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Voltage Divider Rule Derivation

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Understanding the Voltage Divider Rule: A Simple Derivation



The voltage divider rule (VDR) is a fundamental concept in electrical engineering. It allows us to easily calculate the voltage across any resistor in a series circuit. Understanding this rule simplifies circuit analysis significantly, saving time and effort in more complex designs. This article provides a clear and step-by-step derivation of the voltage divider rule, illustrated with practical examples.


1. The Foundation: Ohm's Law and Series Circuits



Before diving into the voltage divider rule, we need a solid understanding of Ohm's Law and the behavior of resistors in series circuits.

Ohm's Law: This states that the voltage (V) across a resistor is directly proportional to the current (I) flowing through it and its resistance (R). Mathematically, V = IR.

Series Circuits: In a series circuit, components are connected end-to-end, forming a single path for current flow. The current is the same through all components in a series circuit. The total resistance (R<sub>T</sub>) is the sum of the individual resistances (R<sub>1</sub> + R<sub>2</sub> + R<sub>3</sub>...).


2. Deriving the Voltage Divider Rule



Consider a simple series circuit with two resistors, R<sub>1</sub> and R<sub>2</sub>, connected to a voltage source V<sub>S</sub>. The current (I) flowing through both resistors is the same, according to the rules of series circuits. Using Ohm's Law, we can express this current as:

I = V<sub>S</sub> / (R<sub>1</sub> + R<sub>2</sub>)

This equation represents the total current flowing through the circuit, determined by the total resistance (R<sub>1</sub> + R<sub>2</sub>) and the source voltage (V<sub>S</sub>).

Now, let's find the voltage across R<sub>1</sub> (V<sub>R1</sub>). Using Ohm's Law again:

V<sub>R1</sub> = I R<sub>1</sub>

Substitute the expression for I from the previous equation:

V<sub>R1</sub> = [V<sub>S</sub> / (R<sub>1</sub> + R<sub>2</sub>)] R<sub>1</sub>

Simplifying this gives us the voltage divider rule for V<sub>R1</sub>:

V<sub>R1</sub> = V<sub>S</sub> [R<sub>1</sub> / (R<sub>1</sub> + R<sub>2</sub>)]

Similarly, the voltage across R<sub>2</sub> (V<sub>R2</sub>) can be derived as:

V<sub>R2</sub> = V<sub>S</sub> [R<sub>2</sub> / (R<sub>1</sub> + R<sub>2</sub>)]


3. Extending the Rule to Multiple Resistors



The voltage divider rule can be easily extended to circuits with more than two resistors in series. For a series circuit with 'n' resistors, the voltage across any resistor R<sub>x</sub> is:

V<sub>Rx</sub> = V<sub>S</sub> [R<sub>x</sub> / (R<sub>1</sub> + R<sub>2</sub> + ... + R<sub>n</sub>)]


4. Practical Examples



Example 1: A 12V battery is connected to two resistors in series: R<sub>1</sub> = 4Ω and R<sub>2</sub> = 8Ω. Find the voltage across R<sub>1</sub>.

Using the voltage divider rule:

V<sub>R1</sub> = 12V [4Ω / (4Ω + 8Ω)] = 4V

Therefore, the voltage across R<sub>1</sub> is 4V.

Example 2: A 24V supply is connected to three resistors: R<sub>1</sub> = 2Ω, R<sub>2</sub> = 4Ω, and R<sub>3</sub> = 6Ω. Find the voltage across R<sub>2</sub>.

V<sub>R2</sub> = 24V [4Ω / (2Ω + 4Ω + 6Ω)] = 8V

The voltage across R<sub>2</sub> is 8V.


5. Key Takeaways



The voltage divider rule simplifies voltage calculations in series circuits.
It's based on Ohm's Law and the properties of series circuits.
The rule can be easily extended to circuits with multiple resistors.
Understanding the derivation helps in applying the rule correctly and efficiently.


Frequently Asked Questions (FAQs)



1. Can the voltage divider rule be used for parallel circuits? No, the voltage divider rule is specifically for series circuits. In parallel circuits, the voltage across each branch is the same as the source voltage.

2. What happens if one of the resistors is open-circuited? If a resistor in a series circuit opens, no current will flow, and the voltage across the open resistor will be equal to the source voltage. The voltage across the other resistors will be zero.

3. Is the voltage divider rule accurate in real-world scenarios? The rule provides a good approximation, but real-world components have tolerances and parasitic effects that can introduce small errors.

4. Can the voltage divider rule be used with AC circuits? Yes, the voltage divider rule applies to AC circuits as well, provided the impedance of the components is used instead of resistance.

5. What are the limitations of the voltage divider rule? The rule is primarily applicable to simple series circuits. It doesn't directly apply to circuits containing dependent sources or more complex topologies. Loading effects from measuring instruments can also affect accuracy.

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