### Automatics

**Chapter 10 Delay Unit and Substitute Transmittance**

**Chapter 10.1 Delay Unit**

**Chapter 10.1.1 Introduction**

**Transmittance of the**

Fig. 10-1

Fig. 10-1

**delay unit**compared to typical transmittance.

The

**delay unit**only delays the signal by

**To**time without changing its shape. It’s hard to give an example. Maybe an echo? Or a

**conveyor belt**on which

**sand**is fed from somewhere out there. If the input is the

**sand level**at the beginning of the conveyor belt, then at the end of the conveyor belt the

**level**will be the same, but with a

**certain delay**

**To**.

Generally, almost every

**dynamic unit**introduces some

**delay**of the

**output signal**relative to the

**input signal**, but it is distorted by its

**inertia**. On the other hand, the delaying unit is a

**pure**delay. It is useful for creating

**substitute transmittance**which the

**second**part of the

**chapter**concerns.

**Chapter 10.1.2 Delay unit with slider and bargraph**

It is a nasty dwarf that mimics you with a **delay** of **To=5 sec**

**Fig. 10-2**The unit restores the state of the slider with a delay of

**To=5 sec**. This can be seen in the movements of the slider and bargraph, as well as in the digital meters.

So its transmittance is:

Fig. 10-3

Fig. 10-3

**Typical**transmittance is the fraction

**G(s)=L(s)/M(s**), where

**L(s)**and

**M(s)**are polynomials. Only the

**delay unit**as an

**exponential function**, does not fall into this category! It looks

**pretty scar**y. Not only is it an

**exponential function**, it’s also a complex number

**s**in power. Don’t worry about it! Treat the

**delay uni**t as something that

**shifts**the input signal by time, here

**To=5sec.**

**Chapter 10.1.3 Delay unit with slider and oscilloscope**The properties of the

**delay uni**t can be seen even better on the oscilloscope

F**ig. 10-4**The output

**y(t**) exactly reproduces the input

**x(**t) from the slider with a

**delay To=5 sec**. Nothing more to add.

**Chapter. 10.2 Substitute Transmittance****Chapter. 10.2.1 Introduction****Fig. 10-5**The equivalent

**transmittance**consists of an

**inertial**and a

**delaying unit**. Many commonly used dynamic units are multi-inertial, especially in the chemical industry. And these, in turn, can be approximated as equivalent

**transfer functions**.

There are only

**3**parameters in it:

**k**– steady-state gain

**T**– time constant of the inertial term

**To**– delay.

This is important for the optimal settings of the

**PID**controller. There are formulas which for the object parameters

**k**,

**T**and

**To**determine the optimal settings of

**Kp**,

**Td**and

**Ti**of the

**PID**controller. We will deal with this in

**chapter 28.**

Let us remember that the

**equivalent transmittance**is only an

**approximation**of the real one.

**Chapter. 10.2.2 Determination of equivalent transmittance parameters based on the unit step response **Equivalent transmittance approximates

**multi-inertial unit**. Let’s find it, for example, for the

**three-inertial unit**. Usually, the determination of the parameters

**k, T1, T2**and

**T3**based on the response to the

**step**is quite complicated.

**Fig. 10-6**The

**animation**shows how to replace the transfer function of the

**three-inertial unit**with the equivalent

**transfer function**based on the response to the

**step**.

Draw a

**right triangle**whose hypotenuse is tangent to the answer

**y(t)**at its

**inflection**point

*****. Determination of the normalized parameters

**T=6.2 sec**and

**To=1.8**

**sec**is obvious in the figure. The

**k=y/x=1**parameter is the gain in which

**y(t)=y=1**, i.e. in a steady state after

**t=28sec**.

**What**is this substitute transmittance for anyway? After all, the best approximation of the transmittance is itself.

**Firstly**

Often the object, which can be a

**tank**in which something is mixed and chemical reactions take place, is not mathematically worked out. So I don’t know its

**transmittance**. In contrast, designating a step response is easy.

**Secondly**

Even if we knew the transmittance of the object, it can be characterized by many parameters. One of the most important tasks of automation is the selection of appropriate

**controller settings**. So that the response to the

**step**was the most optimal in some respect. For example, the shortest

**control time**with relatively small

**oscillations**. This will be discussed in

**chapter 28**.

Wise people have long worked out the selection of

**3**settings of

**Kp, Ti, Td**of the

**PID**controller for various combinations of

**k, T, To**of equivalent transmittance. Then they selected the appropriate tables for these

**3**combinations – i.e. the appropriate equivalent transfer function. Our task is only to identify

**k, T, To**as in

**Fig. 10-5**and select the

**controller**parameters from the table.

Of course, you have to be aware that this is a rather

**“rough”**method. After all, if we draw the tangent slightly differently, the parameters will be different. Especially The one that makes it extremely maliciously difficult to control.

In the next section, we will compare the response of the tested and substitute transmittance. Will the difference be big?

***inflection**point of the function–> where the function “bends the other way”. The definition cries out to heaven for vengeance, but it couldn’t be simpler.

**Chapter 10.2.3 Comparison of test and substitute transmittance based on step response**

**Fig. 10-7**The unit step

**x(t)**is fed to the input of the tested

**three-inertial**

**unit**and equivalent transmittance.

I didn’t expect the same answers. They differ, and quite a lot. Let us remember, however, that the parameters of the

**PID**controller are ideally suited for the equivalent transmittance . And this means that the parameters of the

**PID**controller for the tested transmittance, although not optimal, will be much better than if we tried to select them in a reasonable time by trial and error.