### Automatics

**Chapter 18. Amplitude-Phase Frequency Characteristics ****Chaptere 18.1 Introduction**You will learn the concept of

**amplitude-phase**frequency characteristics in the most natural way. The

**input**of the

**dynamic unit**will be

**sinusoids**with frequencies from

**0**to

**infinity**, and practically from

**very small**to a specific

**large**. The

**output sine**wave will have the same

**frequency**, but a different

**amplitude**(usually lower) and a different

**phase**(usually delayed) than the input

**one**. The

**amplitude**and

**phase**of the

**open**system determined in this way for the entire

**frequency range**contain easy-to-read information about the

**stability**of the

**closed**system.

The

**frequency**response of an

**open**system is relatively easy to determine. You can do it “live” object in a facility without fear that you will lose control over the industrial installation. After all, it is an open system, and therefore stable.

If you know

**complex numbers**, you will plot the

**amplitude-phase characteristic**by substituting

**s=jω**to

**G(s)**and drawing successive values of the expression

**G(jω)**for various

**ω**pulsations on the

**complex plane**.

**Chapter 18.2 Rotating vector as a model of sinusoidal time charts**

**Cosines**and

**sinusoids**are often found in automatics, electrical engineering, acoustics…

**Chapter 18.2.1 A rotating vector as a cosine 1cos(ωt)**

We’ll start with the

**cosine**.

**Fig.18-1**

**Fig. 18-1a**

cosine

**x(t)=1cos(ωt)**for

**ω=1/sec**as

**harmonic**motion on the

**x axis**

**Fig. 18-1b**

A rotating

**cosine**of lenght

**1**vector

A

**vector**of length

**1**has an initial state

**[1,0]=1+j0**and rotates at a rotational speed of

**ω=+1/sec**(that is, with period

**T=2π sec**and counter-clockwise). It is described by the complex equation

**z=1*exp(jωt)***

And now the most important! Projecting the end of the end of vector onto the

**x-axis**is the animation in

**Fig. 18-1a**!

**Fig. 18-1c**

Time chart

**x(t)=1cos(ωt)**

The above

**3**drawings describe the harmonic motion

**x(t)=1cos(ωt)**, which is exactly the same phenomenon. But the version of the rotating vector as

**z=1*exp(jωt)**is the most intuitive model of the

**cosine**function. Here the amplitudes

**A**and the phase shifts

**φ**are best visible!

*****I suggest the chapter “

**Complex**function

**exp(jωt)”**as a rotating vector in the course

**“Rotating Fourier Series” chapter 2**in the

**Home Page with Courses**. If you are less proficient with

**numbers**and

**complex**functions, take a look at the course

**“Complex Numbers”**.

**Fig. 18-2 **

**Rys. 18-2a**

sine

**x(t)=1sin(ωt)**for

**ω=1/sec**as

**harmonic**motion on the

**x axis**

**Fig. 18-2b**

A rotating

**sine**of lenght

**1**vector

The vector has an

**initial**state

**[0,-1]**otherwise

**0 -1j=-1j**and rotates with a rotational speed of

**ω=1/sec**

(i.e. with a period of

**T=2π sec**.) It is described by the equation

**z=-1j*exp(jωt)**or otherwise

**z=1*exp(jωt-90º)**. The projection of the

**end**of

**vecto**r moves on the

**x-axis**in

**Fig. 18-2a**according to the equation

**x(t)=1sin(ωt)**.

**Fig. 18-2c**

time chart

**x(t)=1sin(ωt)=1cos(ωt-90º)**

All this describes the harmonic motion

**x(t)=1sin(ωt)**. Note that the delay

**φ=-90º**of sine

**1sin(ωt)**relative to

**1cos(ωt)**is best seen in the rotating vectors

**Fig. 18-2b**and

**Fig. 18-1b**.

**Chapter 18.2.3 Rotating Delayed Cosine Vector 1cos(ωt-45º)**

**Fig.18-3****Fig. 18-3a**

Cosine **x(t)=1cos(ωt-45º)** for **ω=1/sec** as harmonic motion on **the x-axis**.**Fig. 18-3b****Unit**-vector rotation for **x(t)=1cos(ωt-45º)**

The vector has an initial state [**1/√2,-1/√2]** or **1/√2,-1j/√2** and rotates with a rotational speed of **ω=1/sec** (i.e. with a period of **T=2π sec**. ) It is described by the equation **z=(-1/√2,-1j/√2)*exp(jωt)** It can be described differently and probably simpler **z=1*exp(jωt-45º). **Its length (module) is** 1. **The **projection** of the** end** of the vector moves on the** x-axis** in **Fig. 18-3a** according to the equation **x(t)=1(ωt-45º)**.**Fig. 18-3c**

time chart** x(t)=1cos(ωt-45º)**

**Chapter 18.2.4 Two rotating vectors 1cos(ωt) and 0.75cos(ωt-45º) i.e. with different amplitude and phase**

We will meet with vectors of** different amplitude** and phase in **chap. 18.5 ** and **18.6**.

