Finding the amplitude of the sum vector

In summary, the amplitude of the sum vector can be found by summing each vector described as a magnitude multiplied by a phase.
  • #1
chilge
8
0

Homework Statement


A signal E(t) is made up of three terms, each having the same frequency but differing in phase:

E(t) = E0cos(ωt) + E0cos(ωt + δ) + E0cos(ωt + 2δ)

It is possible to find the amplitude of the sum vector by summing each vector described as a magnitude multiplied by a phase. The sum will therefore contain three terms. You can simplify this to express the sum as a real number times a phase. The real number is the amplitude of the sum vector. Make a plot of the amplitude of the sum vector as a function of δ as δ varies from 0 to 2∏.


Homework Equations



A signal can be represented as the real part of a complex number z,
z = Aei(wt+∅) = A[cos(wt+∅) + jsin(wt+∅)]


The Attempt at a Solution



E(t) = E0cos(ωt) + E0cos(ωt + δ) + E0cos(ωt + 2δ)
E(t) = E0exp(iωt) + E0exp(i(ωt + δ)) + E0exp(i(ωt + 2δ))
E(t) = E0exp(iωt) * (1 + exp(iδ) + exp(i2δ))
E(t) = E0cos(ωt) * (1 + exp(iδ) + exp(i2δ))

I can't think of a way to simplify it any further to put it in the form real number * a phase term. Am I missing something?
 
Physics news on Phys.org
  • #2
First of all, this equation really makes no sense:
chilge said:
Aei(wt+∅) = A[cos(wt+∅) + jsin(wt+∅)]
because you are using two different notations for the imaginary unit in the same expression. If you use i to denote the imaginary unit, you should have:
Aei(wt+∅) = A[cos(wt+∅) + i sin(wt+∅)]

Second, between the first and second steps in your attempt at a solution, you incorrectly applied that formula. You have changed a real expression into an expression with a real part and a non-zero imaginary part.

If you do some simple algebra you should see that you can represent cos(x) as
cos(x) = (eix+e-ix)/2
Alternatively, you could write cos(x) as the real part of eix, e.g.
cos(x)+cos(y) = Re[eix + eiy]

Try applying those formulae and see if you can make any more progress. And feel free to post again if you get stuck. But you should make a better attempt before I give you any more hints.
 
Last edited:
  • #3
chilge said:

Homework Statement


A signal E(t) is made up of three terms, each having the same frequency but differing in phase:

E(t) = E0cos(ωt) + E0cos(ωt + δ) + E0cos(ωt + 2δ)

It is possible to find the amplitude of the sum vector by summing each vector described as a magnitude multiplied by a phase. The sum will therefore contain three terms. You can simplify this to express the sum as a real number times a phase. The real number is the amplitude of the sum vector. Make a plot of the amplitude of the sum vector as a function of δ as δ varies from 0 to 2∏.


Homework Equations



A signal can be represented as the real part of a complex number z,
z = Aei(wt+∅) = A[cos(wt+∅) + jsin(wt+∅)]


The Attempt at a Solution



E(t) = E0cos(ωt) + E0cos(ωt + δ) + E0cos(ωt + 2δ)
E(t) = E0exp(iωt) + E0exp(i(ωt + δ)) + E0exp(i(ωt + 2δ))
E(t) = E0exp(iωt) * (1 + exp(iδ) + exp(i2δ))
[STRIKE]E(t) = E0cos(ωt) * (1 + exp(iδ) + exp(i2δ))[/STRIKE]

I can't think of a way to simplify it any further to put it in the form real number * a phase term. Am I missing something?

You can not use the same notation for the real E(t) and the complex one. Write E(t)=Re(Z) where Z=E0exp(iωt) + E0exp(i(ωt + δ)) + E0exp(i(ωt + 2δ)), and make it equal to Aexp(i(ωt+θ)), where A and θ ( both real) are to be determined.

