Voltage Division Rule | Voltage in a Series Circuit

One of the basic concepts of circuit analysis is the voltage division rule. It can be used with all series and combination circuits.

A voltage divider is always present in a series circuit. Each resistance in a series circuit receives the same amount of current.

As a result, the voltage drop across each resistor is proportional to the ohmic value of the resistor.

The current flowing in a series circuit is proportional to the overall resistance offered by the circuit.

According to ohm’s law, the voltage across each resistor is equal to the current flowing through the circuit multiplied by the resistance value.

Voltage division rule is very famous because it is used for voltage divider circuits. Voltage divider circuit has very wide applications in electrical systems. This circuit works with a simple method and comes with a very simple calculation.

You will often find or even use this circuit for many applications. A voltage divider circuit is able to produce a specific percentage of the maximum voltage from the circuit to our liking.

Voltage divider circuit shares the same circuit as a series circuit. The current in series circuit is always the same for every resistance it passes through. As opposed to parallel circuits, the voltage drop for each resistor is different from each other depending on its resistance.

The total current in the series circuit is divided by total resistance in the circuit.

What is Voltage Division Rule

According to the Voltage Division Rule,

The entire voltage supplied across a series connection of numerous resistors is proportionally divided across the resistors.

This indicates that the voltage drop will be greatest across the resistor with the highest resistance value. Similarly, it will be the smallest for the resistor with the lowest resistance value.

The voltage drop across individual resistance in a series connected resistance is calculated using the Voltage Division Rule. This rule holds true for both AC and DC circuits.

However, in an AC circuit, the impedance value should be regarded rather than the resistance value for obvious reasons.

Voltage in a Series Circuit

The voltage across each resistor can be calculated with the famous Ohm’s Law. It states that the voltage across a resistor is the product between the total current passing that resistor and its resistance.

Before we go on to the voltage division rule, we must understand how voltage, current, and resistance are used to determine the output. We can use the simple example below.

Given that a circuit has n resistor,

voltage division rule

The total resistance of the series circuit will be:

    \begin{align*}R_{eq}&=R_{1}+R_{2}+...+R_{n}\end{align*}

The total current of the series circuit will be:

(1)   \begin{align*}I=\frac{V}{R_{eq}}\end{align*}

The voltage across the R1 will be:

(2)   \begin{align*}V_{1}=I\times R_{1}\end{align*}

The voltage across the R2 will be:

(3)   \begin{align*}V_{2}=I\times R_{2}\end{align*}

The voltage across the Rn will be:

(4)   \begin{align*}V_{n}=I\times R_{n}\end{align*}

Substituting (1) into (2) makes

    \begin{align*}V_{1}=V\times\frac{R_{1}}{R_{eq}}\end{align*}

Substituting (1) into (3) makes

    \begin{align*}V_{2}=V\times\frac{R_{2}}{R_{eq}}\end{align*}

Substituting (1) into (4) makes

    \begin{align*}V_{n}=V\times\frac{R_{n}}{R_{eq}}\end{align*}

Where:

Vn = voltage drop across the n-th resistor

Rn = resistance of the n-th resistor.

The sum voltage drop across the n series resistors is equal to the ratio of total current divided by equivalent resistance of the resistors. From the formula above we conclude that,

The voltage drop across an n-th resistor is the product between input voltage and the resistance of the n-th resistor divided by equivalent series resistance.

Read also : laplace transform definition

Voltage Division Rule Calculation

Consider a circuit with three resistors R1, R2, and R3 connected in series and a voltage V applied across it for a better understanding.

voltage division rule

The voltage drop across each resistance is what we’re looking for. The voltage drop across resistances R1, R2, and R3 is represented by VR1, VR2, and VR3, respectively.

