Determination of Line Constants from Impedance Measurements

* The notation Z₀ (characteristic impedance) instead of Z_K (iterative impedance) is used because a uniform line is under consideration.

Hence,

and

From equation 51, Z₀ = sqrt(Z_OCZ_SC) and can readily be found from open-circuit and short-circuit impedance measurements. Also, from equation 82 tanh γl = sqrt(Z_sc/Z_oc). These equations can be used to find the line constants as follows:

For an open-wire line 197 miles long, Z_oc = 672 - j214 or 705 /-17.65° ohms, and Z_sc = 695-j75.5 or 699/-6.2° ohms. Thus, Z_o = sqrt((705/-17.65°)(699/-6.2°)) = 702/-11.9° ohms, and these measurements are at 1000 cycles. The value of tanh γl = V(699/-6.2°)/(705/-17.65°) = 0.996/+5.725° = 0.991 + 0.0992.

The next step involves finding αl and βl from the expression for tanh γl. From hyperbolic trigonometry, tanh γ = tanh (α + jβ) = A+jB,

and

Thus, tanh 2αl = (2 x 0.991)/(1 + 0.991² + 0.0992²) = 0.995. From tables of hyperbolic functions, 2αl = 2.99, and αl = 1.495 nepers. Hence for the 197-mile line, a = 1.495/197 = 0.0076 neper per mile or 0.0076 x 8.686 = 0.0659 decibel per mile. Similarly, tan 2βl = (2 x 0.0992)/[1 - (0.991² + 0.0992²)] = -24.32. From tables of circular functions, 2βl = 807.65°, and βl = 403.83°. Of course, this fact must be determined from a study of the frequency and the velocity of propagation. At 1000 cycles, for a line 200 miles long and a velocity of propagation of about 180,000 miles per second, βl would be somewhere in the fifth quadrant. The value in radians is (βl = 408.5°/57.3° = 7.04 radians, and β = 7.04/197 = 0.0357 radian per mile. Thus, γ = α + jβ = 0.0076 + j0.0357 = 0.0365 /77.98°. From equation 88,

Hence, R = 10.4 ohms resistance per mile, and L = 23.4/(6.28 x 1000) = 0.00373 henry per mile. From equation 89, γ/Z_o = y = G + jωC = 0.0365 /77.98°/702/-11.9° = 0.000052/89.88⁰ = 0.000000104 +

j0.000052. Hence, G = 0.000000104 mho or 0.104 micromho per mile, and C = 0.000052/(6.28 x 1000) = 0.00000000828 farad or 0.00828 microfarad per mile.

The calculations just made were for an artificial line designed to have characteristics similar to those of a non-pole pair side circuit of a 104-mil, hard-drawn copper, open-wire line with 12-inch spacing, the constants of which arc listed in Table IV. The open-circuited and short-circuited tests used in the calculations were made on this laboratory artificial line.