You need to transform the factor 1 + exp(iδ) + exp(i2δ) into the exponential form, 1 + exp(iδ) + exp(i2δ)=Cexp(iψ). For that, expand the exponentials exp(iδ)=cosδ+isinδ. Note that exp(i2δ)=[exp(iδ)]2. Collect the real and imaginary parts of the sum, and find the amplitude and phase.

ehild
 
Last edited:
  • #4
the solution is much easier if you factor out E0exp(iωt)exp(iδ):

[tex]\hat Z(t)=E_0 e^{i \omega t}+E_0 e^{i (\omega t+\delta)}+E_0 e^{i (\omega t+2\delta)}=E_0 e^{i (\omega t+\delta)} (e^{-i\delta}+1+e^{i\delta})[/tex]

But [itex] e^{-i\delta}+1+e^{i\delta}=1+2\cos \delta[/itex], a real quantity. That makes

[tex]\hat Z(t)=E_0 (1+2\cos \delta)e^{i (\omega t+\delta)} [/tex]

ehild
 
  • #5


Thank you for your question. First, let's clarify that the signal E(t) is a time-dependent quantity that represents the amplitude of a wave at a given time t. So, E(t) is not a vector, but rather a scalar quantity.

To find the amplitude of the sum vector, we can represent each term in the signal E(t) as a complex number with magnitude and phase. Using the Euler's formula, we can write:

E(t) = E0cos(ωt) + E0cos(ωt + δ) + E0cos(ωt + 2δ)
= E0 * Re[exp(iωt)] + E0 * Re[exp(i(ωt + δ))] + E0 * Re[exp(i(ωt + 2δ))]
= E0 * Re[exp(iωt) * (1 + exp(iδ) + exp(i2δ))]

Now, we can see that the term inside the Re[] function is a complex number with magnitude E0 and phase (ωt + δ + 2δ). So, we can rewrite the signal E(t) as:

E(t) = E0 * Re[E0 * exp(i(ωt + δ + 2δ))]

The amplitude of the sum vector is then simply E0, the magnitude of this complex number. As δ varies from 0 to 2∏, the phase term in the exponential function will also vary, resulting in a plot of the amplitude of the sum vector as a function of δ.

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

1. What is the definition of amplitude?

The amplitude of a wave or vector is the maximum displacement or distance from the equilibrium position. It is a measure of the strength or intensity of the wave or vector.

2. How do you calculate the amplitude of a sum vector?

To find the amplitude of a sum vector, you need to first calculate the amplitudes of the individual vectors. Then, use the Pythagorean theorem to find the magnitude of the resultant vector. The amplitude of the sum vector is equal to the magnitude of the resultant vector.

3. Can the amplitude of a sum vector be negative?

No, the amplitude of a vector is always a positive value as it represents the maximum displacement from the equilibrium position. If the resultant vector has a negative direction, it will be represented by a negative sign in front of the magnitude.

4. How does the amplitude of a sum vector change when two vectors are added together?

The amplitude of the sum vector will depend on the amplitudes and directions of the individual vectors. If the two vectors are in the same direction, the amplitude of the sum vector will be the sum of the individual amplitudes. If they are in opposite directions, the amplitude of the sum vector will be the difference between the two amplitudes.

5. What is the unit of measurement for amplitude?

The unit of measurement for amplitude will depend on the unit used to measure the individual vectors. For example, if the individual vectors are measured in meters, the amplitude of the sum vector will also be in meters.

Similar threads

  • Advanced Physics Homework Help
Replies
3
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
814
  • Introductory Physics Homework Help
Replies
12
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
841
  • Introductory Physics Homework Help
Replies
2
Views
2K
  • Advanced Physics Homework Help
Replies
13
Views
3K
  • Atomic and Condensed Matter
Replies
2
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
5K
  • Advanced Physics Homework Help
Replies
1
Views
8K
  • Advanced Physics Homework Help
Replies
1
Views
2K
Back
Top