Please do not take this the wrong way. You need a bit more experience and understanding in order to interpret this notation well.
e^(jwt) = exp(jwt) or e^(-jwt) = exp(-jwt) is talking about mathematical notation. exp(jwt) = cos(wt) + j*sin(wt). If you do not understand that and how to get the equivalent for exp(-jwt), you should not go any further until you understand the mathematics in terms of elementary exponential and trigonometric functions and complex notation. There is no point in moving on to electrical applications unless you are willing to settle for monkey see, monkey do.
This notation is usefully applied to electrical circuits and other applications because derivatives of exponential functions are the same functions except for scale factors. If you used sines and cosines, you would get similar (sine wave) functions but shifted in time. It turns out, in the long run, that the conversion to imaginary exponentials makes life simpler than using mixed sine and cosine expressions.
For most lumped linear circuits, analysis consists of solving a linear differential equation (DE) with constant coefficients driven by a sine wave. The conversion to complex notation converts the solution of the DE into the solution of a polynomial algebraic equation.
On a daily basis, this NG is a forum seemingly for escapees from the school playground, with gratuitous and offensive personal remarks originating from even those who have not been part of a conversation and who could have no reason, other than uncontrolled infantile emotions, for interjecting as they do.
Is such public and international demonstrations really the way forward; the way for PR for the future of Ham Radio. I say, "No!".
I cannot see how something that is a technical pursuit with traditions of gentlemanly behaviour could possibly give way to the childish sneering that is typical of this NG."
Really Blair, the number of times you appear here with the intention of flinging infantile insults says more about you than perhaps you had hoped for.
-- It is wrong in the interpretation which appears to be based on real numbers. You have it right but in light of your reaction to Salmon Egg's comments, it seems that you really don't know why. He was simply pointing out that you should know that and what he said is absolutely correct. Your reaction to his comment was rather childish.
Look up Euler's equation and the whole concept of complex numbers
e^-jwt =cos(wt)-jsin(wt) which happens to be 1/e^jwt
No, it is based on looking at the direction of rotation- CCW or CW. Taking CCW as +ve (the normal convention), CW is -ve. So a CCW rotation increases the argument whereas a CW rotation decreases it. Around time the original post was made Anonymous (under one of his many aliases) had been posting of negative frequency and the context was clear to all but him. His confusion is clear enough, especially as he recently added the word "size" into his argument (as in "his case", not angle).
Well, that is normal of for him. Over the years, I've tried to educate this fellow (as have many others) and he always reacts like this. Sometimes I just leave him to rant on, he seems happier that way.
I'm sure you mean:
e^jwt = cos (wt) + jsin (wt)
Oh, I remember showing the sock puppeteer that on one of the newsgroups. He has previously claimed that division isn't valid in complex numbers.
Unless you want him stomping all over alt.engineering.electrical, I suggest you let him stew.
Thank-you for your extended reply. Unfortunately you have missed the boat. The point of my posts was to draw attention to the heinous error, "is a value that decreases as t increasing" which was published by a teacher of mathematics in Britland.
Unfortunately you have responded to one of my stalkers and not to me.
He is Nathan Hull with the CB Handle of G7KUJ and posts with the id of "+7IiMjEePDrUlHMcutxEQw.user.aioe.org". Many of his posts are parodies of what I post and for many months he has masqueraded as me, and usually adopts the pseudonym that I do shortly after I change pseudonym because of his harassment, in this case "Anonymous".
Nathan Hull is one sick, sick kiddie.
I post with the id of "Z3IpgFh83JnnDIiU15n1gQ.user.aioe.org".
However, I note that you feel sorry for him, as do I.
I think I can see where your misunderstanding arises. (In Britland, this is easily covered by KS2, so see if you can sit it on a class given by a competent maths teacher)
If there had been previously some rotation CCW, then a rotation CW would represent a decrease BUT ONLY UNTIL THE ANGLE CAME BACK TO ZERO, at which point the angle would then grow in a negative direction, INCREASING.
However, this is all by the way. In the function e^(-jwt) there NEVER WAS ANY CCW ROTATION, so that a statement to the effect, " 1/(e^jwt) which is a value that decreases as t increasing" is just plain hogwash and pig-ignorance and would be dangerous if made by a schoolteacher.
