- #1
Natalie Johnson
- 40
- 0
Hi,
I am going around in circles, excuse the pun, with phasors, complex exponentials, I&Q and polar form...
1. A cos (ωt+Φ) = Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt)
Right hand side is polar form ... left hand side is in cartesian (rectangular) form via a trignometric identity?
2. But then sometimes I read...
A cos (ωt+Φ) in polar form has corresponding cartesian form of Bcos(ωt)+Csin(ωt), which is fine to understand because this cartesian form gives X and Y coordinates on a cartesian coordinate axes of a vector in that axes.
3. But point 1 and 2 are different, how can Acos (ωt+Φ) in polar represent Bcos(ωt)+Csin(ωt) in cartesian but also be equal to Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt) via a trignometric identity ---> Is it because Acos(Φ) and Asin(Φ) are constants and therefore also B and C? Might be obvious but I need to ask for my own sanity of seeing so much different ways its written.
What about if B and C are not constants due to the phase changing with time Φ(t)?
I am further purplexed by notation used for complex sinusoids.
3. Acos (ωt+Φ) can be represented as the real part of Aei(ωt+Φ)= Acos(ωt+Φ) + iAsin(ωt+Φ)
but from point 1, the right hand side of this equation can be then re-written with the trigometric identity in point 1, expanding it into 4 terms which removes the phase from the argument and giving constants, like in point 3. So why cannot it not be written without the Φ in the argument on the right hand side and use different constants
Aei(ωt+Φ)=Bcos(ωt)+iCsin(ωt)
I am going around in circles, excuse the pun, with phasors, complex exponentials, I&Q and polar form...
1. A cos (ωt+Φ) = Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt)
Right hand side is polar form ... left hand side is in cartesian (rectangular) form via a trignometric identity?
2. But then sometimes I read...
A cos (ωt+Φ) in polar form has corresponding cartesian form of Bcos(ωt)+Csin(ωt), which is fine to understand because this cartesian form gives X and Y coordinates on a cartesian coordinate axes of a vector in that axes.
3. But point 1 and 2 are different, how can Acos (ωt+Φ) in polar represent Bcos(ωt)+Csin(ωt) in cartesian but also be equal to Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt) via a trignometric identity ---> Is it because Acos(Φ) and Asin(Φ) are constants and therefore also B and C? Might be obvious but I need to ask for my own sanity of seeing so much different ways its written.
What about if B and C are not constants due to the phase changing with time Φ(t)?
I am further purplexed by notation used for complex sinusoids.
3. Acos (ωt+Φ) can be represented as the real part of Aei(ωt+Φ)= Acos(ωt+Φ) + iAsin(ωt+Φ)
but from point 1, the right hand side of this equation can be then re-written with the trigometric identity in point 1, expanding it into 4 terms which removes the phase from the argument and giving constants, like in point 3. So why cannot it not be written without the Φ in the argument on the right hand side and use different constants
Aei(ωt+Φ)=Bcos(ωt)+iCsin(ωt)