Thevenin's and Norton's Theorem

Thevenin’s theorem

Thevenin’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTh in series with a resistor RTh, where VTh is the open-circuit voltage at the terminals and RTh is the input or equivalent resistance at the terminals when the independent sources are turned off.

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. If we're dealing with passive components, this is true. However, there are some components which are nonlinear: that is, their opposition to current changes with voltage and/or current. As such, we would call circuits containing these types of components, nonlinear circuits.

Thevenin's Theorem is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit (called the “load” resistor) is subject to change, and re-calculation of the circuit is necessary with each trial value of load resistance, to determine voltage across it and current through it.

In finding the thevenin's resistance RTh, we need to consider two cases:
 
Case 1: If the network has no dependent sources, we turn off all independent sources. Rth is the inpit resistance of the network looking between terminals a and b, as shown below:           

Case 2: if the network has dependent sources. As with Superposition, dependent sources are not to be turned off because they are controlled by circuit variables.


Sample problem: Determine the thevenin’s equivalent  circuit between the terminals A&B For the circuit shown :

Answer:
First we are about to find the thevenin’s resistance. To find the thevenin’s resistance we remove the resistance R_L and open circuit the AB terminals. Then we remove the voltage source and short circuit it.

We can easily find the thevenin’s resistance now.

R^th = (5//10 + 3)

Now we have to find the thevenin’’s voltage. For this we remove the load resistance R_L.

 

Applying nodal equation method to point c,

(V-30)/5  + (V-0)/10  = 0

V = 20V

From figure 11.3 you can see that,

V_A = V_C

V_A = V^TH

Therefore V = V^TH

V^TH = 20V

NORTON'S THEOREM

In 1926, about 43 years after Thevenin published his theorem, E. L. Norton, an American engineer at Bell Telephone Laboratories, proposed a similar theorem.

Edward Lawry Norton was an accomplished Bell Labs engineer and scientist famous fordeveloping the concept of the Norton equivalent circuit.

Norton's theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source I_N in parallel with a resistor R_N where I_N  is the short-circuit current  through the terminals and R_N is the input or equivalent resistance at the terminals when the independent sources are turned off.

Norton resistances are equal; that is,

To find the Norton current I_N, we determine the short-circuit current flowing from a to b in both circuits.

Original Circuit

Norton Equivalent Circuit

  Since the two circuits are equivalent. Thus,

 Dependent and independent sources are treated the same way as in Thevenin's theorem. Observe the close relationship between Norton's and Thevenin's theorems. Rn = Rth, and: 

 This is essentially source transformation. For this reason, source transformation is often called Thevenin-Norton Transformation.


Sample Problem:

Find the Norton’s equivalent circuit across A-B terminals for the circuit shown.

 Answer:

First we remove the 10Ω resistor and short circuit the terminals A&B. 

The current flowing through the short circuited terminals is called the Norton’s curren I_N.

To find the I_N we apply nodal equation for point C

(V – 30)/5 + V/10 + V/3 = 0

V = 180/19

Ohms law to 3Ω resistance

I = V/R

I_N = (180/19) / 3

I_N = 60/19

This is the Norton’s current.

Now we are about to find the Norton’s resistance. Note that this is equal to the thevenin’s resistance also.

R_N = (10//5) + 3

R_N = 19/3Ω

TO KNOW MORE ABOUT THEVENIN'S AND NORTON'S THEOREM:

 

Circuit Theorems

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 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.

 Linearity Circuit

A linear circuit is one whose output is linearly related (or directly proportional) to its input.

A linear circuit is a special system whose output is linearly related. Linear circuits are very useful in modeling devices and very well understood. 

To learn more about Linearity:

SUPERPOSITION

• The superposition theorem eliminates the need for solving simultaneous linear equations by considering the effect on each source independently.

•  To consider the effects of each source we remove the remaining sources; by setting the voltage sources to zero (short-circuit representation) and current sources to zero (open-circuit representation).

