Thevenin's Theorem 

Thevenin's Theorem States that it is possible to simplify any circuit, no mattter 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 Superposition theorem, where all the underlying equation must be linear.

Thevenin's theorem is useful in analyzing power system and other circuits where one particular resistor  in the circuit ( called the load resistor) is subjected to change, and re construct the circuit with calculation depend on what method you are gonna use.

example:



In this example I decide to choose R2 as the LOAD resistance in the circuit.  We temporarily remove R2(load resistance) from the circuit and reducing what's left to an equivalent circuit composed of a single voltage source and series resitance. The load resistance can be reconnected to this Thevenins equivalent circuit and calculation carried out as if the whole network were nothing but a simple series circuit.

This should be the picture of a Thevenin's equivalent circuit

The Thevenin equivalent circuit, if correctly derived, will behave exactly the same as the original circuit formed by B1, R1, R3, and B2. In other words, the load resistor (R2) voltage and current should be exactly the same for the same value of load resistance in the two circuits. The load resistor R2 cannot “tell the difference” between the original network of B1, R1, R3, and B2, and the Thevenin equivalent circuit of EThevenin, and RThevenin, provided that the values for EThevenin and RThevenin have been calculated correctly.

The advantage in performing the “Thevenin conversion” to the simpler circuit, of course, is that it makes load voltage and load current so much easier to solve than in the original network. Calculating the equivalent Thevenin source voltage and series resistance is actually quite easy. First, the chosen load resistor is removed from the original circuit, replaced with a break (open circuit):

Next, the voltage between the two points where the load resistor determined. Use whatever analysis methods are at your disposal to do this. In this case, the orignal circuit with the load resistor removed is nothing more than a simple series circuit with the voltage Vth, and so we can also determine the voltage across the open load terminals by applying the OHM's law and KVL.

there are the results:



the voltage between the two load connection points can be figured from the voltage and one of the resistor's voltage drops, and come out to 11.2 V which is our thevenins voltage.



to find the Thevenins series resistance for our equivalent circuit, we need the original circuit, then we remove the power source (independent sources) which is the same method of superposition theorem
The voltage source will be SHORTED and current sources will be OPEN circuit. And then we can figure out the resistance from one load terminal to other.



with the load resistor (2Ω) attached between the connection points, we can determine voltage acroos it and current through it as though the whole network were nothing more than a simple series circuit:


in the example notice that the voltage and current of R2 (8V and 4A) are identical to other method of analysis. The voltage and current figures for the thevenin series resistance and the thevenin source(total) do not apply to any component in the original circuit. Thevenins theorem is only useful for determining what happens to a single resistor in a network: the load.

REFLECTION:
The Thevenin's theorem is a way to reduce a network to an equivalent circuit composed of a single voltage source (Vth),series resistance (Rth), and the series load.

In using the thevenins theorem :
-find thevenins source voltage by removing the load resistor from the original circuit and calculate the voltage across the open connection points where load is connected.
-find the thevenins resistance by removing all power sources in the original circuit by shorting voltage sources and current sources open circuit.
-analyze the voltage and current for the load resistor following the rules for series circuits.

the other case in using the thevenin theorem is a circuit the contains a DEPENDENT source. which our professor did not tackle yet.. thanks for reading :)

WEEK 9 : SUPERPOSITION THEOREM


SUPERPOSITION is an algorithmic circuit analysis method with a prescribed procedure like the node voltage or loop current method.

The Superposition is applied only in circuits with 2 or more independent source in which a final solution can be built as the additive of two or more partial solutions 

Superposition only applies to a linear circuit, which are made of circuit elements. It does not apply in non linear circuits.

The superposition principle states that the voltage across the voltage or current in a element in an linear circuit is the voltage across (current through) that element due to each independent source acting alone.

The Superposition helps us to solve the circuit with many independent source but we just have to remember the things 
to use superposition. 

1.  Consider one independent source at a time while all other independent source are turned off. For voltage source, the source will have a short circuit while the current sources are opened.

2. Dependent Sources are left the same because they are control by circuits variables.

Steps in using superposition theorem:
1. Turn off all independent source except for one source. Then find the output depending on what you will use to simplify the circuit, it could be nodal or mesh analysis .

