A resistor is the most basic element for an electric circuit. This element can be used for converting current from a voltage and vice versa. The resistor is often used to adjust the current and voltage in a circuit. Resistor is also a passive element.

Even a resistor is the most basic element, if a circuit has a complicated combination of several resistors, you may find difficulty to analyze the circuit.

May it be series-connected resistors or parallel-connected resistors, we will learn how to solve them.

Series and parallel resistors can be represented by a single resistance Req. This will help us very well to analyze a circuit.

No matter how complex it is, the resistors will follow Ohm’s law and Kirchhoff’s circuit laws.

## Resistors in Series

We can say resistors connected in series if they are connected together in a single wire.

The current has to flow through all the resistors from the first to the end resistor and back to the source terminal.

All the resistors connected in series will have a common current with the same value flowing through all of them. The current that flows through the first resistor must flow through all the rest resistors.

Say, we have a circuit with the terminal A-B as the source terminal and three resistors *R _{1}*,

*R*, and

_{2}*R*, respectively as illustrated below,

_{3}The mathematical equation is:*I _{R1}* =

*I*=

_{R2}*I*=

_{R3}*I*

_{AB}## Equivalent Resistance for Series Resistors

After looking at the equations above, we can replace several resistors into a single resistor with “__equivalent resistance__”.

Say we have two, three, or more resistors connected together in a series connection, their equivalent resistance *R _{eq}* is the sum of all the resistors.

The more we connect resistors into a series circuit, the more resistance we get.

What is the equivalent resistance? We can say:

Equivalent Resistanceis a single resistance that represents the resistances of any resistors connected without changing the value of current and voltage in the circuit.

This total resistance is generally known as the Equivalent Resistance and can be defined as; “a single value of resistance that can replace any number of resistors in series without altering the values of the current or the voltage in the circuit“.

Then the equation given for calculating total resistance of the circuit when connecting together resistors in series is given as:

## Series Resistor Equation

Analyzing a series resistor circuit can be done with Kirchhoff’s laws like before.

Consider a single-loop circuit in Figure.(1) as the example of series connection.

Figure 1. Series resistors |

The two resistors are in series since the same current *i* flows in both of them.

Applying Ohm’s law to each resistor, we get

(1) |

Apply KVL to the loop in the clockwise direction, we obtain

(2) |

Combining Equations.(1) and (2) gives

(3) |

or

(4) |

And Equation.(3) can be expressed as

(5) |

Assuming those two resistors can be expressed by an equivalent resistor *R _{eq}*; so,

(6) |

Hence in Figure.(1) can be replaced by the equivalent circuit in Figure.(2). Those two are equivalent because they have the same values of voltages and current at the terminal a-b.

Figure 2. Series resistor equivalent circuit |

An equivalent circuit like Figure.(2) is very useful in simplifying the analysis of a circuit. In general,

The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.

For N resistors in series then,

(7) |

In order to determine the voltage in each resistor in Figure.(1) then we substitute Equation.(3) into (1) and get

(8) |

## Series Resistor Combination

Looking from the Equation.(6), we can simplify some examples here. Note that equivalent resistance for series resistors is the algebraic sum of the individual resistances.

Here we have two resistors with identical resistances. The *R _{eq}* for two resistors is equal to 2

*R*, for three resistors is equal to 3

*R*, and so on.

Here is another example. We have two resistors with different resistances.

The *R _{eq}* for two resistors is equal to

*R*+

_{1}*R*, for three resistors is equal to

_{2}*R*+

_{1}*R*+

_{2}*R*, and so on.

_{3}One thing to always remember:

The equivalent resistance

Rfor series resistors is_{eq}always greaterthan the largest resistance of the connected resistor in a circuit.

You can check it by yourself easily.

## Series Resistor Voltage

Even if we have the series resistor voltage equation in Equation.(8), we will learn how to get it and how to use it.

If we have a circuit above and need to know the voltages for each resistor, we need to find the Req first.

Remember to find the Req first if we have multiple resistors connected in a circuit to make the calculation easy.

From Equation.(7) we conclude that*R _{eq}* = 1Ω + 2Ω + 3Ω = 6Ω

Using Ohm’s law, we get current as:

I = V / R = 6V / 6Ω = 1 A

And now we have the current, let us find the voltages for each resistor.

For a note,

The value of the voltage source in a circuit is equal to the

sum of the voltage dropor the potential differences of the resistors.

Summary,

V=_{ab}V+_{R1}V+_{R2}V_{R3}

Using Ohm’s law again:

V= I x_{R1}R= 1A x 1Ω = 1V_{1}V= I x_{R2}R= 1A x 2Ω = 2V_{2}V= I x_{R3}R= 1A x 3Ω = 3V_{3}

This proves that Vab = *V _{R1}* +

*V*+

_{R2}*V*= 6V and the value we get from that picture.

_{R3}## Voltage Divider Circuit

The source voltage v is divided among the resistors in direct proportion to their resistances; the larger the resistance, the larger the voltage drop.

This is called the **principle of voltage division** and the circuit in Figure.(1) is called a voltage divider.