**Fig. 18-4****Fig. 18-4a**

Rotating vector **z=1exp(jωt)****Fig. 18-4b**

Rotating vector** z=0.75exp(jωt-30º)****Fig. 18-4c****2** functions **1cos(ωt)** and **0.75cos(ωt-30º)****Application**

The vectors show the **amplitudes** and the **delay ϕ=-30º** cosine better than the **time chart**. Also remember that for **full information**, pulsation must be given with vectors – **here ω=1/sec**!

**Chap. 18.3 Determination of the amplitude-phase characteristic of the proportional unit**

We’ll start with the **simplest dynamic** unit . i.e. **proportiona**l.

Fig. 18-5**Ampliltude-phase** characteristic of the** Proportional** unit.

For this unit **G(s)=**1 at any time there is **x(t)=y(t)**. So the output **sine** wave **y(t)** is equal to the input **sine** wave **x(t)** for each **ω** pulsation. Therefore, the **amplitude-phase** characteristic is so simple that it is difficult! It comes down to one point **(+1.0)**.

**Chap. 18.4 Determination of the amplitude-phase characteristics of the inertial unit.****Chap. 18.4.1 Introduction**

**So far**, we mainly used

**time analysis**of dynamic

**units**. We gave as input

**x(t)**unit

**step**,

**Dirac impulse**or

**sawtooth**. Then the answer

**y(t)**was information about the dynamic properties of this

**unit**. I want the reader to

**associate**the transmittance parameters

**G(s)**with the appropriate

**time charts**. Of course, knowledge of

**complex numbers**will be useful for this, but it is not absolutely necessary. I just mentioned that

**s**of

**G(s)**is a

**complex number**and not to worry too much about it. Therefore, one of the most basic concepts in automation has not appeared so far:

**Amplitude-Phase Characteristics**

It is an extension of the notion of

**frequency band**. As input, we give a

**sine**wave whose

**frequency**slowly changes, theoretically from

**zero**to

**infinity**, and in practice it is only, for example,

**30 frequencies**from

**very small**to

**high**.

The answer

**y(t)**is also a

**sine time chart**with the same

**frequency**, but with a

**different**amplitude and

**phase**.

These quantities, i.e.

**amplitude**and

**phase**, for different

**ω**, shown in one

**diagram**(e.g. Fig. 18-5), create an

**amplitude-phase characteristic**. Unlike the “ordinary”

**band**it also contains information about the

**phase**of the sine

**signal**.

We will now determine the amplitude-phase characteristic for the inertial term with the following parameters:

**K=1**

**T=1 sec**

We will

**star**t with a very small pulsation

**ω=0.31 1/sec**(T=20 sec!) and

**end**with

**ω=10.06 1/sec**(T=0.63 sec). The

**green**

**inpu**t

**sine**wave

**x(t)**always has a constant amplitude

**Xm=1**. For each

**ω**pulsation, we will determine the amplitude

**Ym**and the phase

**φ**of the

**red**output signal

**y(t)**in vector form.

**Chapter 18.4.2 ω=0.31/sec** (T=20 sec)

**Fig. 18-6**The

**input**is a

**green**sine wave

**x(t**) with an amplitude of

**Xm=1**and

**ω=0.31/sec**(T=20 sec)

The

**output**

**red**sine wave

**y(t)**has an amplitude

**Ym=0.95**and a delay

**φ=-17.5°**. Sine wave

**y(t)**is delaying behind by

**φ=-17.5°**relative to the

**x(t)**. The same

**inusoids**are also shown as

**vectors**. The

**phase**and the

**amplitude**of the

**red**sine wave

**y(t)**are much better visible here. Parameters

**Ym**and

**φ**are determined as late as possible (here after about

**25**seconds). Then the

**sine**wave is already “calm down”. At the very beginning, we have a

**transition state**in which the sine wave y(t) with the “elbow” is not really a sine wave yet. As you will see later, transients are more pronounced at

**higher**frequencies.

**Chapter 18.4.3 ω=0.63/sek (T=10 sek)**

**Fig.18-7**

The input is a sine wave** x(t)** with amplitude **Xm=1** and **ω=0.63/sec** (T=10 sec). The delay **y(t)** increased at **φ=-32°** and the amplitude decreased **y(t)** at **Ym=0.85**.

