First-Order Transfer Functions in Process Control

A transfer function is the ratio of the output to the input of a system. The system response is determined from the transfer function and the system input. A Laplace transform converts the input from the time domain to the spatial domain by using Laplace transform relations. The transformed spatial input is multiplied by the transfer function to get the output in the spatial domain. It is then converted back to the time domain using an inverse Laplace transform.

This Demonstration shows the responses to a first-order transfer function. You can vary the magnitude of the input and the time constant

τ

. The time constant is related to how long a system takes to reach the new steady state. The input can be a step function, a ramp, or a sinusoidal input.

Details

Step Input and Response

For a step input of magnitude

a

,

s=

space domain

t=

time domain

x(t)=a

x(s)=

a

s

y(s)=

Ka

s(τs+1)

y(t)=Ka1-

-t

τ

e



τ=0.632ΔSS

time constant

ΔSS=

change in steady states

Ramp Input and Response

For a ramp input of magnitude

a

,

x(t)=at

x(s)=

a

2

s

y(s)=

Ka

(τs+1)

2

s

y(t)=Kaτ

-t

τ

e

-1+Kat

Sinusoidal Input and Response

ω=2πf

angular frequency, rate of change of the function

x(t)=asin(ωt)

x(s)=

(aω)

2

s

+

2

ω

y(s)=

Kaω

(

τ

s

+1)(

2

s

+

2

ω

)

y(t)=

Kaωτ

2

ω

2

τ

+1

-t

τ

e

+

Ka

2

ω

2

τ

+1

sin(ωt+ϕ)

ϕ=-

-1

tan

(ωt)

phase shift, negative value represents a delay

time shift

=-

360

ϕ

p

amplitude ratio

=

A

B

Permanent Citation

Simon M. Lane

"First-Order Transfer Functions in Process Control"
http://demonstrations.wolfram.com/FirstOrderTransferFunctionsInProcessControl/
Wolfram Demonstrations Project
Published: May 6, 2014