Nodal Analysis - Dependent Voltage Source (5-Nodes)


Solve the circuit with the nodal analysis and determine  I_x.
Nodal analysis - circuit with dependent voltage source

Solution
1) Identify all nodes in the circuit. Call the number of nodes  N.
There are five nodes in the circuit:

Total number of nodes- Nodal Analysis
Therefore  N=5
2) Select a reference node
The best option is the node in the bottom because it is connected to both voltage sources.
The reference node - Nodal analysis

3) Assign a variable for each node whose voltage is unknown.
There are four nodes beside the reference node:
Other nodes

Node III and Node IV are connected to the reference node through voltage sources. Therefore, their node voltages can be determined by the voltage sources.
 V_3=2I_x and  V_4=V_s=3V.
Node voltages
 I_x is the current of  R_1. The Ohm's law can be used to write  I_x in terms of the node voltages. Thus,  I_x=\frac{V_3-V_2}{R_1}=V_3-V_2. Substituting  V_3=2I_x,  I_x=2I_x-V_2 \to I_x=V_2 (Eq. 1).
Therefore,  V_3=2V_2.
4) Write down KCL equations.
We only need to write a KCL equation for Node I and Node II:
Node I:  I_s + \frac{V_1-V_3}{R_2} +\frac {V_1}{R_4}=0 . Substituting  V_3=2V_2 and known variables,
  3V_1-2V_2=-2V (Eq. 2).

Node II:  -I_s+\frac{V_2-V_3}{R_1}+\frac{V2-V_4}{R_5}=0.  \to -1+V_2-2V_2+\frac{V_2}{2}-\frac{3}{2}=0 \to V_2=-5V.
Substituting in Eq. 2:
 V_1=-4V.

Now, we need to determine the required quantities. Eq. 1 implies that
 I_x=V_2=-4A.

One thought on “Nodal Analysis - Dependent Voltage Source (5-Nodes)

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