### Automatics

**Chapter 8 Integrating Unit with Inertia**

**Chapter. 8.1 Introduction**

**Transmittance of the integral unit with inertia.**

Fig. 8-1

Fig. 8-1

Do you remember the

**integral unit**? The step

**x(t)**at the input caused the output signal

**y(t)**to increase to

**infinity**with a

**constant speed**. I emphasize. With a

**constant speed**from the beginning of the step

**x(t)**.

The integral

**unit**

**with inertia**also tends to

**infinity**. But he does it a little differently. At the beginning of the step,

**x(t)**“accelerates” starting from a velocity

**V=0**and ending with a certain fixed

**V=const**. So it is a more accurate approximation of, for example,

**an actuator**than a

**ideal integrating**unit.

**Chapter 8.2 k=1 T=3 sec with slider and bar graph**

Again, we’ll start with the** bargraph** to get you acquainted with the dynamics.

The **inertial unit** is a series connection of the integrating **1/sTi** and the **i**nertial** 1/(1+sT)**. Here **Ti=1sec** and **T=3sec**

**Fig. 8-2**The slider is initially set to

**0**. I gave it

**x(t)=+0.1=max**. You will observe waveforms similar to those for the previously tested

**integrating unit**. You also need to give

**0**to the input to

**stop**the

**y(t)**output signal. A

**digital meter**will be useful for the

**0*** setting. If you managed to stop

**y(t)**(does not

**increase**or

**decrease**), then a typical feature of

**PI**or

**PID**controllers will be revealed. The fixed output of the

**yr(t)**controller is

**different**from

**zero**, although the input is

**zero**!

The waveforms are similar to the

**integral unit**, but not the same! I hope you can see the

**inertia**. Especially when you enter

**+max**and in a moment

**-max**. You will notice that the signal continues to increase for a short time even though

**x(t)=0**and even

**decreases**! The phenomenon of

**inertia**will be clearer in the next experiment with the

**oscilloscope**.

*It is difficult to set an exact

**0**on a digital meter. Rather, it is a signal close to

**0**, e.g.

**x(t)=+0.002**. This means that

**y(t)**will

**“almost stop”**i.e.

**y(t)**is increasing

**very slowly**.

**Chapter 8.3 Ti=1 sec and T=3 sec with unit step and oscilloscope**The

**integrating**unit with

**inertia**is a series connection of the

**integrating**unit

**1/sTi**and the

**inertial**unit

**1/(1+sT)**.

**Ti=1sec**and

**T=3sec.**

**F****ig. 8-3**The input is a unit step

**x(t)=1**. The signal

**yp(t)**is after the integrator,

**y(t)**is the

**output**. At the beginning, the inertia

**T=3 sec**is clearly visible. In the steady state, both signals grow at the same speed. The

**integrating**unit with

**inerti**a is an example of the so-called

**astatic**system. At

**non-zero x(t)**, the signal

**y(t)**i

**ncreases**or

**decreases**. For this integrating term

**Ti=1 sec**as the time after which

**yp(t)**becomes equal to

**x(t)**. This is also the

**time**after which, in the steady state,

**y(t)**will increase by the step value

**x(t)=1**, i.e.

**Ti=1 sec.**

In

**steady**state, i.e. after approximately

**18 sec**, the output signal

**y(t)**is delayed by

**T=3 sec**.

**Chapter 8.4 Ti=1 sec T=3 sec with a single square pulse and an oscilloscope**

The **inertial integrating** unit is acted upon by a **single rectangular pulse**.

**Fig. 8-4**

This is a more accurate approximation of the **actuator** – a **motor** with a** gearbox** as an **ideal integrating** element whose** arm** can control the** valve** position. This example is discussed in **Chapter 4. Integrating Unit**. Here, however, we also take into account **inertia**. The black **x(t)** input is the **voltage** at the **motor** and the **red** output **y(t)** is the valve stem **position**. Needless to say, **inertia**, consisting mainly of mechanical** inertia** + some **electrical inertia** – **inductance**, “spoils” the **quality** of the device. In an actuator, it would be difficult to extract the **yp(t)**-signal directly after the **integrator.** This is only possible in the model as in **Fig. 8-4**.

**Chapter 8.5 Integrating Unit with inertia k=1 T=1.25 sec with positive and negative rectangular pulse and oscilloscope**We will see how the

**actuator**works by “searching” for its

**position**.

I.e. it turns once in one direction and once in the other until it finds the position set by the regulator and stops. Therefore, the

**inertial integrating**unit is acted upon by a

**positive**and

**negative**rectangular impulse.

**Fig. 8-5**

The **blue** **yp(t)** is an** ideal actuator** without** inertia**. The **red y(t)** is the **real **actuator. The actuator from **Fig. 8-4** with inertia **T=3 sec** was bought by a **poor** customer. Now the client is** Lord**, who wants the actuator to respond quickly. That’s why I sold the actuator with lower inertia **T=1.25 sec**. The effects are visible. The **actuator** will reach the set speed **faster**. Therefore, it will also reach the set **y(t)** faster. **First**, a positive pulse set the **valve** to** y(t)=8,** the actuator stood for a while (because that’s what e.g. the regulator wanted) and then a negative pulse set the valve to **y(t)=4**.