Nodal Analysis

Nodal Analysis

What is nodal analysis?

• In this method, we set up and solve a system of equations in which the unknowns are the voltages at the principal nodes of the circuit. From these nodal voltages the currents in the various branches of the circuit are easily determined.

Techniques for circuit analysis

• Nodal analysis -> Based on KCL 
• Mesh analysis -> Based on KVL 

Steps in the nodal analysis method are: 

• Count the number of principal nodes or junctions in the circuit.

• Number the nodes N₁, N₂, . . . , N_n and draw them on the circuit diagram. Call the voltages at these nodes V₁, V₂, . . . , V_n, respectively.

• Choose one of the nodes to be the reference node or ground and assign it a voltage of zero.

• For each node except the reference node write down Kirchoff's Current Law in the form "the algebraic sum of the currents flowing out of a node equals zero". (By algebraic sum we mean that a current flowing into a node is to be considered a negative current flowing out of the node.)

     SAMPLE FIGURE:

     EXAMPLE : the node to the right KCL yields the equation:

 

                       Ia + Ib + Ic = 0

• Express the current in each branch in terms of the nodal voltages at each end of the branch using Ohm's Law (I = V / R).

Nodal Analysis with Voltage Sources:

• CASE 1 : the voltage source is connected between two non-reference nodes, the two non-reference nodes form a generalized node or super node, we apply both KCL and KVL to determine the node voltages.

• CASE 2 : 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.

Mesh Analysis

Mesh Analysis.

What is a Mesh Analysis?

Mesh analysis provides another general procedure for analyzing circuits, using mesh currents as the circuit variables. Using mesh currents instead of element currents as circuit variables is convenient and reduces the number of equations that must be solved simultaneously. Recall that a loop is a closed path with no node passed more than once. A mesh is a loop that does not contain any other loop within it.

Nodal analysis applies KCL to find unknown voltages in a given circuit, while mesh analysis applies KVL to find unknown currents. Mesh analysis is not quite as general as nodal analysis because it is only applicable to a circuit that is planar. A planar circuit is one that can be drawn in a plane with no branches crossing one another; otherwise it is nonplanar. A circuit may have crossing branches and still be planar if it can be redrawn such that it has no crossing branches.

To understand mesh analysis, we should first explain more about what we mean by a mesh.

In Fig. 3.17, for example, paths abefa and bcdeb

are meshes, but path abcdefa is not a mesh. The current through a mesh is known as mesh current. In mesh analysis, we are interested in applying KVL to find themesh currents in a given circuit.

In this section, we will apply mesh analysis to planar circuits that do not contain current sources. In the next sections, we will consider circuits with current sources. In the mesh analysis of a circuit with n meshes, we take the following three steps.

Steps to Determine Mesh Currents:

1. Assign mesh currents i1,i2,...,into the n meshes.
2. Apply KVL to each of then meshes. Use Ohm’s law to
3.  express the voltages in terms of the mesh currents.
4. Solve the resulting n simultaneous equations to get the 
5. mesh currents.

 

Wye - Delta

Wye-Delta Transformations

The Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ.

Delta and Wye Circuits

Basic Wye-Delta Trasformation

The transformation is used to establish equivalence for networks with three terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for complex as well as real impedances.

Equations for the transformation from Δ-load to Y-load 3-phase circuit

The general idea is to compute the impedance  at a terminal node of the Y circuit with impedances ,  to adjacent node in the Δ circuit by

where  are all impedances in the Δ circuit. This yields the specific formulae

Equations for the transformation from Y-load to Δ-load 3-phase circuit

The general idea is to compute an impedance  in the Δ circuit by

where  is the sum of the products of all pairs of impedances in the Y circuit and  is the impedance of the node in the Y circuit which is opposite the edge with . The formula for the individual edges are thus

Simplification of Networks

Resistive networks between two terminals can theoretically be simplified to a single equivalent resistor (more generally, the same is true of impedance). Series and parallel transforms are basic tools for doing so, but for complex networks such as the bridge illustrated here, they do not suffice.

The Y-Δ transform can be used to eliminate one node at a time and produce a network that can be further simplified, as shown.

The reverse transformation, Δ-Y, which adds a node, is often handy to pave the way for further simplification as well.