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Thought-Process to Discover Knowledge

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mathsComplex NumbersAlgebra of Complex Numbers

Complex Number: Modeling sine waves

This page provides proofs, illustration, and explanation to how complex number is used to model sine waves and elements interacting with sine waves.



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It is quite easy to introduce the following in abstract definitions:
 •  complex numbers,
 •  the arithmetic operations (addition, multiplication, powers, etc.) and
 •  properties (commutative, distributive, etc.)

Apart from understanding the abstract concepts, it will immensely benefit to understand application scenarios. This topic explains this.

One has to apply his or her mind to understand. Hope you enjoy exercising your brains.

The figure shows a sine wave.sine wave function illustration It is the plot for `f(x)=sin(1.1x)`

The figure shows the sine wave `sin(1.1x)` in blue. In comparison, the sine wave in orange has different magnitude and phase. Phase is the horizontal shift the wave is having as shown in the figure.sine wave with amplitude and phase illustration The curve in orange is for `f(x)=2.5sin(1.1x+.7)`.

This plot has `r=2.5` and `theta = 0.7 text(radian)`.
Complex number `2.5(cos(0.7)+ i sin(0.7))` represents the wave. We will explain this in next few pages.

A sine wave with an amplitude and phase can be equivalently represented as sum of a sine wave and a cosine wave of different amplitudes and `0` phase.

This is shown with a plot of sine and sum of (sin and cos) in the figure (explained in the next page).

Mathematical Proof:
`r sin (omega x + phi)` Substituting sin(a+b) formula
`quad quad = a sin(omega x) + b cos(omega x)`
Where `a = r cos phi` and `b= r sin phi`
Substituting `cos(omega x ) = i sin(omega x)`
`r sin (omega x + phi)`
`quad quad = a sin(omega x) + ib sin(omega x)`
`quad quad = (a+ib) sin(omega x)`

Inverse of that, sum of sine and cosine waves of different amplitudes make a sine wave of a certain phase.

Prove
`a sin(omega x) + b cos(omega x)`
`= r sin (omega x + phi)`
sine wave with phase as a sum of sine and cosine without phase

The figure shows three plots. The sine wave in green equals the sum of sine waves in orange and purple.sine wave with phase as a sum of sine and cosine without phase The sine wave in green is `2.5sin(1.1x+.7)`

The sine wave in orange is `2.5cos(.7)sin(1.1x)`

The sine wave in purple is `2.5sin(.7)cos(1.1x)`

It can be verified that the first equals the sum of latter two at every point.

It is easy to prove that the addition of two sine waves of different phase results in another sine wave.

Prove
`r_1 sin (omega x + phi_1) + r_2 sin (omega x + phi_2) `
`= a_3 sin(omega x) + b_3 cos(omega x)`

prove that
`a_3 = a_1 + a_2`
`b_3 = b_1 + b_2`

This uses only trigonometry to prove the results. The complex number models this addition.

The sine waves, when passing through passive elements (resistors, capacitors, inductors) in AC circuits, undergo change in amplitude and phase. The passive elements are modeled as complex numbers.

The change in amplitude is given by `r` and the change in phase is given by `theta`, then the passive element is given as a complex number `r(cos theta + i sin theta)`.

It is easy to prove that the interaction of sine wave with an element that modifies amplitude and phase, is complex multiplication.
`r_1 sin (omega x + phi_1)` is the sine wave
`(r_2, phi_2) ` is the element given as an ordered pair.

Multiplication is given by
Prove
`r_1 sin (omega x + phi_1) xx (r_2, phi_2) `
`= r_1r_2 sin (omega x + phi_1+ phi_2)`
`= a_3 sin(omega x) + b_3 cos(omega x)`

prove that
`a_3 = a_1a_2 - b_1b_2`
`b_3 = a_1b_1 + a_2b_2`

This uses only trigonometry to prove the results. The complex number models this.

Electrical current, Sine waves (having constant frequency) of different amplitude and phase is mathematically modeled with complex numbers.

Electrical passive elements that affect the amplitude and phase of sine waves is mathematically modeled with complex numbers.

Complex arithmetics (addition, multiplication, etc.) models the interaction between the sine waves and the elements.

Application Details:
 •  Sine wave is modeled as `(a + bi)sin(omega x)`
 •  sine wave addition `[(a_1+a_2)+ i(b_1 + b_2)]sin(omega x)`
 •  sine wave interaction with an element as multiplication `[(a_1a_2 - b_1b_2) + i(a_1b_1 + a_2b_2)] sin(omega x)`

Complex number arithmetic is defined for these applications
 •  Complex number `a + ib`
 •  Complex number addition `(a_1+a_2) + i(b_1 + b_2)`
 •  Complex number multiplication
`(a_1a_2 - b_1b_2) `
`quad quad + i(a_1b_1 + a_2b_2)`

Solved Exercise Problem:

Complex number is used to model which of the following?

  • Sine waves of different frequencies, different amplitudes, same phases
  • Sine waves of same frequencies, different amplitudes, same phases
  • Sine waves of same frequencies, same amplitudes, different phases
  • Sine waves of same frequencies, different amplitudes, different phases
  • Sine waves of same frequencies, different amplitudes, different phases

The answer is 'Sine waves of same frequencies, different amplitudes, different phases'

Solved Exercise Problem:

Complex number is used to model which of the following?

  • Systems that modify frequencies and phase
  • Systems that modify amplitudes and phase
  • Systems that modify amplitudes and phase
  • Systems that modify frequencies and amplitudes

The answer is 'Systems that modify amplitudes and phase'

                            
slide-show version coming soon