darth sidious wrote,..
The above quote coming from the thread "Who Loves an Ice Cream Cone"
Hello Darth,
cis(theta) comes from a Taylor Series although I think form memory it is actually a Maclaurin Series expansion of the e^x. Replace x with j(theta). Group real and imaginary terms and factor out j. Now expand using Taylor Series (Maclaurin Series) cos(theta) and sin(theta). The even terms correspond with cos while the odd terms correspond with sin...that is to say that they are identical to those obtained by expanding e^x, substituting in j(theta), simplifying and grouping terms, so e^j(theta) = cos(theta) + jsin(theta).
The angles can be in either radians or degrees, same applies when expressed in the format...A cos(wt + x) where A is a real number and x is an angle in either radians or degrees.
In the case of Complex Fourier Series, the limits of the integral being a measure of the period of g(t) and the exponent within the integrand g(t)e^jwnt are all in radians. Sometimes it is easier to always stay with radians, but I prefered using degrees when solving circuit problems containing reactive components.
Ron.
Ooh yes, good stuff!
Some older books use cis(theta), I can't remember if it was/is mathematically correct to use degrees with cis. It's not mathematically correct to use degrees with e^j{theta}, but many engineers DO! (Myself included
The above quote coming from the thread "Who Loves an Ice Cream Cone"
Hello Darth,
cis(theta) comes from a Taylor Series although I think form memory it is actually a Maclaurin Series expansion of the e^x. Replace x with j(theta). Group real and imaginary terms and factor out j. Now expand using Taylor Series (Maclaurin Series) cos(theta) and sin(theta). The even terms correspond with cos while the odd terms correspond with sin...that is to say that they are identical to those obtained by expanding e^x, substituting in j(theta), simplifying and grouping terms, so e^j(theta) = cos(theta) + jsin(theta).
The angles can be in either radians or degrees, same applies when expressed in the format...A cos(wt + x) where A is a real number and x is an angle in either radians or degrees.
In the case of Complex Fourier Series, the limits of the integral being a measure of the period of g(t) and the exponent within the integrand g(t)e^jwnt are all in radians. Sometimes it is easier to always stay with radians, but I prefered using degrees when solving circuit problems containing reactive components.
Ron.