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Notes to teachers and students:

Phasors! For those who understand them, they are of incomparable value for the study of elementary and advanced topics. For those who misunderstand them, they are a constant source of confusion and are of no apparent use. So, let's understand them.

The conceptual leap from the complex number e j θ to the phasor e j ( ω t + θ ) comes in "Phasor Representation of Signals" . Take as long as necessary to understand every geometrical and algebraic nuance. Write the MATLAB program in "Exercise 6" to fix the key ideas once and for all. Then use phasors to study beating between tones, multiphase power, and Lissajous figures in "Beating between Tones" through "Lissajous Figures" . We usually conduct a classroom demonstration of beating between tones usingtwo phase-locked sources, an oscilloscope, and a speaker. We also demonstrate Lissajous figures with this hardware.

"Sinusoidal Steady State and the Series RLC Circuit" and Light Scattering by a Slit" on sinusoidal steady state and light scattering are too demanding for freshmen but are right on target for sophomores. Thesesections may be covered in a sophomore course (or a supplement to a sophomore course) or skipped in a freshman course without consequence.

In the numerical experiment in "Numerical Experiment (Interference Patterns)" , students compute and plot interference patterns for two sinusoids that are out of phase.


In engineering and applied science, three test signals form the basis for our study of electrical and mechanical systems. The impulse is an idealized signal that models very short excitations (like current pulses, hammer blows, pile drives, and light flashes). The step is an idealized signal that models excitations that are switched on and stay on (like current in a relay that closes or a transistor that switches). The sinusoid is an idealized signal that models excitations that oscillate with a regular frequency (like AC power, AM radio, pure musical tones, and harmonic vibrations). All three signals are used in the laboratory to design and analyze electrical and mechanical circuits, control systems, radio antennas, and the like. The sinusoidal signal is particularly important because it may be used to determine the frequency selectivity of a circuit (like a superheterodyne radio receiver) to excitations of different frequencies. For this reason, every manufacturer of electronics test equipment builds sinusoidal oscillators that may be swept through many octaves of Orequency. (Hewlett-Packard was started in 1940 with the famous HP audio oscillator.)

In this chapter we use what we have learned about complex numbers and the function e j θ to develop a phasor calculus for representing and ma- nipulating sinusoids. This calculus operates very much like the calculus wedeveloped in "Complex Numbers" and "The Functions e x and e " for manipulating complex numbers. We ap- ply our calculus to the study of beating phenomena, multiphase power, series RLC circuits, and light scattering by a slit.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
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