VR1, VR2, and VR3 should be proportional to R1, R2, and R3 according to the Voltage Division Rule. As a result, we may write

(1)   \begin{align*}V_{R1}=I\times R_{1}\end{align*}

(2)   \begin{align*}V_{R2}=I\times R_{2}\end{align*}

(3)   \begin{align*}V_{R3}=I\times R_{3}\end{align*}

However, the total voltage V across terminals x-y must equal the sum of the voltage drops across each resistance. As a result, we can write,

    \begin{align*}V=V_{R1}+V_{R2}+V_{R3}\end{align*}

We get by multiplying the values of VR1, VR2, and VR3 from (1), (2), and (3).

    \begin{align*}V=\frac{I}{R_1+R_2+R_3}\end{align*}

    \begin{align*}I=\frac{V}{R_1+R_2+R_3}\end{align*}

The voltage distribution across the individual resistance can be found using (1), (2), and (3), as shown below.

    \begin{align*}V_{R1}=V\frac{R_1}{R_1+R_2+R_3}\\V_{R2}=V\frac{R_2}{R_1+R_2+R_3}\\V_{R3}=V\frac{R_3}{R_1+R_2+R_3}\end{align*}

Though the above voltage distribution was estimated using three resistors in series, the method can be used with any number of series linked resistances in a DC circuit or impedance in an AC circuit.

Voltage Division Rule Formula

The voltage division rule formula for “n” series connected resistances is shown below.

    \begin{align*}V_{R1}=V\frac{R_1}{R_1+R_2+R_3+...R_n}\\V_{R2}=V\frac{R_2}{R_1+R_2+R_3+...R_n}\\V_{R3}=V\frac{R_3}{R_1+R_2+R_3+...R_n}\\..............................\\..............................\\V_{Rn}=V\frac{R_n}{R_1+R_2+R_3+...R_n}\\\end{align*}

Pay close attention to the formula above. If we wish to find voltage across any one of the resistances (say R1), we multiply the total voltage (V) by the ratio of another resistance (R1) and total resistance (R1+ R2+ R3+……+ Rn).

Voltage Division Rule Formula Example

For a better explanation, observe the circuit example below:

1.Circuit below consists of 2 resistors with different resistance.

voltage division rule

The total resistance or equivalent resistance is

    \begin{align*}R_{eq}&=R_{1}+R_{2}\\&=3+2\\&=5\Omega\end{align*}

The total current is

    \begin{align*}I&=\frac{V}{R_{eq}}\\&=\frac{5}{5}=1A\end{align*}

The voltage drop across the R1 will be

    \begin{align*}V_{1}&=I\times R_{1}\\&=V\times\frac{R_{1}}{R_{eq}}\\&=5\times\frac{3}{5}\\&=3V\end{align*}

The voltage drop across the R2 will be

    \begin{align*}V_{2}&=I\times R_{2}\\&=V\times\frac{R_{2}}{R_{eq}}\\&=5\times\frac{2}{5}\\&=2V\end{align*}

2.Circuit below consists of 3 resistors with the same resistance.

voltage division rule

The equivalent resistance will be:

    \begin{align*}R_{eq}&=R_{1}+R_{2}+R_{3}\\&=3+3+3\\&=9\Omega\end{align*}

The total current will be

    \begin{align*}I&=\frac{V}{R_{eq}}\\&=\frac{9}{9}=1A\end{align*}

The voltage drops for each resistor will be

    \begin{align*}V_{1}=V_{2}=V_{3}&=V\times\frac{R_{1}}{R_{eq}}\\&=9\times\frac{3}{9}\\&=3V\end{align*}

Since the resistances are the same, the voltage drops for each resistor are the same.

3.Circuit below is one we call a voltage divider circuit. We only add a terminal on one resistor to get desired voltage.

voltage division rule

Get straight to the point, the Vout will be

    \begin{align*}V_{out}&=V_{2}\\&=V\times\frac{R_{2}}{R_{1}+R_{2}}\\&=5\times\frac{3}{5}\\&=3V\end{align*}

Frequently Asked Questions

What is the voltage division rule formula?

Voltage division rule formula for n-th resistor is
Vn = Vx(Rn/Req)

How do you use the voltage division rule?

Voltage division rule is used for series circuits such as voltage divider circuits. It calculates the voltage drop of a desired resistor in a series circuit.

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