Nodal Analysis And Mesh Analysis In AC circuit

Since we know that Kirchhoff's Law is applicable to the AC circuit. We will also apply the Nodal Analysis and Mesh Analysis in analyzing an ac circuits.

STEPS TO ANALYZE AC CIRCUITS:
1. Transform the circuit to the phasor or frequency domain.
2. Solve the problem using circuit techniques(Nodal/Mesh analysis)
3. Transform the resulting phasor to the time domain.


~We've been through about Nodal Analysis and Mesh Analysis in my past blogs. We will just recall it.

Nodal Analysis provide a general procedure for analyzing circuits using node voltages as the circuit variables.

Steps to Determine Node Voltages:

1. Select a node as the reference node, Assign voltages v1, v2, . . . . . , 

vn-1 to the 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 currents in terms of node voltages.

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

Nodal Analysis 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 super node, 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 this case we will solve a problem in which the three(capacitor,conductor,resistor) elements are included. Also with the rectangular form and polar form since the phasor and frequency domain was being applied. 


Example:

The unknown for this problem is Io, in order to get Io we must transform first the inductor and capacitor into the impedance. 

@node Vo,

But Io= (25-Vo) / 2000, so substitute to get the Vo and also the answer must be in polar form.

We can get now the Io using Ohm's Law.

MESH ANALYSIS;

A Mesh is a loop that does not contain any other loop within it.

STEPS TO DETERMINE MESH CURRENTS:
1. Assign mesh currents I1, I2,... In to the n meshes.
2. Apply KVL to each  of the n meshes. Use ohm's law to express the voltages in terms of the mesh currents.
3. Solve the resulting n simultaneous equations to get the mesh currents.

~We  will apply mesh to ac;

Example: 

 

Source:

- Fundamental of Electric Circuits by Alexander and Sadiku

- Google

- Youtube

Phasor Relationship For Circuit Elements

Impedance and Admittance

Impedance Z – of a circuit is the ratio of the phasor voltage to the phasor current, measured in Ohms(Ω).

               Z=V/I                    or           V=ZI

Admittance Y – is the reciprocal of impedance, measured in siemens(s).

               Y=1/Z = I/V

                              Impedances and admittances of passive elements.

Elements	Impedance	Admittance
R (Ω)	Z=R	Y=1/R
L (H)	Z= jωL	Y=1/jωL
C (F)	Z=1/jωC	Y=jωC

That three elements(R,L,C) only the Resistances are real and the rest were imaginary and the reactances. The Capacitor is the positive reactance while the Inductor is the negative reactance.

Example:

We can transform the imaginary elements into an impedance by the use of shown equations.

1H

Z= jωL = j1x10 = j10

1F

Z=1/jωC = 1/j10x1 = -j0.1

In order to proceed and get the unknown values, we must first identify if what concept should be done in this problem. Since, circuit analysis was all about analyzing the different character of a circuit in any ways, we will use the basic law which we’ve been through in a few months “The Kirchhoff’s Law” was already discuss here in the past topics. We will just recall some important matters.

Kirchhoff’s Law has two parts, the Kirchhoff’s voltage Law and Kirchhoff’s current Law.

Kirchhoff's voltage law, states that the algebraic sum of all the voltages around a closed circuit equals zero. 

Kirchhoff's current law, states that the algebraic sum of all the currents entering and leaving a node is equal to zero. 

Looking at the circuit, the KVL must be applied since the sum of all the voltage drops in a closed circuit will equal the voltage source if only we’ll combine all of the impedances in order to have a single loop. And here we can apply the time domain converted into phasor domain.

Z= 1 + (1/j10 + 1/-j0.1 + 1/1)^-1 = 1.01010 – j0.1 = 1.015∠ -5.653

We can solve now the unknown which is the current using Ohm’s Law;

I=V/R = 2∠0/ 1.015∠ -5.653

I = 1.9704∠5.653 = 1.9704cos(10t+5.65) A

SINUSOIDS AND PHASORS

A Sinusoid is a signal that has the form of the sine or cosine function.

