**Rotating Fourier Series3.**

**Summation of rotating vectors exp (jωt)**

**Chapter 3.1 What is this for?
**To make it easier to understand the summation of

**cos (ωt)**functions, more broadly –

**sinusoidal**functions.

And this matches

**Fourier Series,**which are the sum of

**sines**and

**cosines**.

**Chapter 3.2 General about summing rotating vectors exp (jωt)
**

**Rys 3-1**

**Rys 3-1a**

Sum of rotating vectors **F(jt)** where:

**c0** is a constant component or otherwise a **rotating** vector with speed **ω=0**

**c1, c2, c3 …** is a **complex number** which is the** initial state** of each spinning **vector c1*exp (jω1t), c2*exp(jω2t), ec3*exp(jω3t) …**

**ω1, ω2, ω3 …** The pulsations of **ω1, ω2, ω3 …** are **arbitrary**! i.e they may or may not be multiples of the first harmonic of **n * ω1**. This is the most general formula for the sum, and we will not analyze this case.

**Rys3-1b**

**The sum of rotating F(jω0t)** vectors with the same pulsation** ω0**.

It is a **vector-complex number** **c = c1 + c2 + c3 + …** rotating at speed **ω0**.

The real number **c0** is a constant component.

The **sum** of the **sinusoids** and **cosine** waves is also a **sinusoid** with the same pulsation **ω0** and the corresponding phase shift** φ**.

It is used in classical **electrical engineering**, where power plants generate electricity with a **constant **frequency **f=50H**.

The subject has already been thoroughly worked out, probably in the **nineteenth century**.

Example with animation in **chapter 3.3**.

**Figure 3-1c**

This is what we will mainly deal with.

**Ie**. the sum of **rotating vectors** with increasing pulsations **1ω0,2ω0, 3ω0 …**

I emphasize that each pulsation is a **multiple** of the fundamental pulsation **1ω0**, not any **ω1, ω2, ω3 …** as in **Rys.3-1a**

Examples with animation in **chapters 3.4 … 7**

**Chapter 3.3 The sum of 2 rotating vectors exp (jωt) with the same pulsation ω0 = 1 / sec
Chapter 3.3.1 **

**1*exp(j1t) -j1exp(j1t)**

The case of

**Fig.3-1b**when

**c0 =0, c1=1, c2=1*exp(-jπ/2)=-j1 and c3=c1+c2=1-j1=√2*exp(-jπ/4)**

The numbers

**c1, c2 and c3**are the i

**nitial states**of the

**spinning**vectors as in

**Rys.3-2**before the animation.

Note that the vector

**c2**lags

**π/2=90 °**with respect to

**c1**.

**Fig. 3-2
**The sum of

**2**rotating vectors as:

**1exp (j1t)-j1*exp(j1t)**or

**(1-1j)exp(j1t)**or

**√2exp(-jπ/4)*exp(j1t)**where

**-π/4=-45º.**

The

**right**vector is the sum of the

**2**

**left**vectors at any moment and has the parameters

**A=√2, ω=1/sec**and

**ϕ=-π/4=-45º**.

At the initial moment, that is before pressing

**“Start”**, the sum of the right vector is correct (add left vectors -> “diagonal of the square”). Then stop the simulation at any time by clicking on the drawing or

**“Start”**and check approximately if it is correct using .

**The most important conclusion.**

When all vectors have the same speed

**ω0**, their sum is also a rotating vector with the same speed

**ω0**, length

**A**and phase

**ϕ**.

**Note 1**

Applies to any number of rotating vectors.

**Note 2**

The relative position of all

**3**rotatingg

**vectors**to each other is

**constant**! This makes it much easier to analyze

**electrical circuits**. When this is not the case as in

**Rys.3-1a**or

**3-1c**, and the vectors rotate at different

**speeds**, the sum vector changes

**length**and

**velocity**! You will find out about this in

**Chapter 3.4**.

**Chapter 3.4 Sum F(jω0t)=1exp(j1t)+1exp (j2t) **

**Chapter 3.4.1 Vector-only version**

We begin to study the sums of rotating vectors with different pulsations.

Vector only means that the ends of the vectors do not draw the trajectory.

