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Why complex numbers are used/introduced in electricity

Started by sgrassi 7 months ago36 replieslatest reply 6 months ago211 views

Hello I hope you are all doing well, in these difficult times. I would like to ask for your help as I have to prepare (for a job interview):

a 10 min excerpt from a bachelor's level course on the following topic: "Explaining why complex numbers are used/introduced in electricity".

They will judge me particularly in the pedagogic aspects, and the ability to simplify and vulgarize the subject. 

I have already a good idea of what I want to say (mainly from reading chapter 30 "Complex Numbers" of the excellent "Digital Signal Processing: A Practical Guide for Engineers and Scientists" and also chapters 31,32,33 : complex Fourier, s- and z- transforms, for completeness).

But I really appreciate to hear few words on your idea on the subject, applications, anecdotes, examples ... any help is appreciated, also if you could point me to (simple) resources such as videos, tutorials ...

I am a bit nervous because I have to present the excerpt via skype and I have never done that before.

Thanks a lot and have a nice day.

Sara

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Reply by Rick LyonsMarch 26, 2020

Hi Sara. A few words from me (for whatever they are worth):

Complex numbers are important because they have a powerful mathematical relationship with the trigonometric functions of sines and cosines. This is critical because ALL electrical signals are sinusoidal signals or sums of multiple sinusoidal signals.

Complex numbers as used in electrical engineering dates back to the late 1890s, when engineers were first dealing with the new science of alternating current (AC) electricity. And complex numbers have been VERY convenient to use in the mathematical analysis of electric circuits since then. That's because complex numbers enable simple and straightforward mathematical representations of a sinusoidal signal's amplitude, frequency, *AND* phase.


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Reply by sgrassiMarch 26, 2020

Hello Richard, thanks a lot for these valuable words. 

I sure will integrate them into my talk (sketch above). 

I also wanted to mention (maybe when I speak about examples of DSP systems) your nice phrase in the preface of your book: "... the future of electronics is DSP ...".

Regards,

Sara

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Reply by Rick LyonsMarch 27, 2020

Hi Sara. If you have a copy of my "Understanding DSP" textbook then I suggest you, when you have the time, have a look at the following web page:

https://www.dsprelated.com/showarticle/1094.php


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Reply by sgrassiMarch 27, 2020

I will, thanks a lot. Regards. Sara

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Reply by drmikeMarch 26, 2020

I think you should start with your summary.  Being nervous is a good thing, it helps you stay alert.  If you can type up a bullet point list of what you intend to say, we can guide you more easily.  It would be too easy to throw 40 years of experience out and it would be useless for you.  

It seems to me you've done your homework.  Be confident and skeptical.  You know what you know, and you might still be wrong.  It's OK!  As one of my co-workers recently said "if you don't make mistakes, you aren't trying".  

I just use complex numbers every day without thinking about it.  That's not much help.  Good luck on your interview!

Mike

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Reply by sgrassiMarch 26, 2020

Hello Mike and all, thanks very much for your answers. 

Here is a (ongoing) sketch of what I want to say. Please tell me if it sound reasonable and if you would add/remove something, or illustrate it differently. For me the main worry if to simplify and vulgarize as much as possible so that it is understandable to the whole commission (the professors who are not all in the field as well as the RHs) more than to show that I master (or not) the subject.

(2 min) what are imaginary/complex numbers. Give an example where they arise (like the child that throws a ball, the equation, and what happens when we ask the time of an impossible height). Conclude saying that complex numbers allow the encoding of two things into one value, and this is very useful for representing physical signals and systems in a very elegant and mathematically tractable way.

(2 min) examples of physical systems that we encounter every day, either digital (transducers that sense a physical quantity like sound or image, ADC, processing, DAC + transducer to bring the signal back to the physical word). Or an analog system, like an RLC notch circuit.  All these are physical systems whose input / output can be represented with complex numbers (because of the sinusoid nature of signals). And also what the system does to the input to transform it to the output (behavior of the system) can be represented with complex numbers (because physical systems are mostly based on differential equations). And thus "complex" representation actually simplifies the maths to deal with physical systems. 

