# Resistors in Series Circuits: Equivalent Resistance

Electrical components can be connected in two ways: either Series connection or Parallel connection. Both types of connections have their advantages and uses. We will start by looking at the series connection.

## Series connection:

Two components will be called in series if the node where they are connected has no any other component connected to it.

Suppose two resistors connected to each other at point b, the head of the second resistor is connected to the tail of the first resistor. Resistors commonly have no head and tails. But when current flows, you may assign them head and tails by assigning positive and negative signs based on voltage drop, as shown. Series circuit provides only one path for the current.

Notice that there is only one path for current to flow. The positive terminal of R2 is connected to negative terminal of R1 and there is no any circuit element connected to that point based on the supposition that current is flowing from point a toward point c. If there is any third circuit component (source or resistor) connected to point b, the R1 and R­2 will no longer be in series with each other.

R1

R2

R3

REquivalent

## Series Circuit Formula:

Up till now, we have defined and try to identify the type of circuit. The next step is to learn, how to find equivalent resistance in a complex circuit? Suppose we have the following complex circuit and we are interested to answer the following questions. What is the total current, the source is providing?

To find out the total current source is providing, first, we need to know how much resistance the source experience? The total resistance we can calculate using the following formula. Here “n” is a total number of resistors connected in series, in our case n=3.

$R_{Equivalent}=R_{1}+R_{2}+\ldots +R_{n}$
$R_{Eq}=R_{1}+R_{2}+R_{3}$

Suppose  $R_{1} R_{2}$ and $R_{3}$ are 20, 40, and 10 $\Omega$ respectively. By putting the values in the above formula, we get:

$R_{Eq}=20 \Omega +40\Omega +10\Omega$
$R_{Eq}=70\Omega$

After finding the equivalent resistance, the circuit can be reduced to the following circuit and find the total current.

$I_{T}=\frac{V}{R_{Eq}}= \frac{140 v}{70 \Omega }$
$I_{T}=2 A$ What is the current in each resistor?

For all the resistors in series circuit, there is always only one path for current to flow. Because of being on the same path, current is always same for all series connected resistors, as following equation represents.

$I_{T}=I_{1}=I_{2}=\ldots =I_{n}$

Here I, I2 and In are current of the current of R, R2 and Rn respectively and n represents the total number of resistors. In our case, the current for R, R2, and R3 are I, I2 and I3 are 2 amperes.
IT = 2 A
I1 = 2 A,
I2 = 2 A,
I3 = 2 A

What is the voltage drop across each resistor?

It is very simple, ohm law is the best tool for it because we have resistance and current of each resistor so we can find the corresponding voltage easily.

V1 = I1 R1 = 2 A x 20Ω = 40 volts
V2 = I2 R2 = 2 A x 40Ω = 80 volts
V3 = I3 R3 = 2 A x 10Ω = 20 volts

$V_{T}=V_{1}+V_{2}+V_{3}$
$V_{T}=40 v+80 v+20 v$
$V_{T}=120 v$

Note that the total voltage drop across the circuit is exactly equal to the voltage applied to the circuit. It also the satisfies the Kirchhoff’s Voltage Law (KVL). The expression for voltage can be generalized like this.

$V_{T}=V_{1}+V_{2}+\ldots +V_{n}$
$V_{T}=I_{1}R_{1}+I_{2}R_{2}+\ldots +I_{n}R_{n}$

The voltage drop in series connection can easily be calculated by using Voltage Divider Rule. 

Sending