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summary of this topic

Algebra of Complex Numbers

Algebra of Complex Numbers

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Modeling sine waves


 »  Algebraic explanation
    →  `r sin(omega x + phi)`

substitute `sin(A+B)` formula
    →  `= r(cos phi sin omega x ``+ sin phi cos omega x)`

substitute `cos omega x = i sin omega x`
    →  `= r(cos phi + i sin phi)sin omega x`

substitute coordinate form of complex number
    →  `=(a+ib)sin omega x`

considering that `omega` is a constant, `sin omega x` can be taken off the expression.
    →  `=a+ib`


 »  Geometrical explanation
    →  `r sin(omega x + phi)`
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.       `2.5 sin(1.1x+.7)`
      `= 2.5 cos(.7)sin(1.1x)`
      ` + 2.5 sin(0.7)cos(1.1x)`


 »  Sine waves of same frequency is modeled with complex number `a+ib`
    →  Complex addition as per sine wave addition
    →  Complex multiplication as per sine wave interaction

Complex Number: Modeling sine waves

plain and simple summary

nub

plain and simple summary

nub

dummy

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.

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


Keep tapping on the content to continue learning.
Starting on learning "Complex number model of 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.

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.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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)`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

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

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 frequencies and amplitudes

The answer is 'Systems that modify amplitudes and phase'

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Progress

It is quite easy to introduce the following in abstract definitions: ;; complex numbers,;; the arithmetic operations (addition, multiplication, powers, etc.) and ;; properties (commutative, distributive, et cetera. ;; 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.;; It is the plot for f of x equals sine 1 point 1 x
The figure shows the sine wave sine 1 point 1 x in blue. ; In comparison, the sine wave in orange has different magnitude and phase. Phase is the horizontal shift of the wave, as shown in the figure. The curve in orange is for f of x equals 2 point 5 sine 1 point 1 x plus point 7. ;; This plot has r=2 point 5 and theta = 0 point 7 radian. ;; Complex number, 2 point 5 cos point 7 plus i sine point 7, represents the function.
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 (sine and cos) in the figure (explained in the next page). ;; Mathematical Proof : r sine omega x + phi ;; Substituting sine a + b formula equals a sine omega x + b cos omega x ;; Where a = r cos phi and b = r sine phi ;; Substituting cos omega x = i sine omega x ;; r sine omega x + phi ;; equals a sine omega x + i b sine omega x ;; equals a+i b sine omega x ;; Inverse of that, sum of sine and cosine waves of different amplitudes make a sine wave of a certain phase. ;; Prove : a sine omega x + b cos omega x ; equals ; r sine omega x + phi ;;
The figure shows three plots. The sine wave in green equals the sum of sine waves in orange and purple. ;; The sine wave in green is 2 point 5 sine (1 point 1 x + point 7 ;; The sine wave in orange is 2 point 5 cos point 7 sine 1 point 1 x ;; The sine wave in purple is 2 point 5 sine point 7 cos 1 point 1 x ;; 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 sine omega x + phi 1 + r 2 sine omega x + phi 2 ;; equals a 3 sine 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 A C 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 sine 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 sine omega x + phi 1 is the sine wave ;; r 2, phi 2 is the element given as an ordered pair. ;; Multiplication is given by : r 1 sine omega x + phi 1 multiplied by r 2, phi 2 ;; equals r 1 r 2 sine omega x + phi 1 + phi 2 ;; equals a 3 sine omega x + b 3 cos omega x ;; We can prove that ; a 3 = a 1 a 2 - b 1 b 2 ;; b 3 = a 1 b 1 + a 2 b 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 + b i sine omega x ;; sine wave addition ; a 1+a 2 + i b 1 + b 2 sine omega x ;; sine wave interaction with an element as multiplication ; a 1 a 2 minus b 1 b 2 + i a 1 b 1 + a 2 b 2 sine omega x ;; Complex number arithmetic is defined for these applications ;; Complex number a + i b;; Complex number addition : a 1 + a 2 + i b 1 + b 2 ;; Complex number multiplication : a 1 a 2 - b 1 b 2 + i a 1 b 1 + a 2 b 2
Complex number is used to model which of the following?
different frequencies
Sine waves of different frequencies, different amplitudes, same phases
same phases
Sine waves of same frequencies, different amplitudes, same phases
same amplitudes
Sine waves of same frequencies, same amplitudes, different phases
same frequencies;
Sine waves of same frequencies, different amplitudes, different phases
The answer is 'Sine waves of same frequencies, different amplitudes, different phases'
Complex number is used to model which of the following?
modify frequencies and phase
Systems that modify frequencies and phase
modify amplitudes
Systems that modify amplitudes and phase
modify frequencies and amplitudes
Systems that modify frequencies and amplitudes
The answer is 'Systems that modify amplitudes and phase'

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