2. Do the in steps for the other sources.

3. Find the total contribution by assuming up the partial results (voltage or current) to find the unknown.



Recollection About:
in superposition theorem, there is only a simple solution find the unknowns, you can use the Voltage divider and current divider.
and by summing the partial results you will come up with the result.
You can also get directly the unknown by using the voltage or current divider with in that loop as long there is the given independent source and a two resistors given.

LINEARITY PROPERTY 

Linear property is the linear relationship between cause and effect of an element. This property gives linear and nonlinear circuit definition. The property can be applied in various circuit elements. The homogeneity (scaling) property and the additivity property are both the combination of linearity property.

The homogeneity property is that if the input is multiplied by a constant k then the output is also multiplied by the constant k. Input is called excitation and output is called response here. As an example if we consider ohm’s law. Here the law relates the input i to the output v. 

mathematically , 

V= IR

If we multiply the input current  i by a constant k then the output voltage also increases correspondingly by the constant k. The equation stands,   

  kiR = kv

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

Using voltage-current relationship of a resistor if

                                     

  v₁ = i₁R       and   v₂ = i₂R

Applying (i₁ + i₂)gives

V = (i₁ + i₂)R = i₁R+ i₂R = v₁ + v₂

We can say that a resistor is a linear element. Because the voltage-current relationship satisfies both the additivity and the homogeneity properties.

We can tell a circuit is linear if the circuit both the additive and the homogeneous. A linear circuit always consists of linear elements, linear independent and dependent sources.

What is linear circuit?

A circuit is linear if the output is linearly related with its input.

The relation between power and voltage is nonlinear. So this theorem cannot be applied in power.

See a circuit in figure 1. The box is linear circuit. We cannot see any independent source inside the linear circuit.

 

The linear circuit is excited by another outer voltage source v_s. Here the voltage source v_s acts as input. The circuit ends with a load resistance R. we can take the current I through R as the output.

Suppose v_s = 5V and i = 1A. According to linearity property if the voltage is multiplied by 2 then the voltage v_s = 10V and then the current also will be multiplied by 2 hence i = 2A.

The power relation is nonlinear. For example, if the current i₁ flows through the resistor R, the power p₁ = i₁²R and when current i₂ flows through the resistor R then power p₂ = i₂²R.

If the current (i₁ + i₂) flows through R resistor the power absorbed

   P₃ = R(i₁ + i₂)² = Ri₁² + Ri₂² + 2Ri₁i₂ ≠ p₁ + p₂

So the power relation is nonlinear. Circuit solution method superposition is based on linearity property.

MESH ANALYSIS

The Mesh Current Method, also know as the Loop Current Method, is quite similar to the Branch current method in that it uses simultaneous equations, Kirchhoff Voltage Law, and Ohms Law to determine unknown currents in a network. It differs from the Branch current method in that it does not use Kirchhoff  Current Law, and it is usually able to solve a circuit with less unknown variables and less simultaneous equation, which is especially nice if you're forced to solve without a calculator

                The first step in the Mesh Current method is to identify “loops” within the circuit encompassing all components. In our example circuit, the loop formed by B1, R1, and R2 will be the first while the loop formed by B2, R2, and R3 will be the second. The strangest part of the Mesh Current method is envisioning circulating currents in each of the loops. In fact, this method gets its name from the idea of these currents meshing together between loops like sets of spinning gears:

The choice of each current's direction is entirely arbitrary, just as in the Branch Current method, but the resulting equations are easier to solve if the currents are going the same direction through intersecting components (note how currents I1 and I2 are both going “up” through resistor R2, where they “mesh,” or intersect). If the assumed direction of a mesh current is wrong, the answer for that current will have a negative value.

The next step is to label all voltage drop polarities across resistors according to the assumed directions of the mesh currents. Remember that the “upstream” end of a resistor will always be negative, and the “downstream” end of a resistor positive with respect to each other, since electrons are negatively charged. The battery polarities, of course, are dictated by their symbol orientations in the diagram, and may or may not “agree” with the resistor polarities (assumed current directions):

 

Using Kirchhoff's Voltage Law, we can now step around each of these loops, generating equations representative of the component voltage drops and polarities. As with the Branch Current method, we will denote a resistor's voltage drop as the product of the resistance (in ohms) and its respective mesh current (that quantity being unknown at this point). Where two currents mesh together, we will write that term in the equation with resistor current being the sum of the two meshing currents.

