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Thevenin's Theorem

This page is all about this simple lines " If two connections lead away from an arbitrary connection of batteries and resistors, then the electrical effects of the circuit in the box on whatever is connected to the box is just the same as if in the box were merely a single battery connected in series to a single resistance".

Thevenin's Theorem

To illustrate it, you might have seen the combination of linear bilateral circuit elements and active sources, regardless of the connection or complexity, connected to a given load RL. Here in this theorem we can replace the the whole complicated circuit in to simple circuit that has a single voltage source of VTH volts and a simple impedance Req in series with the voltage source, across the two terminals of the load RL. Lets see how its done....

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What is Thevenin's Theorem?

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Thevenin's theorem is popularly known for analyzing the power systems and other complicated circuits where we could determine the value of load resistance which helps to calculate voltage and current across it. It states that
"Any combination of linear network that is bilateral having circuit elements connected with active sources, regardless of the complexity, applied across a given load RL, may be replaced by a simple two terminal network that consists of a single voltage source VTH in series with a single impedance RTH, connected across the two terminals having load resistance RL. The VTH is the open circuit voltage measured at the two terminals of interest, with load impedance ZL removed. Here voltage VTH is Thevenin's equivalent voltage and RTH is the equivalent impedance".

How to Find Thevenin's Equivalent Circuit?

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The Thevenin's equivalent circuit is the electrical equivalent circuit of resistances connected across the load resistance. To get the Thevenin equivalent circuit we need to first remove the power supply connected across the original circuit, voltage sources should be short circuited, while current sources should be open circuited and total resistance should be determined between the open connection points RTH. This equivalent circuit simplifies the entire circuit into a circuit of a single voltage source, series resistance and series load.

Thevenin's Equivalent Circuit


Thevenin Voltage

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We often see the circuit when open circuited i.e terminals when not connected to anything, there would be no flow of current in the circuit but there would the voltage across the terminals what we call open circuit voltage. The Thevenin voltage is an ideal voltage source that would be equal to voltage when open circuited at the terminals.
Thevenin Voltage 
Thevenin Voltage

Thevenin's and Norton Equivalent Circuit

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Thevenin's & Norton Equivalent Circuit
Thevenin and Norton Equivalent Circuit

In the circuits, Thevenin voltage and Norton current can be expressed as 
VTH = IN RN
or
VTH = IN RTH
Where VTH is Thevenin equivalent voltage, IN is Norton current and RTH = RN (RN is Norton resistance)

From the Thevenin and Norton's equivalent circuit we could determine 
  • Thevenin resistance is equal to and Norton resistance (RTH = RN)
  • Thevenin voltage VTH is equal to Norton current IN times Norton resistance RN (VTH = IN RN)
  • Norton current is equal to Thevenin voltage divided by Thevenin resistance.

How to Simplify into Thevenin and Norton Circuit?

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Lets see how to simplify the simple circuit into Thevenin and Norton's circuit:
Step 1: First open the terminal AB, where we need current or voltage and calculate VAB using mesh method. Hence we get
VAB = VTH

Step 2: 
Short the terminal AB and determine the short circuit current (ISC = IN)

Step 3:
 Now calculate RTH = $\frac{V_{TH}}{I_{SC}}$ = RN
           
            Equivalent resistance across AB is 
RAB = RTH

Step 4:
 Draw the Thevenin or Norton equivalent circuit as required and calculate the load current.

Thevenin's Examples

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Lets see some examples on Thevenin's circuit:

Example 1:

Calculate the equivalent voltage for the given Thevenin circuit:

Thevenin's Example

Solution:


Open the terminals AB and then calculate the voltage across AB,

- VAB + 18 + 3 $\times$ 0 + 2 $\times$ 6 = 0

VAB = 30 V = VTH
Example 2:  

Determine the VTH, RTH and norton equivalent IL in the thevenin circuit:

Thevenin's Problem

Solution:

Short the terminal AB,

Applying mesh analysis,

- 20 + 2I + 2 (I - Isc) = 0

4I - 2 Isc = 20

2I -Isc = 10

Now short the terminal AB. Applying mesh analysis,

-20 + 2I + 2 (I-Isc) = 0

4I - 2Isc = 20

2I - Isc = 10.......0

2 (Isc - I) + 2 Isc - 10 = 0

2Isc - I = 5 ...... (2)

By equations (1) and (2), we get Isc = 20/3 A = IN

Now RTH = RN = $\frac{V_{TH}}{I_{sc}}$ = $\frac{20 \times 3}{20}$ = 3 $\Omega$

The thevenin equivalent circuit is given by

IL = $\frac{V_{TH}}{R_{TH} + R_L}$

    = $\frac{20}{3 + 1}$

    = 5A

The Norton equivalent circuit is determined using current division formula

IL = $\frac{R_N}{R_N + R_L}$ $\times$ IN

   = $\frac{3}{3 + 1}$ $\times$ $\frac{20}{3}$

   = 5 A.
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