Line Impedance Calculator: GMD, GMR and Carson’s Equations

Geometry and bundles

Centre spacings give $D_m=(D_{ab}D_{bc}D_{ca})^{1/3}$. For a regular bundle of $n>1$ subconductors, adjacent spacing $s$ gives circumradius $b=s/[2\sin(\pi/n)]$ and equivalent radius $(n b^{n-1}r)^{1/n}$. Use supplied magnetic GMR for $r$ in inductance and physical radius in capacitance. Resistance is divided by $n$. For a single solid nonmagnetic round conductor only, GMR is $r e^{-1/4}$; do not impose this on stranded ACSR.

Carson earth return

Carson (1926), expressed in SI as Theodoulidis (2015), attached s1 file page 2, Eqs. (1)–(2): $z_{ij}=r_i\delta_{ij}+j\omega\mu_0[\ln(D'_{ij}/D_{ij})+I_c]/(2\pi)$. Self terms replace $D_{ii}$ by magnetic GMR. $D'_{ij}$ is distance to the reflected conductor. The integral is

The logarithmic wavenumber substitution uses a 601-point base trapezoidal rule, with extra points for wide horizontal separations, bounds from dimensionless wavenumber $e^{-22}$ to $45/(Hq)$, and $q=\sqrt{\omega\mu_0/\rho}$. A doubled-resolution probe checks convergence. The modified option is its low-argument approximation $z_{ij}=r_i\delta_{ij}+k[\pi/4+j\ln(D_e/D_{ij})]$, $k=\omega\mu_0/(2\pi)$ and $D_e=2e^{1/2-\gamma}/q$. Units are explicitly converted to Ω/km. Iowa State EE653 s2 pp. 23–31 explains the truncation and neutral reduction. Power frequency alone does not bound truncation error. Both methods are evaluated at the current inputs. The modified option is accepted only when every real and imaginary entry of both the reduced phase and sequence matrices agrees with the integral within 5%. Relative differences use the integral entry magnitude, with a denominator floor of 0.000001 Ω/km for nearly zero components. This 5% comparison screen is a tool policy, not a universal error bound or a standard requirement.

Grounded-wire reduction, capacitance and sequences

For series impedance, $Z_{abc}=Z_{pp}-Z_{pn}Z_{nn}^{-1}Z_{np}$. Electrostatic coefficients use $P_{ij}=\ln(D'_{ij}/D_{ij})/(2\pi\epsilon_0)$, with physical radius on the diagonal. Grounded-wire elimination is applied to P, then inverted: $C_{abc}=(P_{pp}-P_{pn}P_{nn}^{-1}P_{np})^{-1}$. This is equivalent to taking the phase block of the full Maxwell capacitance matrix at $V_n=0$. The electrostatic image construction is derived here from the line-charge potential and the zero-potential earth boundary.

Full transposition averages the three cyclic permutations of the reduced Z and C matrices. With $a=e^{j2\pi/3}$ and $A=[1,1,1;1,a^2,a;1,a,a^2]$, $Z_{012}=A^{-1}Z_{abc}A$, ordered zero, positive, negative (Kersting, 4th ed., Ch. 4; Iowa State s2 pp. 36–40). Positive-sequence L and C are diagonal entries; off-diagonal sequence coupling is retained. The GMD estimates are $L=\mu_0\ln(D_m/\mathrm{GMR}_b)/(2\pi)$ and $C=2\pi\epsilon_0/\ln(D_m/r_b)$.

Feasible bundle-spacing curves

The sweep uses the same strict conductor-diameter, ten-envelope-radius phase clearance, earth clearance and grounded-wire clearance rules as the selected geometry. For each bundle size its circumradius is $s/[2\sin(\pi/n)]$. The limiting circumradius is the minimum of $D_{min}/10-r$, $h_{min}/10-r$, and, when included, $D_{pn,min}/10-r-r_n$. Adjacent spacings remain within 0.05–1.5 m and exceed conductor diameter. Curve endpoints stay just inside the strict limits. Infeasible portions are omitted; if no bundle has an interval, only the single-conductor baseline remains and an input message explains what to change.

• Three straight, parallel phase conductors with identical conductor data and identical regular bundles; one optional single grounded wire.
• Homogeneous nonmagnetic earth and air, with soil displacement current neglected. Sinusoidal steady state, using the $e^{j\omega t}$ convention.
• Constant supplied AC resistance and magnetic GMR at the chosen operating condition. Manufacturer rows use 75 °C / 60 Hz; no temperature or skin-effect correction is added. Custom rows must supply suitable values.
• Bundle centres and equivalent radii approximate the field outside each bundle. Equal current sharing and at least ten bundle-envelope radii of separation are required. The adjacent-spacing formula is exact for the self GMR of a regular bundle under equal sharing; interphase fields use bundle centres.
• The optional wire is ideally multigrounded with zero longitudinal voltage drop. Full transposition means equal lengths in cyclic phase positions while this wire remains fixed.
• Electrostatic capacitance assumes a perfectly conducting earth boundary. The magnetic earth-return integral treats finite earth conductivity separately.

Power frequency 40–70 Hz; soil resistivity 10–10,000 Ω·m; conductor heights 1–80 m; horizontal coordinates within ±60 m. Use representative mean heights along a span when sag matters. Close bundles, overlapping conductors and conductors too close to ground are rejected.

The full numerical Carson integral covers the advertised geometry, frequency and soil ranges under the stated physical assumptions. Modified Carson is restricted by the 5% phase-and-sequence R/X comparison screen described in the method; results are withheld and the user is directed to the integral if it fails. High conductors, wide spacings or low soil resistivity can invalidate the approximation even at 50 or 60 Hz. Small differences from the IEEE benchmark are expected if the integral is selected. Published IEEE matrices are rounded; the reference tolerance is 0.00008 Ω/km. Untransposed lines generally have coupled sequence networks.

Effective $R=\Re(Z)$ and $L=\Im(Z)/\omega$ include the selected earth and neutral return model. These are per-unit-length steady-state parameters, not a broadband line model. Physical AC resistance and GMR remain fixed when frequency changes; update custom data where needed.

No underground cable insulation, concentric neutral, sheath bonding, earth stratification, magnetic soil, corona, lightning, surge propagation, ampacity, clearance compliance, footing impedance or sag calculation. The tower support is schematic and is not part of the electromagnetic model. Installation work belongs to a qualified person.

Bundle-spacing sweeps use these same validity constraints for every plotted point. A selected single-conductor geometry can be valid even when no twin, triple or quad bundle fits. Unsupported bundle curves are omitted rather than assigned an artificial minimum sweep interval.