Sinusoids represented by Phasors

[salman213]

1. Find the sinusoids represented by these phasors:
(a) I = -3 + j4 A


3.

well to convert first to polar they got the following

they didnt show their steps:

I = 5 <126.87


then they said

i(t) = 5 cos(wt + 126.87)

What I don't get is the fact that how they got 126.87

to get 5 i just take 3 and 4 sum their squares and take the square root.

to find the angle i would do tan-1(y/x) = tan-1(4/-3)=-53.13

ok since its tangent you may say u can add 180 and it is the same answer which is what they did -53.13 +180=126.87

but my question is WHY. why do they add the 180 if its the same thing. Whats the use. I would just leave my final answer as

i(t) = 5cos(wt - 53.13)

why add 180?

 

[Defennder]

It's easily seen if you sketch the Argand diagram for -3+j4. Remember that the angle you want is measured from the x-axis on the right to the complex number "vector". And that is given by 180 - 53.13.

 

[piercas]

I would explain that the reason we add 180 degrees to the angle in this case is because we are dealing with a complex number in the form of a phasor. A phasor is a vector representation of a sinusoidal function, where the magnitude represents the amplitude and the angle represents the phase shift.

In this case, the phasor I = -3 + j4 A can be converted to polar form as I = 5 <126.87, where 5 is the magnitude and 126.87 is the angle in degrees. However, when we convert back to the time-domain, we need to consider the direction of the vector in relation to the x-axis.

In the time-domain, the cosine function represents the x-component of the phasor, and the sine function represents the y-component. In this case, the x-component is negative and the y-component is positive, which means that the angle is in the second quadrant. However, the inverse tangent function only gives us the angle in the first or fourth quadrant, so we need to add 180 degrees to get the correct angle in the second quadrant.

Therefore, the correct representation of the sinusoid would be i(t) = 5 cos(wt + 126.87), which is equivalent to i(t) = 5 cos(wt - 53.13). Both representations are correct, but adding 180 degrees ensures that we are representing the correct phase shift in the time-domain.