Series Circuit Formula

A series circuit is one of the most important concepts of electrical and electronics courses. In a series circuit, all the components are sequentially arranged and connected with each other to form a single current path. The installed total resistance is the sum of all individual resistors’ resistances. Hence, the total voltage drop is also the sum of the individual voltage drops across the respective resistors. This article will cover the equations, simplifications, and uses of series circuit systems.

What is Series Circuit?

Series circuit is an electrical circuit configuration where components are linked to each other end-to-end, creating a single path for the current flow. Thus, the current in the circuit will flow through every component successively. In a series circuit, the total resistance, which is the sum of the individual resistances of the components in the circuit, is increased as the number of resistors connected in the circuit increases.

• This is because the current goes on through each resistor, one after the other, then the resistance it will face is the sum of the resistances of each resistor.
• This is because the voltage drop in each element on which current is flowing is constant and proportional to the element’s resistance.
• A key characteristic of the series circuit is that if one part or component fails or is removed, the circuit would be broken so that no current will flow through it.

Series Circuit Formula

In a series circuit, all the current moves through each individual component because there is only one unidirectional path for the current movement. In a series circuit, the total resistance (R_s) is the sum of the individual resistance of the resistors (R₁, R₂, R₃,…R_n):

R_s = R₁ + R₂ + R₃ + … + R_n

The total voltage drop (V_s) in a series circuit is equal to the sum of the individual voltage drops (V₁, V₂, V₃,…V_n):

V_s = V₁ + V₂ + V₃ + … + V_n

Derivation of Series Circuit formula

In a series circuit, the current is the same at every point because there is only one path for the current to flow. We can use this fact to derive the formula for the total resistance in a series circuit.

Let’s assume we have a series circuit with n resistors, each with R₁, R₂, R₃, …, R_n resistance. The total resistance of the circuit, which we will denote as R_T, is the sum of the individual resistances:

R_T = R₁ + R₂ + R₃ + … + R_n

We can derive this formula by considering the voltage drop across each resistor. The formula gives the voltage drop across a resistor:

V = IR

Where V is the voltage drop, I is the current, and R is the resistance. Since the current is the same at every point in the circuit, we can use the same current I in each voltage drop formula. Therefore, the voltage drop across each resistor is:

V₁ = I * R₁

V₂ = I * R₂

V₃ = I * R₃

…

V_n = I * R_n

The total voltage drop across the entire circuit is the sum of the voltage drops across each resistor:

V_T = V₁ + V₂ + V₃ + … + V_n

Substituting the voltage drop formulas for each resistor, we get:

V_T = I × R₁ + I × R₂ + I × R₃ + … + I × R_n

We can factor out the current I from each term to get:

V_T = I × (R₁ + R₂ + R₃ + … + R_n)

The total resistance R_T is defined as the ratio of the total voltage drop V_T to the current I:

R_T = V_T / I

Substituting the expression for V_T, we get:

R_T = I × (R₁ + R₂ + R₃ + … + R_n) / I

The current I cancels out, leaving us with the formula for the total resistance in a series circuit:

R_T = R₁ + R₂ + R₃ + … + R_n

This equation shows that the total resistance of a series circuit is straightforward addition of the individual resistances. This is an important property of series circuits because it allows us to obtain the total resistance of a circuit by adding the individual resistances of its elements.

Three Rules of Series Circuit

The Three rules of the series circuit are as follows:

Current in Series Circuits

The current (I) in a series circuit is the same at every point. Ohm’s law can be applied to determine the current through a resistor when the voltage and resistance are known:

I = V / R

For example, if a resistor has a voltage of 9V across it and a resistance of 3kΩ, the current through the resistor can be calculated as follows:

I = 9V / 3kΩ = 0.003A or 3mA

Voltage in Series Circuits

The voltage drop across each resistor in a series circuit is directly proportional to the size of the resistor. The voltage drop (V_R) across a resistor in a series circuit can be calculated using Ohm’s law:

V_R = I × R

For example, if the current through a resistor is 3mA and the resistance is 3kΩ, the voltage drop across the resistor can be calculated as follows:

V_R = 0.003A × 3kΩ = 9V

Resistance in Series Circuits

The total resistance in a series circuit equals the sum of the individual resistors. This is because the current flowing through each resistor is the same. The formula for calculating the total resistance in a series circuit is:

R_s = R₁ + R₂ + R₃ + … + R_n

For example, if a circuit has three resistors in series with resistances of 4 ohms, 8 ohms, and 2 ohms, the total resistance can be calculated as follows:

R_s = R₁ + R₂ + R₃

R_s = 4 ohms + 8 ohms + 2 ohms

R_s = 14 ohms

Applications of Series Circuit

Series circuits have various applications, including:

• Lighting Circuits: The series circuit is present in the lighting boards, where the bulbs are connected in series. An event such as the burning out of one bulb will result in the electrical circuit, which will, in turn, interrupt the connection, and all bulbs will be turned off. This feature is often seen in older holiday light strings.
• Voltage Dividers: A series circuit is common where a voltage divider circuit is used, where the divider output voltage is a fraction of the input voltage. This technique is accomplished by building a series circuit with a couple of resistors, resulting in the voltage across one resistor.
• Temperature Sensors: A concrete example of a series circuit is temperature sensors, where the change in resistance of a resistor that changes with temperature (thermistor) is measured to determine the true value of temperature. There are two thermistors in the circuit. The thermistor is placed both in series with the fixed resistor and in the voltage divider circuit.
• Battery Packs: Series circuits are a tool to increase the battery’s total output voltage. The voltage gets steadily higher as you connect the batteries in series. This way, a high output voltage is achieved, resulting in many applications, including electric cars and mobile electronics.
• Electrical Safety Devices: Series circuits include fuses and circuit breakers. The latter are devices placed in the circuit that break it whenever a short circuit or overload is detected to protect the components.

