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.

Ohm's Law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

In circuit analysis, three equivalent expressions of Ohm's law are used interchangeably:

Each equation is quoted by some sources as the defining relationship of Ohm's law, or all three are quoted, or derived from a proportional form, or even just the two that do not correspond to Ohm's original statement may sometimes be given.

The interchangeability of the equation may be represented by a triangle, where V (voltage) is placed on the top section, the I (current) is placed to the left section, and the R (resistance) is placed to the right. The line that divides the left and right sections indicate multiplication, and the divider between the top and bottom sections indicates division (hence the division bar).

Other versions

with  the element of path along the integration of electric field vector E. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation:

=================================================================

Nodal Analysis.

In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage between "nodes" 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 or mesh analysis using Kirchhoff's voltage law. 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 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 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.

More Info About Nodal Analysis.

=================================================================

Parallel Circuits:

        A parallel circuit is a circuit in which the resistors are arranged with their heads connected together, and their tails connected together. The current in a parallel circuit breaks up, with some flowing along each parallel branch and re-combining when the branches meet again. The voltage across each resistor in parallel is the same. 
The total resistance of a set of resistors in parallel is found by adding up the reciprocals of the resistance values, and then taking the reciprocal of the total: 
equivalent resistance of resistors in parallel: 
1 / R = 1 / R1 + 1 / R2 + 1 / R3 +... 


Example:

1/Req=R1//R2//R3

=1/10+1/2+1/1

=625ohms

   Resistors are said to be connected in Series, when they are daisy chained together in a single line. Since all the current flowing through the first resistor has no other way to go it must also pass through the second resistor and the third and so on. Then, resistors in series have a Common Current flowing through them as the current that flows through one resistor must also flow through the others as it can only take one path.


   Resistors are said to be connected in Series, when they are daisy chained together in a single line. Since all the current flowing through the first resistor has no other way to go it must also pass through the second resistor and the third and so on. Then, resistors in series have a Common Current flowing through them as the current that flows through one resistor must also flow through the others as it can only take one path.

Kirchhoff Laws

There are two Kirchhoff laws the KVL and KCL, which kirchhoffs volatage and current law.

-kirchhoff current law states that the algebraic sum of the currents entering a node is zero

as Sir jay taught as, The current entering a node is equal to the current leaving 

current leaving=current entering 

here how to solve kcl 

i1+(-i2)+i3+i4+(-i5)=0

i1+i3+i4=i2+i5

Kirchhoff Voltage Law

KVL is based on the principle of the conservation of energy

Kirchhoff's voltage law states that the algebraic sum of all voltages around a closed path is zero.

here's how to solve KVL

We can start with the voltage source and go clockwise around the loop as shown; then voltages would be -v+v2+v3,-v4, and +v5, in that order. If we reach branch 3, the positive terminal is met first; hence, we have +v3. For Branch , we reach the negative terminal; hence -v4, thus, KVL yields.

-v1+v2+v3-4+v5=0

v2+v3+v5=v1+v4

it is interpreted as :

Sum of Voltage drops = Sum of voltage rises.

Reflection:

As for now im still trying hard to identify how many nodes are in a circuits in times of complicated circuits. And the one cant also understand is by deriving a dependent source which the equation can be more tricky and confusing. As for now we will have our quiz and i hope i can answer that sir jay will give to us. because last meeting he gave as a seat work and its kinda confusing because the resistors and sources have no description given, and so when sir jay answers the seat work. it's not really hard to answer, its just that it was a trick when you loop a kvl or a kcl.