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Analytical formulas (theory)

lineforge's Phase 1 solvers are closed-form approximations valid over published geometric ranges. They run in microseconds and serve two roles:

  1. Fast path for the 7 standard PCB geometries. Most users never need the bitmap kernel.
  2. Cross-validation reference for the Phase 2/3 numerical kernels. Every bitmap-rasterized standard geometry must agree with the closed-form answer to within 1%.

Microstrip: Hammerstad-Jensen (1980)

Effective permittivity:

\[ \varepsilon_{\text{eff}} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2}\left(1 + \frac{10}{u}\right)^{-ab} \]

where \(u = W/H\) and:

\[ a = 1 + \frac{1}{49}\ln\!\frac{u^4 + (u/52)^2}{u^4 + 0.432} + \frac{1}{18.7}\ln\!\left(1 + (u/18.1)^3\right) \]
\[ b = 0.564 \left(\frac{\varepsilon_r - 0.9}{\varepsilon_r + 3}\right)^{0.053} \]

Characteristic impedance (zero-thickness strip):

\[ Z_{0,\text{air}} = \frac{\eta_0}{2\pi} \ln\!\left(\frac{f}{u} + \sqrt{1 + \frac{4}{u^2}}\right), \qquad f = 6 + (2\pi - 6)\,e^{-(30.666/u)^{0.7528}} \]
\[ Z_0 = \frac{Z_{0,\text{air}}}{\sqrt{\varepsilon_{\text{eff}}}} \]

Finite-thickness Wheeler correction replaces \(W\) with \(W_{\text{eff}} = W + \Delta W\) before substitution.

Validity: \(0.05 \le W/H \le 20\), \(\varepsilon_r \le 128\). Outside this range, lineforge emits an out_of_range warning and the user should fall back to the bitmap kernel (Phase 2).

Reference: E. Hammerstad and Ø. Jensen, Accurate Models for Microstrip Computer-Aided Design, IEEE MTT-S 1980; IPC-2141A Appendix A.

Symmetric stripline: Cohn (1954) / Wadell

Wide-strip closed form:

\[ Z_0 = \frac{\eta_0}{\sqrt{\varepsilon_r}} \cdot \frac{1}{4}\,\frac{1}{\dfrac{W}{B - T} + \dfrac{C_f'}{\pi}} \]

where the fringing capacitance term:

\[ C_f' = \frac{B}{\pi(B - T)} \left[ m\,\ln\!\frac{m+1}{m-1} - \ln\!\frac{m^2 - 1}{4} \right],\quad m = \frac{2B}{B - T} \]

Validity: \(W/(B-T) > 0.35\), \(T < 0.25\,B\).

Reference: S. B. Cohn, Characteristic Impedance of the Shielded-Strip Transmission Line, IRE Trans. MTT, July 1954; Wadell §3.4.

CPWG: Wen (1969) / Wadell

Elliptic-integral formula:

\[ \varepsilon_{\text{eff}} = \frac{1 + \varepsilon_r\,(K(k')/K(k))(K(k_1)/K(k_1'))} {1 + (K(k')/K(k))(K(k_1)/K(k_1'))} \]
\[ Z_0 = \frac{60\pi}{\sqrt{\varepsilon_{\text{eff}}}} \cdot \frac{1}{K(k)/K(k') + K(k_1)/K(k_1')} \]

where:

\[ k = \frac{W}{W + 2S}, \qquad k_1 = \frac{\tanh(\pi W/4H)}{\tanh(\pi(W + 2S)/4H)} \]

and \(K(\cdot)\) is the complete elliptic integral of the first kind, evaluated in lineforge via scipy.special.ellipk.

Validity: \(W/H \ge 0.05\). Conductor thickness ignored when \(T \ll W,S\).

Reference: C. P. Wen, IEEE MTT-S 1969; Wadell §3.6.4; IPC-2141A.

Differential pairs: IPC-2141A coupling correction

For edge-coupled microstrip and stripline differential pairs, lineforge uses the empirical IPC-2141A coupling correction on the single-trace Z₀:

Microstrip: \(Z_{\text{odd/even}} = Z_0\,(1 \mp 0.48\,e^{-0.96\,S/H})\)

Stripline: \(Z_{\text{odd/even}} = Z_0\,(1 \mp 0.347\,e^{-2.9\,S/B})\)

Differential and common-mode impedances follow from:

\[ Z_{\text{diff}} = 2\,Z_{\text{odd}}, \qquad Z_{\text{common}} = Z_{\text{even}}/2 \]

For broadside-coupled stripline, lineforge uses the parallel-plate Wadell §6.5 formula with finite-thickness correction.

Caveat: The IPC-2141A coupling exponential is empirical and loses accuracy for tightly coupled pairs (\(S/H < 0.5\)). For exact differential analysis with arbitrary geometries, use the Phase 4 direct odd/even-mode bitmap solver.

References: Wadell §6; IPC-2141A.

Bibliography

  • B. Wadell, Transmission Line Design Handbook, Artech House 1991.
  • IPC-2141A, Design Guide for High-Speed Controlled Impedance Circuit Boards.
  • E. Hammerstad and Ø. Jensen, IEEE MTT-S 1980.
  • S. B. Cohn, IRE Trans. MTT, July 1954.
  • C. P. Wen, IEEE MTT-S 1969.