Laplace Transforms and Transfer Functions

Laplace transform solution in time domain solution in s domain inverse Laplace transform problem in time domain transform solution in original way of thinking solution in transform way

Laplace transform

solution in time domain

solution in s domain

inverse Laplace transform

problem in

time domain

transform

solution in original way

of thinking

solution in transform way of thinking

inverse transform

problem in

original way

of thinking

linear differential equation

time domain solution

Laplace transformed eq

(algebra eq.)

Laplace solution

easy

time domain

difficult

frequency domain (Laplace domain)

solution in original way

of thinking

solution in transform way of thinking

inverse transform

problem in

original way

of thinking

𝑞𝑖𝑛

𝑞𝑜𝑢𝑡

𝑅𝐿

ℎ

𝑑

Input flowrate

Output flowrate

R

𝑒𝑖𝑛(𝑡)

𝑖(𝑡)

inlet air signal

𝐹𝑎 = 𝑝𝑖𝑛𝐴 =

flexible diaphragm

flow direction

spring with capacitance,Cm stem and diaphragm plate are the moving mass, m seal with resistance,Rm

applied force

Fluid temperature, Ta (actual temperature)

The liquid in the bulb expands as in its temperature increases, forcing more liquid up the small tube The height of liquid is a measure of the liquid in the bulb

Bulb temperature, Tm

(measured temperature)

∆ℎ(𝑡)

𝑞 𝑖𝑛 (𝑡)

𝑞𝑜𝑢𝑡(𝑡)

𝑑

Input flowrate

Output flowrate

𝑡 0

Positive displa-cement pump

Tm

Ta

inlet air signal

𝐹𝑎 = 𝑝𝑖𝑛𝐴 =

flow direction

applied force

System

Output

0.5 0

2 4 6 8 10 12 14

Input current i ( t )

0.5 0

2 4 6 8 10 12 14

final value of x ( t )=8 mm

control valve position x ( t )

𝑥 𝑡 = 8 + 8.04𝑒−𝑡𝑐𝑜𝑠⁡ (10𝑡+174.30)

inlet air signal

𝜃0

𝑓

Frequency (rad/sec)

10 20 30 40 50 60

Frequency (rad/sec) -90

-80 -70 -50

-30 -20 -10 0

-40

-60

gain

phase angle

algebra

time domain

operational calculus

frequency domain (Laplace domain)

𝜏𝑠𝐻(𝑠) + 𝐻(𝑠) = 𝐺𝑄𝑖𝑛(𝑠)

𝑄𝑖𝑛 𝑠 =𝐾

𝑠

𝑞𝑖𝑛 𝑡 = 𝐾 𝑖𝑓 𝑡 ≥ 00 𝑖𝑓 𝑡 < 0

𝜏𝑑ℎ(𝑡)

𝑑𝑡 + ℎ(𝑡) = 𝐺𝑞𝑖𝑛(𝑡)

ℎ 𝑡 = 𝐺𝐾 1 − 𝑒−𝑡𝜏

𝐻 𝑠 = 𝐺𝐾

𝑠(𝜏𝑠 + 1)