Electrical: Reduce the expression to the form V_mcos(wt+theta)

[Northbysouth]

Homework Statement

Reduce the Expression:

15sin(wt-45°) + 5cos(wt-30°) + 10cos(wt-120°)

to the form V_mcos(wt+θ)

Homework Equations

The Attempt at a Solution

My theta value at the end isn't coming out right.

My first step was to put make sure each term was in terms of cosine, hence:

15cos(wt-45°-90°) + 5cos(wt-30°) + 10cos(wt-120°)

Which in phasor form is:

15∠-135 + 5∠-30 + 10∠-120

Then I used cos and sin to find the real values and the j values

15cos(-135) + j15sin(-135) + 5cos(-30) + j5sin(-30) + 10cos(-120) + j10sin(-120)

= -11.28 -j21.77

Which in a phasor format is:

24.52∠62.6

24.53 = (11.28²+ 21.77²)^1/2

62.6 = arctan(21.77/11.28)

But apparently the solution is:

24.52∠-117.39°

What am I missing?

Any help would be appreciated. Thank you

 

[gneill]

Note the signs of the complex components of your phasor sum. In what quadrant will the phasor lie?

 

[Northbysouth]

Are you saying that because I have -11.28 - j21.77 as negatives that the phasor will be in the third quadrant and consequently the angle made with the postive x-axis will be 180-62.6?

 

[gneill]

Are you saying that because I have -11.28 - j21.77 as negatives that the phasor will be in the third quadrant and consequently the angle made with the postive x-axis will be 180-62.6?

The phasor will be in the third quadrant, yes. You can either add 180 or subtract 180, but generally the convention is to choose the result with a magnitude less than or equal to 180. Alternatively, employ the atan2(y,x) function rather than the atan function (if your calculator has it); it automatically sorts out the quadrant from the given y,x data.

 

[Nickie]

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Your first step of converting each term to cosine is correct. However, when finding the phasor form, you need to use the angle associated with the cosine term, not the angle associated with the sin term. This means that for the first term, instead of using -135 degrees, you should use -45 degrees. Similarly, for the third term, you should use -60 degrees instead of -120 degrees. This will result in the correct solution of 24.52∠-117.39°.