                              There are two parts of sinusoid, the Sinusoidal Current and Sinusoidal Voltage. Sinusoidal current is usually referred to as alternating current. Such a current reverses at regular time intervals and has alternately positive and negative values. Circuits driven by sinusoidal current or voltage sources are called ac circuit. 

Sinusoidal voltage,

v(t) = Vmsinωt

where;

Vm= the amplitude of the sinusoid

ω = the angular frequency in radian/s

ωt = the argument of the sinusoid

Sample equation to determine it's label;

  6cos(200t + 15° )

Amplitude- 6

Phase angle- 15°

Angular Frequency- 200t

A Phasor – is a complex number that represents the amplitude and phase of a sinusoid.

2 PHASES

* IN PHASE

*OUT OF PHASE

IN PHASE,

The same;

*Time

*Period

*Frequency

OUT OF PHASE,

It's either have the same amplitude or not.

~Sinusoids are easily expressed in terms of phasors, in which are more convenient to work with than sine and cosine function.

Sinusoid-Phasor Transformation 

Time Domain representation	Phasor Domain Representation
Vmcos(ωt + ɸ )	Vm ∠ ɸ
Vmsin(ωt + ɸ )	Vm ∠ ɸ - 90 °
Imcos(ωt + 0 )	Im∠0
Imsin(ωt + 0 )	Im∠ 0 - 90 °

To transform the Time domain into the Phasor domain, the time domain is in the rectangular form of

 z = x + jy,

where in x is the real part of z and y is the imaginary part.

The equation going to polar form for the amplitude;

Square root of x squared plus y squared

For the Phase angle;

arctan(y divided by x) 

Example; 

5 + j2
= 5.39∠ 21.80°

Since all of these was all about the currents and voltages, in getting the value of each of them we go through graphing in a sinusoidal form. So between that two, there must be leading and lagging. 

Looking at the figure, the voltage leads the current since leading is when a sinusoid peaks first in time and it is closer to the reference axis. And the current here is lagging.

Thevenin's and Norton's Theorem

Thevenin's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load. The qualification of “linear” is identical to that found in the Superposition Theorem, where all the underlying equations must be linear (no exponents or roots).

• 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.
• Steps to follow for Thevenin's Theorem:
• (1) Find the Thevenin source voltage by removing the load resistor from the original circuit and calculating voltage across the open connection points where the load resistor used to be.
• (2) 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.
• (3) 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.
• (4) Analyze voltage and current for the load resistor following the rules for series circuits.

Example:

This figure is an example that can be solved by the Thevenin's Theorem. Since R2 is a load, it can be remove temporarily in order to get the RTH and VTH. :) 

SUPERPOSITION Theorem

The superposition theorem for electrical circuit states that for a linear system the response (voltage or current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedances.

To ascertain the contribution of each individual source, all of the other sources first must be "turned off" (set to zero) by:

1. Replacing all other independent voltage sources with a short circuit (thereby eliminating difference of potential i.e. V=0; internal impedance of ideal voltage source is zero (short circuit)).
2. Replacing all other independent current sources with an open circuit (thereby eliminating current i.e. I=0; internal impedance of ideal current source is infinite (open circuit)).

Example:

 Find R2;

The figure has two(2) voltage source, in order to get the voltage across R2, we need to get first the voltage that being supplied on the other loop by deactivating/turning off the other source.

As you can see, the B2 is already turned off, therefore we can solve it using the voltage division principle.
-The Voltage division can be seen on the past blog for more information!

After that, the B1 must be turned off and B2 is now on in order to get the voltage on the other loop.

 

 I'm sure, we can get now the total voltage that R2 have.

TAKE NOTE: When there is Three(3) or more sources in a circuit, only 1 must be turned on and the rest is disable.

Mesh Analysis

A Mesh is a loop that does not contain any other loop within it.

~ a loop can be a mesh, but a mesh can't be a loop.

Same as the nodal analysis, a mesh analysis have also a steps in getting the equation but they differ for some aspects which is the Nodal analysis talks about the nodal voltages while the Mesh analysis talks about mesh currents. 

~ a mesh current is quite similar to the Branch Current method in that it uses simultaneous equations, Kirchhoff's Voltage Law, and Ohm's Law to determine unknown currents in a network.

