Question

Consider Figure 2.9 of the text. Let the nodal equations in matrix form be given by Eq. (2.4.1) of the text.

A. The element ${\mathbf{Y}}_{\mathbf{11}}$ is given by

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\mathbf{0} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{j}\mathbf{13} \\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{}\mathbf{-}\mathbf{j}\mathbf{7} \end{gathered}$$

B. The element${\mathbf{Y}}_{\mathbf{31}}$ is given by

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\mathbf{0} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{-}\mathbf{j}\mathbf{7}\mathbf{} \\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{}\mathbf{j}\mathbf{1} \end{gathered}$$

A.The admittance matrix is always symmetric square.

(a) False

(b) True

Short answer

Answer

A. Therefore, the correct option is (c) $-j7$

B. Therefore, the correct option is (a)

C. Therefore, the correct option is (b) True

Step-by-step solution

 Step 1: Define the bus admittances.

Consider that the $Y_{kk}$ is a bus admittance for diagonal elements, where localid="1652524671936" $K=1,2,...N.N$is a total number of buses.

COnisder localid="1652524194450" $Y_{kn}$is a bus admittance for off diagonal elements.

This admittance values are calculated by,

${\mathbf{Y}}_{\mathbf{kk}}\mathbf{=}\mathbf{sum}\mathbf{}\mathbf{of}\mathbf{}\mathbf{admittances}\mathbf{}\mathbf{connected}\mathbf{}\mathbf{to}\mathbf{}\mathbf{bus}\mathbf{}\mathbf{k}$ ……. (1)

${\mathbf{Y}}_{\mathbf{kn}}\mathbf{=}\left(\mathrm{sum}\mathrm{of}\mathrm{admittances}\mathrm{connected}\mathrm{between}\mathrm{buses}k\mathrm{and}n\right)$ ……. (2)

Determine the admittance for subpart A.

Consider the given figure is shown below.

From equation (1) the bus adimmtance $Y_{11}$is calculated using Figure 2.9.

$$\begin{gathered} Y_{11}=y_{11}+y_{12} \\ =-j10+j3 \\ =-j7S \end{gathered}$$

Thus the option (c) is correct.

Determine the admittance for subpart B.

From equation (2) the bus adimmtance$Y_{31}$is calculated using Figure 2.9.

$Y_{31}=0$

There is no connection between the bus 1 and bus 3 seen from the diagram.

Thus the option (a) is correct.

Determine the order of admittance matrix.

From equation (2.4.1), it is clear that the order of the admittance matrix is $N\times N$, henceadmittance matrix is always symmetric square.

Thus the option (b) is correct.