Nodal Analysis – Dependent Current Source

Deploy nodal analysis method to solve the circuit and find the power of the dependent source.


Solution
I. Identify all nodes in the circuit. Call the number of nodes .
The circuit has 4 nodes:

Therefore, .

II. Select a reference node. Label it with reference (ground)
symbol.

All nodes have the same number of elements. We prefer to select one of the nodes connected to the voltage source to avoid having to use a supernode.

III. Assign a variable for each node whose voltage is unknown.
We label the remaining three nodes as shown above.

IV. If there are dependent sources in the circuit, write down equations that express their values in terms of other node voltages.
There is one dependent source, which is a current controlled current source. We need to write in terms of node voltages. is the current passing through the – resistor. Applying the Ohm’s law,
.
Hence, .

V. Write down a KCL equation for each node.
Node : .
(Eq. 1).

Node : .
Please note that we avoid using all unknowns except node voltages. Using in this KCL equation introduces an unnecessary unknown to the equations set. Substituting and rearranging results in:
(Eq. 2).

Node has a voltage source connected to. Therefore, . Substituting this in Eq. 1 and Eq.2 leads to
.
By solving the system of equations,
and .

Now, we need to find the voltage across the dependent current source and the current passing through it. Lets start with .
.

Assuming positive terminal placed on the node of , the voltage across the dependent current source is . The current flowing through the dependent current source is . Therefore the power of the dependent current source is . Because the current direction and the voltage polarity is in accordance with the passive sign convention and the power is negative, the dependent current source is supplying power.