Why is it QUITE SIMILAR TO THE BRANCH CURRENT?
-  it is quite similar, then of course it is also quite different but it depends on how the mesh is being ISOLATED.


There are steps in determining mesh currents same as nodal analysis, there are steps to determine nodal voltage in order to form an equation.

STEPS TO DETERMINE MESH CURRENTS:
1. Assign mesh currents I1, I2,... In to the n meshes.
2. Apply KVL to each  of the n meshes. Use ohm's law to express the voltages in terms of the mesh currents.
3. Solve the resulting n simultaneous equations to get the mesh currents.

Example:

As you observe the figure, the current flow counter clockwise but wherever the mesh currents will flow, it's direction is arbitrary and does not affect the validity of the solution.

~ as a class, we are more prefer to have the mesh currents direction clockwise because for us it is more convenient and easier to analyze.

Mesh analysis with current sources

Case 1:

When a current source exists only in one mesh

Case 2:

When a current source exists between two meshes and that is SUPERMESH.

What is supermesh?

A SUPERMESH results when two meshes have current source in common.

For more information about mesh, just watch this:

Chapter III. Continuation of Nodal Analysis and Wye-Delta Transformation

As we go through the "Nodal Analysis" topic which includes the KCL and KVL and the SUPERNODE, we come up with a lot of problem solving then we have two method in solving/analyzing the circuit, which is the short cut method and the long way method.

The short cut method,

•  The nodal voltage must be determined.

•   it gives the adjacent of the resistors into voltage.

The long way method, 

• The nodal voltages must be determined.

• The flow current of the current must be assigned.

 

#REMINDERS
     ~ The current leaving is positive, and the current entering is negative.

WYE - DELTA TRANSFORMATION

The wye - delta transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network.

•  wye    -   Y
• delta    -   Δ

~ I think why the wye-delta transformation are made because usually in a circuit(complicated design) have a form of  Y and Δ in order to get the total resistance.

What is Nodal Analysis

Nodal Analysis provide a general procedure for analyzing circuits using node voltages as the circuit variables.

Steps to Determine Node Voltages:

1. Select a node as the reference node, Assign voltages v1, v2, . . . . . , 

vn-1 to the 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 currents in terms of node voltages.

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

Nodal Analysis 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 super node, 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 every different kind of circuits when in comes to analyzing and solving, we should practice the steps of determining the node voltages and must able to observe if what kind of case the circuit is.

To understand more, watch this video :)

There are some instances that a voltage source is in between to the two non-reference node in a loop, that's SUPERNODE.
In circuit theo, a super-node is a theoretical construct that can be used to solve a circuit. This is done by viewing a voltage source on a wire as a point source voltage in relation to other point voltages located at various nodes in the circuit, relative to a ground node assigned a zero or negative charge.

The application that needs to study:
- KVL/KCL (on my CHAPTER 2 blog)
- Cramer's rule

Our chapter 2 journey in Electric circuit!

There are some instances that it's hard for us to solve/interpret a circuit especially when your knowledge is just revolves around Ohm's Law because the circuit have it's different behavior that needs to analyze. But thanks to Gustav Robert Kirchhoff for having the Kirchhoff's Law(KCL & KVL). But for now, let me introduce to you the voltage division and current division.

Anyway, voltage and current division allow us to simplify the task of analyzing a circuit. :)

Voltage division allows us to calculate what fraction of the total voltage across the series string of resistors is dropped across any one resistor. 

 The figure above is a sample of a circuit that can be defined by using voltage divider, the formula of that is:

 So, you can solve already the simple circuit that I've shown to you by having that formula. :)

Next,
Current division, it allows us to determine how the current flowing into a node is split between various parallel resistors.

Observe this circuit,

In this circuit, the resistors are in parallel connection. So we can apply here the current divider because that's what it takes. Here is the formula:

Let's go back to KVL and KCL.
 
 Kirchhoff's voltage law, states that the algebraic sum of all the voltages around a closed circuit equals zero. 
 Another way of stating kirchhoff's voltage law is, the sum of all the voltage drops in a closed circuit will equal the voltage source.

