Nodal analysis Totally Explained

Everything about Nodal Analysis totally explained

In electrical engineering, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit. It is used to solve for the voltages and currents at any point in a circuit without working through many individual KCL or KVL rules.
This method is very powerful as many different circuit elements can be modeled. Active circuit elements such as operational amplifiers can be added to the analysis. These elements can be as simple or complicated as desired to achieve the fidelity needed in the simulation. For example: A number of different transistor models are available that can be used in the nodal analysis. The only requirement is that the elements are linear.

Method

Label all the nodes in the circuit (for example 1, 2, 3...), and select one to be the "reference node." It is usually most convenient to select the node with the most connections as the reference node.
Assign a variable to represent the voltage of each labeled node (for example V₁, V₂, V₃...). The values of these variables, when calculated, will be relative to the reference node (for example the reference node will be 0V).
If there's a voltage source between any node and the reference node, by Kirchhoff's voltage law, the voltage at that node is the same as the voltage source's. For example, if there's a 40 V source between node 1 and the reference node, node 0, V₁ = 40 V.
Note any voltage sources between two nodes. These two nodes form a supernode. By Kirchhoff's voltage law, the voltage difference at these two nodes is the same as the voltage source. For example, if there's a 60 V source between node 1 and node 2, V₁ − V₂ = 60 V.
For all remaining nodes, write a Kirchhoff's current law equation for the currents leaving each node, using the terminal equations for circuit elements to relate circuit elements to currents. For example, if there's a 10 Ω resistor between nodes 2 and 3, a 1 A current source between nodes 2 and 4 (leaving node 2), and a 20 Ω resistor between nodes 2 and 5, the KCL equation would be (V₂ − V₃)/10 + 1 + (V₂ − V₅)/20 = 0 A.
For all sets of nodes that form a supernode, write a KCL equation, as in the last step for all currents leaving the supernode, for example sum the currents leaving the nodes of the supernode. For example, if there's a 60 V source between nodes 1 and 2, nodes 1 and 2 form a supernode. If there's a 40 Ω resistor between nodes 1 and the reference node, a 2 A current source between nodes 1 and 3 (leaving node 3), and a 30 Ω resistor between nodes 2 and 4, the KCL equation would be (V₁ − 0)/40 + (−2) + (V₂ − V₄)/30 = 0 A.
The KCL and KVL equations form a system of simultaneous equations that can be solved for the voltage at each node.

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