•  The current through, or voltage across, a portion of the network produced by each source is then added algebraically to find the total solution for current or voltage.

•  The only variation in applying the superposition theorem to AC networks with independent sources is that we will be working with impedances and phasors instead of just resistors and real numbers.

•   The superposition theorem is not applicable to power effects in AC networks since we are still dealing with a nonlinear relationship.

•  It can be applied to networks with sources of different frequencies only if the total response for each frequency is found independently and the results are expanded in a nonsinusoidal expression.

•  One of the most frequent applications of the superposition theorem is to electronic systems in which the DC and AC analyses are treated separately and the total solution is the sum of the two.

 

When a circuit has sources operating at different frequencies, the separate phasor circuit for each frequency must be solved independently, and the total response is the sum of time-domain responses of all the individual phasor circuits. Superposition Theorem applies to AC circuits as well. For sources having different frequencies, the total response must be obtained by adding individual responses in time domain.



These are some Superposition Technique for sources having different frequencies.

All sources except DC 5-V set to zero

  All sources except 10cos(10t) set to zero

All sources except 2 sin 5t set to zero

vo= v1+ v2+ v3

Consider the following example:

The principle of superposition helps us to analyze a linear circuit with more than one independent source by calculating the contribution of each independent source separately. However, to apply the superposition
principle, we must keep two things in mind:


1. We consider one independent source at a time while all other independent sources are turned off. This implies that we replace every voltage source by 0 V (or a short circuit), and every current source by 0 A (or an open circuit). This way we obtain a simpler and more manageable circuit.

2. Dependent sources are left intact because they are controlled by circuit variables. With these in mind, we apply the superposition principle in three steps.


 Steps to Apply Superposition Principle:
 
1. Turn off all independent sources except one source.

2. Repeat step 1 for each of the other independent sources.

3. Find the total contribution by adding algebraically all the contributions due to the independent sources.

 SUPERPOSITION

Transform a voltage source in series with impedance to a current source in parallel with impedance for simplification or vice versa.

Consider the following example: Calculate the current Io.

  

If we transform the current source to a voltage source, we obtain the circuit shown in Fig. (a).

By current division,

A source transformation is the process of replacing a voltage source vs in series with a resistor R by a current source is in parallel with a resistor R, or vice versa.

Voltage Source Transformation

 We will first go over voltage source transformation, the transformation of a circuit with a voltage source to the equivalent circuit with a current source. In order to get a visual example of this, let's take the circuit below which has a voltage source as its power source:

Using source transformation, we can change or transform this above circuit with a voltage power source and a resistor, R, in series, into the equivalent circuit with a current source with a resistor, R, in parallel, as shown below:
 

We transform a voltage source into a current source by using ohm's law. A voltage source can be changed into a current source by using ohm's formula, I=V/R.

Current Source Transformation

We will now go over current source transformation, the transformation of a circuit with a current source to the equivalent circuit with a voltage source. In order to get a visual example of this, let's take the circuit below which has a current source as its power source:

Using source transformation, we can change or transform this above circuit with a current power source and a resistor, R, in parallel, into the equivalent circuit with a voltage source with a resistor, R, in series, as shown below:

We transform a current source into a voltage source by using ohm's law. A voltage source can be changed into a current source by using ohm's formula, V= IR. 

 

 

TO KNOW MORE ABOUT SOURCE TRANSFORMATION:

 

 

MESH ANALYSIS

Mesh Analysis

Mesh analysis is a method that is used to solve planar circuits for the currents  at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. Mesh analysis use Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution.

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.

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

Steps to Determine Mesh Currents:

1. Assign mesh currents 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.

Mesh Analysis with Current Sources

Two possible cases:

• CASE 1 : When a current source exists only in one mesh.

• CASE 2 : When a current source exists between two meshes:  We create a SUPERMESH by excluding the current source and any elements connected in series with it.

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