# 0.8 Complex numbers

 Page 1 / 2
This module is part of a collection of modules that address engineering applications of PreCalculus. The collection is intended for use by students enrolled in a special section of MATH 1508 (PreCalculus) for preengineers at the University of Texas at El Paso.

## Introduction

It is essential that engineers master the concept of complex numbers because the important role that complex numbers play in a variety of application areas. In this module applications in the field of electric circuits are provided.

## Alternating current (ac) electric circuits

Earlier we introduced a number of components that are typically found in common electric circuits. These included voltage sources, current sources and resistors. We also observed that the behavior of an electric circuit could be predicted by using several laws from Physics, including Ohm’s Law and Kirchoff’s Laws.

In this laboratory exercise, we will introduce two additional components of electric circuits: the inductor and the capacitor. These elements are typically found in electric circuits which involve sinusoidally varying voltage or current sources. These circuits are called alternating current or AC circuits. AC circuits abound in the physical world. The voltage and current that power household appliances comes from AC sources.

Figure 1 shows the plot for a sinusoidally varying waveform that represents the output of an AC voltage source. Such a waveform could also be used to represent the current that is supplied by an AC current source. It is important to note that the waveform has a repetitive or periodic nature.

In the figure, we note that the amount of time that occurs between successive maxima of the sinusoidal waveform is equal to the period . The angular frequency of the waveform is denoted by the symbol ω and is defined in terms of the period by the equation

$\omega =\frac{2\pi }{T}\text{rad}/s$

If we denote the amplitude as V max , then we can express the sinusoidal waveform for the voltage mathematically as

$v\left(t\right)={V}_{\text{max}}\text{cos}\left(\omega t+{\theta }_{v}\right)$

Here the instantaneous value of the voltage is measured in the units volts. The term θ v is called the phase angle of the sinusoidal waveform. It is measured in degrees. Its usage and importance in the analysis of AC circuits will be discussed later in the course during the study of trigonometry.

Inductors and capacitors are found in circuits of all types and designs, so their understanding is critical to the education of an engineer or scientist. One important distinction between resistors and these two new components (inductors and capacitors) is that they are analyzed using different mathematic techniques. In the case of a resistor, it was quite easy to determine the relationship between the current, voltage and resistance present in a circuit by means of simple algebra. In the case of the inductor and the capacitor, we will see that we must expand our knowledge of mathematics particulary in the are of complex numbers to analyze circuits that contain inductors and capacitors.

what are the products of Nano chemistry?
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
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution