DELTA TO WYE TRANSFORMATION

Why Do We Need to Use Wye to Delta/Delta to Wye Transformations?

-It is because situations often arise in circuit analysis when the resistors are neither parallel or series. See the picture below for example.

3 Terminal Arrangements Commonly Used in Power System

L earnings:

Through this process, I was able combine resistors. resistors can be given immediately as series or parallel. By the other examples we have encounter more complex problems that requires you to transform the resistors connected neither in parallel or series into delta-to-wye or wye-to-delta depending on how they connected given in a diagram.

MESH ANALYSIS

| Mesh Current Analysis Circuit |

• Nodal analysis was developed by applying KCL at each non-reference

node.

• A mesh is a loop which doesn't contain any other loops within it.

• Loop (mesh) analysis results in a system of linear equations which

must be solved for unknown currents.

•Powerful analysis method which applies KVL to find unknown currents.

•It is applicable to a circuit with no branches crossing each other.

| What is a Mesh ? |

A mesh is a loop which does not contain any other loops within it.

| Steps of Mesh Analysis |

Step1: Identify the number of basic meshes.

Step 2: Assign a current to each mesh.

Step 3: Apply KVL around each loop to get an equation in terms of the loop currents.

Step 4: Solve the resulting system of linear equations.

As an example

V1 + I1 R1 + (I1 - I2) R3 = 0

I2 R2 + V2 + (I2-I1) R3 = 0

| Cases to be considered for Mesh Analysis |

Case I

A current source exists only in one mesh

set i2 = −5A and write a mesh equation for the other mesh

Case II

A current source exists between two meshes

exclude the circle part

a supermesh was created by

excluding the current source and any

elements connected in series with it,

| What is a Supermesh ? |

A supermesh results when two meshes have a (dependent or independent) current source in common.

LEARNINGS:

I learned that in Mesh analysis it is a method that is used to solve planar circuits for the currents at any place in the circuit. The difference between mesh and loop is that the mesh is a loop which does not contain any other loops within it while loop is any continuous path available for current flow from a given point in a circuit and back to that same point in the circuit from the opposite direction that it left from without crossing or retracing it`s own path. I learned also the steps and cases of mesh analysis which most important to know on how to determine and solve the mesh analysis.

SUPER POSITION THEOREM

Superposition

Superposition theorem is one of those strokes of genius that takes a complex subject and simplifies it in a way that makes perfect sense.
A theorem like Millman's certainly works well, but it is not quite obvious why it works so well. Superposition, on the other hand, is obvious.

The strategy used in the Superposition Theorem is to eliminate all but one source of power within a network at a time, using series/parallel analysis to determine voltage drops (and/or currents) within the modified network for each power source separately.

Then, once voltage drops and/or currents have been determined for each power source working separately, the values are all “superimposed” on top of each other (added algebraically) to find the actual voltage drops/currents with all sources active.

example

Since we have two sources of power in this circuit, we will have to calculate two sets of values for voltage drops and/or currents, one for the circuit with only the 28 volt battery in effect. . .

When re-drawing the circuit for series/parallel analysis with one source, all other voltage sources are replaced by wires (shorts), and all current sources with open circuits (breaks). Since we only have voltage sources (batteries) in our example circuit, we will replace every inactive source during analysis with a wire.

Analyzing the circuit with only the 28 volt battery, we obtain the following values for voltage and current:

Analyzing the circuit with only the 7 volt battery, we obtain another set of values for voltage and current:

Applying these superimposed voltage figures to the circuit, the end result looks something like this:

Currents add up algebraically as well, and can either be superimposed as done with the resistor voltage drops, or simply calculated from the final voltage drops and respective resistances (I=E/R). Either way, the answers will be the same.

SOURCE TRANSFORMATION

SOURCE TRANSFORMATION is simplifying a circuit solution, especially with mixed sources, by transforming a voltage into a current source, and vice versa. Finding a solution to a circuit can be difficult without using methods such as this to make the circuit appear simpler.

The circuits in this set of problems consist of independent sources, resistors and a meter. In particular, these circuits do not contain dependent sources. Each of these circuits has a series-parallel structure that makes it possible to simplify the circuit by repeatedly

Performing source transformations:

-Replacing series or parallel resistors by an equivalent resistor.

-Replacing series voltage sources by an equivalent voltage source.

-Replacing parallel current sources by an equivalent source source.

LINEARITY PROPERTY

Linearity

-is the property of an element describing a linear relationship between cause and effect.

-The property is a combination of homogeneity(scaling) property and the additivity property.

The homogeneity property requires that if the input (also called the excitation) is multiplied by a constant, then the output (also called the response) is multiplied by the same constant

if v = iR ⇒ kv = kiR

The additivity property requires that the response to a sum of inputs is the sum of the responses to each input applied separately

v1 = i1R, v2 = i2R ⇒ v = (i1 + i2)R = v1 + v2

THEVINEN THEOREM AND NORTON EQUIVALENT

-Any linear electrical network with voltage and current sources and only resistances can be replaced at terminals A-B by an equivalent voltage source V_th in series connection with an equivalent resistance R_th.

Norton

- Consist of a single current source and a single parallel resistor.

