• smallwa
In summary, the conversation discusses simplifying the function x(t)=5cos(wt)+5cos(wt+120)+5cos(wt-120) into standard sinusoidal form using phasors. It is determined that the result is 0 and the corrected function is x(t)=0.
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.

## 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

Last edited:
smallwa said:

## Homework Statement

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

ehild

ehild said:

ehild

sorry,corrected

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

ehild

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.

## What is the phasor addition rule?

The phasor addition rule is a mathematical rule used in signal processing to add together multiple sinusoidal signals that have the same frequency. It allows us to simplify complex calculations and analyze the overall behavior of the signals.

## How is the phasor addition rule used in signal processing?

The phasor addition rule is used to combine multiple sinusoidal signals with the same frequency, such as in AC circuit analysis. It allows us to determine the magnitude and phase of the resulting signal, which can help us understand the behavior of the system.

## What is a phasor?

A phasor is a representation of a sinusoidal signal using a complex number. The magnitude of the complex number represents the amplitude of the signal, and the angle represents the phase shift of the signal. Phasors are used in signal processing to simplify calculations and analysis.

## What is the difference between the phasor addition rule and the vector addition rule?

The phasor addition rule and the vector addition rule are similar in that they both involve adding complex numbers. However, the phasor addition rule is specific to signals with the same frequency, while the vector addition rule can be used for any type of vector addition.

## Can the phasor addition rule be applied to non-sinusoidal signals?

No, the phasor addition rule is only applicable to sinusoidal signals with the same frequency. Non-sinusoidal signals require different methods for addition and analysis.

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