<< Chapter < Page Chapter >> Page >
V = I R size 12{V=I`R} {}

Photograph of a resistor.

Instead of a single resistor, electric circuits often employ multiple resistors. Let us consider the case where we have two resistors that are denoted as R 1 and R 2 .

The two resistors can be connected in an end-to-end manner as shown in Figure 2 (a).

Series and parallel connections of resistors.

Resistors connected in this manner are said to be connected in series . We may replace a series connection of two resistors by a single equivalent resistance, R eq . From an electrical standpoint, the single equivalent resistance will behave exactly the same as the combination of the two resistors connected in series. The equivalent resistance of two resistors connected in series can be calculated by summing the resistance value of each of the two resistors.

R eq = R 1 + R 2 size 12{R rSub { size 8{ ital "eq"} } =R rSub { size 8{1} } +R rSub { size 8{2} } } {}

Let us now consider the case where the two resistors are placed side-by-side and then connected at both ends. This situation is depicted in Figure 2 (b) and is called a parallel combination of resistors. Whenever resistors are connected in this manner, they are said to be connected in parallel. The equivalent resistance of two resistors connected in parallel obeys the following relationship.

1 R eq = 1 R 1 + 1 R 2 size 12{ { {1} over {R rSub { size 8{ ital "eq"} } } } = { {1} over {R rSub { size 8{1} } } } + { {1} over {R rSub { size 8{2} } } } } {}

Once this quantity is calculated, one may easily take the reciprocal of the result to obtain the value of R eq .

The rule governing the determination of the equivalent resistance for the series connection of more than two resistors can be expanded to accommodate any number ( n ) of resistors. For n resistors connected in series, the equivalent resistance is equal to the sum of the n resistance values.

In addition, the rule governing the determination of the equivalent resistance for the parallel connection of more than two resistors can be expanded. For n resistors connected in parallel,

1 R eq = 1 R 1 + 1 R 2 + 1 R n size 12{ { {1} over {R rSub { size 8{ ital "eq"} } } } = { {1} over {R rSub { size 8{1} } } } + { {1} over {R rSub { size 8{2} } } } + dotsaxis { {1} over {R rSub { size 8{n} } } } } {}

It is clear to see that one’s ability to determine the equivalent resistance of parallel resistors depends upon one’s ability to add fractions. The following exercises are included to reinforce this idea.

Example 1: Two resistors are connected in series. The value of resistance for the first resistor is 5 Ω, while that of the second is 9 Ω. Find the equivalent resistance of the series combination.

R eq = 5 Ω + 9 Ω = 14 Ω size 12{R rSub { size 8{ ital "eq"} } =5` %OMEGA +9` %OMEGA ="14"` %OMEGA } {}

Example 2: Consider the two resistors presented in Example 1. Let the two resistors now be connected in parallel. Find the equivalent resistance of the parallel combination.

1 R eq = 1 5 Ω + 1 9 Ω size 12{ { {1} over {R rSub { size 8{ ital "eq"} } } } = { {1} over {5` %OMEGA } } + { {1} over {9` %OMEGA } } } {}

The lowest common denominator is 45 Ω. So we incorporate it into our analysis.

1 R eq = 9 45 Ω + 5 45 Ω size 12{ { {1} over {R rSub { size 8{ ital "eq"} } } } = { {9} over {"45"` %OMEGA } } + { {5} over {"45"` %OMEGA } } } {}
1 R eq = 14 45 Ω size 12{ { {1} over {R rSub { size 8{ ital "eq"} } } } = { {"14"} over {"45"` %OMEGA } } } {}
R eq = 45 Ω 14 = 3 . 21 Ω size 12{R rSub { size 8{ ital "eq"} } = { {"45"` %OMEGA } over {"14"} } =3 "." "21"` %OMEGA } {}

Example 3: Three resistors of values 2 kΩ, 3 kΩ, and 5 kΩ are connected in parallel. Find the equivalent resistance.

1 R eq = 1 2, 000 Ω + 1 3, 000 Ω + 1 5, 000 Ω size 12{ { {1} over {R rSub { size 8{ ital "eq"} } } } = { {1} over {2,"000"` %OMEGA } } + { {1} over {3,"000"` %OMEGA } } + { {1} over {5,"000"` %OMEGA } } } {}

The lowest common denominator for the fractional terms is

LCD = 2 × 3 × 5 × 1, 000 Ω = 30 , 000 Ω size 12{ ital "LCD"=2 times 3 times 5 times 1,"000"` %OMEGA ="30","000"` %OMEGA } {}

We rewrite the original equation to reflect the lowest common denominator

1 R eq = 15 30 , 000 Ω + 10 30 , 000 Ω + 6 30 , 000 Ω size 12{ { {1} over {R rSub { size 8{ ital "eq"} } } } = { {"15"} over {"30","000"` %OMEGA } } + { {"10"} over {"30","000"` %OMEGA } } + { {6} over {"30","000"` %OMEGA } } } {}
1 R eq = 31 30 , 000 Ω size 12{ { {1} over {R rSub { size 8{ ital "eq"} } } } = { {"31"} over {"30","000"` %OMEGA } } } {}

R eq = 30 , 000 Ω 31 = 968 Ω size 12{R rSub { size 8{ ital "eq"} } = { {"30","000"` %OMEGA } over {"31"} } ="968"` %OMEGA } {}

Summary

This module has presented how to add fractions using the lowest common denominator method. Also presented is the relationship among voltage, current and resistance that is known as Ohm’s Law. Examples illustrating the use of the lowest common denominator method to solve for equivalent resistances of parallel combinations of resistors are also provided.

Exercises

  1. Consider a 10 kΩ and a 20 kΩ resistor. (a) What is the equivalent resistance for their series connection? (b) What is the equivalent resistance for their parallel connection?
  2. Consider a parallel connection of three resistors. The resistors have values of 25 Ω, 75 Ω, and 100 Ω. What is the equivalent resistance of the parallel connection?
  3. Consider a parallel connection of four resistors. The resistors have values of 100 Ω, 100 Ω, 200 Ω and 200 Ω. What is the equivalent resistance of the parallel connection?
  4. Conductance is defined as the reciprocal of resistance. Conductance which is typically denoted by the symbol, G, is measured in the units, Siemens. Suppose that you are presented with two resistors of value 500 Ω and 1 kΩ. What is the conductance of each resistor?
  5. The equivalent conductance of a parallel connection of two resistors is equal to the sum of the conductance associated with each resistor. What is the equivalent conductance of the parallel connection of the resistors described in exercise 4?
  6. What is the equivalent resistance of the parallel connection of resistors described in exercise 5?
  7. Suppose that you are presented with 2 resistors. Each resistor has the same value of resistance (say, R). Derive an expression for the equivalent resistance of their parallel connection.
  8. Three resistors with resistance values of 100 kΩ, 50 kΩ, and 100 kΩ are connected in parallel. What is the equivalent resistance? (Hint: You may use the result of Exercise 2 to simplify your work.)
  9. Four resistors, each with a value of 10 Ω, are connected in parallel. What is the equivalent conductance of the parallel connection? What is the equivalent resistance of the parallel connection?
  10. A 30 Ω resistance is connected in series with a parallel connection of two resistors, each with a value of 40 Ω. What is the equivalent resistance of this series/parallel connection?

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution
BUGAL Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Math 1508 (laboratory) engineering applications of precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11337/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Math 1508 (laboratory) engineering applications of precalculus' conversation and receive update notifications?

Ask