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**Thévenin’s and Norton’s Theorems**

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**Objective of Lecture State Thévenin’s and Norton Theorems.**

Chapter 4.5 and 4.6 Fundamentals of Electric Circuits Demonstrate how Thévenin’s and Norton theorems can be used to simplify a circuit to one that contains three components: a power source, equivalent resistor, and load.

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Thévenin’s Theorem A linear two-terminal circuit can be replaced with an equivalent circuit of an ideal voltage source, VTh, in series with a resistor, RTh. VTh is equal to the open-circuit voltage at the terminals. RTh is the equivalent or input resistance when the independent sources in the linear circuit are turned off.

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**Circuit Schematic: Thévenin’s Theorem**

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**Definitions for Thévenin’s Theorem**

Linear circuit is a circuit where the voltage is directly proportional to the current (i.e., Ohm’s Law is followed). Two terminals are the 2 nodes/2 wires that can make a connection between the circuit to the load.

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**Definitions for Thévenin’s Theorem**

+ Voc _ Open-circuit voltage Voc is the voltage, V, when the load is an open circuit (i.e., RL = ∞W).

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**Definitions for Thévenin’s Theorem**

Input resistance is the resistance seen by the load when VTh = 0V. It is also the resistance of the linear circuit when the load is a short circuit (RL = 0W).

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**Steps to Determine VTh and RTh**

Identify the load, which may be a resistor or a part of the circuit. Replace the load with an open circuit . Calculate VOC. This is VTh. Turn off all independent voltage and currents sources in the linear 2-terminal circuit. Calculate the equivalent resistance of the circuit. This is RTh. The current through and voltage across the load in series with VTh and RTh is the load’s actual current and voltage in the original circuit.

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Norton’s Theorem A linear two-terminal circuit can be replaced with an equivalent circuit of an ideal current source, IN, in parallel with a resistor, RN. IN is equal to the short-circuit current at the terminals. RN is the equivalent or input resistance when the independent sources in the linear circuit are turned off.

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**Circuit Schematic: Norton’s Theorem**

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**Definitions for Norton’s Theorem**

Short-circuit current Isc is the current, i, when the load is a short circuit (i.e., RL = 0W).

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**Definitions for Norton’s Theorem**

Input resistance is the resistance seen by the load when IN = 0A. It is also the resistance of the linear circuit when the load is an open circuit (RL = ∞W).

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**Steps to Determine IN and RN**

Identify the load, which may be a resistor or a part of the circuit. Replace the load with a short circuit . Calculate ISC. This is IN. Turn off all independent voltage and currents sources in the linear 2-terminal circuit. Calculate the equivalent resistance of the circuit. This is RN. The current through and voltage across the load in parallel with IN and RN is the load’s actual current and voltage in the original circuit.

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Source Conversion A Thévenin equivalent circuit can easily be transformed to a Norton equivalent circuit (or visa versa). If RTh = RN, then VTh = RNIN and IN = VTh/RTh

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**Voltage Polarity and Current Flow**

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**Value of Theorems Simplification of complex circuits.**

Used to predict the current through and voltage across any load attached to the two terminals. Provides information to users of the circuit.

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Example #1

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Example #1 (con’t) Find IN and RN

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**Example #1 (con’t) Calculation for IN**

Look at current divider equation: If RTh = RN= 1kW, then IN = 6mA

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**Why chose RTh = RN? Suppose VTh = 0V and IN = 0mA**

Replace the voltage source with a short circuit. Replace the current source with an open circuit. Looking towards the source, both circuits have the identical resistance (1kW).

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**Source Transformation**

Equations for Thévenin/Norton Transformations VTh = IN RTh IN = VTh/RTh RTh= RN

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**Example #1: Norton’s Theorem**

IN is the current that flows when a short circuit is used as the load with a voltage source IN = VTh/RTh = 6mA

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**Example #1: Norton’s Theorem**

RN is the resistance of the linear circuit when the power sources in the original circuit are turned off (VTh is replaced with a short circuit).

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**Example #1: Norton’s Theorem**

The Norton equivalent circuit is:

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**Check: Thévenin Theorem**

VTh is the voltage across the load when an open short circuit is used as the load with a current source VTh = IN RTh = 6V

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Check: Iload and Vload

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Example #2 Simplification through Transformation

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Example #2 (con’t)

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Example #2 (con’t) Find Req to obtain a Norton equivalent circuit

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**Example #2 (con’t) Current Source to Voltage Source RTh = 3W**

VTh = 0.1A (3W) = 0.3V 0.3V

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Example #2 (con’t) 0.3V

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**Example #2 (con’t) Voltage Source to Current Source RTh = 2W**

IN = 3V/2W = 1.5A

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**Example #2 - Solution 1 Simplify to Minimum Number of Current Sources**

0.3V

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**Example #2 (con’t) Voltage Source to Current Source RTh = 6W**

IN = 0.3V/6W = 50.0mA 0.3V

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Example #2 (con’t)

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Example #2 (con’t) Current Sources in Parallel Add

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**Example #2 - Solution 2 Simplify to Minimum Number of Voltage Sources**

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Example #2 (con’t) Transform solution for Norton circuit to Thévenin circuit to obtain single voltage source/single equivalent resistor in series with load.

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PSpice

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Example #2 - Solution 1

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Example #2 – Solution 2

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Summary Thévenin and Norton transfomrations are performed to simplify a circuit for analysis and design. Two techniques were described. Examples using the source transformation technique were given.

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