You can certainly evaluate a quotient by multiplying by unity (a unity that is the quotient of the original denominator over itself), but evaluating the quotient by the action of multiplication is not the action of division.
How would you evaluate (253 + 16j) / (4 +3j) by a dividing process? Would you divide the 253 by the 4 in the first instance or the 16j by the 3j?
Division is only valid for vector quantities when those vectors are related by simple colinearity.
I an never rude, nor have I ever used these International Fora to broadcast hogwash.
It seems almost inconceivable that the contributors to the gangrenous degeneration that is alt.electrical.engineering would have such a primitive understanding of the level of arithmatic required to pass an examination written for 6 year olds!
I append some text to disambiguate.
Copyright 1999 G.A.Evans. All Rights Reserved.
OK, you're still with me - good! I am pleased that you are determined to make the effort to improve yourselves. I was a bit worried that my audience would consist of only negative CB whingers, who never get anywhere, and who gripe persistently that we Radio Hams never do anything to encourage newcomers. In fact, such is the bigotry of such whingers in this NG, that I half expected them to interpret my opening comments in Lesson 1 as a polemic against CBers. How wrong can you be!
Where do we go from here? OK. I'm going to take you through some essential ideas in maths, and then I'm going to leave the maths behind for a while and start talking about electrical matters. (It's electricity that matters for Radio Hams!) My first topic on electrical matters will be "Energy"; this should give you advanced notice to enable you to think about what energy means to you.
For the moment, though, let's get back to the maths.
Some of the topics I am going to cover may seem to you to be far too simple, but let's get the foundations secure before we build the walls. The topics I have selected come from my experience as a part-time tutor in adult education, dealing with the common problems that my students come up with.
Here's the road map :- Addition, Multiplication, Finish Off Addition, Subtraction, Negative Numbers, Division, Fractions And Decimals, Brackets. Algebraic Puzzles.
At this point, you will have enough under your belt to solve electrical formulae, I will stress time and time again that THERE IS NOTHING NEW IN MATHS - most of what you will come across will be a different way of writing down something that you already know. The different way is a shorthand to make life easy. But remember! If at any time you are confused by the new short-hands, you can always go back and work in the older, simpler ways, (although it might take you longer, you'll still get the correct answer!)
Suppose that you've got three eggs (and continuing my pun of "X's" for eggs, you will have XXX.) Someone now gives you another five XXXXX. How many have you got? Well, you put your three, XXX, next to the new five, XXXXX giving you XXXXXXXX, and you count them again. This is the simple rule for addition. You had three, you add another five, giving you eight. You don't need to understand how to do adding, because after putting them together, you can count them again from scratch.
OK. Suppose that your weekly income is £200, and you get a raise of £37 pounds per week, the same principle applies - put down 200 ticks, then write down a further 37, and count them all from scratch - you're doing addition.....
This soon gets tedious, so let's devise a shorthand to make life easier - but remember! You don't HAVE to use the shorthand, because it's just another way of writing down something that you already know - you can carry on putting down two lots of ticks next to each other, and count them all again from scratch.
The short-hand is "5 + 3", where the "+" is known as a "Plus Sign". When you see this Plus Sign ("+"), it means take the number on the left side, and write down that many ticks. Then take the number on the right side, and write down THAT many ticks, right next to the original group of ticks. Then count how many ticks you've got altogether.
So, let's recap... you can see in the example TWO short hands, firstly the use of "5" instead of "XXXXX", and then the use of "+" to read as the instruction to do an addition. OK. I bet you've never seen that presented like that before, but you see, even addition, which you did right in the early days at school, is a convenient way OF WRITING DOWN SOMETHING ELSE WHICH YOU ALREADY KNEW ABOUT.
Now, whether you do 5 + 3 :- XXXXX then XXX to give XXXXXXXX or 3 + 5 :- XXX then XXXXX to give XXXXXXXX you get the same result. This effect is known in the trade as "Commutativeness", which means "switching around". The word has the same origin as the "commutator" in an electric motor which "switches around" the coils in the armature.
OK, in the early days, one concept per lesson is more than enough. I need to digress into multiplication in the next lesson, before I can finish with addition.