Short Transmission Lines
Demonstrative Video
Short transmission line
TL having its length less than 80 km is considered as a short transmission line.
In short transmission line capacitance is neglected because of small leakage current
Other parameters (resistance and inductance) are lumped in the transmission line.
Equivalent Circuit of Short TL
\[\begin{aligned}
I_{S} & =I_{R}\\
V_{s} & =V_{R}+I_{R}\cdot Z\\
Z & = R+jX
\end{aligned}\]
Phasor Diagram
\[\begin{aligned}
OD^{2}&=O G^{2}+G D^{2}\\
&=(O F+F G)^{2}+(G C+C D)^{2}
\end{aligned}\]
\[\begin{aligned} V_{S}^{2}&=\left(V_{r} \cos \phi_{r}+I R\right)^{2}+\left(V_{r} \sin \phi_{r}+I X\right)^{2} \\ V_{S}&=\sqrt{\left(V_{r} \cos \phi_{r}+I R\right)^{2}+\left(V_{r} \sin \phi_{r}+I X\right)^{2}} \end{aligned}\]
Power factor of the load measured at the sending end is \[\cos \phi_{s}=\frac{O G}{O D}=\frac{O F+F G}{O
D}=\frac{V_{r} \cos \phi_{r}+I R}{V_{S}}\]
If \(V_r\) be the reference phasor then, \[V_{r}=V_{r}<0^{\circ}=V_{r}+j 0\]
For lagging PF \(I=I\angle-\Phi_{r}=\operatorname{Icos} \Phi_{r}-j I \sin \phi_{r}\)
For leading PF \(I=I\angle+\Phi_{r}=\operatorname{Icos} \Phi_{r}+j I \sin \phi_{r}\)
For unity PF, \(I=I\angle0^{\circ}=I+j 0^{\circ}\)
The line impedance is given by \(Z=R+j X\)
Sending end voltage is \[V_{s}=V_{r}+Z I\]
For lagging power factor, \[\begin{aligned} V_{S}=&\left(V_{r}+j 0\right)+(R+j X)\left(I \cos \phi_{r}-j \operatorname{I\sin} \phi_{r}\right) \\ &=\left(V_{r}+I \operatorname{R\cos} \phi_{r}+I X \sin \phi_{r}\right)+\left(I X \cos \phi_{r}-I R \sin \phi_{r}\right) \\ V_{S}&=\sqrt{\left[\left(V_{r}+I R \cos \phi_{r}+I X \sin \phi_{r}\right)^{2}+\left(I X \cos \phi_{r}-I R \sin \phi_{r}\right)^{2}\right]} \\ \tan \delta & = \frac{I X \cos \phi_{r}-I R \sin \phi_{r}}{V_{r}+I R \cos \phi_{r}+I X \sin \phi_{r}} \end{aligned}\]
ABCD Constants of Short TL
\[\begin{aligned} I_{S} & =I_{R}\\ V_{s} & =V_{R}+I_{R}\cdot Z\\ Z & = R+jX \end{aligned}\]
$$\boxed{ \left[\begin{array}{c} V_S \\ I_S \end{array}\right]=\left[\begin{array}{ll} 1 & Z \\ 0 & 1 \end{array}\right]\left[\begin{array}{c} V_R \\ I_R \end{array}\right] }$$
Voltage Regulation
Rise in \(V_R\) expressed in % of FL voltage, when FL at a specified pf is removed keeping \(V_s\) constant
At full load, \[\left|V_{R,F.L}\right|=\left|V_{r}\right|\]
At no load, \[\left|V_{R, N.L}\right|=\left|V_{S}\right|\]
$$\boxed{ \% \text { Regulation }=\frac{\left|V_{R, N L}\right|-\left|V_{R, F L}\right|}{\left|V_{R, F L}\right|} \times 100 }$$
greatest for lagging pf
least or -ve for leading pf
Line Efficiency
\[\text {Efficiency}=\frac{\text {Power delivered at the receiving end}}{\text {power delieverd at the sending end }+\text { losses}}\]