Nodal Analysis with Voltage Source

Nodal Analysis with Voltage Source 

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.

Two cases:

• Case 1: 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.

• CASE 2: If the voltage source (dependent or independent) is connected between two non-reference nodes, the two non-reference nodes form a Super Node. Apply KCL and KVL to determine the node voltages.

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

Properties of Super Node:

1. The voltage source inside the super node provides a constraint equation needed to solve for the node voltages.
2. A super node has no voltage of its own.
3. A super node requires the application of both KCL and KVL.
   
   
   
   
   In figure 2,  Apply the 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.

Nodal Analysis

What is Nodal Analysis?

Nodal analysis provides a general procedure for analyzing circuits using node voltages as the circuit variables.

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.

By 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). 

For instance, for a resistor, I_branch = V_branch * G, where G (=1/R) is the admittance (conductance) of the resistor.

 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.

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.

In nodal analysis, we are about to find the node voltages. Given a circuit with n nodes without voltage sources, the nodal analysis of the circuit involves taking the following steps:

1. Select a node as the reference node. Assign voltages to the remaining 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

Some Example of Wye Delta

EXAMPLE PROBLEM OF WYE TO DELTA TRANSFORMATION 

Find the resistance shown by the meter !

Let's convert the R₁, R₂, R₃ wye network to a delta network. This conversion is the best choice for simplifying this network.

First, we do the Wye to delta conversion, then we notice the instances of paralleled resistors in the simplified circuit.

{Wye to delta conversion for R1, R2, R3 }

Gy: = 1/R1+1/R2+1/R3;

Gy = [95m]

RA =R1*R2*Gy;

RB =R1*R3*Gy;

RC=R2*R3*Gy;

Req:=Re-plus (Re-plus(R6,RB), (Re-plus(R4,RA)+Re-plus(R5,RC)));

RA=[76]

RB=[95]

RC=[190]

Req=[35]

EXAMPLE PROBLEM OF DELTA TO WYE TRANSFORMATION

Here we go through an example that utilizes a delta to wye transformation, along with series and parallel resistance equations.

Wye-Delta Transformation

What is Wye-Delta Transformation?
The Y-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. The Y, spelled out as wye, can also be called T or star; the Δ, spelled out as delta, can also be called triangle, Π (spelled out as pi), or mesh. Thus, common names for the transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π. 


Wye to Delta Conversion:

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 impedance. 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 impedance.

Equations for Wye-Delta:

Delta to Wye Conversion:

Equation for Delta to Wye Conversion:

The equations can be presented in an alternate form based on the total resistance (Rd) of R₁, R₂, and R₃ (as though they were placed in series):

Rd = R₁+R₂+R₃ 

and:

R_A = (R₁*R₃) / Rd

R_B = (R₂*R₃) / Rd

R_C = (R₁*R₂) / Rd

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. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899.

Arthur Edwin Kenelly