Nodal Analysis - Dependent Voltage Source

Use nodal analysis method to solve the circuit and find the power of the $3\Omega$ - resistor.

Solution

I. Identify all nodes in the circuit.
The circuit has 3 nodes as shown below.

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 $N$ .
The circuit has 4 nodes:

Therefore, $N=4$ .

Nodal Analysis - Dependent Voltage Source (5-Nodes)

Solve the circuit with the nodal analysis and determine $I_x$ .

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

Nodal Analysis - Supernode

Solve the circuit with nodal analysis and find $I_x$ and $V_y$ .

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

Nodal Analysis Problem with Dependent Voltage and Current Sources

Solve the circuit with the nodal analysis and determine $i_x$ and $V_y$ .

Solution
1) Identify all nodes in the circuit. Call the number of nodes $N$ .
The circuit has 5 nodes. Therefore, $N=5$ .

Reference Node and Node Voltages

Reference Node
In circuits, we usually label a node as the reference node also called ground and define the other node voltages with respect to this point. The reference node has a potential of $0 V$ by definition. The following symbol is used to indicate the reference node:

The Reference Node Symbol

As mentioned, the selection of the reference node is arbitrary. However, a wise selection can make the solving easier. As a general rule, it is usually chosen to be

Voltage Divider - Voltage Division Rule

The voltage division rule (voltage divider) is a simple rule which can be used in solving circuits to simplify the solution. Applying the voltage division rule can also solve simple circuits thoroughly. The statement of the rule is simple:

Voltage Division Rule: The voltage is divided between two series resistors in direct proportion to their resistance.

It is easy to prove this. In the following circuit

Voltage Divider

the Ohm's law implies that
$v_1(t)=R_1 i(t)$ (I)
$v_2(t)=R_2 i(t)$ (II)

Problem 1-16: Voltage Divider

Find $V_x$ (or $v_x(t)$ ) and $I_x$ (or $i_x$ ) using voltage division rule.
a)

b)

c)

d)

Solution

a)

Voltage divider: $V_x=\frac{5\Omega}{2\Omega+5\Omega}\times 14 V=10 V$
Ohm's law: $I_x=\frac{V_x}{5 \Omega}=2 A$

Ideal Independent Sources

1) Ideal Independent Voltage Sources
An ideal independent voltage source is a two-terminal circuit element where the voltage across it
a) is independent of the current through it
b) can be specified independently of any other variable in a circuit.
There are two symbols for ideal independent voltage source in circuit theory:

Symbol for Constant Independent Voltage Source

a) The current source keeps the current of the loop $2A$ and the voltage source keeps the voltage across the current source $3v$ as shown below.