Characteristic Impedance and Surge Impedance Loading

Demonstrative Video

Characteristic or Surge Impedance, \(Z_c\)

  • If load impedance \(Z_{L}=\dfrac{V_{R}}{I_{R}}=Z_{c}\) is terminated with \(Z_c\), the reflected voltage wave is zero \(\left(V_{R} - I_{R}Z_{c} = 0 \right)\)

  • A line terminated in its \(Z_c\) is called infinite line

  • The incident wave under this condition cannot distinguish between termination and infinite continuation of the line

  • Overhead line: value 400 \(\Omega\), phase angle 0 to -\(15^{0}\)

  • Underground cables: \(1/10^{th}\) of the OH line

  • \[Z_{c}=Z_{s}=\sqrt{\dfrac{j\omega L}{j\omega C}}=\sqrt{\dfrac{L}{C}},~\mbox{a pure resistance}\]
    The term Surge impedance is used for surges (lightning or switching) or transmission line without losses

Derivation and explanation

  • Long TLs have the distributed lumped \(L\) and \(C\).

  • When TLs are energized, capacitance feeds the reactive power to the line, and the inductance absorbs the reactive power.

  • The amount of reactive power in MVAR range depends on the capacitive reactance and the energized line voltage.

  • \[\mathrm{MVAR_C} = \dfrac{kV^2}{X_C}\]
    Mathematically, the expression of MVAR produced is written as,

  • TL also uses reactive power to support their magnetic field.

  • The strength of the magnetic field depends on the magnitude of the current and its natural reactance.

  • \[\mathrm{MVAR_L} = \mathrm{I^2X_L}\]
    uses or absorbs by TL is, The expression of

  • In SIL, reactive power production is equal to reactive power uses by the transmission line.

  • This reactive power balance relation is written as,

\[\begin{aligned} I^{2} X_{L}&=\frac{V^{2}}{X_{C}} \\ X_{L} X_{C}&=\frac{V^{2}}{I^{2}} \\ \frac{\omega L}{\omega C}&=\frac{V^{2}}{I^{2}} \\ \sqrt{\frac{\omega L}{\omega C}}&=\frac{V}{I} \\ \frac{V}{I}&=\sqrt{\frac{L}{C}} \\ Z_{c}&=\sqrt{\frac{L}{C}} \end{aligned}\]

Surge Impedance Loading (SIL)

  • SIL is an important parameter in PS when an issue arises related to prediction of maximum loading capability of TL

  • The maximum \(3\phi\) active power transfer capability of a TL is called the SIL.

  • When dealing with high frequencies or with surges due to lightning, losses are often neglected and then the surge impedance becomes important

  • defined as power delivered by a line to a purely resistive load equal in value to the surge impedance of the line

When so loaded, the line supplies a current of

\[\left|I_{L}\right|=\dfrac{\left|V_{L}\right|}{\sqrt{3}Z_{s}}\]
\[\mbox{SIL} =\sqrt{3}\left|V_{L}\right|\dfrac{\left|V_{L}\right|}{\sqrt{3}Z_{s}}\]
$$\boxed{ \mathrm{SIL}=\frac{\left|V_L\right|^2}{Z_S} }$$
\[\begin{aligned} \mbox{SIL} & =V_{L}^{2} \sqrt{\frac{C}{L}} \\ \mbox{SIL} & \propto V_{L}^{2} \\ \mbox{SIL} & \propto \sqrt{C} \\ \mbox{SIL} & \propto \frac{1}{\sqrt{L}} \end{aligned}\]