For the Thevinen equivalent voltage

For the Norton equivalent current

LEARNINGS:

Thevenin's Theorem is a way to reduce a network to an equivalent circuit composed of a single voltage source, series resistance, and series load.

Find the Thevenin source voltage by removing the load resistance from the original circuit and calculating voltage across the open connection points where the load resistor used to be.
Find the Thevenin resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points.
Draw the Thevenin equivalent circuit, with the Thevenin voltage source in series with the Thevenin resistance. The load resistor re-attaches between the two open points of the equivalent circuit.
Analyze voltage and current for the load resistor following the rules for series circuits.

In norton's theorem the only thing that is changed is the unknown, which the current across the current load, which is only the alternate method of thevenins theorem.

Capacitors

Capacitors are components designed to take advantage of this phenomenon by placing two conductive plates (usually metal) in close proximity with each other. There are many different styles of capacitor construction, each one suited for particular ratings and purposes.

finding the equivalent capacitance

IN PARALLEL

IN SERIES

Inductor

Inductor is an electrical component that stores energy in magnetic field.

The inductor is made of a coil of conducting wire.

In an electrical circuit schematics, the inductor marked with the letter L.

The inductance is measured in units of Henry [L].

Inductor reduce current in AC circuits and short circuit in DC circuits.

Inductors in series

For several inductors in series the total equivalent inductance is:

L_Total = L₁+L₂+L₃+...

Inductors in parallel

For several inductors in parallel the total equivalent inductance is:

$\frac{1}{L_{Total}}=\frac{1}{L_{1}}+\frac{1}{L_{2}}+\frac{1}{L_{3}}+...$

Inductor's voltage

$v_L(t)=L\frac{di_L(t)}{dt}$

Inductor's current

$i_L(t)=i_L(0)+\frac{1}{L}\int_{0}^{t}v_L(\tau)d\tau$

Energy of inductor

$E_L=\frac{1}{2}LI^2$

MAXIMUM POWER TRANFER

The Maximum Power Transfer Theorem is another useful Circuit Analysis method to ensure that the maximum amount of power will be dissipated in the load resistance when the value of the load resistance is exactly equal to the resistance of the power source. The relationship between the load impedance and the internal impedance of the energy source will give the power in the l

The power transferred from a source or circuit to a load is maximum when the resistance of the load is made equal or matched to the internal resistance of the source or circuit providing the power to the load.

Let,
V= EMF supplied to the load.
RL = Load resistance.
Ri = Internal resistance of the source.

I = Current flowing through the load, internal resistance and the source of the circuit.
PL = Power transferred to the load.
Pi = Power dissipated at internal resistances.

Then,

Power transferred to the load = PL = I²RL

or,
$P_L$ = $\left( \dfrac{V}{R_i + R_L}\right)^2 \times R_L$ = $\dfrac{V^2}{\frac{R_i^2}{R_L}+2R_i +R_L}$

Now using the theorems of Differential calculus , If we keep the RL variable and want to calculate the maximum value of PL then we need to differentiate the PL with respect to RL and equate it with zero.
Thus,
Under Maximum power transfer to load condition:
$\dfrac{d}{dR_L}P_L = \dfrac{d}{dR_l}\dfrac{V^2}{ \frac{R_i^2}{R_L}+2R_i +R_L}= 0$
or,
$-\dfrac{R_i^2}{R_L^2} +1= 0$

or,

$R_i = R_L$

LEARNINGS:

Maximum power transfer theorem is applied in radio electronics; for example: In Antenna Signal amplifier for radio and TV receivers; and various other fields where maximum performance is required but the maximum efficiency is not desired for example to match an Amplifier with a Loudspeaker to yield maximum power to the speaker and thus produce maximum sound.

First Order Circuits

First order circuits are circuits that contain only one energy storage element (capacitor or inductor), and that can, therefore, be described using only a first order differential equation. The two possible types of first-order circuits are:

1. RC (resistor and capacitor)

2. RL (resistor and inductor)

RL and RC circuits is a term we will be using to describe a circuit that has either a) resistors and inductors (RL), or b) resistors and capacitors (RC).

An RL Circuit has at least one resistor (R) and one inductor (L). These can be arranged in parallel, or in series. Inductors are best solved by considering the current flowing through the inductor. Therefore, we will combine the resistive element and the source into a Norton Source Circuit. The Inductor then, will be the external load to the circuit. We remember the equation for the inductor:

If we apply KCL on the node that forms the positive terminal of the voltage source, we can solve to get the following differential equation:

An RL parallel circuit

LEARNINGS:

An RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance .

CIRCUITS

Huwebes, Oktubre 16, 2014

| Mesh Current Analysis Circuit |

Superposition

SOURCE TRANSFORMATION is simplifying a circuit solution, especially with mixed sources, by transforming a voltage into a current source, and vice versa. Finding a solution to a circuit can be difficult without using methods such as this to make the circuit appear simpler.

THEVINEN THEOREM AND NORTON EQUIVALENT

Capacitors

Inductor

Inductors in series

Inductors in parallel

Inductor's voltage

Inductor's current

Energy of inductor

First Order Circuits