### Automatics

**Chapter 3 Inertial UnitChapter 3.1 Introduction**This is the simplest

**dynamic**unit except

**Proportional**. Unit step response is no longer instantaneous. You may not know the concept of transfer function-transmittance

**G(s)**yet, but after completing this chapter you will associate its parameters with time courses. To treat the transfer function

**G(s)**as a kind of

**amplification**, no

**differential**or

**operator calculus**is needed. Maybe that was too strong a word. It would be better that way. Knowledge of higher mathematics is not absolutely necessary to understand automatics, but it is very helpful.

Fig. 3-1

Fig. 3-1

**Inertial**unit parameters:

–

**K**-steady state gain

–

**T**– time constant

We will do some experiments with different

**T**time constants.

We will do some experiments with different input signals

**x(t)**. The output signals

**y(t)**will be observed on the bargraf or on the oscilloscope.

**Chapter 3.2 K=1 T=5 sek, step x(t) from slider, y(t) bargraf**

**Fig****. 3-2**Response to a unit step of the inertial unit

**K=1**and

**T=5sec**

The

**x(t)**input signal was set with a 0-1 step on the slider.

You can follow the waveforms:

**analog -x(t)**on the slider and bargraph

**digitally**

**-y(t)**on gauges

Digital meters are especially useful for slow changes at the end of the waveform.

**Note**

The

**second**is only accidentally associated with the complex number

**s**at the time constant

**T**.

What if we increase the time constant to

**T=10 sec**?

**Chapter 3.3 K=1 T=10 sek, step x(t) from slider, y(t) bargraf**

**Fig. 3-3**Response to a unit step of the inertial unit

**K=1**and

**T=10sec**

Compare with waveform

**Fig. 3-2**. The “heaviness” of the transmittance is clearly

**2**times greater.

**Chapter 3.4 K=1 T=5 sek, step x(t) from slider, y(t) oscilloscope**

**Fig. 3-4**Response to a unit step of the inertial element

**K=1**and

**T=5sec**

The input signal

**x(t)**is a step unit that occurred in

**12**seconds.

The output signal

**y(t)**has the highest rate of

**increase**at the beginning. Then it increases, but at a decreasing speed. In a steady state, it reaches the value

**y=1**.

The waveform clearly shows what is what in the transmittance

**G(s)**in

**Fig. 3-4**.

**1**in the transmittance numerator

**G(s)**is the steady-state gain

**K=y/x=1**.

**5**in the transfer function denominator is the time constant

**T=5**seconds.

This is the time after which the steady state

**y(t)=1**would occur if the growth rate was still the same as at the beginning. The signal would then grow like a

**tangent**– a dotted line. The state

**y(t)=x(t)=1**would be reached after

**T=5sec=17sec-12sec.**

You can read the times

**17sec**and

**12sec**from the time axis t.

**Chapter 3.5 “Smoothing” action of the inertial unit K=1, T=5sec**

We “swing” the slider. So we will check how the inertial term behaves to fast changes of the input signal **x(t)**.

**Fig. 3-5**“Smoothing” action of the inertial unit

**Note:**

There is a relatively large inertia

**T=5sec**. Therefore, the output signal

**y(t)**is hardly similar to the input one.

It will become more like when:

– the

**x(t)**signal is slowly changing

– the time constant

**T**will be smaller, e.g.

**T=0.2sec**

**Chap. 3.6 Comparison of two different inertial units**

The same unit step **x(t**) will be given to **2** different inertial units.

This time **x(t)** comes from a signal generator that is more precise than the previous **slider**.

**Fig. 3-6**The inputs of two inertial units, in which parameters

**3**and

**7**are “replaced”, are supplied with a unit step

**x(t)**.

In

**Fig. 3-4**we had an inertial unit in which

**K=1**and

**T=5sec**. When

**K=1**then

**y(t)=1**. We also showed how to calculate T=5sec from the graph.

Now we calculated

**K1=7**and

**T1=3 sec**for the upper inertial term and

**K2=3**and

**T2=7 sec**for the lower one in the same way.

**Chapter 3.7 Dirac pulse**

**Chapter. 3.7.1 Introduction**

Another signal used in the control theory is the so-called

**Dirac pulse–> x(t) = δ(t)**. Its characteristic feature is that it lasts

**infinitely short**but has an

**infinitely high**value. However, the energy of this impulse, i.e. the impulse field, is equal to

**unity**, i.e. it is

**finite**.

An approximation can be, for example, a power plant that supplies an electric kettle with all its power (e.g.

**3600 MW**), but for a

**very short time**. The water in the kettle will heat up to

**+100°C**because in a

**few nanoseconds**, finite energy will be delivered (and not a lot at all!), and the kettle will not be destroyed!

For us, the kettle got hot immediately! We think we are dealing with

**Dirac’s**Ideal pulse.

A

**mechanical**example of a

**Dirac pulse**is a hammer blow. It also takes a very short time and during this time the finished work of driving the nail will be done.

