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 (2), 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.