Converting Complex Impedance to Euler Form: Is it Applicable to Just Cos?

[wateveriam]

1. I have a complex Ohm question in which u(t) is given as Umax*$\sqrt{2}$cos($\varpi$+$\varphi$), i know how to convert from trigo to euler form if i have both sin and cos but this doesn't. Is it possible to convert just a cos to Euler form ?





3. Since it is a complex impedance i tried to reason that if we calculate the rms intensity we simply take the real part and leave out the imaginary part, so i by-part the conversion. Is it ok to do that ?

 

[ehild]

cos(ωt +φ) = Re[e^{i(ωt +φ)}]. One calculates with the exponential form, then takes the real part of the result to get the average power, and it comes out that the rms voltage or current is 1/√2 times of the amplitude of the exponential form.

ehild

 

[wateveriam]

Thank you, i understand now, so the given formula gives the numerical result of the Urms and the imaginary part is implied from the angle?

 

[ehild]

I do not get you. In the original formula, the time dependence should be U(t) = U_max cos (ωt+φ), and Umax =√2*U_rms. The Euler form is U₀e^i(ωt+φ), with U₀=U_max.

ehild

 

[wateveriam]

Oh i meant i(t) = cos(ωt+φ) + j*sin(ωt+φ) so the imaginary part of i(t) is j*sin(ωt+φ) so for U=Ri(t), we take the real part and get U = R*cos(ωt+φ) and hence the imaginary part is implied by (ωt+φ), i.e, it is not written out but we can easily get it to be j*sin(ωt+φ). Is that not correct ?

 

[ehild]

It is about right, but the imaginary part is defined as sin(ωt+φ), without j. But it is multiplied by j when you write the trigonometric form of the complex number. Any complex number is of the form

z=x + j*y, x is called the real part and y the imaginary part.

ehild

 

[wateveriam]

Thank you i handed in the work yesterday and it seems i somehow got it right, if only the numerical values. Cheers :D