Carson’s formulas are

$\displaystyle\mathbf{Z_{ii-g}}$ | $\displaystyle=r_{i}+j2\omega ln\left(\frac{2h_{i}}{gm{r}_{i}}\right)+4\omega(P% +jQ)$ | (1) | ||

$\displaystyle\mathbf{Z_{ij-g}}$ | $\displaystyle=j2\omega ln\left(\frac{D_{ij}}{d_{ij}}\right)+4\omega(P+jQ)$ | (2) |

where

Z_{ii–g} is the self-impedance of conductor i with ground return.

Z_{ij–g} is the mutual impedance between conductors i and j
with common ground return.

gmr_{i} is the effective radius (or geometric mean radius) of
conductor i in centimeters.

h_{i} is the height of conductor i in centimeters.

r_{i} is the internal resistance of conductor i.

d_{ij} the distance between conductors i and j in centimeters.

D_{ij} the distance between conductor i and the image of conductor
j in centimeters.

$\omega$ is 2$\pi$f, where f is the frequency in cycles per second.

Obviously, the self-impedance Z_{ii–g} and mutual impedance
Z_{ij–g} can be decomposed into their real and imaginary components

$\displaystyle\mathbf{Z_{ii-g}}$ | $\displaystyle=R_{ii-g}+jX_{ii-g}$ | (3) | |||

$\displaystyle\mathbf{Z_{ij-g}}$ | $\displaystyle=R_{ij-g}+jX_{ij-g}$ | (4) |

Collecting terms in Equation 1 and Equation 2 and comparing to Equation 3 and Equation 4, it is apparent that

$\displaystyle R_{ii-g}$ | $\displaystyle=r_{i}+4\omega P$ | (5) | ||

$\displaystyle R_{ij-g}$ | $\displaystyle=4\omega P$ | (6) | ||

$\displaystyle X_{ii-g}$ | $\displaystyle=2\omega ln\left(\frac{2h_{i}}{gm{r}_{j}}\right)+4\omega Q$ | (7) | ||

$\displaystyle X_{ij-g}$ | $\displaystyle=2\omega ln\left(\frac{D_{ij}}{d_{ij}}\right)+4\omega Q$ | (8) |

The P and Q terms in the preceding equations are defined by Carson as an infinite series expressed in terms of two parameters, call them k and $\theta$. The form of P and Q are the same for Equation 1 and Equation 2. However, the value of k and $\theta$ differ. For self impedances

$\displaystyle k$ | $\displaystyle=4\pi{h}_{i}\sqrt{2\lambda f}$ | (10) | ||

$\displaystyle\theta$ | $\displaystyle=0$ | (11) |

For mutual impedances

$\displaystyle k$ | $\displaystyle=2\pi{D}_{ij}\sqrt{2\lambda f}$ | (12) | ||

$\displaystyle\theta$ | $\displaystyle=\frac{{cos}^{-1}({h}_{i}+{h}_{j})}{{D}_{ij}}$ | (13) |

where

$\lambda$ is the earth conductivity in ab℧/cm^{3}.

$\theta$ is the angle defined in Figure 1.

Figure 1 defines the line geometry associated with Equation 10 through Equation 13.

The first few terms of the expansion of P and Q follow:

$\displaystyle P=$ | $\displaystyle\frac{\pi}{8}-k\frac{\cos\theta}{3\sqrt{2}}+{k}^{2}\frac{\cos% \left(2\theta\right)\left(0.6728+\ln\left(\frac{2}{k}\right)\right)}{16}+{k}^{% 2}\frac{\theta\sin\left(2\theta\right)}{16}$ | (14) | ||

$\displaystyle+{k}^{3}\frac{\cos\left(3\theta\right)}{45\sqrt{2}}-{k}^{4}\frac{% \pi\cos\left(4\theta\right)}{1536}$ | ||||

$\displaystyle Q=$ | $\displaystyle-0.0386+\frac{1}{2}ln\left(\frac{2}{k}\right)+k\frac{\cos\theta}{% 3\sqrt{2}}-\pi{k}^{2}\frac{\cos\left(2\theta\right)}{64}$ | (15) | ||

$\displaystyle+{k}^{3}\frac{\cos\left(3\theta\right)}{45\sqrt{2}}-{k}^{4}\frac{% \sin\left(4\theta\right)}{384}-{k}^{4}\frac{\cos\left(4\theta\right)\left(1.08% 95+\ln\left(\frac{2}{k}\right)\right)}{384}$ |

Clarke (3) states that Equation 14 and Equation 15 exhibit less than one percent error for values of k up to one. Table 1 shows the wide applicability of these expressions for fundamental and harmonic analysis of power systems by examining values of k for a range of geometries, frequencies, and resistivities.

Distance | Frequency | Earth Resistivity | k |
---|---|---|---|

100 ft | 60 Hz | 10 Ω/m^{3} |
0.4196 |

660 Hz | 1.3916 | ||

1020 Hz | 1.7300 | ||

60 Hz | 100 Ω/m^{3} |
0.1327 | |

660 Hz | 0.4401 | ||

1020 Hz | 0.5471 | ||

60 Hz | 1000 Ω/m^{3} |
0.0419 | |

660 Hz | 0.1391 | ||

1020 Hz | 0.1730 |

100 ft - Large double circuit transmission tower

10 Ω/m^{3} - Resistivity of swampy ground

100 Ω/m^{3} - Resistivity of average damp earth

1000 Ω/m^{3} - Resistivity of dry earth

At 60 Hz, it is common practice to ignore the higher order terms of the expansion of P and Q, i.e. let

$\displaystyle P$ | $\displaystyle=\frac{\pi}{8}$ | ||

$\displaystyle Q$ | $\displaystyle=-0.0386+\frac{1}{2}ln\left(\frac{2}{k}\right)$ |

This practice effectively decouples the series impedance from the conductor’s height above ground.
According to Wagner and Evans (4), this omission tends to overstate the
computed resistance and understate the computed reactance. At commercial frequencies and low earth
resistivities ($\rho$=10), the first order approximations may introduce resistance errors in the
neighborhood of 10 per cent. Under similar circumstances, self reactance errors rarely exceed one
per cent. However, mutual reactance errors are more volatile. For $\rho$=10, f=60, and
D_{ij}=200 feet, the low order approximation of Q understates the mutual
reactance by much as 4 per cent. At higher harmonics, these tendencies are magnified.