# Transmission Line Impedance Calculation

## Derivation of Transmission Line Sequence Impedance

Accurate calculation of the series impedance requires a complete description of a particular transmission line. This includes the height above the ground of each conductor bundle, the number of conductors per bundle, the size and resistance per unit length of each conductor, the position of each conductor within the bundle and the resistivity of the earth.

The work by Carson and others [1] can then be used to evaluate the self series-impedance per unit length of each phase bundle and earth wire. It is also a relatively simple task to calculate the mutual impedance per unit length between all pairs of phase bundles or earth wires, as shown below.

where the current and voltage on phases A, B and C are defined by IA, IB, IC and VA, VB, VC respectively. Similarly, ZA, ZB, ZC are the self impedances, and ZAB, ZAC, ZBA, ZBC, ZCA, and ZCB are the mutual impedances.

This impedance matrix excludes the currents and voltages on the earth wires, which have been implicitly included through routine manipulation of complex matrices. For a balanced (or symmetrically transposed) circuit, the diagonal terms of the impedance matrix (conductor self impedances) are identical. The same is true for the off-diagonal elements (mutual coupling parameters).

This matrix is often transformed into sequential components by:

where I0, I1, and I2 are the zero, positive and negative sequence currents, while V0, V1, and V2 are the zero, positive and negative sequence voltages. The sequence impedance associated with each of these voltages and currents is given by:

where ZS is the nominal self impedance of the circuit and ZM the nominal mutual coupling impedance, and they can be respectively expressed as:

## Positive Sequence Resistance

Since our online applications presently only consider the positive sequence impedance, it is possible to use approximations that are based on the geometric mean values. For instance, the positive sequence resistance at a specific operating temperature can be defined by:

It is then necessary to account for the skin effect using a polynomial approximation, where a0, a1, a2, a3 and a4 are calculated from the effective ratio of the conductor thickness to the conductor diameter, and f is the system frequency.

It is then necessary to account for the magnetic heating developed within steel reinforced aluminium conductors, where there are an odd number of aluminium layers. This can be achieved using methods such as [2] or the simplified current and conductor-temperature dependent approaches, as documented in [3].

## Positive Sequence Reactance

The positive sequence inductive reactance of a three-phase conductor system is given by:

where Deq is the geometric mean distance between the phases and DSL is the geometric mean radius between the conductors of one phase. It should be noted that this equation is occasionally defined by the “inductive reactance spacing factor” and the “inductive reactance at 1m spacing” by conductor manufacturers.

The geometric mean distance between phases is defined by the distances between the phases (dab, dbc, dca):

Similarly, the geometric mean radius for a single conductor can be derived by multiplying the conductor radius by 0.7788. Conversely, the geometric mean radius of a conductor bundle requires a solution to the following, where dkm is the distance between conductors k and m within the bundle:

## Positive Sequence Susceptance

The positive sequence susceptance of a three-phase transmission line can be calculated from the following equation.

## References

[1] H.W. Dommel, "Electromagnetic Transients Program Reference Manual", (EMTP Theory Book) Portland, OR, Bonneville Power Administration, Aug 1986.
[2] J.S. Barret, et al "A new Model of AC Resistance in ACSR Conductors" IEEE Transactions on Power Systems, Vol. PWRD-1, No. 2, April 1986
[3] TNSP Co-operative Charter Plant Rating Working Group. (2009). TNSP Operational Line Ratings
[4] Glover and Sarma, “Power System Analysis & Design” Boston, PWS Publishing, 1994.