# 0.8 Complex numbers

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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 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
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution