Expression of AM Cosine Wave using Phasors

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In summary, an AM cosine wave can be expressed in the form of A1cos(w1*t+phi1) + A2cos(w2*t+phi2)+A3cos(w3*t+phi3) where w1<w2<w3 by using phasors and replacing the sine and cosine functions with exponential expressions and collecting terms with the same angular frequency.
  • #1
yoyo
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An AM cosine wave is represented by x(t)=12+7*sin(pi*t-(1/3)*pi)]*cos(13*pi*t). Use phasors to show that x(t) can be expressed in form of:

A1cos(w1*t+phi1) + A2cos(w2*t+phi2)+A3cos(w3*t+phi3) where w1<w2<w3.

I am really stuck with this. don't know where to start. can someone please help me out?
 
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  • #2
yoyo said:
An AM cosine wave is represented by x(t)=12+7*sin(pi*t-(1/3)*pi)]*cos(13*pi*t). Use phasors to show that x(t) can be expressed in form of:

A1cos(w1*t+phi1) + A2cos(w2*t+phi2)+A3cos(w3*t+phi3) where w1<w2<w3.

I am really stuck with this. don't know where to start. can someone please help me out?

Use that

[tex] cos(\omega t +\phi)= \frac{e^{i(\omega t + \phi)}+e^{-i(\omega t + \phi)}}{2} [/tex]

and

[tex] sin(\omega t +\phi)= \frac{e^{i(\omega t + \phi)}-e^{-i(\omega t + \phi)}}{2i} [/tex]

Replace the sin and cos functions in x(t) with the exponential expressions, do all the multiplications, collect the terms with the same angular frequency and see what you get.

ehild
 
  • #3


I would like to first clarify that AM cosine waves are commonly used in communication systems to transmit information through radio waves. These waves are a combination of a high frequency carrier wave and a lower frequency modulating wave.

Now, to express the given AM cosine wave using phasors, we need to first understand what phasors represent. Phasors are complex numbers that can be used to represent the amplitude and phase of a sinusoidal signal at a specific frequency. In other words, they represent the magnitude and phase of a sinusoidal signal in the frequency domain.

Let's break down the given AM cosine wave into its components using phasors. We can write the expression as:

x(t) = 12 + 7*sin(pi*t - (1/3)*pi) * cos(13*pi*t)

= 12 + 7 * (1/2j) * (e^(j*pi*t) - e^(-j*pi*t)) * (e^(j*13*pi*t) + e^(-j*13*pi*t))

= 12 + (7/2j) * (e^(j*14*pi*t) - e^(j*12*pi*t) - e^(j*26*pi*t) + e^(j*24*pi*t))

= 12 + (7/2j) * (e^(j*14*pi*t) - e^(j*12*pi*t)) - (7/2j) * (e^(j*26*pi*t) - e^(j*24*pi*t))

Now, we can use the phasor representation to write the above expression as:

x(t) = 12 + A1 * e^(j*14*pi*t) + A2 * e^(j*12*pi*t) + A3 * e^(j*26*pi*t) + A4 * e^(j*24*pi*t)

where A1 = (7/2j), A2 = -(7/2j), A3 = -(7/2j), A4 = (7/2j)

We can further simplify this expression by converting the complex exponentials to cosines using Euler's formula:

e^(j*wt) = cos(wt) + j*sin(wt)

Therefore, our final expression using phasors becomes:

x(t) = 12 + A1 * cos(14*pi*t) + A2 * cos(12*pi
 

1. What is the purpose of using phasors in expressing AM cosine waves?

Phasors are complex numbers that represent the magnitude and phase of a sinusoidal function. Using phasors allows us to simplify the mathematical representation of AM cosine waves, making it easier to analyze and manipulate.

2. How do you convert an AM cosine wave into its phasor form?

To convert an AM cosine wave into its phasor form, we use the Euler's formula e^(jx) to represent the cosine function. This results in a complex number with a magnitude and phase, which is the phasor representation of the wave.

3. What information can we obtain from the phasor representation of an AM cosine wave?

The phasor representation of an AM cosine wave provides us with the amplitude and phase of the wave. It also allows us to easily calculate the instantaneous amplitude and phase of the wave at any given time.

4. What is the relationship between the phasor representation and the time-domain representation of an AM cosine wave?

The phasor representation and the time-domain representation of an AM cosine wave are interrelated through the use of the Fourier transform. The phasor representation captures the frequency and phase information of the wave, while the time-domain representation shows the amplitude and time information.

5. Can phasors be used to analyze other types of waves besides AM cosine waves?

Yes, phasors can be used to analyze any type of sinusoidal wave, including AM sine waves, FM waves, and even non-electrical waves such as sound waves. This is because the fundamental principles of phasors, such as representing magnitude and phase, apply to all sinusoidal functions.

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