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**Contents**show

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 resistors are multiple resistors connected together in a single path.

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

Keep in mind, resistors are used to limit current in the circuit and linked to Ohm’s law (I=V/R) where higher resistance will make the current lower.

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_{2}, and R_{3}, respectively as illustrated below,

The mathematical equation is:

**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 below as the example of series connection.

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

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

Combining two equations above gives

Or

We replace R_{1} + R_{2} into R_{eq} as its equivalent resistance

and we get

Hence the circuit above can be replaced by the equivalent circuit below. Those two are equivalent because they have the same values of voltages and current at the terminal a-b.

An equivalent circuit above 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,

Looking back to the circuit above with two resistors, R_{1} and R_{2}, we can calculate their corresponding voltage.

In order to determine the voltage in each resistor, we substitute

Into

To get

**Series Resistor Combination**

Looking from the equivalent resistance equation above, 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 2R, for three resistors is equal to 3R, and so on.

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

The R_{eq} for two resistors is equal to

for three resistors is equal to

and so on.

One thing to always remember:

The equivalent resistance R

_{eq}for series resistors is always greater than 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 above, 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 R_{eq} first.

Remember to find the R_{eq} first if we have multiple resistors connected in a circuit to make the calculation easy.

From series resistance equation we conclude that

Using Ohm’s law, we get current as:

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 drop or the potential differences of the resistors.

Voltage drop in series combined together is the voltage source applied to the circuit.

Summary,

Using Ohm’s law again:

This proves that

and the value we get from that picture.

**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 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 behavior 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 voltage divider example is shown below.

For easier explanation, we will only use two resistors R_{1} and R_{2} connected in series. We use 10V voltage source **V _{i}**, 4Ω and 6Ω resistors, and put an extra wire to R

_{2}as

**V**.

_{o}We can use the voltage divider formula to find the **V _{o}**. The mathematical equation is:

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 to form a voltage divider circuit as shown below.

Hence, the mathematical voltage divider 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_{N}) in series with the voltage source **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 resistor 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 a voltage divider circuit.

**Series Resistors Example**

For better understanding let us review the examples below:

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 R_{eq}:

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

Current:

To find the current we use Ohm’s law. Hence, the current would be 1 A.

Voltage drop:

Using Ohm’s law again: