2.1 Carson’s Equations

Carson’s formulas are

𝐙𝐢𝐢-𝐠\displaystyle\mathbf{Z_{ii-g}} =ri+j2ωln(2higmri)+4ω(P+jQ)\displaystyle=r_{i}+j2\omega ln\left(\frac{2h_{i}}{gm{r}_{i}}\right)+4\omega(P% +jQ) (1)
𝐙𝐢𝐣-𝐠\displaystyle\mathbf{Z_{ij-g}} =j2ωln(Dijdij)+4ω(P+jQ)\displaystyle=j2\omega ln\left(\frac{D_{ij}}{d_{ij}}\right)+4\omega(P+jQ) (2)

where

Zii–g is the self-impedance of conductor i with ground return.

Zij–g is the mutual impedance between conductors i and j with common ground return.

gmri is the effective radius (or geometric mean radius) of conductor i in centimeters.

hi is the height of conductor i in centimeters.

ri is the internal resistance of conductor i.

dij the distance between conductors i and j in centimeters.

Dij the distance between conductor i and the image of conductor j in centimeters.

ω\omega is 2π\pif, where f is the frequency in cycles per second.

Obviously, the self-impedance Zii–g and mutual impedance Zij–g can be decomposed into their real and imaginary components

𝐙𝐢𝐢-𝐠\displaystyle\mathbf{Z_{ii-g}} =Rii-g+jXii-g\displaystyle=R_{ii-g}+jX_{ii-g} (3)
𝐙𝐢𝐣-𝐠\displaystyle\mathbf{Z_{ij-g}} =Rij-g+jXij-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

Rii-g\displaystyle R_{ii-g} =ri+4ωP\displaystyle=r_{i}+4\omega P (5)
Rij-g\displaystyle R_{ij-g} =4ωP\displaystyle=4\omega P (6)
Xii-g\displaystyle X_{ii-g} =2ωln(2higmrj)+4ωQ\displaystyle=2\omega ln\left(\frac{2h_{i}}{gm{r}_{j}}\right)+4\omega Q (7)
Xij-g\displaystyle X_{ij-g} =2ωln(Dijdij)+4ωQ\displaystyle=2\omega ln\left(\frac{D_{ij}}{d_{ij}}\right)+4\omega Q (8)

2.1.1 Approximation of P and Q in Carson’s Equations

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 θ\theta. The form of P and Q are the same for Equation 1 and Equation 2. However, the value of k and θ\theta differ. For self impedances

k\displaystyle k =4πhi2λf\displaystyle=4\pi{h}_{i}\sqrt{2\lambda f} (10)
θ\displaystyle\theta =0\displaystyle=0 (11)

For mutual impedances

k\displaystyle k =2πDij2λf\displaystyle=2\pi{D}_{ij}\sqrt{2\lambda f} (12)
θ\displaystyle\theta =cos-1(hi+hj)Dij\displaystyle=\frac{{cos}^{-1}({h}_{i}+{h}_{j})}{{D}_{ij}} (13)

where

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

θ\theta is the angle defined in Figure 1.

Figure 1: Transmission Line Geometry
Illustrates the geometry of an arbitrary overhead transmission transmission line configuration. Includes the dimensions and angles between conductors and their images required by Carson's equations.

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:

P=\displaystyle P= π8-kcosθ32+k2cos(2θ)(0.6728+ln(2k))16+k2θsin(2θ)16\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)
+k3cos(3θ)452-k4πcos(4θ)1536\displaystyle+{k}^{3}\frac{\cos\left(3\theta\right)}{45\sqrt{2}}-{k}^{4}\frac{% \pi\cos\left(4\theta\right)}{1536}
Q=\displaystyle Q= -0.0386+12ln(2k)+kcosθ32-πk2cos(2θ)64\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)
+k3cos(3θ)452-k4sin(4θ)384-k4cos(4θ)(1.0895+ln(2k))384\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}

2.1.2 Accuracy of Approximations to P and Q

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.

Table 1: Range of the Constant k
Distance Frequency Earth Resistivity k
100 ft 60 Hz 10 Ω/m3 0.4196
660 Hz 1.3916
1020 Hz 1.7300
60 Hz 100 Ω/m3 0.1327
660 Hz 0.4401
1020 Hz 0.5471
60 Hz 1000 Ω/m3 0.0419
660 Hz 0.1391
1020 Hz 0.1730
  • 100 ft - Large double circuit transmission tower
    10 Ω/m3 - Resistivity of swampy ground
    100 Ω/m3 - Resistivity of average damp earth
    1000 Ω/m3 - Resistivity of dry earth

  • 2.1.3 Use of First Order Approximations to P and Q

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

    P\displaystyle P =π8\displaystyle=\frac{\pi}{8}
    Q\displaystyle Q =-0.0386+12ln(2k)\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 (ρ\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 ρ\rho=10, f=60, and Dij=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.