Tracing the left loop of the circuit, starting from the upper-left corner and moving counter-clockwise (the choice of starting points and directions is ultimately irrelevant), counting polarity as if we had a voltmeter in hand, red lead on the point ahead and black lead on the point behind, we get this equation:

 -28 + 2 (I1 - I2) + 4I1 = 0

Notice that the middle term of the equation uses the sum of mesh currents I1 and I2 as the current through resistor R2. This is because mesh currents I1 and I2 are going the same direction through R2, and thus complement each other. Distributing the coefficient of 2 to the I1 and I2 terms, and then combining I1 terms in the equation, we can simplify as such:

-28 + 2 (I1 - I2) + 4I1 = 0                  Original Form of Equation

..Distributing to terms within parenthesis..
-28 + 2I1+ 2I2 + 4I1 = 0

..Combining like terms..

-28 + 6I1 + 2I2 = 0                          Simplified Form of Equation

At this time we have one equation with two unknowns. To be able to solve for two unknown mesh currents, we must have two equations. If we trace the other loop of the circuit, we can obtain another KVL equation and have enough data to solve for the two currents. Creature of habit that I am, I'll start at the upper-left hand corner of the right loop and trace counter-clockwise:

-2 (I1 +I2) +7 - 1I2 = 0

Simplifying the equation as before, we end up with:

-2I1 - 3I2 +7 = 0

Now, with two equations, we can use one of several methods to mathematically solve for the unknown currents I1 and I2:

-28 + 6I1 + 2I2 = 0

-2I1 - 3I2 + 7 = 0

..rearranging equation for easier equation..

6I1 + 2I2 = -28

-2I1 - 3I2 = -7

Solutions:

I1 = 5 A

I2 = -1 A


CASES TO BE CONSIDERED FOR MESH ANALYSIS

CASE 1
  A current source exists only in one mesh

 

CASE 2

A current source exist between two meshes

 

We can write one mesh equation by considering the current source. The current source  is related to the mesh currents at by

i1=i2+1.5

In order to write the second mesh equation, we must decide what to do about the current source voltage. Notice that there is no easy way to express the current source voltage in terms of the mesh currents. In this example, illustrate two methods of writing the second mesh equation.

Apply KVL to the supermesh corresponding the current source. Shown in blue, this supermesh is the perimeter of the two meshes that each contain the current source. 

 

Apply KVL to the supermesh to get 

9i1+3i2+6i2-12=0 or 9i1+9i2=12

This is the same equation that was obtained using method 1. Applying KVL to the supermesh is a shortcut for doing three things.

1. labeling the current source voltage as v

2. applying KVL to both meshes that contain the current source

3. eliminating v from the KVL equations

In summary, the mesh equation are 

i1=i2+1.5

and 

9i1+9i2=12

by simplying this, the equation results to 

i1=1.4167 A          and      i2=-83.3 mA

Reflection:

In this topic : Mesh analysis , i have learned that it is use to solve planar circuits for the currents at any place in the circuit. this means that there no wires crossing to each other. The difference between a mesh and a loop is that, the mesh is a loop which does not contain any other loops within it while a loop is any continuity path that is available to a current flow from a given circuit. But so far in answering the mesh analysis, it is still challenging because it is hard for me to get equations in a given circuit especially in using matrix solving. I am troubling in using matrix because of lack of practice. That's why I have practice in order have correct answer to the problem.

DELTA TO WYE CIRCUITS

In many circuits applications, we encounter components connected together in one of two ways to form a three terminal network: the "Delta", or Δ (also known as the "Pi," or  π)

configuration, and the "Y" (also known as the "T") configuration.

a

It is possible to calculate the proper values of resistors necessary to form one kind of network ( Δ or Y) that behaves identically to the other kind, as analyzed from the terminal connections alone. That is, if we had two separate resistor networks, one  Δ and one Y, each with its resistor hidden from view, with nothing but three terminals (A,B, and C) exposed for testing, the resistor could be sized for the two networks so that there would be no way to electrically determine one network apart from the other. In other words, equivalent  Δ and Y networks behave identically.

