Chapter 4, Problem 1P

(a)

To determine

Calculate the number of branches in the circuit.

Answer to Problem 1P

The number of branches in the circuit is 12 branches.

Explanation of Solution

Given data:

Refer to Figure P4.1 in the textbook.

The given circuit is modified as shown in Figure 1.

A branch is defined as a single electrical device or elements.

In Figure 1, there are 12 branches in the circuit. They are, $25\text{mA}$, $2\text{V}$, $R_1$, $R_2,$$R_3$, $R_4$, $R_5$, $R_6$, $R_7$, $R_8$, $18v_{\Delta }$, and $5i_{\Delta }$.

That is, 2 branch with a dependent source, 2 branches with independent sources, and 8 branches with resistors.

Conclusion:

Thus, the number of branches in the circuit is 12 branches.

(b)

To determine

Find the number of branches where the current is unknown.

Answer to Problem 1P

The number of branches in which the current unknown is 11 branches.

Explanation of Solution

Given data:

Refer to Figure 1 in Part (a).

In Figure 1, the only one known current in the circuit which is the $25\text{mA}$ current source. And the current is unknown in all other branches. Therefore, the current unknown in 11 branches.

Conclusion:

Thus, the number of branches in which the current unknown is 11 branches.

(c)

To determine

Find the number of essential branches in the circuit.

Answer to Problem 1P

The number of essential branches in the circuit is 10 essential branches.

Explanation of Solution

Given data:

Refer to Figure 1 in Part (a).

Essential branch: It is defined as a path that attaches essential nodes without passing through an essential node.

The essential branches in the circuit are, $25\text{mA}$, $R_1$, $R_2,$$R_3$, $R_6$, $R_7$, $18v_{\Delta }$, $5i_{\Delta }$, $\left(R_8-2\text{V}\right)$, and $\left(R_1-R_2\right)$.

In the circuit, $\left(R_8-2\text{V}\right)$ and $\left(R_1-R_2\right)$ are essential branches with two elements and the rest of the eight branches are essential branches that contains single element.

Therefore, the circuit has 10 essential branches.

Conclusion:

Thus, the number of essential branches in the circuit is 10 essential branches.

(d)

To determine

Find the number of essential branches where the current is unknown in the circuit.

Answer to Problem 1P

The number of essential branches where the current unknown is 9 essential branches.

Explanation of Solution

Given data:

Refer to Figure 1 in Part (a).

Refer to Part (c), the circuit has 10 essential branches.

In Figure 1, the current is known only in the essential branch that containing the $25\text{mA}$ current source. And the current is unknown in all the other 9 essential branches.

Therefore, in 9 essential branches the current is unknown.

Conclusion:

Thus, the number of essential branches where the current unknown is 9 essential branches.

(e)

To determine

Find the number of nodes in the circuit.

Answer to Problem 1P

The number of nodes in the circuit is 7 nodes.

Explanation of Solution

Given data:

Refer to Figure 1 in Part (a).

Node: It is defined as a connection point between two or more branches.

In Figure 1, the nodes present in the circuit are, $a$, $b$, $c$, $d$, $e$, $f$, and g. That is, in Figure 1 three are identified by dotted rectangular boxes, two are identified by dotted triangles, and two are identified by dotted diamonds. Therefore, 7 nodes are present in the circuit.

Conclusion:

Thus, the number of nodes in the circuit is 7 nodes.

(f)

To determine

Find the number of essential nodes in the circuit.

Answer to Problem 1P

The number of essential nodes in the circuit is 5 essential nodes.

Explanation of Solution

Given data:

Refer to Figure 1 in Part (a).

Essential node: It is a node that joins three or more electrical element or devices

In Figure 1, the essential nodes present in the circuit are $a$, $c$, $d$, $e$, and g. That is, in Figure 1 three are identified with dotted rectangular boxes and two are identified with the dotted triangles. Hence, there are 5 essential nodes in the circuit.

Conclusion:

Thus, the number of essential nodes present in the circuit is 5 essential nodes.

(g)

To determine

Find the number of meshes present in the circuit.

Answer to Problem 1P

The number of meshes present in the circuit is 6 meshes.

Explanation of Solution

Given data:

Refer to Figure 1 in Part (a).

Mesh: It is defined as a closed loop path that has no any other smaller loops present inside.

In Figure 1, the closed loops present in the circuit are $\left(25\text{mA}-R_6\right)$, $\left(R_6-R_3-18v_{\Delta }\right)$, $\left(18v_{\Delta }-R_4-R_7\right)$, $\left(2\text{V}-R_5-R_7-R_8\right)$, $\left(R_1-R_2-R_3-R_4-R_5\right)$, and $\left(5i_{\Delta }-R_1-R_2\right)$. Therefore, there are 6 meshes present in the circuit.

Conclusion:

Thus, the number of meshes present in the circuit is 6 meshes.