- (2 min) speak about sinusoid signal (maybe show the RF spectrum say that some of these signals we can hear/see, other serve to carry information, ... and show/hear a sinusoid) frequency amplitude and phase, and how we can encode this in complex form (magnitude and phase, leaving the frequency fixed).

- Speak about the response of a system (plot of magnitude and phase). For an input sinusoid (at a given frequency, it changes the magnitude and phase, to produce an output sinusoid. These "change" (behavior of the system) can also be represented with a complex number. 

- (2 min) example of an RLC notch filter: application (e.g. remove 60 Hz hum), picture, diagram, response plot, maybe hear a signal in / out. Then explain how to solve it with differential equation (just show that the equations can be complicated, not explain all) and then show how it is simpler to solve it using complex numbers. 

- (2 min) summary of the main points.

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Reply by shafie7March 26, 2020

Hello Sara,

Hope all is well.

My view is that all numbers are a language to count something and keep track of "goods".

Greek and Roman mathematicians believed in positive integer numbers, e.g. 1, 2, 3, 4, 5, and so on.

Indian mathematicians added Zero, nothingness or empty, to the concept of integer numbers, e.g. 0, 1, 2, 3, 4, 5 and so on.

Chinese mathematicians added the concept of negative integer numbers, e.g. …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, and so on.

It is fuzzy, but somewhere along the line, mathematicians added the concept of non-integer or rational numbers, e.g. ½, ¾, etc., which fills in between all of the above numbers.And irrational numbers such as “pi” or “e” which are non-integer but never ends in decimal points, continues for ever, and never boring or repeating.  Another way of thinking about "pi" is that it embeds everyone's birth date, passport id, etc.

If you were to group all of the above numbers into one category, the category is called “Real” Numbers in mathematics nowadays. Perhaps, “Real” represents what is tangible to human senses or understandings.

In high school, we are still teaching that there are no even roots of any negative numbers. For instance, the square root of any negative number does not exist.

Yet, when we further study mathematics, we learn that we can calculate square root of any negative number. For instance, square root of negative one is “i” or “j” if you are an electrical engineer. In other words, the even root of any number yield, what is known to be an Imaginary number, “i”.Therefore, any Complex number can be represented or split by a Real plus an Imaginary parts; i.e. 2+i3.

Best regards,

Shahram

https://ortenga.com/

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Reply by sgrassiMarch 26, 2020

Hello Shahram. Very nice explanation. Will use part of it in the first 2 min. 

Thanks an regards,

Sara


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Reply by shafie7March 26, 2020

Sara,

Based on Mike and Rick feedback, we can think of Imaginary numbers enable us/electrical engineers to keep track of "negative frequencies" or "Spectrum".

Complex numbers as whole represent "Phasor" notation, which replaces "differential equations" with respect to time, with "jW", for Maxwell and Circuit equations, therefore the solution to differential equation can be calculated "Algebraically" instead of "Integral" using Fourier Transform.  

In other words, "Phasor" or Complex notations enable us to simplify solving differential equations.

Best regards,

Shahram

https://ortenga.com/

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Reply by sgrassiMarch 27, 2020

And thanks also for this part. I will also use it in my talk, When I give the example of an eléectric circuit to show that is easy to do the calculation with complex numbers, so that instead of differential equations I have algebraic. 

Regards, Sara

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Reply by ab3apMarch 26, 2020

Sara,

Another topic to consider is the I/Q (in-phase/quadrature) representation of a waveform where Q is sampled 90 degrees from I.  With two sample streams, each ADC can operate at about halfrate from a single sampler.  Because of the 90 degrees offset, think of real/imag axes, and suddenly complex math where real and imag axes are 90 deg offset can be used to manipulate two streams of relatedsamples.  See, for example, https://www.dsprelated.com/showarticle/192.php

Good luck with your presentation.

Mike M

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Reply by sgrassiMarch 26, 2020

I thought about, especially that in my last project was the software-defined-radio where I/Q data is very useful. 

But finally I decided to do something simpler (see my sketch in the other answer). I have only ten minutes, and I think it is better that the talk stay simpler. 

But maybe I can put it at the end (when I cannot do harm anymore) when I recall domains/applications and mention some more.

