# Automatics

**Chapter 9 Differentiating Unit with Inertia**

**Chapter. 9.1 Introduction**

**Transmittance of the**

Fig. 9-1

Fig. 9-1

**Differentiating Unit with Inertia**.

Do you remember the ideal differentiator in

**Chapter**

**6**? The linearly increasing signal

**x(t**) at the input caused a step

**y(t**) at the output with an amplitude proportional to the

**rate**of increase

**x(t)**. The

**Differentiating Unit with Inertia**works similarly. It also measures the

**rate**of increase

**x(t**), but it does so with some

**inertia**. It can be said that calculating the

**rate**of increase

**x(t)**, i.e. calculating the

**derivative y(t)=x'(t)**will take him some time. Not so immediately as in ideal

**Chapter 9.2 Differentiating Unit with Inertia Td=2sec T=0.5sec, sawtooth with oscilloscope**

**Fig. 9-2**Differentiating Unit with Inertia

**Td=2 sec**

**T=0.5 sec**.

For this unit test with, the best solution is

**x(t)**increasing linearly with the

**speed**(derivative!)

**1/sec**. The formula for

**x(t)**is given in the diagram. Check if it is correct, e.g. for

**t=0**and

**t=7 sec**. The

**output signal**calculating the speed

**x(t)**settled down after about

**five tim**e constants

**T**, i.e. after

**5*0.5 =2.5 sec**. So

**T**proves the quality of the

**speedometer**. A

**cheap**speedometer will give you the exact speed after, for example,

**2.5 seconds**, as in the example, and a good one after, for example,

**0.5 seconds**. And what is

**Td=2 sec**? Go back to the ideal differentiator for a moment–>

**chapter 5**, i.e. without inertia. There

**Td=1 sec**and it was the time after which the output

**y(t)**equaled the saw

**x(t)**. Here the definition is similar, only it applies to a steady state, e.g. after

**3 sec**. Here, after

**Td=2 sec**, the signal increment

**x(t)**also equals

**y(t)=2**. Note that the larger the

**Td**, the greater the differentiation

**intensity**.

**Chapter 9.3 Differentiating Unit with Inertia Td=2, T=0.5 sec, rectangular pulse with oscilloscope**

**Fig. 9-3**Instead of a saw, there is a

**rectangular pulse**at the input

Compare this waveform with the analogous one, only for the ideal differentiating unit ->

**Fig. 5-9**

**chapter 5**. There, at the

**positive**slope of the step

**x(t)**, the speed

**y(t)**was

**infinitely**large. Ideal showed this as a

**Dirac**impulse. Here, the

**real**one showed

**y(t)**as a rate of

**4/1sec**when

**t=3.5sec**.

From the plot y(t) we can obtain the parameters of the differentiator:

–

**T=3.5-3=0.5sec**(from the green tangent)

–

**Td=4/05sec=2/sec**

**Conclusion**-rectangular pulse has more difficult interpretation than a ramp.

.