- #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) = E

_{0}cos(ωt) + E

_{0}cos(ωt + δ) + E

_{0}cos(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 = Ae

^{i(wt+∅)}= A[cos(wt+∅) + jsin(wt+∅)]

## The Attempt at a Solution

E(t) = E

_{0}cos(ωt) + E

_{0}cos(ωt + δ) + E

_{0}cos(ωt + 2δ)

E(t) = E

_{0}exp(iωt) + E

_{0}exp(i(ωt + δ)) + E

_{0}exp(i(ωt + 2δ))

E(t) = E

_{0}exp(iωt) * (1 + exp(iδ) + exp(i2δ))

E(t) = E

_{0}cos(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?