Week 10: Thevenin and Norton's Theorem

Thevenin's Theorem

What is Thevenin'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). If we're dealing with passive components (such as resistors, and later, inductors and capacitors), this is true. However, there are some components (especially certain gas-discharge and semiconductor 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.

Simple Steps to Analyze Electric Circuit through Thevenin’s theorem.

* Open the load resistor

* Calculate / measure the Open Circuit Voltage. This is the Thevenin Voltage (V_TH)

* Open Current Sources and Short Voltage Sources.

* Calculate /measure the Open Circuit Resistance. This is the Thevenin Resistance (R_TH)

* Now, Redraw the circuit with measured open circuit Voltage (V_TH) in Step (2) as voltage Source and measured open circuit resistance (R_TH) in step (4) as a series resistance and connect the load resistor which we had removed in Step (1). This is the Equivalent Thevenin Circuit of that Linear Electric Network or Complex circuit which had to be simplified and analyzed. You have done.

* Now find the Total current flowing through Load resistor by using the Ohm’s Law IT = V_TH/ (R_TH + R_L)

Example:

Find V_TH, R_THand the load current flowing through and load voltage across the load resistor in fig (1) by using Thevenin’s Theorem.

Solution:

* Step 1

Open the 5kΩ load resistor (Fig 2)

Step 2

Calculate / measure the Open Circuit Voltage. This is the Thevenin Voltage (V_TH). Fig (3)

We have already removed the load resistor from figure 1, so the circuit became an open circuit as shown in fig 2. Now we have to calculate the Thevenin’s Voltage. Since 3mA Current flows in both 12kΩ and 4kΩ resistors as this is a series circuit because current will not flow in the 8kΩ resistor as it is open.

So 12V (3mA  x 4kΩ) will appear across the 4kΩ resistor. We also know that current is not flowing through the 8kΩ resistor as it is open circuit, but the 8kΩ resistor is in parallel with 4k resistor. So the same voltage (i.e. 12V) will appear across the 8kΩ resistor as 4kΩ resistor. Therefore 12V will appear across the AB terminals. So,

V_TH = 12V 

Step 3

Open Current Sources and Short Voltage Sources. Fig (4)

Step 4

Calculate /measure the Open Circuit Resistance. This is the Thevenin Resistance (R_TH)

We have Reduced the 48V DC source to zero is equivalent to replace it with a short in step (3), as shown in figure ()  We can see that 8kΩ resistor is in series with a parallel connection of 4kΩ resistor and  12k Ω resistor. i.e.:

8kΩ + (4k Ω || 12kΩ) ….. (|| = in parallel with)

R_TH = 8kΩ + [(4kΩ x 12kΩ) / (4kΩ + 12kΩ)]

R_TH = 8kΩ + 3kΩ

R_TH = 11kΩ

Step 5

Connect the R_THin series with Voltage Source V_TH and re-connect the load resistor. This is shown in fig (6) i.e. Thevenin circuit with load resistor.

Step 6

Now apply the last step i.e. calculate the total load current & load voltage as shown in fig 6.

I_L = V_TH/ (R_TH + R_L)

= 12V / (11kΩ + 5kΩ) → = 12/16kΩ

I_L= 0.75mA

And

V_L = I_Lx R_L

V_L = 0.75mA x 5kΩ

V_L= 3.75V

Norton's Theorem

What is Norton's Theorem?

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

Contrasting our original example circuit against the Norton equivalent: it looks something like this:

. . . after Norton conversion . . .

Steps to follow for Norton's Theorem:

• (1) Find the Norton source current by removing the load resistor from the original circuit and calculating current through a short (wire) jumping across the open connection points where the load resistor used to be.
• (2) Find the Norton 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 Norton equivalent circuit, with the Norton current source in parallel with the Norton 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 parallel circuits.

Reflection/Learnings:

Thevenin's Theorem pretty much  simplifies an entire circuit no matter how complex into an equivalent circuit with only one Voltage source (Vth). In Thevenin's theorem, We also change one resistor into an open circuit.  It does this because we need to measure the open circuit voltage of the circuit. In Norton's theorem the same concept is pretty much applied but instead of turning one load into an open circuit and measuring the open circuit voltage, We turn it into a short circuit and then we measure the short circuit current. In this section however, I have only discussed the case 1 of Thevenin's Equivalent and in the next section, I will discuss the case 2 of Thevenin's Equivalent as well as explain both cases.

Video:

Introduction to Thevenin's Theorem

Introduction to Norton's Theorem

Engineering problems are under-defined, there are many solutions, good, bad and indifferent. The art is to arrive at a good solution. This is a creative activity, involving imagination, intuition and deliberate choice.

--Ove Arup