Sabado, Agosto 9, 2014

Kirchoffs Circuit Law

application of analysis of network in circuits

Kirchoffs First Law – The Current Law, (KCL)

Kirchoffs Current Law or KCL, states that the “total current or charge entering a junction or node is exactly equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within the node“. In other words the algebraic sum of ALL the currents entering and leaving a node must be equal to zero, I(exiting) + I(entering) = 0. This idea by Kirchoff is commonly known as the Conservation of Charge.

Kirchoffs Current Law

kirchoffs current law
 Here, the 3 currents entering the node, I1, I2, I3 are all positive in value and the 2 currents leaving the node, I4 and I5 are negative in value. Then this means we can also rewrite the equation as;

I1 + I2 + I3 – I4 – I5 = 0
The term Node in an electrical circuit generally refers to a connection or junction of two or more current carrying paths or elements such as cables and components. Also for current to flow either in or out of a node a closed circuit path must exist. We can use Kirchoff’s current law when analysing parallel circuits.

Kirchoffs Second Law – The Voltage Law, (KVL)

Kirchoffs Voltage Law or KVL, states that “in any closed loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop” which is also equal to zero. In other words the algebraic sum of all voltages within the loop must be equal to zero. This idea by Kirchoff is known as the Conservation of Energy.

Kirchoffs Voltage Law

kirchoffs voltage law
Starting at any point in the loop continue in the same direction noting the direction of all the voltage drops, either positive or negative, and returning back to the same starting point. It is important to maintain the same direction either clockwise or anti-clockwise or the final voltage sum will not be equal to zero. We can use Kirchoff’s voltage law when analysing series circuits.
When analysing either DC circuits or AC circuits using Kirchoffs Circuit Laws a number of definitions and terminologies are used to describe the parts of the circuit being analysed such as: node, paths, branches, loops and meshes. These terms are used frequently in circuit analysis so it is important to understand them.

Common DC Circuit Theory Terms:

  • • Circuit – a circuit is a closed loop conducting path in which an electrical current flows.
  • • Path – a single line of connecting elements or sources.
  • • Node – a node is a junction, connection or terminal within a circuit were two or more circuit elements are connected or joined together giving a connection point between two or more branches. A node is indicated by a dot.
  • • Branch – a branch is a single or group of components such as resistors or a source which are connected between two nodes.
  • • Loop – a loop is a simple closed path in a circuit in which no circuit element or node is encountered more than once.
  • • Mesh – a mesh is a single open loop that does not have a closed path. There are no components inside a mesh.
Note that:
    Components are said to be connected in Series if the same current flows through component.
    Components are said to be connected in Parallel if the same voltage is applied across them.

A Typical DC Circuit

kirchoffs circuit law

Kirchoffs Circuit Law Example No1

Find the current flowing in the 40Ω Resistor, R3
kirchoffs law example
The circuit has 3 branches, 2 nodes (A and B) and 2 independent loops.
Using Kirchoffs Current Law, KCL the equations are given as;
At node A :    I1 + I2 = I3
At node B :    I3 = I1 + I2
Using Kirchoffs Voltage Law, KVL the equations are given as;
Loop 1 is given as :    10 = R1 x I1 + R3 x I3 = 10I1 + 40I3
Loop 2 is given as :    20 = R2 x I2 + R3 x I3 = 20I2 + 40I3
Loop 3 is given as :    10 – 20 = 10I1 – 20I2
As I3 is the sum of I1 + I2 we can rewrite the equations as;
Eq. No 1 :    10 = 10I1 + 40(I1 + I2)  =  50I1 + 40I2
Eq. No 2 :    20 = 20I2 + 40(I1 + I2)  =  40I1 + 60I2
We now have two “Simultaneous Equations” that can be reduced to give us the values of I1 and I2 
Substitution of I1 in terms of I2 gives us the value of I1 as -0.143 Amps
Substitution of I2 in terms of I1 gives us the value of I2 as +0.429 Amps
As :    I3 = I1 + I2
The current flowing in resistor R3 is given as :    -0.143 + 0.429 = 0.286 Amps
and the voltage across the resistor R3 is given as :    0.286 x 40 = 11.44 volts
The negative sign for I1 means that the direction of current flow initially chosen was wrong, but never the less still valid. In fact, the 20v battery is charging the 10v battery.