From the time chart, at first glance, it seems that **ω** is the same as before. But the time base has changed! Here and in the following experiments, the experiment lasts **30** seconds instead of** 60** seconds as before.

**Chapter 18.4.4 ω=1.26/sec (T=5 sek)**

**F****ig. 18-8**

**Xm=1**and

**ω=1.26/sec**(T=5 sec)

Delay

**y(t)**increased at

**φ=-52°**and the

**y(t)**amplitude decreased at

**Ym=0.61**

Chapter. 18.4.5 ω=2.51/sek (T=2.5 sek)

Chapter. 18.4.5 ω=2.51/sek (T=2.5 sek)

**Fig. 18-9****Xm=1** and **ω=2.51/sec** (T=2.5sec)

Delay** y(**t) increased at **φ=-68**° and the amplitude decreased** y(t)** at **Ym=0.37**

**Chapter. 18.4.6 ω=5.03/sek (T=1.25 sek)**

**Fig. 18-10**

The** red** arrow **y(t)** continues to **delay** and **decrease**. At higher **frequencies**, the **transient** state of the **sine wave** is more clearly visible at the **beginnin**g of the **waveform.**

**Chapter. 18.4.7 ω=10.06/sek (T=0.63 sek)**

**Fig. 18-11**

The **red** arrow **y(t)** continues to **delay** and **decrease**. At higher frequencies, the transient state of the sine wave is more clearly visible at the beginning of the **red** sine.

This completes the first stage of determining the **amplitude-phase** characteristics of the** inertial** unit **1/(1+sT)**.

**Chap. 18.4.8 Amplitude-phase characteristics of the tested inertial unit**

Let’s make one common **drawing** from successive **phasor graphs** (arrows).**Fig. 18-12**A common drawing is

**Fig. 18-12a**. The

**green**vector is a symbol of

**6**sinusoidal input signals

**x(t)**with pulsations

**ω1…ω6**and amplitude

**Xm=1**. The remaining

**6**

**red**vectors are the corresponding

**sinusoidal**responses. There were only

**6**pulsations here. What if there were more of them, e.g. a

**hundred**, or let’s go all the way – a

**million**? We will then obtain

**Fig. 18-2b**, in which the

**end**of the vector draws a

**semicircle**.

Take another look at the next drawing.

**The ends of all vectors are visible as a**

Fig. 18-13

Fig. 18-13

**red**semicircle. Added

**x,y**axes. Seemingly similar, but how to treat the

**red**response vector to a sinusoidal input signal

**x(t)**with

**ω0=0**pulsation?

**Firstly**, we did not do such an experiment, and

**secondly**, what is

**zero**pulsation or frequency? Treat it as a

**very small**pulsation! E.g. corresponding to the period

**T=1 year**. All year you sit in front of the oscilloscope and watch the sine wave! You start on

**January 1**from the level

**x(t)=0**. After the first quarter, you have a signal

**x(t)=+1**, etc. You will agree that

**y(t)**will also be slow and with almost

**zero**shift

**φ**. So small that you can take

**φ=0**. The amplitude will also be

**1**.

Notice that we don’t have a

**green**vector

**x(t)**. It was only needed for didactic reasons. Moreover, even

**red**vectors are unnecessary. Here we have kept only one of their representatives for pulsations

**ω=0.31/sec**. A semicircle alone is enough as the ends of various vectors.

The

**amplitude-phase characteristic**is a generalization of the concept of frequency

**response**(this, in turn, is a generalization of the concept of

**gain**). From it, you can read not only the

**gain**for a given

**pulsation**, but also the

**phase**. Here, for example, you can see that for

**ω=0.31/sec**, the gain

**K=0.95**and the phase shift

**φ=-17.5°**.

Each

**inertial unit**with gain

**K=1**starts with pulsation

**ω=0**from the point

**(+1.0)**and ends at

**ω=infinity**at the point

**(0.0)**.

If the time constant

**T**was

**twice**as large, i.e.

**G(s)=1(1+2s)**, then for

**ω=0.31/se**c there would be

**K=0.85**and

**φ=-32°**.

For a different gain, e.g. for

**K=3**, the beginning would be

**(+3.0)**but the end would be the same -> also

**(0.0)**. The semicircle would be

**3**times

**bigger**.

**Chap. 18.5 Determination of the amplitude-phase characteristic of the integral unit****Chap. 18.5.1 Integral unit as an example of astatic**

The inertial and proportional **unit** are the so-called static **units**. They show the **beginning** for **ω=0** and the end for **ω=∞**. And how it is for the astatic **unit**, e.g. integrating, you will see for yourself in a moment. We will study the integrating **unit**, similarly to the inertial **unit**.