Thanks and regards,

Sara

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Reply by jbrowerMarch 26, 2020

Sarah-

If they ask you that, you might start by saying that complex numbers started as a math invention (Euler), and now are used everywhere, not just electricity.  Aerodynamics, thermodynamics, quantum mechanics, etc. Then dive into electricity :-)

-Jeff

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Reply by sgrassiMarch 26, 2020

Good point. I will for sure speak of Euler and his identity (Swiss mathematician, being in Switzerland is a must :-) 

In the first 2 min when I introduce complex numbers, and then again when I says is used in representing systems (not only electrical but everywhere) then I will use your words.

Thanks a lot,

Sara

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Reply by WFlaMarch 26, 2020

Sara, take a look,when is it possible, on this analogy by Mr B.P. Lathi .

It isn't enough , but a complement to the informations already given...

Good Look , GOD bless your work !

complex numbers analogy.jpg

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Reply by sgrassiMarch 27, 2020

Lovely ! Thanks a lot. This will be a great way to conclude de last part of the talk (the RLC notch filter: show how it is simpler to solve it using complex numbers.)

Regards,

Sara

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Reply by TreefarmerMarch 26, 2020

One good tutorial which deals with the Fourier Transform also gives a guide to complex numbers. https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/  See particularly the last equation under Appendix: Projecting onto cycles which uses color to help you understand the parts of the equation.

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Reply by sgrassiMarch 27, 2020

Hello and thanks, it looks good, I will definitely go through it.

Best regards, Sara

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Reply by orestesMarch 26, 2020

While I agree with most of the comments already made, I would like to add, or emphasize, that complex numbers were introduced by Steinmetz in his theory of AC circuits because they can convert complicated problems into simpler, algebraic ones. i.e.: if you express voltages and currents in an AC circuit by means of phasors, the V-I relations on every circuit element become algebraic, and you can treat the whole circuit like a purely-resistive one.


As a bonus, you get the concept of impedance (although this was introduced by Heaviside), resonance, signal spectra, frequency response...

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Reply by sgrassiMarch 27, 2020

Hello, thanks a lot. I will add it to my talk, when I give an example of electrical circuit, to show that with complex numbers we can simplify the calculation (algebraic instead of differential equations).

Best regards,

Sara

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Reply by chalilMarch 27, 2020

Hi Sara, adding to many valuable comments, here is my 5c:

in general, complex numbers are 2-dimensional extension of real numbers. complex numbers further extended to vector, quaternion and/or more. 

for signal processing, complex numbers are intuitive to handle harmonic and transient relationships with of parameters such as amplitude, frequency, phase, envelope, energy.

complex numbers make transforms possible from one domain to another without loosing information. inverse transforms are also simple in complex domain. possibility of lossless real transforms are limited as there is one dimension.    

hope it helps.

Chalil


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Reply by AlanTipperMarch 27, 2020

Here is my "elevator pitch" on why complex numbers are important in Electricity.

Electrical engineering can be broadly partitioned into Communications, Command and Control. Modern Communications needs to be understood in terms of frequency spectra and modulation constelations, complex numbers are fundamental to manipulating the resulting fourier transforms. Many command structures are implemented in DSP where IQ manipulation of complex numbers is essential and control systems cannot be fully understood without the use of Laplace, z transform and state space eigen systems all of which require familiarity with complex number spaces. So complex numbers are absolutely crucial to understanding modern applications of electricity.

Alan.

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Reply by sgrassiMarch 27, 2020

Hello Alan, thanks for your input. I will add it to my presentation, either at the introduction, the conclusion or both. Best regards. Sara

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Reply by JohnEhlersMarch 27, 2020

Sara:  You already have plenty of inputs, and here is my take to "vulgarize" the topic.

Electricity involves the relationship of charge and magnetism.  These are orthogonal properties.  That is, one can have a magnetic field without a charge and conversely one can have an electric field without a magnetic field.  However, experiments showed they react to each other with time.  For example, moving a conductor through a magnetic field produces current.  The conventional number system can be thought of as a straight line between minus infinity and plus infinity.  However, complex numbers comprise a plane of orthogonal components.  This number system with orthogonal components is used to describe the interaction between the orthogonal electric and magnetic fields.  This is exactly what Maxwell's equations do.