Application of Kirchoffs Circuit Laws

These two laws enable the Currents and Voltages in a circuit to be found, ie, the circuit is said to be “Analysed”, and the basic procedure for using Kirchoff’s Circuit Laws is as follows:
1. Assume all voltages and resistances are given. ( If not label them V1, V2,… R1, R2, etc. )
2. Label each branch with a branch current. ( I1, I2, I3 etc. )
3. Find Kirchoff’s first law equations for each node.
4. Find Kirchoff’s second law equations for each of the independent loops of the circuit.
5. Use Linear simultaneous equations as required to find the unknown currents.
As well as using Kirchoffs Circuit Law to calculate the various voltages and currents circulating around a linear circuit, we can also use loop analysis to calculate the currents in each independent loop which helps to reduce the amount of mathematics required by using just Kirchoff's laws. In the next tutorial about DC Theory we will look at Mesh Current Analysis to do just that.
(C) http://www.electronics-tutorials.ws/dccircuits/dcp_4.html 

Linggo, Agosto 3, 2014

network analysis (c) Larson Edwards Falvo Elementary Linear Algebra




simple Netwrok analysis

Networks composed of branches and junctions are used as models in many diverse fields
such as economics, traffic analysis, and electrical engineering.
In such models it is assumed that the total flow into a junction is equal to the total flow
out of the junction. For example, because the junction shown in Figure 1.9 has 25 units
flowing into it, there must be 25 units flowing out of it. This is represented by the linear
equation
x1+ x2 = 25.
Because each junction in a network gives rise to a linear equation, you can analyze
the flow through a network composed of several junctions by solving a system of linear
equations. This procedure is illustrated in Example 5.


Sabado, Hulyo 5, 2014

POLYNOMIAL CURVE FITTING

This is used when there are given points and you need to find the polynomial to see the whole graph..

1st find the system of the polynomial by substituting points to
y= a(sub 0)+a(sub1)x + a(sub2) x^2 +a(sub3) x^3+.......a(subn)x^n
the number of points minus one determine the highest degree you  of x in
y= a(sub 0)+a(sub1)x + a(sub2) x^2 +a(sub3) x^3+.......

solve using any methos you want to find  a(sub 0),a(sub1),a(sub2),a(sub3)...a(subn)
after you find those values... write it in the form of p(x)..
like this given
a(sub 0)=12
a(sub1)=2
a(sub2)=3
a(sub3)=5
p(x)= 12+2x + 3x^2 +5x^3 so this is the polynomial function

Sabado, Hunyo 28, 2014

Gaussian Jordan Method in solving in Solving System of Equations
(read the previous posts so you'll not be lost)

Tip:
dodge making the rows with fraction values
you can eliminate them by not just only multiplying a scalar.. you can add a multiple of other equation to another equatiion

Steps:
Put system of Equation to matrix form
You need to put the matrix to echelon form using row operations
you must leave each row only one variable ( diff variables each row)
after that put to system and write coordinate form of the solution

(1,4,-2)




Biyernes, Hunyo 20, 2014

matrix
a rectangular array of number
its size is row x column or m x n
\begin{bmatrix}1 & 9 & -13 \\20 & 5 & -6 \end{bmatrix}. -------> this is a 2 x 3 matrix
matix with same number of columns and rows is called square matrix

2 types of matrix coefficient and augmented
coefficient matrix-- matrix which only the coefficient of the variables in the given system is written
x+9y-13z=2
20x+5y+-6z=9
\begin{bmatrix}1 & 9 & -13 \\20 & 5 & -6 \end{bmatrix}.
augmented matrix-- matrix which the constant is written
x+9y-13z=2
20x+5y+-6z=9






Elementary  Row Operations
used in system of equation to solve for solutions
1. Interchange two rows (or columns).
2. Multiply each element in a row (or column) by a non-zero number.
3. Multiply a row (or column) by a non-zero number and add the result to another row (or column).

