The operator j and complex numbers.

[Herald1360]

Originally Posted by G8HQP Dave

Not quite. Low deviation FM is asymptotically equivalent to DSB with the carrier 90 degrees out of phase. Higher deviations have other sideband elements at various phases.

Yes. Somewhat OT but for a year or so in the '70s you could do very well in the FM section on VHF Field day by sitting on a Welsh mountain using low deviation and operating down in the "SSB" end of 2m. You got a lot of comments about poor carrier suppression but you got the contact and the points. There was initially nothing in the rules about where you used what mode and it was only your transmission mode that determined the class you were in.

I guess you could make a connection in the sense that the radio club I was in and j could be thought of as tricksy operators.

And I always like to refer to j as being imaginary because it's off at a right angle to reality.......

[emeritus]

I last used the j-notation in earnest in the mid-1970's, when I was designing the input and output matching networks for broadband stripline amplifiers. I suppose there must be programs that do it these days, but then it involved plotting real and imaginary impedance co-ordinates on a Smith chart and the use of a ruler and compasses to determine what they looked like at the end of a given length of transmission line.

[Synchrodyne]

As a sidebar item, although pertinent to the topic in hand, an interesting and engaging read is the book: “An Imaginary Tale – The Story of √-1”, by Paul J. Nahin (1).

Basically it covers the history of i (√-1), touching upon some of its applications. There is a part-chapter (17 pages) on "Complex Numbers in Electrical Engineering”. ]It does mention Steinmetz’ landmark 1893 paper, “Complex Quantities and Their Use in Electrical Engineering”, and includes the quotation from it: “We are coming more and more to use these complex quantities instead of using sines and cosines, and we find great advantage in their use for calculating all problems of alternating currents, and throughout the whole range of physics. Anything that is done in this line is of great advantage to science.” Within GE, Steinmetz was knows as “the wizard who generated electricity from the square root of minus one”.

Also mentioned is a “famous electronic circuit that works because of √-1”, namely the phase-shift oscillator made famous by a couple of gentlemen by the names of Hewlett and Packard who developed a variable frequency AF oscillator, which was then used by Disney to generate sound effects in the movie “Fantasia”.

Anyway, it is evident that the operator j became mainstream in electrical engineering at a very early stage, and even before electronic engineering was established as a derivative branch.

(1) Published 1998 by Princeton University Press; ISBN 0-691-02795-1. This was one of several like books of the period dealing with significant mathematical constants, all of which seem to have been inspired by the earlier and much better known work by Petr Beckmann, “A History of Pi”.


Cheers,

[dsergeant]

Originally Posted by Herald1360

Yes. Somewhat OT but for a year or so in the '70s you could do very well in the FM section on VHF Field day by sitting on a Welsh mountain using low deviation and operating down in the "SSB" end of 2m. You got a lot of comments about poor carrier suppression but you got the contact and the points. There was initially nothing in the rules about where you used what mode and it was only your transmission mode that determined the class you were in.

I guess you could make a connection in the sense that the radio club I was in and j could be thought of as tricksy operators.

And I always like to refer to j as being imaginary because it's off at a right angle to reality.......

And you probably remember Bracknell ARC at the time when we did that as G4BRA/P from Stokenchurch. I remember it well as I was part of the team (although maybe not in favour of the methods used). The change in rules afterwards became known as the G4BRA exclusion clause.

All good stuff this j theory. Being taught it all in sixth form and uni (G3KMI who you also might remember) it is all second nature for me.

73 Dave G3YMC

[Craig Sawyers]

Then there is the wonderful identity exp (j x pi) = -1

[kalee20]

Absolutely... One of the most stunning results in all mathematics.

Though, in this thread, we have progressed from 2+3j times 4-5j giving 23+2j (simple enough!) -

To 2.718 raised to the power j3.142 and getting (according to my calculator): -0.9999 -0.00008157j

Which is, as near as dammit, -1.

[Herald1360]

Originally Posted by Craig Sawyers

Then there is the wonderful identity exp (j x pi) = -1

Isn't there an e missing there?




e exp jπ or e exp iπ

[Tyso_Bl]

Had it explained to me that i stood for an imaginary number, and that j was substituted instead as i was already allocated in electronics, when I understood that it all made more sense somehow, algebra, just more fun.

[kalee20]

Originally Posted by Herald1360

Isn't there an e missing there?
e exp jπ or e exp iπ

No - exp(x) is accepted notation, albeit not used very frequently, for e to the power x.

Thus exp(2) = e² = 7.389 approx., and exp(jπ) = -1

I do find that identity quite fascinating, two transcendental numbers i.e. ones that have an infinite number of digits that can't be represented as a fraction, become -1 using a very simple equation. I suppose it's linked by time constant (e to the something) and the reactive bit of frequency (2*pi*j*f). That's the low pass filter sorted!

