Can someone aid in my understanding how author dervied this equation?

[Lucifer12]

The highlighted equation. I just can, not think of a way to decompose the equation into getting A and B without getting rid of j. Any ideas?

 

[GregLocock]

Where does the -2j come from in the first line?

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Greg Locock


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[rb1957]

the last line looks like vector summation. but I wonder if B is in the "j" direction ? would that make it A^2+(jB)^2 = A^2-B^2 ??

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[JStephen]

Haven't figured it out, but I think his third equation is wrong, and he meant to set A=2acos(wt) and B=2asin(wt) which makes the last equation work.

 

[rb1957]

but what about the "-j" term ? doesn't that change the bracket to sin(wt)-j*cos(wt) ?

another day in paradise, or is paradise one day closer ?

 

[JStephen]

You can go from the second equation to the third by assuming A= -2aj and B= -2aj(j)= 2a. But when I substitute that into the last equation, I get zero, not 2a. Either I'm doing that wrong, or one of the last three equations is wrong.

 

[robyengIT]

JStephen : for me is B = -2aj^2 and not B=2a and substituting I get 2aj * (1+j^2)^0.5

 

[JStephen]

I was assuming j=i=square root of -1 so (1+j^2) is zero. If you take j as a real number, then the first two equations disagree.

 

[Lucifer12]

To answer the question of first poster.

This is the equation it comes from originally. I was thinking that if -j rotated the complex clockwise by 0.5pi vectorialy but even then the components don't add up since j is a necessary coefficient in de moviere theorem!

Another issue is getting rid of j when the terms inside the bracket are multiplied by j which eliminates j on sine term but not on the cosine one.
The amplitude is bothersome indeed.

 

[rb1957]

e^(i*x) = cos(x) +i*sin(x)

so e^(-ikx) - e^(ikx) = cos(-kx) + i*sin(-kx) - (cos(kx) + i*sin(kx)) = (cos(-kx) - cos(kx)) + i*(sin(-kx) - sin(kx))
cos(-kx) = cos(kx)
sin(-kx) = -sin(kx)
so we get = -2i*sin(kx) which is what you have.


another day in paradise, or is paradise one day closer ?

 

[Lucifer12]

Its the yellow equation that I can, not derive I know how that is derived. Anyway shall I assume an error on account of author?

 

[rb1957]

I can't see how it can be correct. How do you change an expression with real and imaginary components into a real number ?

another day in paradise, or is paradise one day closer ?

 

[Lucifer12]

I have a gut feeling the author is using complex vector geometry in the argand plane or in polar coordinates. I will think into it otherwise I am moving on. It will be bothersome for a bit.

 

[IRstuff]

Perhaps you should post the actual reference, rather than a picture of a part of it.

TTFN (ta ta for now)
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[Lucifer12]

Thank you all for the help.