- #1

- 25

- 0

- Thread starter ffleming7
- Start date

- #1

- 25

- 0

- #2

Dale

Mentor

- 30,276

- 6,737

- #3

- 243

- 1

ie.

[tex] e^{ix} = cos(x)+isin(x) [/tex]

In engineering we deal A LOT with sinusoidal signals with phases. Since 'i' is just a phase shift of pi/2, we can represent phases as a "complex quantity". This allows us to convert all types of sinusoidal signals into exponentials rather than trig, where you need big trig identities to simplify or manipulate.

In physics and engineering, a lot of processes and theories deal with differential equations. There solutions are in the form of (real) trig equations ie. asin(x) + bcos(x). But again, these are too long and tedious to deal with so we put it complex exponential form and 'remember' that the solution is always the real part of exp(ix).

That is the very very basics of it, it off course is much more complicated and hopefully I havn't miss-explained parts of it. Someone will correct me though :)

- #4

- 27

- 0

Use it all the time... it's no joke. Seriously.

- #5

- 25

- 0

wherever we have to deal with frequency, 'i' comes into picture.

also for your information, in electrical engineering most of the time, we write 'j' instead of 'i'. (i thought this would add to your curiousity) I don't know the exact reason, but may be because we use 'i' for representing current, thats why.

- #6

- 1,789

- 4

Another reason you will like to use the complex exponential in expressing solutions to linear differential equations is its relation to eigenvalue problems. Specifically,

[tex]D_{x}(e^{j\omega x}) = je^{jx}[/tex]

[tex]D_{x}^{2}(e^{j\omega x}) = -\omega^{2}e^{jx}[/tex]

Sure enough you could do this with sinusoids too (in fact thats what you're doing right now), but doing it with complex exponentials is a whole lot neater. Quite a few problems of interest in the time domain involve the solution of eigenvalue equations, where these properties are often used.

Finally, even though all this is sufficient reason to use complex exponentials and more generally complex numbers, both the Laplace and Fourier Transforms have kernels that are complex exponentials and are members of a more general class of transforms whose variants are heavily used in signal processing, control systems and lot of other allied areas of electrical engineering and mathematics.

And yes, you encounter j in EE rather than i, because historically [itex]i(t)[/itex] has been used to denote "instantaneous" current, as Varun points out. (http://www.perl.com/doc/manual/html/lib/Math/Complex.html differs...apparently, i has also been used to denote intensity :tongue:)

- #7

stewartcs

Science Advisor

- 2,177

- 3

That's correct. A lower case "i" is reserved for current in EE.

wherever we have to deal with frequency, 'i' comes into picture.

also for your information, in electrical engineering most of the time, we write 'j' instead of 'i'. (i thought this would add to your curiousity) I don't know the exact reason, but may be because we use 'i' for representing current, thats why.

CS

- #8

- 61

- 0

- #9

- 5

- 0

Phasors, Impedance, Complex Power... list goes on.

- Last Post

- Replies
- 3

- Views
- 5K

- Last Post

- Replies
- 3

- Views
- 2K

- Replies
- 7

- Views
- 2K

- Replies
- 2

- Views
- 3K

- Replies
- 2

- Views
- 2K

- Last Post

- Replies
- 2

- Views
- 1K

- Last Post

- Replies
- 29

- Views
- 5K

- Last Post

- Replies
- 2

- Views
- 4K

- Replies
- 4

- Views
- 1K

- Replies
- 1

- Views
- 2K