What is Nodal Analysis : Circuit with Example

In an electric or electronic circuit, a node is used to connect different components to each other. According to KCL (Kirchoff’s Current Law), the current that enters into a node should leave from a node. In the circuit, each point on the node has the same voltage, so this voltage is known as the node voltage. It is the voltage disparity between the node at an arbitrary location & the ground point. The nodes within real circuits can be made up of wires, but the voltage is not the same all over the node. To solve any electrical circuit, there are two techniques are used mesh analysis and nodal analysis. In nodal analysis, we use the node voltages along with the ground and is also known as the node voltage technique. This article discusses an overview of nodal analysis with an example.

What is Nodal Analysis?

Nodal analysis is one kind of method used in electrical networks to analyze circuits through node voltages like the circuit variables. An alternate name of this method is Node-Voltage Method. The main features of nodal analysis include the following.

• Nodal Analysis depends on the KCL which is known as Kirchhoff’s Current Law.
• If the nodes in the circuit are ‘n’ then ‘n-1’ simultaneous equations will be there to solve.
• The voltage on all nodes in the circuit can be obtained by solving ‘n-1’ equations.
• The total nodal equations are equivalent to total nonreference nodes that can be obtained.

In nodal analysis, there are two kinds of nodes available like reference and nonreference.

• A node in the network which performs like a reference point to remaining nodes is known as the reference node or Datum Node.
• A node in the network which includes an exact node voltage is known as a non-reference node. For instance, node 1 & 2 are the non-reference nodes

Again, reference nodes are classified into two types like chassis ground and earth ground.

• Chassis Ground is a kind of reference node that works like an ordinary node for above one circuit.
• In any network, once earth potential is utilized like a reference then this kind of reference node is known as Earth Ground.

Step by Step Procedure

The step-by-step procedure of nodal analysis is used to solve nodal analysis in an electrical circuit or network.

• In Step-1, recognize the main nodes & select one of them as a reference node. So, this node is treated like the Ground.
• In Step-2, mark the node voltages with respect to the ground terminal from all the main nodes apart from the reference node.
• In Step-3, at all the main nodes excluding the reference node, write nodal equations. This equation can be attained by applying KCL first & after that Ohm’s law.
• In Step-4, to get the node voltages, we need to solve the nodal equations which are achieved in Step-3.

So by using node voltages, we can discover the current flowing throughout any part & the voltage across any part that is there in the given circuit through node voltages.

Electrical Sources

Electrical sources are classified into two types namely independent & dependent.

An independent electrical source gives a set value of current or voltage that is connected to the circuit. These sources are nothing but batteries & power supplies. Here, the power supply gives a stable set value while batteries provide a stable fixed value eventually without recharging them.

A dependent source is nothing but a current or voltage source whose value mainly depends on a current or voltage value anywhere within the circuit. These sources are used to analyze amplifiers and the main characteristics of amplifiers are current gain (Ai) & voltage gain (AV).

Linear Dependent Sources

Linear-dependent sources are classified into four types like following.

• Voltage-controlled voltage source
• Current-controlled voltage source
• Current-controlled current source
• Voltage-controlled current source

In a voltage-controlled voltage source, the o/p is ‘V’, AV is the voltage gain & VCD is the parameter being detected. The equation given below is allied with this linear dependent source.

V = Av VCD

In a current-controlled voltage source, the o/p is V; the resistance is RM & IC is the parameter being detected. The below equation can be allied through a current-controlled voltage source.

V = RvM IC

In the current-controlled current source, the o/p is ‘I’ & ‘Ai’ is the current gain & IC is the parameter being detected. The following mentioned equation can be allied through a current-controlled current source.

I = AI Ic

In voltage-controlled current source, the o/p is ‘I’, GM is the conductance & VCD is the parameter being detected. The below equation can be connected through a voltage-controlled current source.

I = GM * VCD

Difference between Mesh and Nodal Analysis

The difference between mesh and nodal analysis includes the following.

In a nodal analysis of an electrical network, the voltages can be observed at a certain branch whereas, in mesh analysis, current values are used in a certain branch of a circuit.

In the electrical network, mesh analysis is one kind of technique, used to solve planar circuits particularly for the currents at any branch. These circuits can be drawn over a flat surface without crossing wires with each other. A more general method like loop analysis is used in any circuit.

