Automatics

Chapter 2 Proportional Unit

Chapter 2.1 Introduction
We will examine the simplest dynamic unit – the proportional unit. The output is related to the input by the relation y(t)=k*x(t). Therefore, the chapter is rather an excuse to get acquainted with the animations that most intuitively represent dynamic units. Then it will be easier for you to understand the mathematical apparatus describing the dynamic units, i.e. differential equations and transfer functions G(s).
In the next lessons, there will be animations with which we proceed similarly:
1. Turn on the video related to the given dynamic unit or automatic control system.
2. The input signal x(t) can come from:
– step signal generator (most common)
– virtual slider – otherwise manual signal.
– linear signal generator – otherwise “saw”
– Dirac impulse generator – otherwise “short hammer blow”,
3. We observe the output signal y(t) with:
– oscilloscope (most common)
– bargraph, i.e. an analog meter,
– digital meter,
Other signals will also be observed, e.g. control error e(t).
We repeat the experiments with different parameters of the object and different input signals. As a result, you will associate the transmittance G(s) with their responses y(t) to the input signal x(t) – most often a unit step. And more generally – with their dynamic properties.

Fig.2-1
Just take a look at the two similar transmittances G1(s) and G2(s) and you’ll immediately know what’s going on. And that without any knowledge of differential and operator calculus! The other thing is that this knowledge won’t hurt.

Chapter 2.2 Video support
The main advantage of this course is the video-supported animations. It is much more imaginative than a simple chart.

Fig. 2-2
Description of buttons and video indicators
-Start clicking starts the animation and turns into a Stop button
-Clock current experiment time, also as a yellow bar
-Simulation time otherwise – the duration of the experiment.
-Full display click to enlarge the screen, click again to reduce etc…
-Stop Clicking stops the simulation and turns it into a Restart button
-Restart as the name suggests.

Chapter 2.3 Animation off the proportional unit G(s)=1

Fig.2-3
Animation of the proportional unit G(s)=1 
Click the “Start” button.
x(t)-input signal visible on the slider and on the digital meter
y(t)-output signal visible on the bargraph and digital meter
The figure shows the initial state in which x(t)=0 and y(t)=0 and therefore the bargraph is invisible. The bargraph is an analog meter in the form of a vertical bar whose height is proportional to the size of the measured signal y(t).
Don’t try to move the slider! The author did it for you and recorded it on video.
The transmittance G(s) is simply the gain k=1. Note that the response of the proportional unit is instantaneous.
And what will be the animation of the term G(s)=2? The indications of the bargraph and the digital meter y(t) will be simply twice as large.

Chapter 2.4 Proportional unit G(s)=1 with slider and oscilloscope
On a two-channel oscilloscope you will observe the x(t) input and the y(t) output. The input signal x(t) is also a “slider swing”.

Fig. 2-4
Waveforms x(t) and y(t) of the proportional unit G(s)=1 on a two-channel oscilloscope.
The oscilloscope shows that until about 17 seconds x(t)=0. Only then did the swinging of the slider begin. For the proportional unit, the reaction to the input is immediate, and when k=1, the output y(t) is a copy of the input x(t).

Chapter 2.5 Conclusion
The reaction of the proportional unit is immediate. On the first approach, almost every dynamic term is a proportional term for us! Even the oscillating unit* discussed later. If you set the oven temperature to +120°C, after e.g. 10 minutes the thermometer will also show +120°C. So the estimated transfer function-transmittance of this unit G(s)=1. Input equals output. Only that we skip the transition state! If you were to watch the time course of this furnace on a 24-hour (not 10-minute) graph and the timeline was the same length as 10 minutes, you wouldn’t see the transition state! For you, the answer to a step would be a step too!
*Note:
Does not apply to the so-called astatic units, e.g. integrating. I’ll come back to the topic