–2 Ω and affects tightly coupled pairs and GCPW more, since more of their field sits in the gaps.
6. Dispersion — Kirschning–Jansen
As the frequency rises, the field moves into the substrate and εeff climbs from its quasi-static value toward εr. Kirschning and Jansen's model, accurate to 0.6% up to 60 GHz, gives:
Dispersive effective permittivity
P(f) = P₁P₂[(0.1844 + P₃P₄)·fn]1.5763, fn = f·h in GHz·mm
The impedance shifts with it. The calculator uses the power–current form Z₀(f) = Z₀·√(εeff,0/εeff(f))·(εeff(f) − 1)/(εeff,0 − 1), which raises Z₀ slightly with frequency.
7. Conductor Loss — Wheeler's Incremental Inductance Rule
At RF the current flows in a skin depth δ = 1/√(πfμ₀σ) at the surface of the trace and the ground. Wheeler showed that the resulting loss equals the change in impedance when every conductor wall recedes by δ/2. The calculator evaluates that derivative numerically on the closed form and scales it to the FEM line:
Incremental inductance rule
Rs = √(πfμ₀/σ) · ∂/∂n = recession of all walls: w − 2n, t − 2n, h + 2n · result in Np/m (× 8.686 for dB)
Rough copper lengthens the current path. Hammerstad's roughness factor raises the loss by up to a factor of two once the RMS roughness Rq exceeds the skin depth:
Roughness factor
Standard ED copper is about 1–2 µm Rq, reverse-treated about 0.5 µm, rolled copper about 0.3 µm
8. Dielectric Loss
Only the part of the field inside the substrate sees its loss tangent. The filling factor (εeff − 1)/(εr − 1) accounts for that:
Dielectric attenuation
k₀ = 2πf/c · result in Np/m
Laminate Dk and Df are not constant. With the frequency-dependent option on, the calculator fits a wideband Debye (Djordjevic–Sarkar) model to the single Dk and Df your datasheet quotes, which is causal and keeps Dk and Df consistent with each other:
Wideband Debye (Djordjevic–Sarkar)
ω₁ = 10⁴ and ω₂ = 10¹² rad/s; ε∞ and Δε are set so that Dk and Df match the datasheet at its quoted frequency
The line library is solved once at the datasheet Dk; at each frequency the quasi-static εeff is moved to the new Dk through the filling factor (εeff − 1)/(εr − 1), and Z₀ follows as 1/√εeff. When you synthesize a width, the target impedance is met at the design frequency rather than at the datasheet frequency.
Dielectric loss grows in proportion to f while conductor loss grows with √f, so on FR-4 (tan δ ≈ 0.02) the dielectric dominates above about 1 GHz. On low-loss laminates the conductor usually dominates up to tens of GHz.
9. S-Parameters of a Line Section
A line of length ℓ with propagation constant γ = α + jβ and impedance Z₀ has the ABCD matrix below. Converting it to S-parameters in a reference impedance Zs shows the insertion loss and the mismatch when Z₀ ≠ Zs:
Line ABCD and S-parameters
10. Validation Against Independent References
The FEM solver is checked against exact analytic solutions where they exist and against the most accurate published closed forms elsewhere. All cases use zero copper thickness, since that is what the references assume, and the same mesh settings the calculator uses. Stripline and grounded CPW have exact solutions (Cohn's and Ghione's conformal mappings), so they test the solver itself, including the odd- and even-mode boundary conditions used for differential pairs. Microstrip has no exact solution; Hammerstad and Jensen quote their single-line formula at about 0.2% and their coupled-line formulas at better than 1%.
