2.3 Series Impedance Computations

This discussion of overhead transmission line series impedance concludes with a brief dicussion of computing contant factors associated with the impedance matrix and reconciling units of measure while evaluating these constants.

2.3.1 Computation of k in the Series Approximation to P and Q

The parameter k appears in the series expansion which approximates the P and Q terms of Carson’s equations (see Equation 10 and Equation 12 of Section 2.1.1 for details). It is of the form

k=4πd2λfk=4\pi d\sqrt{2\lambda f} (20)


λ\lambda is the earth conductivity in ab℧/cm3.

d is a distance in centimeters.

This can be rewritten in terms of readily available quantities (i.e. commonly published units) by substituting earth resistivity (Ω/m3) for conductivity and distance in conductor separation units for distance in centimeters as follows

k=4πd(uCScm)2λf(λρ)ρk=4\pi d(u_{CS}\to cm)\sqrt{\frac{2\lambda f(\lambda\to\rho)}{\rho}} (21)


uCS is conductor separation unit. In the US, conductor separation is usually measured in feet.

uCS\to cm is the number of centimeters per conductor separation unit.

λρ\lambda\to\rho is a constant converting ab℧/cm3 to Ω/m3.

Assuming that the frequency and resitivity are constant for any set of impedance computations the bulk of the expression

4π(ucscm)2λf(λρ)ρ4\pi(u_{cs}\to cm)\sqrt{\frac{2\lambda f(\lambda\to\rho)}{\rho}} (22)

is a constant which is computed once then stored for reuse.

2.3.2 Constants in the P and Q Terms of Carson’s Equations

After P and Q are computed, the terms 4ω\omegaP and 4ω\omegaQ in Equation 5 through Equation 8 of Section 2.1.1 produce impedances in units of abΩ/cm. If impedances are expressed in Ω/uLL, these terms expand to

4ω(uLLcm)(abΩΩ)P4\omega(u_{LL}\to cm)(ab\Omega\to\Omega)P (23)


4ω(uLLcm)(abΩΩ)Q4\omega(u_{LL}\to cm)(ab\Omega\to\Omega)Q (24)


uLL is line length unit. In the US, line length is usually measured in miles.

uCR\to cm is the number of centimeters per line length unit.

abΩΩ\Omega\to\Omega is a constant converting abΩ to Ω, i.e. 1×10-9.

Assuming that the frequency is constant, both P and Q are multiplied by the same factor

42πf(ullcm)(abΩΩ)4\cdot 2\pi f(u_{ll}\to cm)(ab\Omega\to\Omega) (25)

The first terms of of the inductive reactance equations (Equation 7 and Equation 8 of Section 2.1) are also multiplied by half of this value, i.e.

22πf(ullcm)(abΩΩ)2\cdot 2\pi f(u_{ll}\to cm)(ab\Omega\to\Omega) (26)

Once again, both of these constants are calculated once then stored.

2.3.3 Unit Conversions Associated With GMR Terms

When the logarithmic term in Equation 7 of Section 2.1 is computed, the conductor’s GMR must be converted to conductor separation units, i.e.

ln(2higmrj)ln\left(\frac{2h_{i}}{gmr_{j}}\right) (27)

is actually evaluated as

ln(2higmrj(uCRuCS))ln\left(\frac{2h_{i}}{gmr_{j}(u_{CR}\to u_{CS})}\right) (28)

where uCR \to uCS converts conductor radius units to conductor separation units. Factoring out a constant in this expression yields

ln(chigmrj)ln\left(\frac{ch_{i}}{gmr_{j}}\right) (29)


c=2uCRuCSc=\frac{2}{u_{CR}\to u_{CS}} (30)

The factor c is also calculated once and stored.