 The sample figure that I've shown to you as KVL sample, there are four voltage drops and one voltage source in the circuit. If the voltages are summed around the circuit as shown, they equal zero.

The voltage source has a sign opposite that of the voltage drops. Therefore, the algebraic sum equals zero.  In another way, the sum of voltage drops will equal the voltage source. 

These two formulas are just the same thing and are equivalent ways of expressing Kirchhoff's Voltage Law. :)


Kirchhoff's current law, states that the algebraic sum of all the currents entering and leaving a node is equal to zero. Another way of stating the KCL is that the total current flowing into a node is equal to the sum of the current flowing out of that node.

Let me tell you something about a Node :)
A node is defined as any point of a circuit at which two or more current paths meet. In parallel circuit, the node is where the parallel branches of the circuit connect.
Then what is a branch? 
A branch  represents a single element such as a voltage source or a resistor that can be found in a loop.
Do you know what is a loop?
Anyway, a loop is any closed path in a circuit. So anything that is a closed path in a circuit is a loop.

To be clearer to you about the KCL, let just watch this video for a further explanation.  :) 

I hope you've learn a lot about it as what I have learn too. Some information that was written/viewed, data and pictures are came from the internet/book, and some are came from my own learning and understanding. Thank you :)

TAKE NOTE: "you will gradually know the characteristics of a circuit by just keep on solving any behavior of a different circuit that can be seen in the book of Alexander Sadiku the author, named Fundamentals of Electric Circuit(fourth edition). Just enjoy solving and you'll find it easier."

See you on the next blog!

The importance of OHM's LAW in Electric circuit

Ohm's Law defines the relationship among three fundamental quantities which are the current, voltage, and resistance Current is directly proportional to voltage and inversely proportional to resistance. 

This may expressed as: V=I/R 

I = current in Amperes 

V = voltage in volts 

R = resistance in ohms 

 

 Few days ago, we discuss how the three fundamental quantities are being applied in a circuit.

The current flow in an electric circuit can be varied by changing either the voltage applied to circuit or the resistance in the circuit. The current changes in exact proportions to the change in the voltage or resistance. If the voltage is increased, the current also increases. If the voltage is decreased, the current also decreases. On the other hand, if the resistance is increased, the current decreases. 

 

 To be continued! :)

See you on the next topic that I might gonna share to you, and its all about KVL/KCL

Knowing the basic laboratory equipments and componets

There are common equipments and components in the laboratory in circuits 1. We should familiarize all of them so that we'll know their uses, capacity and what are the purpose of these objects. We are given 10 elements named Capacitor, IC, Transistor, Inductor, Resistor, Digital Multimeter, Power supply and Current source.

• A capacitor stores electric charge and is mainly used with resistors in timing circuits. It also acts as a filter by passing alternating current (AC), and blocking direct current (DC). The charge of a capacitor is measured in units called Farads. 

• An integrated circuit (IC), sometimes called a chip or microchip, is a semiconductor wafer on which thousands or millions of tiny resistors, capacitors, and transistors are fabricated. An IC can function as an amplifier, oscillator, timer, counter, computer memory, or microprocessor. A particular IC is categorized as either linear (analog) or digital, depending on its intended application. 
•  A transistor is a semiconductor device used to amplify and switch electronic signals and electrical power.
•  An inductor is a passive electronic component that stores energy in the form of a magnetic field. In its simplest form, an inductor consists of a wire loop or coil.
•  Resistor is an electrical component that reduces the electric current.
•  A multimeter is used to make various electrical measurements, such as AC and DC voltage, AC and DC current, and resistance.

• A power supply is an electronic device that supplies electric power to an electrical load.   

CAPACITOR

IC (INTEGRATED CIRCUIT)

TRANSISTOR

DIGITAL MULTIMETER

INDUCTOR

POWER SUPPLY

RESISTOR

We usually see those instruments everywhere, but we didn't know their importance in our life. The different function they have could help a lot to us as a human being but thanks to those who invented them. I think by knowing those objects, we can use it now in accordance to its purpose and we need more experience through it so we are flexible where to put them and what it could bring about. 

  

 TO BE CONTINUED! :)