### Automatics

**Chapter. 6 Double Inertial Unit**

**Chapter 6.1 Introduction**

Fig. 6-1

Fig. 6-1

**Transmittance**of the

**Two-inertial**unit with gain

**K**and time constants

**T1**and

**T2**

Previously, an example of an

**Inertial Unit**was a furnace. At the beginning, the

**temperature**increases the fastest, then the temperature speed gradually decreases (but the temperature itself continues to increase!). Eventually it will stop growing and reach

**a steady-maximum**state, e.g.

**+150 °C**. At the beginning, the temperature rises the

**fastest?**Is it? If you look closely, nothing happens at first. That is, the

**initial rate**of temperature increase is

**zero**. Only then the temperature gradually accelerates and at some point the

**speed**reaches its maximum. (

**Velocity**that is a derivative of temperature, not the

**temperature itself**!). From that moment, the

**speed**decreases until it becomes

**zero**and the temperature reaches a

**steady-maximum**value.

So the

**Inertial Unit**as a furnace model is only its first approximation. The

**Two Inertial Unit**is more accurate. But not overly accurate, because as you will see in a moment, the initial speed, although

**not maximum**, is not

**zero**either.

**Chapter 6.2 Slider as input and bar graph as output**

**Fig. 6-2**A

**Two Inertial Uni**t as a serial connection of two

**Inertial ones**.

**yp(t)**-indirect signal from the

**first inertial**unit.

“Waving” the

**slider**and observing the

**bar graph**is only a preliminary familiarization with the object. You may notice a gradual “acceleration” of the temperature. But you’ll definitely notice a feature of the

**Two-Inertial**that the

**Inertial**didn’t have.

The

**+maximum**slider input is given, and when

**y(t)**is the most

**accelerated**, –

**minimum**is given. You will surely notice that the output signal continues to grow, although the input is not

**only zero**but

**even negative**!

This is also true in a real oven. The temperature continues to rise even when the power supply is interrupted.

When observing

**digital meters,**you will notice that the

**intermediate signal**

**yp(t)**“leads” the

**output signal y(t**).

You will notice more interesting things, including the inflection point, by examining the waveforms on the oscilloscope.

**Chapter 6.3 Input signal x(t) as a unit step**

**Fig. 6-3**

The “transition” signal **yp(t)** after the first inertia clearly precedes the second inertia **y(t)** – i.e. the output of the **Double-Inertial Unit**. The acceleration effect is also visible. Around second **6**, the rate of increase of **y(t)** (i.e., the derivative of **y'(t)**) is the highest. Here is the so-called **inflection** point of the function **y(t)** After **29** seconds we have a steady state **y(t)=1**. Then the velocity **y(t)**, i.e. the derivative **y'(t)** is zero.

**Chapter 6.4 Input signal x(t) as a single rectangle pulse.**

**Fig. 6-4**

At time** 8…8.7 sec**, the output signal **y(t)** increases even though the input signal** x(t)** has fallen to zero!

At this time, **yp(t)>y(t**) and this is what causes this phenomenon.

**Chapter 6.5 Multi-Inertial Units**

Look at **Fig. 6-1**. The denominator has two factors and is a **two-inertial** unit. If it had three factors, it would be **three-inertial**. And if there were many factors, it would be **multi-inertial**. The larger this “many”, the slower the initial velocity of the output signal **y(t)**. Many processes in the industry, especially in the chemical industry, are **multi-inertial** units. In **chapter 10** you will learn that they are approximated by the so-called **substitute transmittance**.