How Can Phasors Represent the Function g(t) in Complex Form?

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Dusty912
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Homework Statement


Given a function g(t)=acosωt + bsinωt, where a and b are constants, show that g(t) is the real part of the complex function: keiΦeiωt for some k and Φ
Remark: the complex expression keiΦ is called a phasor. If we know that g(t) has the form kcos(ωt+Φ) then we need know only the constants k and Φ-the amplitude and the phase- to know the function g. Hence we can use the phasor keiΦ as a notation for the function g(t)=keiΦeiωt

Homework Equations


Euler's formula eiωt= cosωt +isinωt

The Attempt at a Solution


Not really sure where to start here except for expanding using euler;s as the first step. any help would be greatly appreciated.
 

Discussion

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okay so then I get k(cosΦt +isinΦt)(cosωt+isinωt)
but what should this be equal too?
 
umm I'm not sure. I didn't know we would be using that symbol. What does φ represent?
 
Dusty912 said:
umm I'm not sure. I didn't know we would be using that symbol. What does φ represent?
I think perhaps you did not understand my question. In your post #3, you had terms like sin(φt). I don't understand how you got those. Please post your working.
 
Sorry I'm kind of lost. Where did φ come from?
 
In my third post I only haveΦ k ω and t
 
yes they look very different. I think that's where the confusion was.
 
But in regards to your earlier quastion. I used euler's formula to obtain sin(Φt) eiΦt=cosΦt +isin(Φt)
right?
 
that sin (Φt) is not apart of that calculation
 
oops, my mistake, than diregard the t's
 
k(cosΦ +isinΦ)(cosω+isinω)
 
k(cosΦ +isinΦ)(cosωt+isinωt)
 
kcosΦcosωt +kisinΦcosωt + kisinωtcosΦ -ksinΦsinωt
 
The part that does not have i's in them? the real parts
 
i'm not sure I want to say 0 or π/2 for Φ but that doesn't seem right.
 
Dusty912 said:
i'm not sure I want to say 0 or π/2 for Φ but that doesn't seem right.
Remember that t is a variable, so the functions are to produce the same value for every value of t. This means that the cos(ωt) term must look the sameinboth functions, and the sin term must look the same too. That givesyou two equations for the coefficients.
 
well if I equate this to g(t)=acosωt+bsinωt than I would get Φ=π/2 and b for the sin's and Φ=0 and a for the cos's or is my algebra wrong on this?
 
ωT)
Dusty912 said:
well if I equate this to g(t)=acosωt+bsinωt than I would get Φ=π/2 and b for the sin's and Φ=0 and a for the cos's or is my algebra wrong on this?
Compare the expression for g with the real part of the complex expression in post #23. Look at the two cos(ωt) terms. If they are to be exactly the same, they must have the same coefficient. That gives you an equation relating a and b to k and φ. Do the same with the two sin(ωt) terms.