| Case | Quantity | Reference | This calculator (FEM) | Difference |
|---|
| Single microstrip vs Hammerstad–Jensen (±0.2%) |
| εr 4.3, w/h 0.5 | Z₀ | 96.40 Ω | 96.43 Ω | +0.03% |
| εr 4.3, w/h 1.9 | Z₀ | 50.74 Ω | 50.70 Ω | -0.08% |
| εr 3.55, w/h 1 | Z₀ | 77.96 Ω | 77.89 Ω | -0.09% |
| εr 10, w/h 1 | Z₀ | 48.82 Ω | 48.82 Ω | -0.01% |
| εr 2.2, w/h 5 | Z₀ | 35.47 Ω | 35.37 Ω | -0.27% |
| Differential microstrip vs Hammerstad–Jensen coupled lines (±1%) |
| εr 10, w/h 1, s/h 0.5 | Z odd | 37.05 Ω | 37.08 Ω | +0.06% |
| εr 10, w/h 1, s/h 0.5 | Z even | 59.04 Ω | 59.01 Ω | -0.05% |
| εr 9.7, w/h 1, s/h 1 | Z odd | 42.70 Ω | 42.71 Ω | +0.01% |
| εr 9.7, w/h 1, s/h 1 | Z even | 55.87 Ω | 55.81 Ω | -0.11% |
| εr 4.3, w/h 1.1, s/h 0.5 | Z odd | 51.57 Ω | 51.63 Ω | +0.13% |
| εr 4.3, w/h 1.1, s/h 0.5 | Z even | 83.05 Ω | 82.94 Ω | -0.14% |
| εr 4.3, w/h 2, s/h 0.5 | Z odd | 38.48 Ω | 38.95 Ω | +1.23% |
| εr 4.3, w/h 2, s/h 0.5 | Z even | 57.39 Ω | 57.27 Ω | -0.20% |
| εr 3.55, w/h 1, s/h 1 | Z odd | 66.24 Ω | 66.28 Ω | +0.06% |
| εr 3.55, w/h 1, s/h 1 | Z even | 88.91 Ω | 88.73 Ω | -0.21% |
| εr 2.2, w/h 3, s/h 1 | Z odd | 44.11 Ω | 44.64 Ω | +1.19% |
| εr 2.2, w/h 3, s/h 1 | Z even | 56.26 Ω | 56.16 Ω | -0.18% |
| Stripline vs Cohn (exact) |
| εr 4.3, w/b 0.25 | Z₀ | 67.52 Ω | 67.53 Ω | +0.02% |
| εr 4.3, w/b 0.5 | Z₀ | 48.47 Ω | 48.45 Ω | -0.03% |
| εr 3.5, w/b 1 | Z₀ | 34.96 Ω | 34.89 Ω | -0.20% |
| Differential stripline vs Cohn coupled lines (exact) |
| εr 4.3, w/b 0.25, s/b 0.25 | Z odd | 53.65 Ω | 53.66 Ω | +0.03% |
| εr 4.3, w/b 0.25, s/b 0.25 | Z even | 80.36 Ω | 80.35 Ω | -0.01% |
| εr 4.3, w/b 0.5, s/b 0.5 | Z odd | 44.98 Ω | 44.98 Ω | -0.02% |
| εr 4.3, w/b 0.5, s/b 0.5 | Z even | 51.71 Ω | 51.68 Ω | -0.06% |
| εr 3.5, w/b 0.3, s/b 0.1 | Z odd | 44.98 Ω | 44.98 Ω | -0.00% |
| εr 3.5, w/b 0.3, s/b 0.1 | Z even | 89.13 Ω | 89.11 Ω | -0.02% |
| εr 4.3, w/b 0.4, s/b 0.2 | Z odd | 42.54 Ω | 42.58 Ω | +0.08% |
| εr 4.3, w/b 0.4, s/b 0.2 | Z even | 64.60 Ω | 64.61 Ω | +0.02% |
| Grounded CPW vs conformal mapping (exact for t = 0, wide grounds) |
| εr 4.3, w 1 mm, s 0.2 mm, h 1.6 mm | Z₀ | 54.89 Ω | 54.89 Ω | -0.01% |
| εr 4.3, w 0.5 mm, s 0.5 mm, h 1 mm | Z₀ | 82.14 Ω | 81.98 Ω | -0.20% |
| εr 9.9, w 0.3 mm, s 0.15 mm, h 0.635 mm | Z₀ | 49.05 Ω | 49.04 Ω | -0.02% |
Against the 14 exact solutions the largest difference is 0.20%. Across all 31 cases the mean difference is 0.15%; the two largest, about 1.2% in the odd mode of wide microstrip pairs, are at the edge of the Hammerstad–Jensen formulas' own stated accuracy. The solver reaches this by extrapolating each capacitance from two mesh densities (Richardson extrapolation), keeping cells small near narrow strips, and placing the open boundary 24 substrate heights away. The 3D full-wave check in step 4 adds an independent test of single microstrip including frequency dependence.
11. Full-Wave Verification — 3D FDTD
The models above are quasi-static or empirical, so step 4 can check them by solving Maxwell's equations directly with the finite-difference time-domain (FDTD) method on a 3D Yee grid. The grid holds the strip between two lumped 50 Ω ports, the substrate, the ground plane at the bottom face, and air above, ending in a convolutional perfectly matched layer (CPML) on the other five faces. The fields leap-frog in time:
Yee update
σ = 2πf₀ε tan δ represents the dielectric loss at the design frequency
A Gaussian pulse drives port 1 and port 2 is a matched load. Voltage probes at evenly spaced planes Δy apart along the middle of the line record V(t), and a Fourier transform gives Vn(f) at each plane. On a uniform line V(y) = Ae−γy + Beγy, so any three neighbouring planes satisfy a recurrence that gives the propagation constant without knowing A or B:
Propagation constant from the probes
Solved in the least-squares sense over all planes · γ = α + jβ
A least-squares fit then gives the forward and reverse amplitudes A and B, and the measured port current I₁ fixes the impedance: Z₀ = (A − B)/I₁ at port 1, with the matching expression at port 2; the calculator averages the two. Because the probes sit away from the ports, the port discontinuities do not enter γ.