There are several equations used to convert one network to the other: 

 

DELTA to WYE CIRCUITS

DELTA to WYE transformations:

                                             X = _   BC   __

                                                   A + B + C

                                            Y = _   AC   __

                                                   A + B + C

                                            Z = _   AB   __

                                                   A + B + C

In general:

                                    Rwye = _ Product of adjacent R's in Δ __

                                                  summation of all resistors in Δ

WYE to DELTA transformations:

  

A =  XY + XZ + YZ_        B =__XY + XZ + YZ                  C =    XY + XZ + YZ     

               X                                       Y                                                 Z

In general:

                                    Rwye = _ summation of cross product in WYE __

                                                                opposite R in WYE

   With    A =  B = C = Rdelta          = RΔ         and          X = Y = Z = Rwye   = Ry

                              RΔ = 3Ry                or             Ry = RΔ/3

 

NODAL ANALYSIS 

In this topic we will tackle about nodal analysis ..

Nodal Analysis provides a general procedure for analyzing circuits using node voltage as the circuits variables. Choosing node voltages instead of voltage elements as circuit variables is convenient and reduces the number of equations one must solve simultaneously.
To simplify matters, we shall assume in this section that circuits do not contain voltage sources. Circuits that contain voltage sources will be analyzed in the next section.
In nodal analysis, we are interested in finding the node voltages given a circuit with n nodes without voltage sources, the nodal analysis of the circuit involves taking the following three steps.

Step to Determine Node Voltages:
1. Select a node as the reference node or a ground. 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 Ohms law to express the branch currents in terms of node voltages.

For example, for 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). Here are some examples:



The current downward out of node 1 depends on the voltage difference V1 - V3 and the resistance in the branch.





In this case the voltage difference across the resistance is V1 - V2 minus the voltage across the voltage source. Thus the downward current is as shown.






In this case the voltage difference across the resistance must be 100 volts greater than the difference V1 - V2. Thus the downward current is as shown.




The result, after simplification, is a system of m linear equations in the m unknown nodal voltages (where m is one less than the number of nodes; m = n - 1). The equations are of this form:

where G₁₁, G₁₂, . . . , G_mm and I₁, I₂, . . . , I_m are constants.


NODAL ANALYSIS WITH VOLTAGE SOURCES

Case !

If a voltage souve is connected between the reference node and a nonreference node, we simply set the voltage at the non reference node equal to the voltage of the voltage source. In fig. 3.7 for example 

V1=10v

Thus our analysis is somewhat simplified by this knowledge of the voltage at this node.

Case 1

If the voltage source( dependent of independent ) is connected between two nonreference nodes, the two nonreference nodes form generalized node or supernode, we apply both KCL and KVL to determine the node voltages.

A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.

In Fig. 3.7, nodes 2 and 3 form a supernode. (We could have more than two nodes forming a single supernode. For example, see the circuit in Fig. 3.14.) We analyze a circuit with supernodes using the same three steps mentioned in the previous section except that the supernodes are treated differently. Why? Because an essential component of nodal analysis is applying KCL, which requires knowing the current through each element. There is no way of knowing the current through a voltage source in advance. However, KCL must be satisfied at a supernode like any other node. Hence, at the supernode in Fig. 3.7,

i1 + i4 = i2 + i3                                                                                   (3.11a)
or
v1 − v2 / 2 + v1 − v3  /4 = v2 − 0 / 8 + v3 − 0 / 6                              (3.11b)

To apply Kirchhoff’s voltage law to the supernode in Fig. 3.7, we redraw the circuit as shown in Fig. 3.8. Going around the loop in the clockwise direction gives

−v2 + 5 + v3 = 0 ⇒ v2 − v3 = 5                                                          (3.12)

Note the following properties of a supernode:

1. The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages.

2. A supernode has no voltage of its own.

3. A supernode requires the application of both KCL and KVL.

SERIES PARALLEL CIRCUIT

In this section we will tackle about the series parallel circuit. This topic is also discussed by our teacher when we take up Physics 2. So we in circuits 1 this topic is really applied especially when we find the voltage,current, and resistance. 

SERIES CIRCUIT

A series circuit in which the resistor is connected or short, so the current has only one path. The current flowing through the resistors. The total resistance of the circuit is found by adding up the resistance value of the resistor. 

R=R1+R2+R3+.....

        

   In this example the circuit is a closed path, so the current will flow in counter clockwise, and according to the series resistance, we should total or add the resistances in the circuit

Req=R1+R2+R3

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.