Elmomc SimplIQ Digital Servo Drives-Bell Getting Started Instrukcja Użytkownika Pobierz pdf (Strona 83)

The SimplIQ for Steppers Getting Started & Tuning and Commissioning Guide

MAN-BELGS (Ver. 1.1)

B.2 Mathematical Models for LTI Systems

LTI systems, like any other system, are modeled by differential or algebraic

equations. The basic one is the differential equation, which means that the

relation between the output,

, and input,

, should obey the differential

equation

() ( ) () ( ) ( ) ()

ubububuyayayaya

n 0

++++=++++

−

ΛΛ

(2)

A simple example is a DC motor in current mode, described by the differential

equation

kuy =

(3)

where

is the current and

the shaft angle. Let us now assume that the input

to the system (2) is

(

)

tu ωsin=

. It can be confirmed by substitution that

() ()()

ωϕ+ω⋅ω= tsinay

(4)

solves (2), that is it is the system output, where

()

ωa

and

()

ωϕ

are the absolute

value and phase, respectively. The dependence of

and

on the frequency

called a Transfer function. For the system of (2), the transfer function is:

() () ()

() ()

ajaj

bjbjbjb

+⋅++

+⋅++⋅+⋅

−

ωω

ωωω

(5)

Note that the expression in (5) yields a complex number. The magnitude of this

number is

)(ω

and its phase is

)(ω

A major property of an LTI system is that its output,

)(ty

, for an input of the

form

(

)

tu ωsin=

() ()()

ωϕ+ω⋅ω= tsinay

, hence, the output is the same as

the input apart from an amplification factor

()

ωa

and time delay

(

)

ωωϕ− /

. The

parameter

is called the frequency of the signal

(and

()

ωa

the

amplitude of

and

()

ωϕ

its phase.

The transfer function is the basic engineering description of a linear system. It

directly describes the frequency response - the way the system responds to a

sinusoidal signal of any frequency. The transfer function is closely related to the

Laplace transform of the system. In fact, the Laplace transform of the system is

obtained by replacing in (2) the expression

with the Laplace variable

. The

Laplace transform of the system described by the differential equation (2) is:

asas

bsbsbsb

+++

++++

−

(6)

1 2 ... 78 79 80 81 82 83 84 85 86 87 88 ... 93 94

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