### Automatics

**Chapter 5 Differentiating Unit**

**Chapter 5.1 Introduction**

Fig. 5-1

Fig. 5-1

**Differential Unit**transfer function

**G(s)**. More strictly –

**Ideal Differential Unit.**

The

**Differential Unit**reacts to the

**rate**of change of the signal

**x(t)**, not to its

**value**! It is the best teaching aid to understand the concept of the

**derivative**of a function, in general

**calculus**. Usually in transfer functions

**G(s)**, the letter

**s**appears in the

**denominator**of the fraction, while in the

**derivative unit**it is an

**ordinary**number (but complex

**s**like any other in this article) in the

**numerator**. (here

**denominator=1**)

With manual control from the

**slider**, the output waveforms of the derivative element are just mean, some pins, etc … This makes it difficult to draw the right conclusions. Therefore, when examining this

**unit**, there will be no diagrams with a

**slider**and a

**bargraph**.

The

**Differential Unit**component gives the output signal

**y(t)**proportional to the rate of change of the input signal

**x(t**), i.e. to its

**derivative**.

**Therefore**, in each experiment, make sure that this is the case, i.e.:

When

**x(t)**is growing

**rapidly**–>

**y(t**) is

**large**

**positive**

When

**x(t)**is constant –>

**y(t)=0**because the

**velocity**is zero

When

**x(t)**is rapidly

**decreasing**–>

**y(t)**is

**large negative**(negative speed)

**Chapter 5.2 Signal x(t) increases linearly when Td=1sec**

**Fig. 5-2**The drawing is at first the same as for the

**integratitng**unit in

**Fig. 4-4**in

**chapter 4**. But there was a

**step**at the input and a

**ramp**at the

**output**, and here it’s the other way around! That is why it is said that

**differentiation**is the

**inverse**of

**integration**and vice versa.

**The**most important

For the

**differentiator**with transmittance

**G(s)=s*Td**when

**Td=1sec**, the

**y(t)**output signal is equal to the rate of change of the

**x(t)**input signal. The speed of change is

**1/second**and is constant. Therefore, the response to this signal is a constant value of

**1**.

The

**rate of change**of any function is the

**derivative**of this function. Here the

**derivative**was easy to calculate, because the rate of change was constant.

The derivative of

**x(t)**is

**y(t)**. Up to the

**third**second,

**y(t)=dx/dt=0**, because

**x(t)=0**, and does not change, and from the

**third**second,

**dx/dt=1**.

Definition of

**differentiation**time

**Td**

After the time

**Td=1sec**, the

**differentiator**signal

**y(t**) equals the increasing input signal

**x(t)**. Note that

**Td**does not depend on the rate of increase

**x(t)**. When the speed increases, the alignment will also occur after

**Td=1sec**, but at a higher level of

**y(t)**!

Let’s repeat the experiment, but with

**twice**the

**slew**rate of the linear signal.

**Fig. 5-3**A

**double**increase in the

**growth rate**of

**x(t)**resulted in a

**double**increase in

**y(t)**.

But equating

**x(t)**with

**y(t)**also happened after

**1 second**. The

**differentiating**unit

**G(s)=s*Td**is like a

**speedometer**of the

**x(t)**signal.

**Chapter 5.3 Signal x(t) “rising, standing and falling”**

A phenomenon known not only in automatics.

**Fig. 5-4**

**0…2 sec**constant

**x(t)=0**therefore

**y(t)=0**(because “speed”

**x(t)=**zero)

**2…4 sec**signal

**x(t)**increases with a constant (positive)

**rate=1/sec**(derivative!)–>

**y(t)=+1**

**4…6 sec**the signal

**x(t)=1**is constant, i.e. it has velocity=0 –>

**y(t)=0**

**6…10 sec**the signal

**x(t)**decreases with a constant (negative) speed

**=-1/sec**–>

**y(t)=-1**

**Chapter 5.4 Signal x(t) with two constant speeds**

**Fig. 5-5**

**Doubling**the speed (derivative) of the input signal

**x(t)**resulted in a

**doubling**of the output signal

**y(t)**.

**Chapter 5.5 Signal x(t) with four constant speeds**

Each time the speed, i.e. the **derivative** of the input signal** x(t)** will increase by the same speed increase

**Fig. 5-6**A

**fourfold**increase in the speed (derivative) of the input signal

**x(t)**resulted in a

**fourfold**increase in the output signal

**y(t)**.

The

**x(t)**signal becomes similar to a

**parabola**(square function) and

**y(t)**to a linear one.

**Roughly**, of course. What if you gave

**x(t)**as a parabola as input? Can you guess what will happen?

**Chapter 5.6 Signal x(t) is a square function**

And if there are **8, 16, ….. 1024** …** infinitely** many of these segments, will we get a perfect **parabola**? Let’s check.

**Fig. 5-7**

The **Differentiating Un****it** confirms the formula for the derivative of a quadratic function known since the **17th** century. Check, for example, that the derivative formula is valid for** t=4**.

**Chapter 5.7 x(t) is the sine wave of sin(t)**

**Fig****. 5-8**

**x(t)=sin(t)**

**y(t)=cos(t)**

The

**sine**signal

**x(t)=sin(t)**was

**differentiated**by

**G(s)**giving a

**cosine**output.

A

**cosine**wave is a

**derivativ**e of a

**sine**wave.

**Chap. 5.8 x(t) is a single rectangular pulse**

**Fig****. 5-9**In

**3**seconds, the input signal

**x(t)**increases as a step. So its

**growth rate**has the value

**+infinity.**

Hence this red pin

**y(t)**is dirac to

**+infinity**. In

**6**seconds, the signal decreases to zero –>negative speed -> red pin

**y(t)**is dirac to

**-infinity**. In the rest of the time (i.e. everywhere except

**3**and

**6**seconds)

**x(t)**is constant. So

**x(t)**has

**zero**velocity. So at this time

**y(t)**is also

**zero**.

**Chapter 5.8 Conclusions****Differentiating Unit **reacts to the **rate** of change of the input signal **x(t), **i.e. to its derivative** x'(t).**

**Fig. 5-10**An example is a

**coil**with inductance

**L**, when the input

**x(t**) is the current

**i(t)**and the output

**y(t)**is the voltage on the coil

**u(t)**.

A typical input signal

**x(t)**for testing most dynamic units is a

**unit step**. For the

**differentiating unit**, however, it is a linearly

**increasing**signal. Then it is easy to obtain the

**Td**parameter from

**Fig. 5-1**.

For a unit step ->

**Fig.5-10**, it would be more difficult. Therefore, increasing signals

**x(t)**of the sawtooth type are best suited for the study of

**differentiators**.