From the explanation above we can see that a single 6V voltage source can provide different voltage drops or potential differences across the resistors.

This behaviour can make a series resistor circuit act as a voltage divider circuit.

This circuit splits the voltage source across to each resistor proportional to their resistances. The voltage is determined by the resistance of the resistor.

The larger the resistance, the larger the voltage drop and vice versa.

Do you remember what we have learned about Kirchhoff’s voltage law? Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero.

The principle of voltage division is used to divide the voltage source *v* proportionally to the resistances in the circuit.

The example of a voltage divider circuit is shown below.

For easier explanation, we will only use two resistors *R _{1}* and

*R*connected in series. We use 10V voltage source

_{2}*V*, 4Ω and 6Ω resistors, and put an extra wire to

_{i}*R*as

_{2}*V*.

_{o}We can use the Equation.(8) to find the Vo. The mathematical equation is:

V=_{o}V= (_{2}R/(_{2}R+_{1}R) )*_{2}V_{i}

We can use more than two resistors for voltage divider circuits. But, the voltage for each resistor will be smaller.

Now let us use three resistors form a voltage divider circuit as shown below.

Hence, the mathematical equation for the voltage across the 6Ω is 3V according to:

This proves what we have concluded before, the more resistances we use, the smaller the voltage drop or potential difference across the resistors we get.

In general, if voltage divider has N resistors (*R _{1}*,

*R*, ….,

_{2}*R*) in series with the voltage source

_{N}*v*, the

*n*th resistor will have a voltage drop of

The voltage divider is used to divide a large voltage to a smaller one.

## Series Resistors Summary

After learning a lot of series resistor explanation, here we try to summarize in short explanation:

- Series resistor is a circuit when we connect multiple resistors in a single wire. We connect the end of the first resistor to the head of the second resistor and so on.
- Series resistors connection has the same value of current.
- The voltage drop across each resistor is proportional to the sum of the resistances and follows Ohm’s law (V = I x R).
- Series resistors circuit acts as voltage divider circuit.

## Series Resistors Example

For better understanding let us review the examples below:

1. We have a circuit with a 20V voltage source, three resistors with 3Ω, 7Ω, and 10Ω. Find the equivalent resistance *R _{eq}*, the current, and the voltage drop for each resistor in the circuit.

__Equivalent resistance Req:__

The *R _{eq}* for series resistors is the sum of all the resistances in the circuit.

*R*=

_{eq}*R*+

_{1}*R*+

_{2}*R*= 3 + 7 + 10 = 20 Ω.

_{3}__Current:__

To find the current we use Ohm’s law (I = V / *R _{eq}*). Hence, the current would be 1 A.

__Voltage drop:__*V _{ab}* =

*V*+

_{R1}*V*+

_{R2}*V*

_{R3}Using Ohm’s law again:

*V*=

_{R1}*I*x

*R*= 1A x 3Ω = 3V

_{1}*V*=

_{R2}*I*x

*R*= 1A x 7Ω = 7V

_{2}*V*=

_{R3}*I*x

*R*= 1A x 10Ω = 10V

_{3}## Frequently Asked Question

### What is a resistor and its uses?

A resistor is a passive electrical element that provides resistance to the circuit. This element is used for reducing the current, tuning electrical signal, voltage divider circuit, and circuit breaker.

### How do you know if a resistor is in series?

A resistor is said in series connection if it is connected to the other element in a single wire end-to-end. Multiple resistors are connected in series if the back of the first resistor is connected to the head of the second resistor and so on.

### Does current drop across a resistor?

The current entering a resistor is the same as the current leaving that resistor. But if you put a resistor in a circuit, according to Ohm’s Law (I = V / R), the total current will be reduced. Still the current in a series resistors circuit will remain the same for each resistor.

### What is the main function of the resistor?

The main function of a resistor is to control the current flowing in a circuit. Some circuit element such as LED and Integrated Circuit require specific current to operate or it will be malfunctioned or destroyed.

**What are the 4 types of resistors?**The types of resistors include:

Thermistor.

Light Dependent Resistor.

Metal Film Resistor.

Carbon Composition Resistor.

Carbon Film Resistor.

Variable Resistor.

Varistor

Wire Wound Resistor.**How are resistors connected in series?**Series Resistor Voltage:

The value of the voltage source in a circuit is equal to the sum of the voltage drop or the potential differences of the resistors (*V*=_{ab}*V*+_{R1}*V*+_{R2}*V*)_{R3}

Series Resistor Current:

The total current in a circuit depends on the total resistances*R*(I = V /_{eq}*R*)._{eq}**What happens when resistors are connected in series?**If we use series resistors, the current will be equal for each resistor but the voltage depends on the resistance value of each resistor.

**Do you add resistors in series?**The total resistance for series resistors is the sum of all of the resistor’s resistance connected together. (

*R*=_{T}*R*+_{1}*R*+_{2}*R*+ ….)_{3}**Is voltage the same in series?**The value of the voltage source in a circuit is equal to the sum of the voltage drop or the potential differences of the resistors.

Summary,*V*=_{ab}*V*+_{R1}*V*+_{R2}*V*_{R3}