Drawing Phasor Representation for v(t)=20cos(200t+45°)+cos(200t)

[noppawit]

Draw the phasor representation for each of the following signals. Also write the signal as one sinusoid, v(t) = 20cos(200t+45°)+cos(200t)

From this equation, do I have to turn to v(t) = A sin (ωt + φ) ? If yes, how can I convert it?

Thank you.

 

[mplayer]

I think its asking to first draw the phasors for each one of the cosine waves in v(t). Then add the cosine waves together and write the equation for v(t) as a single sinusoid in the form you listed: A sin (ωt + φ). When drawing the phasors you'll probably want to convert each part of the signal from the time-domain to frequency domain, A∠φ. Hope that helps.

 

[Mene]

Yes, you can convert the given equation to the form v(t) = Asin(ωt + φ) by using the trigonometric identity cos(a+b) = cos(a)cos(b) - sin(a)sin(b). In this case, A = 20 and ω = 200. To find φ, we can use the fact that cos(45°) = sin(45°) = 1/√2. Therefore, the equation becomes v(t) = 20cos(200t)cos(45°) - 20sin(200t)sin(45°) + cos(200t). Simplifying this, we get v(t) = 10√2cos(200t) - 10√2sin(200t) + cos(200t). Now, we can write this as v(t) = 10√2sin(200t + 45°) + cos(200t). This is now in the form v(t) = Asin(ωt + φ), where A = 10√2, ω = 200, and φ = 45°.

To draw the phasor representation, we can plot a vector with length A = 10√2 at an angle φ = 45° on the complex plane. This represents the first term 10√2sin(200t + 45°). Then, we can plot a vector with length 1 at an angle 0°, representing the second term cos(200t). The resultant phasor representation will be the vector sum of these two vectors, which will have a length and angle determined by the addition of the two individual vectors.

In summary, the phasor representation for v(t) = 20cos(200t+45°)+cos(200t) is a vector with length A = 10√2 at an angle φ = 45° plus a vector with length 1 at an angle 0°. This can also be written as v(t) = 10√2sin(200t + 45°) + cos(200t).