Difference between Series circuit and Parallel circuits

The difference between series circuit and parallel circuits are as follows:

Series Circuit	Parallel Circuit
Components are connected end-to-end in a line	Components are connected parallel to each other
All components have the same current flowing through them	The current through each component is different
If one component breaks down, the whole circuit burns out	Other components will function even if one component breaks down
Components share one electrical node with their nearest neighbor	Components share two common nodes
Total resistance is equal to the sum of individual resistors	Total resistance is less than the smallest individual resistor

Solved Examples on Series Circuit Formula

Example 1: A series circuit consists of a 12V battery, a 4Ω resistor, and a 6Ω resistor. Find the total resistance, current, and voltage drop across each resistor.

Solution:

1. Total resistance in a series circuit is equal to the sum of the individual resistances:

Total resistance, R_T = R₁ + R₂ = 4Ω + 6Ω = 10Ω

2. Using Ohm’s law, the current in the circuit is:

Current, I = V / R_T = 12V / 10Ω = 1.2A

3. The voltage drop across each resistor can be calculated using Ohm’s law:

Voltage drop across 4Ω resistor, V₁ = I × R₁ = 1.2A × 4Ω = 4.8V 

Voltage drop across 6Ω resistor, V₂ = I × R₂ = 1.2A × 6Ω = 7.2V 

The total voltage drop across the circuit is the sum of the individual voltage drops:

V_T = V₁ + V₂ = 4.8V + 7.2V = 12V

Example 2. A series circuit has a 120V power source, a 10Ω resistor, and a 15Ω resistor. Calculate the current flowing in the circuit and the voltage drop across each of the resistor.

Solution:

Total resistance in the series circuit:

R_T = R₁ + R₂ = 10Ω + 15Ω = 25Ω 

Using Ohm’s law, the current in the circuit is:

I = V / R_T = 120V / 25Ω = 4.8A 

The voltage drop across each resistor can be calculated using Ohm’s law:

Voltage drop across 10Ω resistor, V₁ = I × R₁ = 4.8A × 10Ω = 48V 

Voltage drop across 15Ω resistor, V₂ = I × R₂ = 4.8A × 15Ω = 72V 

The total voltage drop across the circuit is the sum of the individual voltage drops:

V_T = V₁ + V₂ = 48V + 72V = 120V 

Example 3. A series circuit has a 220V power source, a 100Ω resistor, and a light bulb with an unknown resistance. If the current in the circuit is 2A, find the resistance of the light bulb.

Solution:

Total resistance in the series circuit:

R_T = V / I = 220V / 2A = 110Ω 

The resistance of the light bulb can be found by subtracting the known resistor from the total resistance:

R_{light bulb} = R_T – R₁ = 110Ω – 100Ω = 10Ω

Practice Questions on Series Circuit Formula

Q1. A series circuit has a 9V battery, a 3Ω resistor, and a 6Ω resistor. Find the total resistance, current, and voltage drop across each resistor. (R_T = 9Ω, I = 1A, V₁ = 3V, V₂ = 6V)

Q2. A series circuit consists of a 15V power source, a 20Ω resistor, and a 30Ω resistor. Calculate the total resistance, current, and voltage drop across each resistor. (R_T = 50Ω, I = 0.3A, V₁ = 6V, V₂ = 9V)

Q3. In a series circuit, the total resistance is 50Ω, and the current is 0.4A. If there are two resistors in the circuit, one with a resistance of 30Ω, find the value of the other resistor. (R₂ = 20Ω)

FAQs on Series Circuit Formula

What is the difference between a series and a parallel circuit?

In a series circuit, components are connected end-to-end, creating a single path for current to flow through them. In a parallel circuit, devices are connected across different paths, allowing current to be distributed separately through each.

What are the three rules of a series circuit?

Three rules of a series circuit: 1) The same current flows through each component. 2) The total voltage is the sum of individual voltage drops. 3) The total resistance is the sum of individual resistances.

Which is more powerful, series or parallel?

Power depends on the application. In a series circuit, power might decrease if one component fails. In a parallel circuit, power can remain stable as each component has its own current path, allowing more independent operation.

Is current equal in a series circuit?

Yes, the same current flows through each component in a series circuit because there is only one path for the current to flow.

What are the disadvantages of a series circuit?

Disadvantages of a series circuit: 1) If one component fails, it can disrupt the entire circuit. 2) Adding more components increases resistance, potentially reducing current flow. 3) The total voltage is divided among components, affecting performance.