Nodal Analysis depends on the KCL (Kirchhoff’s Current Law) application. If the circuit has n-nodes then there will be ‘n-1’ instantaneous equations to resolve. By solving these equations, we can get all the node voltages. The amount of nonreference nodes is equivalent to the number of Nodal equations that can be acquired.

Nodal Analysis Circuit

In the following circuit, the nodal analysis method is generally used to determine the voltage through KCL at a node. In this circuit, extraordinary nodes are V1, V2, V3 & V3 is in reference to GND.

In the following circuit, voltages like V1, V2 & V3 are extraordinary nodes whereas V3 is ground.

Apply Kirchoff current law at nodes 1 & 2 to the above circuit

At node 1

i1+i2+i3 = 0 ……(1)

i1= (V1-Vbs)/R1+R2

i2= V1/R3

i3 = V1-V2/R4

Thus, substitute these current values in the above equation (1)

((V1-Vbs)/R1+R2) + (V1/R3) + (V1-V2/R4) = 0

(V1/ R1+R2)-(Vbs/R1+R2)+ (V1/R3)+(V1/R4)- (V2/R4)

V1((1/ R1+R2)+1/R3+1 /R4) – (V2/ R4) = Vb/R1+R2

At node 2

i4+i5+i6 = 0 ……(2)

i4=V2-V1/R4

i5=V2/R5

i6=I0

Substitute these current values in the above equation (2)

((V2-V1)/R4)+ (V2/R5)+ I0

(-1/R4)V1+ (1/R4 + 1/R4)V2= I0

Simultaneous equation solution

a11V1+a12V2 = b1

a21V1+a22V2 = b2

Example Problem

Find the node voltage ‘V1’ by applying nodal analysis in the following circuit.

i1+i2+i3 = 0

(V1-10/100)+(V1/100)+(V1-5/200+100)

                                                      (V1/100 -10/100) + V1/100+ (V1-5/200+100)

(V1/100+V1/100+V1/300) –(10/100)-(5/300)

V1(1/100+1/100+1/300) –(10/100)-(5/300)

(3+3+1/300)V1 =  (30/300+5/300)

(7/300)V1 = 30+5/300

V1 = (35/300)x(300/7)

V1 = (35/7)

V1 = 5V

Nodal Analysis with Current Source

In the following example circuit, the nodal analysis using current sources is discussed.

Ex: For the following circuit, calculate the nodal voltage

There are three nodes in the following circuit where one is a reference node and the remaining two are nonreference nodes like node 1 & node2.

Step1:

In step 1, node voltages are denoted with v1 & v2, and also branch currents directions can be marked with respect to the reference nodes.

Step 2:

In this step, apply Kirchoff current law to two nodes like node1 and node2
When Kirchoff current law is applied to node1 in the above circuit

i1 =i2+i3……(1)

Similarly, at node 2

i2+i4 = i1+i5….(2)

Step 3:

When ohms law is applied to KCL equations

At node1, apply ohms law to KCL equations

i1 =i2+i3

5 = (V1-V2/4) + (V1-0/2)

Once the above equation is simplified then

20 = 3V1-V2……(3)

At node2, apply ohms law to KCL equation2

i2+i4 = i1+i5

(V1-V2/4)+10 = 5 + V2-0/6

Simplifying the above equation we get

60 = -3V1+5V2 ….(4)

Solve the equations like 3 & 4 to obtain the v1 & v2 values

Using elimination method

20 = 3V1-V2

60 = -3V1+5V2

4V2 = 80

V2= 20

When V2 = 20 is substituted in equation (3) then we can get

20 = 3V1-V2

20 = 3V1-20 => V1 = 40/3 = 13.3 V

Therefore, node voltages like v1 = 13.33 Volts & v2 = 20 Volts.

Similarly, nodal analysis with a voltage source can also be calculated

Thus, this is all about an overview of Nodal analysis, nodal analysis examples, etc. It is one kind of circuit analysis that works with Kirchhoff’s Current Law & node equations to solve voltage values within the circuit wherever the circuit diagram does not include any conductor lanes crossing. This nodal analysis can be applied for concluding the voltage at every node with respect to a reference node that is normally known as ground wherever the voltage at the ground terminal is equivalent to 0 Volts. Here is a question for you, what is mesh analysis?