Question

(a) Consider a medium-length transmission line represented by a nominal $\tau \tau$circuit shown in Figure 5.3 of the text. Draw a phasor diagram for lagging power-factor condition at the load (receiving end).

(b) Now consider a nominal T circuit of the medium-length transmission line shown in Figure 5.18. First, draw the corresponding phasor diagram for lagging power-factor load condition. Then determine the ABCDparameters in terms of Yand Zfor the nominal T circuit and for the nominal $\tau \tau$circuit of part (a).

Short answer

(a) The phasor diagram is

(b) The phasor diagram is

The ABCD parameter for T network is

$$\begin{gathered} A=\left[1+\frac{ZY}{2}\right] \\ B=Z\left[1+\frac{ZY}{4}\right] \\ C=Y \\ D=\left[1+\frac{ZY}{2}\right] \end{gathered}$$

For pi network, the ABCD parameters are,

$$\begin{gathered} A=\left[1+\frac{ZY}{2}\right] \\ B=Z \\ C=Y\left[1+\frac{ZY}{4}\right] \\ D=\left[1+\frac{ZY}{2}\right] \end{gathered}$$

Step-by-step solution

Given data.

Draw the diagram in figure 5.3.

Draw the figure 5.18.

Determine the formula for sending end voltage and current.

Write the formula of sending end voltage in terms of transmission line parameters.

${\mathbf{V}}_{\mathbf{S}}\mathbf{=}{\mathbf{AV}}_{\mathbf{R}}\mathbf{+}{\mathbf{Bl}}_{\mathbf{R}}$ ……. (1)

Here, $V_S$is the sending end voltage, $V_R$is the receiving end voltage, and $l_R$is the transmission line current.

Write the formula of sending end current in terms of transmission line parameters.

${\mathbf{l}}_{\mathbf{S}}\mathbf{=}{\mathbf{CV}}_{\mathbf{R}}\mathbf{+}{\mathbf{Dl}}_{\mathbf{R}}$ ……. (2)

Here, $l_S$is the sending end current, $V_R$is the receiving end voltage, and $l_R$is the receiving end current.

Determine the phasor diagram.

(a)

In the diagram (a), take$V_R$(receiving end voltage) as the reference.

For lagging load,$V_R$will lead the$l_R$(receiving end current).

Now,$l_{CR}$and$l_{CS}$are the currents through capacitor,$l_S$is sending end current and$V_S$is sending end voltage.

From the diagram,

$$\begin{gathered} V_S=V_R+(R+j{X}_L)\overline{l}............\left(1\right) \\ \overline{l}=l_R+l_{CR}...........\left(2\right) \\ {l}_S=\overline{l}+l_{CS}..............\left(3\right) \end{gathered}$$

Using equations (3), (4) and (5), draw the phasor diagram as below:

Determine phase diagram for the lagging power factor and the ABCD parameters in terms of Y and Z for the nominal T circuit and for the nominal   circuit.

(b)

In the diagram (b), take (receiving end voltage) as the reference.

For lagging load, will lead the (receiving end current).

Now, is the current through capacitor, is the voltage across the capacitor, is sending end current and is sending end voltage.