Norton equivalent resistance formula

Explore the Norton equivalent resistance formula, its significance in circuit analysis, derivation steps, and a practical calculation example.

Understanding the Norton Equivalent Resistance Formula

The Norton equivalent resistance formula is an important concept in electrical engineering, enabling a simplified representation of complex circuits. This article will discuss the significance of the Norton equivalent resistance formula, its theoretical background, and the steps to derive it.

The Norton Equivalent Circuit

A Norton equivalent circuit is a current source in parallel with a resistor, representing a linear, two-terminal network. The concept is based on the principle that any linear, bilateral network can be replaced by an equivalent current source and a resistor connected in parallel. The main advantage of using the Norton equivalent circuit is that it simplifies complex circuit analysis by converting the circuit into a more manageable form.

Norton’s Theorem

Norton’s Theorem is a powerful analytical tool in electrical circuit analysis, stating that any linear, two-terminal network can be replaced by an equivalent current source in parallel with a resistor. The theorem was developed by E. L. Norton, an American engineer working at Bell Labs, in 1946. The theorem is closely related to Thévenin’s theorem, which represents a linear, two-terminal network as a voltage source in series with a resistor.

Deriving the Norton Equivalent Resistance Formula

1. Begin by short-circuiting the voltage sources in the original network and open-circuiting the current sources.
2. Next, identify the terminals of interest (usually marked as A and B).
3. Apply an external current source or voltage source between the terminals of interest, and calculate the resultant current or voltage.
4. Using Ohm’s law, calculate the equivalent resistance (R_N) between the terminals of interest. The formula is given by:

R_N = V_OC / I_SC

where V_OC is the open-circuit voltage and I_SC is the short-circuit current.

Key Points to Remember

• The Norton equivalent resistance formula is essential for simplifying complex electrical circuits.
• Norton’s theorem is the basis for the Norton equivalent resistance formula and states that any linear, two-terminal network can be replaced by an equivalent current source in parallel with a resistor.
• The equivalent resistance in the Norton equivalent circuit can be found by applying an external source between the terminals of interest and using Ohm’s law to calculate the resistance.
• The Norton equivalent resistance formula is closely related to Thévenin’s theorem, which represents a linear, two-terminal network as a voltage source in series with a resistor.

In conclusion, the Norton equivalent resistance formula is a crucial tool in electrical circuit analysis, allowing engineers to simplify complex networks and perform calculations more efficiently. Understanding the theorem’s background and derivation steps helps to ensure accurate application in practical situations.

Example of Calculating Norton Equivalent Resistance

Let’s consider a simple electrical circuit consisting of two resistors (R₁ = 4Ω and R₂ = 6Ω) and a voltage source (V_S = 10V) connected in series. We will calculate the Norton equivalent resistance for this circuit.

1. First, deactivate the voltage source by replacing it with a short circuit. The original circuit now consists of two resistors connected in series.
2. Next, identify the terminals of interest (A and B). In this example, they are the open ends of the resistors R₁ and R₂.
3. Now, apply an external current source or voltage source between the terminals of interest. In this case, we will use a 1A current source (I_ext) connected between terminals A and B.
4. Calculate the resultant voltage across the terminals A and B. Since the resistors R₁ and R₂ are connected in series, the total resistance (R_T) is the sum of the individual resistances:

R_T = R₁ + R₂ = 4Ω + 6Ω = 10Ω

Now, use Ohm’s law to calculate the voltage across the terminals:

V_AB = I_ext × R_T = 1A × 10Ω = 10V

5. Finally, calculate the Norton equivalent resistance (R_N) using the formula:

R_N = V_AB / I_ext = 10V / 1A = 10Ω

In this example, the Norton equivalent resistance of the given circuit is 10Ω. With this information, we can now represent the original circuit as a Norton equivalent circuit, simplifying further analysis.