**Conclusion**There are no ideal

**Dirac Pulses**in nature. There are only

**real**approximations of it.

**Chapter 3.7.2 Inertial unit K=1 T=1sek with the real Dirac pulse**

**Fig. 3-7**Inertial unit response to (almost)

**Dirac**pulse

**δ(t)**

A pulse with an amplitude of

**A=10**and a duration of

**tp=0.1sec**is only an approximation of the

**ideal Dirac**pulse

**δ(t)**. You can see how the signal grows rapidly over time

**tp**. Then it drops because there is

**zero**in the input. This is how

**fast**charging through the resistor

**R**of the capacitor

**C**and its discharge looks like.

**Chapter 3.7.3 Inertial unit K=1 T=1 sec with “more ideal” Dirac pulse**

The previous **Dirac** with an amplitude of **10** lasted **0.1 seconds**. Since we are unable to give a perfect **Dirac**, let’s at least give something closer to the ideal. i.e. pulse with an amplitude of **100** that will last **0.01 sec**. Notice that its energy, or pulse area, is also 1.

**Fig. 3-8**

**Inertial unit**response to a (more ideal)

**Dirac**pulse

**δ(t)**

The pulse is

**10**times higher (i.e.

**10**0 and exceeds the scope of the oscilloscope) and

**10**times narrower. Its field=

**1**, or energy, is the same as before. It’s still not perfect, although

**Dirac’s**appearance is

**textbook**.

**Chapter 3.8 Why do we need these Diracs?**

Especially since a **unit step** is technically easier to implement than a **Dirac pin**. In the **unit step**, it is enough to enter the maximum power per input, e.g. **100 kW**. In an almost perfect **Dirac**, this is Power Station **4,000 MW** for a few nanoseconds. There are technical ways to deliver a short pulse of high power without the use of a power plant, e.g. **laser power **supplies, **radar** stations … But this is no longer simple.

Exactly. Why this **Dirac**? I’ll run a little ahead. It turns out that the transfer function **G(s)** is simply the **Laplace transformation** of the response **y(t)** to the **Dirac** pulse. That’s all for now. A special chapter will be devoted to **G(s**) and **Laplace** transforms.

**Chap. 3-9 Two seemingly different inertial units**

**Fig. 3-9**The same unit step

**x(t)**acts simultaneously on the

**lower**and

**upper**inertial units. The reactions, i.e.

**y1(t)**and

**y2(t**) are identical. So both

**inertial units**are also identical. But in the upper member you can’t see the parameters

**K=2**and

**T=3sec**. To see them, make the upper denominator

**1+…**So divide the upper numerator and denominator by

**7**. It turns out that both fractions

**G1(s)**and

**G2(s)**are the same

**1/(1+3*s)**!

This l

**ower**transmittance is in the

**normalized**version. It is easy to read the parameters

**K=2**and

**T=3sec**. All basic dynamic units from

**chapters. 2…9**are presented in a

**normalized**version. Especially in

**chapter 7 Oscillatory Unit.**

**Chapter 3.10 Typical inertial units****RC Circuit**We’ll start with an example of an electric-

**RC**circuit. When I apply a voltage unit step

**x(t)**to the input, the voltage

**y(t)**with such a waveform will appear at the output.

**Fig. 3-10**It is an

**Inertial**unit with parameters

**k**and

**T, k=1**because in steady state

**y(t) = x(t)**When, for example,

**R = 100 kΩ**and

**C = 10 µF**, then

**T = R*C = 100,000 Ω * 10* 0.000001F = 1 second**

**DC motor**

If I put

**step DC voltage**to the input, then its start-up is approximately typical for the

**inertial unit**. The input is

**voltage**and the

**output**is rotational

**speed**. Initially, the speed is zero, then it increases all the time, but the speed increments become smaller and smaller. Eventually it will reach its maximum value, depending of course on the input voltage and electro-mechanical parameters. The time constant also depends on these parameters – mainly on mechanical inertia, resistance and inductance. Stop! We’ve gone too far into details.

**Bathtub**with the

**stopper removed**

The

**input**is the “immediate” opening of the tap (i.e. indirectly the flow)

The

**output**is a water level

The level is low at the beginning. So the pressure and drain are also low. The inflow is greater than the outflow and the level is rising. However, it will increase more and more slowly because with increasing level, the outflow increases (higher pressure). After some time, when the

**inflow**is

**equal**to the

**outflow**, the

**level**will stabilize.

If we consider the differential equation of this phenomenon in more detail, the outflow obviously increases with height, but not

**proportionally**to the

**level**, but to the

**square root of the level**. At small levels, however, proportionality can be assumed, and then it is already a classic Inertial unit.

**Chapter 3.11 Conclusion**

Just as the **Proportional unit** is the first approximation of almost every dynamic term, the **Inertial unit** is the second, more accurate approximation. After all, almost* every member in response to a step will reach some constant value after some time! We will then assume that it is an** inertial unit** with steady state gain and a time constant **4…5 times** smaller than the signal settling time. Of course, this will be a poor approximation, but it’s better than the **proportional unit** approximation***almost** – because it does not apply to, for example, the **Integral Unit**.