A phasor can be thought of as a vector, representing a complex number, rotating around the polar

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A phasor can be thought of as a vector, representing a complex number, rotating around the polar plane at a certain frequency in radians/second. The projection of such a vector onto the real axis gives a cosine with a certain amplitude and phase. This problem will show the algebra of phasors which would help you with some of the trigonometric identities that are hard to remember.

(a) When you plot y(t) = Asin(Ω0t) you notice that it is a cosine x(t) = Acos(Ω0t) shifted in time, i.e.,

y(t) = Asin(ot) = A cos(No(t – A;)) = x(t – A;)


how much is this shift Δt? Better yet, what is Δθ = ΩθΔt or the shift in phase? One thus only need to consider cosine functions with different phase shifts instead of sines and cosines.

(b) From above, the phasor that generates 

r(t) = A cos(Qot) is Ae10 so that r(t) = Re[Ae³°e%ot].


The phasor corresponding to the sine y(t) should then be Ae-jπ/2. Obtain an expression for y(t) similar to the one for x(t) in terms of this phasor.

(c) From the above results, give the phasors corresponding to -x(t) = -Acos(Ω0t) and -y(t) = -sin(Ω0t). Plot the phasors that generate cos, sin, -cos and -sin for a given frequency. Do you see now how these functions are connected? How many radians do you need to shift in positive or negative direction to get a sine from a cosine, etc.

(d) Suppose then you have the sum of two sinusoids, for instance z(t) = x(t) + y(t) = Acos(Ω0t) + Asin(Ω0t), adding the corresponding phasors for x(t) and y(t) at some time, e.g., t = 0, which is just a sum of two vectors, you should get a vector and the corresponding phasor. For x(t), y(t), obtain their corresponding phasors and then obtain from them the phasor corresponding to z(t) = x(t) + y(t).

(e) Find the phasors corresponding to

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Related Book For  book-img-for-question

Signals and Systems using MATLAB

ISBN: 978-0128142042

3rd edition

Authors: Luis Chaparro, Aydin Akan

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