Voltage Divider Rule

In this tutorial, I am going to introduce a top-notch technique for solving series circuits. You can solve a series circuit easily by applying Ohm's law. But Ohm's law has a limitation. You have to know the current flowing through the circuit. This method allows you to find out the voltage across any series component without knowing the current. This method is derived from two very common circuit solving laws, that is Ohm's law and KVL. In this article, I am going to derive the expression of the voltage divider rule as well.

Explanation:

It is only applicable to series circuits where the current remains the same throughout the circuit. Consider a series circuit given below.

Let,

v1 = voltage across R1

v2 = voltage across R2

v3 = voltage across R3

i = total current across the circuit

According to KVL,

v = v1 + v2 + v3

v = iR1 + iR2 + iR3

v = i (R1 + R2 + R3)

i = v/ (R1 + R2 + R3) ….. equation 1

If you want to evaluate voltage across R2, then using Ohm's law

v2 = iR2

i = v2/R2 ….. equation 2

Look at equation 1 and equation 2. Substitute i.

v2/R2 = v/ (R1 + R2 + R3)

v2 = (R2*v) / (R1 + R2 + R3)

Similarly,

v1 = (R1*v) / (R1 + R2 + R3)

v3 = (R3*v) / (R1 + R2 + R3)

This another simple method called the voltage division rule. And the circuit is called a voltage divider because the total voltage is divided into resistances. The larger the resistance, the larger the voltage drop across that resistance.

Solved Examples:

In the previous article (resistors in series), I analysed and solved a series circuit using Ohm's law. In this tutorial, I am going to solve the same examples with the help of the voltage divider rule.

Example 1:

A 12 V source is connected with these resistances: 1kΩ, 2kΩ and 4kΩ. How much current flows through the 4kΩ resistance?

I solved this problem in my previous tutorial.

Another way to solve this problem.

The total resistance will be the sum of all individual resistances.

RT = R1 + R2 + R3
RT = (1+2+4) kΩ
RT = 7 kΩ

Voltage across 4kΩ or R3 = v3

v3 = (R3/RT).V

v3 = (4k/7k).12

v3 = 6.85V

Current flows through R3

i = v3/R3

i = 6.85/4k

i = 1.7mA

Note: The current 1.7mA is the total current flowing through the series circuit.

Series Resistors | Equivalent Series Resistance | Formula Of Series Resistance

Current & Voltage Through Series Resistors: Analysis of Series Circuits:

Before starting my tutorial on series-connected resistors, I would like to recall series circuits.

“Two elements connected in such a way that they share a single node exclusively.”

Similarly, two resistors are connected in series, when they share a single node exclusively.

Properties Of Series Connected Resistors:

• Current remains the same in every series-connected resistor
• Another property of the series circuit is the voltage division property. The voltage is divided proportionally to all resistors. The larger the resistance value, the greater will be the voltage drop across that resistance.

Formula Of Resistance In Series:

Apply Ohm's law to the circuits below,

Circuit 1:

The voltage drop across R1 is va

va = iR1

The voltage drop across R2 is vb

vb = iR2

The total voltage across circuit the is the sum of voltage drops across R1 and R2

v1 = va + vb … equation 1

Circuit 2:

Voltage across Req

v2 = iReq … equation 2

Observations and Calculations:

Have a close look at both circuits. v1 and v2 are the same that is 10V. I take same the same voltage sources, and the current flowing through the circuit remains the same. The difference is the number of resistances in the circuits. Circuit 1 has R1 and R2. and Circuit 2 has only Req.

Look at circuits 1 and 2,

v1 = v2

va + vb = v2

iR1 + iR2 = iReq

From observations and calculations, we can conclude that

Req = R1 + R2

Figure: Electrically equivalent circuits

Implying that we can replace R1 and R2 with a single equivalent resistor. Similarly, we can replace N series resistors with a single equivalent resistor. This is because Req has the same voltage drop across terminals a & b. Also, current and power relationships in the equivalent circuit will remain the same.

The equivalent resistance of a series-connected resistor is the algebraic sum of all the individual resistances.

Req = R1 + R2 + R3 + …..RN

Analysis Of Some Confusing Series Circuits:

Circuit 1:

Circuit 1 is a series circuit. All 5 components are serially connected.

Circuit 2:

It is not a series circuit.

Circuit 3:

It is also a series circuit. It contains multiple sources and resistors. All are connected in series with each other.

Circuit 4:

It is not a series circuit.

Solved Examples:

Example 1:

A 12 V source is connected with these resistances: 1kΩ, 2kΩ and 4kΩ. How much current flows through the 4kΩ resistance?

Solution:

Figure: Draw the circuit with the help of given data

Before calculating the current, you have to find out the total or equivalent resistance of the circuit. After evaluating the total or equivalent resistance, apply Ohm's Law to find out the current through the circuit.

The total resistance will be the sum of all individual resistances.

RT = R1 + R2 + R3

RT = (1+2+4) kΩ

RT = 7 kΩ

Current remains the same  in a series circuit

Current through the 4kΩ is:

i = v/RT

i = 12/7

i = 1.7 mA

Example 2:

From the following data, evaluate the resistance value that should be connected in series to limit the load current to 5mA.

RL = load resistance = 250Ω

RS = series resistance =?

IT = total current required by the load resistance

First of all, draw a circuit and label it with the given data.

Figure: Circuit diagram for example 2

RT = total resistance = RS + RL = RS + 250

RS = RT -250

IT = V/RT

5*10-3 = 5/RT

RT = 1000Ω

RS = 750Ω

Conclusion:

I discussed series-connected resistors in detail. As far as practical applications are concerned, resistors in series are frequently used to limit the voltage. Such circuits are called voltage divider circuits. In complex circuits, we frequently use this technique to simplify circuit analysis.
 There is another way of solving series circuits without knowing the current values. This method is known as Voltage Divider Rule.