A zero-thickness strip on a Yee grid behaves wider than drawn, because the field is singular at its edge. The calculator applies the standard one-third rule: the outermost grid line sits one third of a cell inside the physical edge, so a strip drawn N cells wide represents a width of (N + ⅔) cells. With it, the full-wave Z₀ and εeff typically land within 1–2% of the bare-strip model even on the coarse mesh.
12. Validity and Limitations
The quasi-TEM model holds while the line is electrically thin. Two limits mark where it breaks down:
Mode limits
Transverse resonance of the strip · onset of strong TM₁ surface-wave coupling
- Solder mask is a uniform conformal layer; real mask is thinner over copper edges and thicker in narrow gaps.
- Differential pairs assume perfect symmetry; dispersion and loss for pairs and GCPW use the microstrip models applied to the FEM values, which is an estimate at high frequency.
- GCPW assumes wide top grounds stitched to the bottom ground with closely spaced vias.
- The ground plane is assumed solid and wide (at least 3w + 6h); nearby copper pour or a cover changes Z₀.
- The substrate is isotropic. Woven-glass laminates are anisotropic, and the εr seen by a microstrip can differ from the datasheet process value; use the "design" εr where the supplier gives one.
- Radiation loss from open ends and discontinuities is not included.
- The full-wave solve models zero-thickness, lossless copper; conductor loss and copper thickness appear only in the model.
Treat the results as a strong starting point and verify critical lines with a TDR or VNA measurement on a test coupon.
References
- Pozar, D.M., Microwave Engineering, 4th ed., Wiley, 2011, ch. 3.
- Gupta, K.C., Garg, R., Bahl, I. & Bhartia, P., Microstrip Lines and Slotlines, 2nd ed., Artech House, 1996.
- Wadell, B.C., Transmission Line Design Handbook, Artech House, 1991.
- Cohn, S.B., “Characteristic Impedance of the Shielded-Strip Transmission Line,” IRE Trans. Microwave Theory Tech., vol. 2, 1954; and “Shielded Coupled-Strip Transmission Line,” vol. 3, 1955.
- Djordjevic, A.R., Biljic, R.M., Likar-Smiljanic, V.D. & Sarkar, T.K., “Wideband Frequency-Domain Characterization of FR-4 and Time-Domain Causality,” IEEE Trans. Electromagn. Compat., vol. 43, no. 4, 2001.
- Steer, M., Microwave and RF Design: Transmission Lines, NC State University, 2019 (coupled-microstrip formulas and Example 5.1).
- Ghione, G. & Naldi, C.U., “Coplanar Waveguides for MMIC Applications: Effect of Upper Shielding, Conductor Backing, Finite-Extent Ground Planes, and Line-to-Line Coupling,” IEEE Trans. Microwave Theory Tech., vol. 35, no. 3, 1987.
- IPC-2141A, Design Guide for High-Speed Controlled Impedance Circuit Boards, IPC, 2004.
- Hammerstad, E. & Jensen, Ø., “Accurate Models for Microstrip Computer-Aided Design,” IEEE MTT-S Int. Microwave Symp. Digest, 1980.
- Kirschning, M. & Jansen, R.H., “Accurate Model for Effective Dielectric Constant of Microstrip with Validity up to Millimetre-Wave Frequencies,” Electronics Letters, vol. 18, 1982.
- Wheeler, H.A., “Formulas for the Skin Effect,” Proc. IRE, vol. 30, no. 9, 1942.
- Hammerstad, E. & Bekkadal, F., Microstrip Handbook, ELAB report STF44 A74169, University of Trondheim, 1975.
- Jin, J.-M., The Finite Element Method in Electromagnetics, 3rd ed., Wiley, 2014.
- Yee, K.S., “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, 1966.
- Taflove, A. & Hagness, S.C., Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed., Artech House, 2005.
- Roden, J.A. & Gedney, S.D., “Convolution PML (CPML): An Efficient FDTD Implementation of the CFS-PML for Arbitrary Media,” Microwave Opt. Technol. Lett., vol. 27, no. 5, 2000.
