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Complex number arithmetic

Find the real part, imaginary part, the magnitude and angle of the complex numbers given by the following expressions.

  1. -1
  2. 1 3 2
  3. 1 2
  4. 3 3

Discovering roots

Complex numbers expose all the roots of real (and complex) numbers. For example, there should be two square-roots, three cube-roots, etc. of any number.Find the following roots.

  1. What are the cube-roots of 27? In other words, what is 27 1 3 ?
  2. What are the fifth roots of 3 ( 3 1 5 )?
  3. What are the fourth roots of one?

Cool exponentials

Simplify the following (cool) expressions.

  1. 2

Complex-valued signals

Complex numbers and phasors play a very important role in electrical engineering. Solving systems for complexexponentials is much easier than for sinusoids, and linear systems analysis is particularly easy.

  1. Find the phasor representation for each, andre-express each as the real and imaginary parts of a complex exponential. What is the frequency (in Hz)of each? In general, are your answers unique? If so, prove it; if not, find an alternative answer forthe complex exponential representation.
    1. 3 24 t
    2. 2 2 60 t 4
    3. 2 t 6 4 t 3
  2. Show that for linear systems having real-valued outputs for real inputs, that when the input is thereal part of a complex exponential, the output is the real part of the system's output to the complexexponential (see [link] ). S A 2 f t S A 2 f t

For each of the indicated voltages, write it as the real part of a complex exponential( v t V s t ).Explicitly indicate the value of the complex amplitude V and the complex frequency s . Represent each complex amplitude as a vector in the V -plane, and indicate the location of the frequencies in the complex s -plane.

  1. v t 5 t
  2. v t 8 t 4
  3. v t t
  4. v t 3 t 4 t 3 4
  5. v t 5 2 t 8 t 2
  6. v t -2
  7. v t 4 2 t 3 2 t
  8. v t 2 100 t 6 3 100 t 2

Express each of the following signals as a linear combination of delayed and weighted step functions andramps (the integral of a step).

Linear, time-invariant systems

When the input to a linear, time-invariant system is the signal x t , the output is the signal y t ( [link] ).

  1. Find and sketch this system's output when the input is the depicted signal .
  2. Find and sketch this system's output when the input is a unit step.

Linear systems

The depicted input x t to a linear, time-invariant system yields the output y t .

  1. What is the system's output to a unit step input u t ?
  2. What will the output be when the input is the depicted square wave ?

Communication channel

A particularly interesting communication channel can be modeled as a linear, time-invariant system. When thetransmitted signal x t is a pulse, the received signal r t is as shown .

  1. What will be the received signal when the transmitter sends the pulse sequence x 1 t ?
  2. What will be the received signal when the transmitter sends the pulse signal x 2 t that has half the duration as the original?

Analog computers

So-called analog computers use circuits to solve mathematical problems, particularly when they involve differential equations. Suppose we are given the following differential equation to solve. t y t a y t x t In this equation, a is a constant.

  1. When the input is a unit step ( x t u t ), the output is given by y t 1 a t u t . What is the total energy expended by the input?
  2. Instead of a unit step, suppose the input is a unit pulse (unit-amplitude, unit-duration) delivered to the circuit at time t 10 . What is the output voltage in this case?Sketch the waveform.

Questions & Answers

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Answers please
Nikki Reply

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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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