[signal]phasor addition rule

[smallwa]

Homework Statement

Define x(t)=5cos(wt)+5cos(wt+120)+5cos(wt-120)

simplify x(t) into the standard sinusoidal form:x(t)= A cos(wt+phase).Use phasors to do the algebra.

Homework Equations

The Attempt at a Solution

i know it needs to use phasor addition rule.
First,i represent x1(t),x2(t),x3(t) by the phasors,and then add them together
however the result is 0,then i don't know to dothanks a lot

 

[ehild]

Homework Statement

Define x(t)=5cos(wt)+5cos(wt=120)+5cos(wt-120)

Correct x(t), please!


ehild

 

[smallwa]

Correct x(t), please!


ehild

sorry,corrected

 

[ehild]

The result is really zero. So write x(t)=0.

ehild

 

[asteriatic]

The phasor addition rule states that when adding two or more sinusoidal functions, you can represent them as phasors and add them geometrically. In this case, x(t) can be represented as the sum of three phasors: x1(t)=5cos(wt), x2(t)=5cos(wt+120), and x3(t)=5cos(wt-120).

To simplify x(t) into the standard sinusoidal form, we first need to find the magnitude and phase of each phasor. The magnitude of each phasor is 5, since the coefficient of the cosine function is 5. The phase of x1(t) is 0, x2(t) is -120, and x3(t) is 120. This is because the argument of the cosine function represents the phase shift, and in this case, the phase shift is 0 for x1(t), -120 for x2(t), and 120 for x3(t).

To add the phasors together, we can use the parallelogram rule. This means drawing each phasor with its magnitude and phase, and then connecting the tails of the phasors to create a parallelogram. The diagonal of the parallelogram represents the sum of the phasors.

In this case, the parallelogram will be a straight line since the phase differences between the phasors are all multiples of 120 degrees. The diagonal of the parallelogram has a magnitude of 5 and a phase of 0, which means the standard sinusoidal form of x(t) is x(t)=5cos(wt).

Therefore, the simplified form of x(t) using the phasor addition rule is x(t)=5cos(wt). This shows that the three cosine functions with phase shifts of 0, -120, and 120 add up to give a single cosine function with no phase shift. This is because cos(wt+0)=cos(wt), cos(wt-120)=cos(wt), and cos(wt+120)=cos(wt).

I hope this helps! Let me know if you have any further questions.