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Reply by sgrassiMarch 27, 2020

Hello John, thanks for your input. 

Best regards, Sara

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Reply by Rick LyonsMarch 27, 2020

Hi Sara. Complex numbers prove the puzzling idea that the product of two negative numbers is a positive number. Have you ever seen this poem:

  A minus times a minus is a plus

  the reason for this we need not discuss.


Using the polar form of complex numbers, one explanation is if

$$-2=2e^{-180\deg.}\ and -3=3e^{-180\deg.}$$

we can say the product of -2 times -3 is

$$-2\cdot-3=2e^{-180\deg.}\cdot3e^{-180\deg.}=6e^{-360\deg.}=6$$

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Reply by sgrassiMarch 27, 2020

Nice proof. Thanks a lot for this one as well. Best regards. Sara

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Reply by TreefarmerMarch 28, 2020

Post deleted by author

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Reply by TreefarmerMarch 28, 2020

Dear sgrassi, You thumbs up every post. But are you getting what you need for the task?

Here look at the diagram attached. Also, consider the powers of i. Go through them on paper. i squared... cubed ... etc. They repeat at some point. Then you can study a Taylor Series.

06 dft expressed in colors stuart riffle

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Reply by sgrassiMarch 29, 2020

Yes I did thumb up, and answered, every comment because it is valauble to me and useful for my task. Each person contributed with his view on the topic.

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Reply by TreefarmerMarch 28, 2020

Dear sgrassi, I used Corel Draw to make this diagram and it is embarrassingly simple, but it might help. As all readers here know, mathematicians regard this as a beautiful equation because it uses five of the most important constants of mathematics, and it uses exponentiation, addition, etc. 07 Eulers Equation.JPG This shows it in a diagram.

Others, please do not berate me with the basics. I'm just trying to reply to this post. I know others are beyond this. Be kind and helpful "in these difficult times."

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Reply by sgrassiMarch 29, 2020

Hello, your drawing is incredibly simple and extremely beautiful. I am using a similar one to illustrate Euler's Formula, but now I have added the point -1, to illustrate the his identity as well. Thanks a lot.

Sara

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Reply by CedronMarch 29, 2020
Hi Sara,

If you have had your presentation already, I hope it went well. I didn't see this until just now.

I would start with the proclamation: "It is the best mathematical tool for the job."

These two articles of mine are meant to help newbies to complex numbers understand how they work, and that they aren't really complicated.

Why they are the best tool has already been answered well by others. As a Mathematician, I disagree that vectors are an extension of complex numbers, they are similar, that is all.

Ced

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Reply by sgrassiMarch 29, 2020

Hello Ced, thanks a lot. I am still working in my presentation. Thanks for the links. And I will definitely use your proclamation as I fully agree with it. Best regards, Sara

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Reply by sgrassiMarch 27, 2020

Hello all. Thanks a lot for your valuable comments and good wishes. They will help me a lot in preparing my talk. 

Have a nice day,

Sara


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Reply by sgrassiApril 12, 2020

Hello all,

- Your comments helped me a lot in preparing the presentation, and I succeed in the interview! Now I have yet a final interview in which I have to give a 15' teaching sequence on the discrete convolution in time and frequency. Maybe I will ask again in another thread...

- Here is the resulting presentation :https://drive.switch.ch/index.php/s/7BIx72gxuWP19d... in case it can be of use. Sorry for the French, I have a draft in English but is not up to date.

- I used basically all your inputs, but specially:

@Cedron:  I started with: Why do we use complex numbers in electrical engineering? If I were to answer with one phrase, I would simply say that “It is the best mathematical tool for the job“

@Rick : I used in slide 8-9-10 "ALL electrical signals are sinusoidal signals or sums of multiple sinusoidal signals", "complex numbers enable simple and straightforward mathematical representations of a sinusoidal signal's amplitude, frequency, *AND* phase" (as well as the systems that modify them). And slide 12-13: "And complex numbers have been VERY convenient to use in the mathematical analysis of electric circuits since then"

@Treefarmer : I use a picture like yours to illustrate Euler's equation in slide 1 and 6

@WFla : to conclude, slide 14, I used the analogy and figure you gave me, and the loved it!

Thanks very much again and have a nice day,

Sara