**Chapter 18.5.2 ω=0.31/sek (T=20 sek)**

**Fig.18-14**

The output sine wave **y(t)** has an amplitude of **3.2** and a phase shift** φ=-90°**. Here’s a vector diagram of it that shows the same thing. The **red** output vector **y(t)** of length **3.2** is also delayed by **φ=-90°** relative to the input **x(t)**.

**Chapter 18.5.3 ω=0.63/sek (T=10 sek)**

**Fig. 18-15**The amplitude decreased twice to

**1.6**, but the phase

**φ=-90°**remained the same!

**Chapter. 18.5.4 ω=1.26/sek (T=5 sek)**

**F****ig. 18-16**The amplitude decreased

**twice**at

**0.8**,

**φ=-90**°.

Chapter 18.5.5 ω=2.51 1/sek (T=2.5 sek)

Chapter 18.5.5 ω=2.51 1/sek (T=2.5 sek)

**Fig. 18-17**The amplitude decreased

**twice**at

**0.4**,

**φ=-90**

**Chapter 18.5.6 ω=5.03/sek (T=1.25 sek)**

**F****ig. 18-18**The amplitude decreased twice at

**0.2, φ=-90°**

**Chapter 18.5.7 Amplitude-phase characteristics of the integral unit**Let’s make

**Fig.18-14…18**one common figure.

**Fig. 18-19**The common one is

**Fig.18-19a**.

**Fig. 18-19b**shows the

**amplitude-phase**characteristic of the

**integral unit**, without arrows and for all

**0<ω<∞**.

**For ω=0**, the gain of the integrating term is

**K=∞**and for

**ω=∞**, it is

**K=0**.

**Chapter 18.6 Static Units**

They have no **integrating** elements. So they do not have single** s** in the denominator **G(s)**. In response to a **unit step**, they give a** steady state**, and do not grow to **infinity** like **astatic ones**. It can be the previously presented inertial **unit**, as well as **multi-inertial**, **oscillatory**, **delayed** etc… Their characteristics, being more **complex** than** inertial**, are no longer **semi-circles**, but something **similar**. The **three-inertial** unit gives, for example, this characteristic.**Fig. 18-20**The characteristics of the

**three-inertial**unit pass through three

**quadrants**. You can guess that

**two-inertial**is only

**two quadrants**, just like the poor thing,

**inertial**only has

**one**. For a certain pulsation

**ω=ωkr**, the output

**sine**wave

**y(t)**of th

**e three-inertial**units is in antiphase to the input sine wave

**x(t)**There is no such thing in the

**two-inertial**term (because it is only in

**2**quadrants), and even more so in the inertial

**unit**.

**Chapter 18.7 Conclusions**

Previously, we treated the transfer function **G(s)**:**-Intuitively** as “something” that transforms the input signal **x(t)** output signal **y(t)**. For example, the inertial term **G(s)=2/(1+3s)** transforms a **unit step** **x(t)=1(t)** into an exponential waveform y(t) tending to **y=2** with a time constant of** 3T**. So you associated **G(s)** with the response to a **unit step**.**– Strictly** as the quotient **G(s)=Y(s)/X(s)** where **Y(s)** is the **Laplace** transform of **y(t)** and **X(s)** is the** Laplace** transform of** x(t)**.

It can be said that both definitions complement each other because:**-intuitive** is not **exact****-strict** is not** intuitive**

Now the transmittance **G(s)** transforms each sine wave** x(t)=1sin(ωt)** with pulsation **ω=0…∞** into a sine wave **y(t)=Asin(ωt-φ)** with amplitude **A** and phase shift **-φ**. An example is **Fig.18-13**. Here the inertial term **G(s)=1/(1+1s)** turns the input sine wave **x(t)=1sin(ωt)** for **ω=0.31/sec** into the output sine wave **y(t)=sin(ωt-φ)** where** A =0.95** and **φ=-17.5º**. Each point of the **amplitude-phase characteristic** corresponds to the pulsation **ω=0…∞**, from which the **amplitude** and **phase** of the output sine wave** y(t)=Asin(ωt-φ)** can be read. In short, the transmittance **G(s)** is the **phase-amplitude characteristic**. It contains information on how to transform each input sine wave **x(t)=1sin(ωt)** in the range **ω=0…∞** into an output sine wave **y(t)=sin(ωt-φ)**. It is a generalization of the frequency response in which there is not only a gain **K** for each **frequency**, but also a phase shift **-φ**.

Note that the definition of** G(s**) as an **amplitude-phase characteristic** is quite strict and intuitive