# Scilab

**12. Xcos-Transmittances, Operator Calculus and Differential Equations**

**Chapter 12.1 Introduction**

In **Chapter 11** we learned the basic dynamic units. The so-called transmittance **G(s)**. This is an extension of the concept of gain **K**. You already associate **G(s)** with its response, e.g. to a **unit step**. And so in the** Inertial** unit** G(s)=K/(1+sT)** you know what the parameters **K** and** T** do. And **s** itself? What the devil? You may skip the topic on your first attempt. It just is and that’s it. And **G(s)** is something that reacts this way and not differently to a **unit step** or a **“ramp”**. A more serious look at **G(s)** is as the gain **K** depending on the frequency of the **sine** wave introduced to the input. A kind of frequency response. In the proportional term **G(s)=K** the relation is simple **y(t)=K*sin(ωt)**. For each **frequency/pulsation**, the gain **K** is constant. Also the sine wave** y(t)** is in phase with s**in(ωt)**. However, for “real” **G(s)**, which is governed by** differential equations**, the **amplitude** and **phase** of the sine wave change with **frequency.** Typically, the amplitude decreases and the phase lags.

You can find more in the **Automatic** course, **chapter 18**.

**Chapter 12.2 Transmittance G(s) and the differential equation****Chapter 12.2.1 Introduction**

What is **s** in the transmittance **G(s)**? In **Operational Calculus** – in other words, in **Laplace Transforms**, each time function **f(t)** corresponds to its transform **F(s)** and vice versa, i.e.** f(t)<==>F(s**).

E.g. for a unit step of **1(t)<==>1/s**.

The **Operational Calculus** has one nice feature**If f(t)<==>F(s)** **then f'(t)<==>s*F(s)**

That is, the transform of the derivative** f'(t)** is the product **s*F(s)**.

If it did not have this feature, there would be no **Operational Calculus**.

While it is impossible to write a simple relation in an object as simple **y(t)/x(t)** (because it would change over time **t**), it can be written as a ratio of **2** transforms **G(s)=y(s)/x(s)**.

**Chapter 12.2.2 Inertial Unit and the differential equation**Fig.12-1

Relation between the differential equation and the transfer function

**G(s)**of the inertial unit

**Fig.12-1a**

Transmittance

**G(s)**of the inertial unit as

**G(s)=y(s)/x(s)=K/(1+st)**

**Fig.12-1b**

The relationship between the operator equation of the inertial unit and its differential equation.

**Chapter 12.2.3**A more complicated transmittance and the differential equation

We have already shown that an object described by a simple differential equation

**K*x(t)=T*y'(t)+y(t)**

corresponds to the transmittance

**G(s)**of the inertial unit with parameters

**K**and

**T**.

The figure below shows how the transmittance is created for a more complicated differential equation, e.g.

**3rd**degree.

**Fig. 12-2**

So we see that the transmittance **G(s)** is a quotient where the numerator **L(s)** and the denominator **M(s)** are polynomials of the appropriate degrees. Usually, **L(s)** has a lower degree than **M(s)** and most often it is only **L(s)=bo**, in addition **bo=1**, and the denominator is the product of polynomials of the **first** **or second** degree.

In a similar way, a transfer function will be created based on a **differential** **equation** of any **degree.**

Let’s check the response of the** G(s)** to the step unit **x(t)** in **Fig. 12-2** with specific parameters of the denominator **M(s)** and the numerator **L(s)**.**M(s)–> a0,a1,a2,a3****L(s)–> b0,b1,b2,b3**

**Fig. 12-3**

We see higher derivatives mixing up over time. Especially since the coefficients for these derivatives are small. What if they were big. Scary to think!

For most transmittances, the steady-state gain **K** can be easily determined. It is simply an free expression because **bo=1.25** in the numerator of the transmittance, assuming that the free expression in the denominator **ao=1**. If **ao** is different from **1**, divide the **numerator** and **denominator** by **ao**.

Of course, this is the case when **G(s)** concerns stable systems. Not ones where any disturbance causes **G(s)** to become a generator. You know a lot about the transmittance response **G(s)** unit step **1(t)**.

The output **y(t=2)=0**, which is obvious, and that in the steady state **y(t)=bo=K**. But what happens in the transition state (“in between”), i.e. for **t=2…15 sec**? The remaining transmittance parameters are responsible for this. These are the coefficients** a1, a2, a3 and b1, b2, b3,** the specific values of which are visible in the transmittance **G(s)** in **Fig. 12-3**