EcHELON Form
form of system of equation in which the a variable in the first is not included in the second up to the last and the variables of the first variable in each equation is one.
 to get echelon arrange the variables first and use elementary  row operatiions to get the echelon form..


ex.of echelon form
x+3y+4z=22
      y+3z=12
           z=3

after that use back subtitution to find the solutions.. substitute the value of the variable on the third equation in the echelon form and substitute the corresponding values to the first to find the variable leading in the first equation.
so z=3 then y+3(3)= 12 then y= 3 then substitute to the first x+3(3)+4(3)=22 then x= 1
so the solution is in this form (x,y,z) then (1,3,3) is the solution..


other way to solve system of equations









Sabado, Hunyo 14, 2014

Inconsistent -----------------------> parallel LINES

this is when a1/a2 = b1/b2 and not equal to c1/c2
when the coefficient in the variables are the same but different constant



http://www.algebra.com/algebra/homework/coordinate/Types-of-systems-inconsistent-dependent-independent.lesson  <---- for more info


This lesson concerns systems of two equations, such as: 


2x + y = 10
3x + y = 13.


The equations can be viewed algebraically or graphically. Usually, the problem is to find a solution for x and y that satisfies both equations simultaneously. Graphically, this represents a point where the lines cross. There are 3 possible outcomes to this (shown here in blue, green, and red):
The two lines might not cross at all, as in 

graph%28+300%2C+200%2C+-20%2C+20%2C+-20%2C+20%2C+x%2C+x%2B10%29+
y = x
y = x + 10.


This means there are no solutions, and the system is called inconsistent. 

If you try to solve this system algebraically, you'll end up with something that's not true, such as 0 = 10. 

Whenever you end up with something that's not true, the system is inconsistent.

Sabado, Hunyo 7, 2014

consistent(dependent)---> coinsiding lines

If the equations on a give equation have they only have one 

solution

a1/a2 is  equal to b1/b2 is  equal to c1/c2<<< construction of the 

coefficient of the variables in the equations

there are infinite solutions... but you need to put parameters

sistema
The graphical solution is any point on the two identical straight lines.
recta
to find the solution of the given system you need to find the value interms of x and y
and replace them with other variable.. like  the solution of given system  : (1-b,b) which y=t... it is most ad visable to change the variable of the rightmost in the equation...

Biyernes, Hunyo 6, 2014

intersecting lines are consistent(independent)??



consistent(independent)---> intersecting lines

If the equations on a give equation have they only have one solution
a1/a2 is not equal to b1/b2 is not equal to c1/c2<<< construction of the coefficient of the variables in the equations

Problem
Find all solutions to the system y – x = 1 and y + x = 3.



First, graph both equations on the same axes.
 The two lines intersect once. That means there is only one solution to the system.

The point of intersection appears to be (1, 2).
Read the point from the graph as accurately as possible.

y – x = 1
2 – 1 = 1
1 = 1

TRUE

(1, 2) is a solution of y – x = 1.
y + x = 3
2 + 1 = 3
3 = 3

TRUE

(1, 2) is a solution of y + x = 3.
Check the values in both equations. Substitute 1 for x and 2 for y. (1, 2) is a solution.
Answer
(1, 2) is the solution to the system y – x = 1 and
y + x = 3.
Since (1, 2) is a solution for each of the equations in the system, it is the solution for the system.

what are you?

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It is the study of lines, planes, and subspaces and their intersections using algebra. Linear algebra assigns vectors as the coordinates of points in a space, so that operations on the vectors define operations on the points in the space. <<< from wikipedia

Linear algebra is the study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of rotations in space,least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering. Confusingly, linear algebra is not actually an algebra in the technical sense of the word "algebra" (i.e., a vector space V over a field F, and so on). <<<<from   http://mathworld.wolfram.com/

linear algebra is studiies about linees.