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
The term Surge impedance is used for surges (lightning or switching) or transmission line without losses \[Z_{c}=Z_{s}=\sqrt{\dfrac{j\omega L}{j\omega C}}=\sqrt{\dfrac{L}{C}},~\mbox{a pure resistance}\]
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.
Mathematically, the expression of MVAR produced is written as, \[\mathrm{MVAR_C} = \dfrac{kV^2}{X_C}\]
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.
The expression of \(Q\) uses or absorbs by TL is, \[\mathrm{MVAR_L} = \mathrm{I^2X_L}\]
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
\[\left|I_{L}\right|=\dfrac{\left|V_{L}\right|}{\sqrt{3}Z_{s}}\]
\[\mbox{SIL} =\sqrt{3}\left|V_{L}\right|\dfrac{\left|V_{L}\right|}{\sqrt{3}Z_{s}}\]
$$\boxed{ \mathrm{SIL}=\frac{\left|V_L\right|^2}{Z_S} }$$
\[\begin{aligned} \mbox{SIL} & =V_{L}^{2} \sqrt{\frac{C}{L}} \\ \mbox{SIL} & \propto V_{L}^{2} \\ \mbox{SIL} & \propto \sqrt{C} \\ \mbox{SIL} & \propto \frac{1}{\sqrt{L}} \end{aligned}\]