[julie_m]

It's all to do with this more general identity:

cos(θ) + j * sin(θ) ≡ exp(j * θ)

which you can prove by expanding the series.

Now when you set θ = π, you get

exp(j * π) = cos(π) + j * sin(π)
= -1 + j * 0
= -1 ∎

[SteveCG]

I know it is a bit off topic but you may remember the fashion for Fractal images about 30 years ago. A lot of these very detailed and intricate patterns were produced by iteratively applying a simple formula of the type

Z := Z * (Z+1) where Z is a complex number

So complex numbers can be deceptively complicated !!

[Radio Wrangler]

Originally Posted by merlinmaxwell

I do find that identity quite fascinating, two transcendental numbers i.e. ones that have an infinite number of digits that can't be represented as a fraction, become -1 using a very simple equation. I suppose it's linked by time constant (e to the something) and the reactive bit of frequency (2*pi*j*f). That's the low pass filter sorted!

Also, two quite real things; Pi and e, manage to convert an imaginary value into a real one. There's a lot going on in that simple identity.

David

[grampy2]

Can recommend those four books about e, i, pi and phi all on my bookshelf. I reread from time to time and understand a bit better each time.

[red16v]

I have a vague recollection that after we had mastered 'j' notation that we moved on to 's' notation. Something along the lines of jwL = sL, implying jw=s.

Does that ring any bells with anyone? I threw all those sorts of notes out when I retired and even then I hadn't looked at it, or required it, since I left college in 1974.

[ParcGwyn]

Laplace transform and the s-domain, I remember the wonderful 3d contours that could be visualised showing all the poles and holes. This landscape could be sectioned along jw to show the frequency response.

Dave

GW7ONS

[wireless_john]

Not to mention the Mandelbrot Set of course !!

[Ed_Dinning]

Hi Gents, for my sins I studied both Electrical and Electronic engineering with large chunks of 3 phase theory. To confound you even further we used "a" operator, which is the cube root or -1.
A mathematician I know tells me there are a whole series of roots of -1, but could not tell me what other letters they were associated with!

"a" operator was actually very useful in 3 phase fault studies.

Ed

[Craig Sawyers]

Originally Posted by wireless_john

Not to mention the Mandelbrot Set of course !!

http://www-03.ibm.com/ibm/history/ib...icons/fractal/

[Synchrodyne]

Originally Posted by Ed_Dinning

Hi Gents, for my sins I studied both Electrical and Electronic engineering with large chunks of 3 phase theory. To confound you even further we used "a" operator, which is the cube root or -1.
A mathematician I know tells me there are a whole series of roots of -1, but could not tell me what other letters they were associated with!

"a" operator was actually very useful in 3 phase fault studies.

A quick search does not bring up any direct evidence as to the origins of "a" and as to whether any of the higher roots of -1 have been assigned operator letters.

Fortescue mentioned the use of the higher roots of -1 to analyse polyphase systems, see: http://www.energyscienceforum.com/fi...ymmetrical.pdf. Accordingly, the cube root of -1 was applicable to three-phase systems.

In the discussion attached to that paper (page 1117), however, a Mr. J. Slepian does refer to a three phase system with individual phase currents that have the relationship of I, aI and a²I, where a = -0.5 + j(0.5√3), which is a cube root of -1. Thus “a” represented a 2*pi/3 rotation.

So that might have been the origin of operator “a”; simply Slepian’s “a” was taken as was. Given that “a” was derived from j, specifically for analysis of three-phase systems, where it had high utility, I imagine that it did not find much use elsewhere. Or, it belonged more in the world of applied mathematics than in pure mathematics. The higher (than cube) roots of -1 might not have had enough utility to justify being given operator symbols.

Steinmetz may have been the originator of the term “j”. He used in the above mentioned paper, which may be read at: https://www.***********/doc/83021093...al-Engineering. His opening approach was interesting. Initially he assigned j as a non-mathematical identifier to distinguish the vertical from the horizontal components of AC vectors. Then he went on to show that anyway j must be equal to √-1, thus stepping into complex numbers, already a well-established field of mathematics.

I think that Steinmetz was also the first to do a full analysis of three-phase AC circuits under symmetrical and asymmetrical loading conditions. Before this, uncertainties about the behaviour of three-phase systems under asymmetrical loading conditions had led to the “avoidance” approach where two-phase, easier to understand, was used for generation and consumption, with three-phase for long-distance transmission, with conversion between two- and three-phase being done by Scott-connected transformers.


Cheers,