Thevenin and Norton Theorems: Circuit Analysis Overview

This document provides a basic description of the Thevenin theorem and the Norton theorem. It includes a step-by-step guide to solving networks using both theorems, which are widely used in circuit analysis. These theorems are particularly useful when you need to determine the current through or voltage across a specific element in a network without solving complex equations.

Thevenin Theorem

The Thevenin theorem states that any two-terminal bilateral linear DC circuit can be replaced by an equivalent circuit consisting of a voltage source and a series resistor.

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Let’s consider an example circuit and derive the load current ($I_L$) using the Thevenin theorem.

Steps to Apply Thevenin Theorem:

1. Find $V_{OC}$ (Open-Circuit Voltage): First, remove the load resistor ($r_L$) and then find the open-circuit voltage ($V_{OC}$) using the following equation:

   $V_{OC} = I \cdot r_3 = \frac{V_s}{r_1 + r_3} \cdot r_3$

2. Find $R_{Th}$ (Thevenin Resistance): Determine the Thevenin resistance of the circuit by replacing the voltage source with a short circuit. If a current source is present, replace it with an open circuit.

3. Find $I_L$ (Load Current): Using the Thevenin equivalent circuit, as shown in Fig. 1(d) (not provided, but assumed), calculate the load current as follows:

   $I_L = \frac{V_{OC}}{R_{Th} + r_L}$

Thevenin Network: Equivalent voltage source in series with internal resistance.

Norton Theorem

The Norton theorem is the converse of the Thevenin theorem. It states that any two-terminal linear circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor.

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Norton Network: Equivalent current source in parallel with internal resistance.

The similarity between the Thevenin and Norton theorems is that the calculation of the internal resistance of the network is the same.

Steps to Apply Norton Theorem:

1. Find $I_{SC}$ (Short-Circuit Current): To determine $I_{SC}$, replace $r_L$ with a short circuit.

   $i = \frac{V_s}{r_1 + \frac{r_2 \cdot r_3}{r_2 + r_3}}$

   $I_{SC} = i \cdot \frac{r_3}{r_3 + r_2}$

2. Find $R_{Th}$ (Thevenin Resistance, which is also $R_{int}$ for Norton): Now, remove the short circuit and deactivate independent sources to determine $R_{int}$ as follows:

   $R_{int} = r_2 + \frac{r_1 \cdot r_3}{r_1 + r_3}$

3. Find $I_L$ (Load Current): According to the Norton theorem, the equivalent source circuit contains a current source in parallel with an internal resistance. The current source is the short-circuit current across the shorted terminals of the load resistor.

   $I_L = I_{SC} \cdot \frac{R_{int}}{R_{int} + r_L}$