The case of **Rys. 3-1c** when **c0=0 c1=1, c2=1 **and** ω0=1/sec**

**Fig. 3-3
F(jω0t)= 1*exp (1jt)+1*exp (2jt) (ω0 = 1 / sec)
**You see complex functions as rotating vectors

**1*exp(1jt), 1*exp(2jt)**and their sum

**1*exp(1jt)+1*exp (2jt)**

You can clearly see the speed of

**1*exp(2jt)**

**twice**as high. Try to stop the animation at different times

**t**and check if the

**right**function is a vector sum of

**2**left ones. Here I am asking for some tolerance for the author’s ineffective use of the animation program.

**Rozdział 3.4.2 Trajektoria czyli wersja “tylko śladowa”**

**Chapter 3.4.2 Trajectory or ” trace only” version
**

**“Trace only”**means that the ends of the vectors from

**Fig.3-2**draw a trace. The vectors themselves are invisible and this trace is the trajectory

**F(jω0t)**. This is the case of

**Fig. 3-1c**when

**c0=0 c1=1, c2=1 and ω0=1/sec**. Only one period of animation is shown. Further rotations follow the same tracks and the animation looks static! Also in the next animations.

Here and further we will limit ourselves to the “only with a trace” version. The “vector” version of this case is the animation in

**Fig. 3-3.**

**Fig. 3-4
**

**F(jω0t)=1*exp(1jt)+1*exp(2jt)**

The

**right**animation is the sum of the

**left**

**2**. Note that

**1*exp(2jt)**“stopped” after the

**first half**period.

But the t

**rajectory**is still rotating, just on the same track!

**Chapter 3.5 F(jω0t)= 1exp(j1t)+0.7exp(j2t)
**

**Fig. 3-5
**

**F(jω0t)=**

**1exp(j1t)+0.7exp(j2t)**

**Chapter 3.6 F(jω0t)=1exp(j1t)+1exp(j2t-π/6)**

**Fys. 3-6
**

**F(jω0t)=1exp(j1t)+1exp(j2t-π/6)**

**φ=-π/6**is the actual delay

**φ=-30º**.

The previous examples had a phase shift of

**φ = 0º**, now one of the components

**1exp(j2t-π /6)**has

**φ**non-zero. The lag component

**1exp (j2t-π/6)**causes a counterclockwise rotation.

Compare with

**Rys. 3-4**, where the second component of j

**2t**has a delay of

**φ=0**. Try to explain it to yourself somehow.

**Chapter 3.7 F(jω0t)=0.3exp(j1t)+0.5exp(j2t-π/6)+0.45exp(j2t+π/4)
**More fancy trajectory then.

**Fig. 3-7
**

**F(jω0t)=0.3exp(j1t)+0.5exp(j2t-π/6)+0.45exp(j2t+π/4)**

Let me remind you that the sum vector is spinning all the time, even after the animation ends after the first

**T**period.

The

**red**point at

**(0,0)**is also the so-called the

**center of gravity**of the

**trajectory**. More on this in the next chapter.

**Rozdział 3.8 Conclusions
**

**Fig. 3-8**

**F(jω0t)**the sum of rotating vectors in which:

**– c0**is a constant component

**or**a formally rotating vector with pulsation

**ω=0**.

It is a real number (and a complex number at the same time!)

**– c1exp (1jω0)**is a vector rotating at speed

**1ω0**

**– c2exp (2jω0)**is a vector rotating at speed

**2ω0**

**– c3exp (3jω0)**is a vector rotating at speed

**3ω0**

…

…

**And**the complex numbers

**c1, c2, c3 …**are the

**initial**states of these rotating vectors.

**Some people**treat the initial state as the beginning of the world, and others as the moment

**t=0**when we start the experiment.

**Take**a moment to look at

**Fig.4-8**in the next

**chapter**.

There are coefficients

**c0=-0.5**,

**c1=0.9-j0.6**,

**c2=0.6 j0.4, and c3=0.4-j0.2**for a particular

**Fourier Series**.

**The**trajectory

**F(jω0t)**is drawn after the period

**T**along the same path. This period corresponds to the first harmonic

**1ω0**and

**T≈6.28sec**.

**The formula**

**Fig.3-7**describes the trajectory for

**ω0=1/sec**. The shape of the trajectory will be exactly the same for

**ω0=2/sec, ω0=3/sec**… Only that it will rotate

**2, 3…**times faster.