BLOG 5 - ELECTRICAL CIRCUIT 1

Nodal analysis

Kirchhoff's current law is the basis of nodal analysis.

In electric circuits analysis, 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 in terms of the branch currents.

In analyzing a circuit using Kirchhoff's circuit laws, one can either do nodal analysis using Kirchhoff's current law (KCL) or mesh analysis using Kirchhoff's voltage law (KVL). Nodal analysis writes an equation at each electrical node, requiring that the branch currents incident at a node must sum to zero. The branch currents are written in terms of the circuit node voltages. As a consequence, each branch constitutive relation must give current as a function of voltage; an admittance representation. For instance, for a resistor, I_branch = V_branch * G, where G (=1/R) is the admittance (conductance) of the resistor.

Nodal analysis is possible when all the circuit elements' branch constitutive relations have an admittance representation. Nodal analysis produces a compact set of equations for the network, which can be solved by hand if small, or can be quickly solved using linear algebra by computer. Because of the compact system of equations, many circuit simulation programs (e.g. SPICE) use nodal analysis as a basis. When elements do not have admittance representations, a more general extension of nodal analysis, modified nodal analysis, can be used.

While simple examples of nodal analysis focus on linear elements, more complex nonlinear networks can also be solved with nodal analysis by using Newton's method to turn the nonlinear problem into a sequence of linear problems.

Steps to determine node voltages:

1. Select a node as the reference node. Assign voltage v1, v2,...., vn-1 to remaining n - 1 nodes. The voltages are referenced with respect to the reference node.

2. Apply KCL to each of the n - 1 non reference nodes. Use Ohm's law to express the branch currents in terms of node voltages.

3. Solve the resulting simultaneous equations to obtain the unknown node voltages.   

• Current flows from a HIGHER POTENTIAL to a LOWER POTENTIAL in a resistor.

i = vhigher - vlower / R

Nodal analysis with voltage sources

Nodal analysis is the method to determine voltage or current using nodes of the circuit. In nodal analysis we choose node voltage instead of element voltages and hence the equations reduces in this process. We have to consider voltage source is not in this circuit. We have to solve a circuit with n nodes without voltage sources. To solve a circuit using nodal analysis method you must have good knowledge about node branch loop in a circuit. If you have no clear idea read the article then come back here. There are three steps to solve a circuit using nodal analysis

1. Select a node as a reference node. Give names v₁, v_2,…. v_n-1 to remaining n-1 nodes. All the voltages are the referenced voltages respecting to the reference node.

1. Apply KCL and KVL to each non reference node. To express the branch currents in terms of node voltages us ohm’s law.

1. Solve the equations to get unknown node voltages.

First step is to select a reference node. It is also called datum node. The reference node commonly called the ground. It has zero potential.

In circuit the reference node is denoted by any of the three symbols in figure 1. Figure 1 (c) is called a chassis ground because it is used in the case chassis act, enclosure as a reference point in the circuit. Figure 1 (a) and (b) are used when the potential of the earth taken as reference. I use symbol (b).

In two cases nodal analysis can be done with voltage sources.

Case 1: If the voltage source (dependent or independent) is connected between two non-reference nodes, the two non-reference nodes form a generalized node or supernode, we apply both KCL and KVL to determine the node voltages.

Case 2: if a voltage source is connected between the reference node and a non-reference node, we simply set the voltage at the non-reference node equal to the voltage of the voltage source in figure 2 for example,

                                   v₁ = 20V

What is supernode?

A supernode is formed by enclosing a (dependent or independent) voltage source connected between two non-reference nodes and any elements connected in parallel with it.

In figure 2 node 2 and node 3 form a supernode. Applying KCL at super node which are node 2 and 3 we get,

                                        i₁ + i₄  = i₂ + i₃

To apply KVL redrawing the figure 2 circuit to figure 3 and going around the loop in the clockwise direction gives,

                       – v₂ + 10 + v₃ = 0

                        Or  v₂ – v₃ = 10      ————————— (ii)

From equation (i),(ii) we will obtain node voltages using any solution method.