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2/ H(s) along the imaginary axis

H(s) evaluated for s = jω or H(jω) is called the frequency response.

3/ Radian frequency and frequency

  • ω is the radian frequency in units of radians/second.
  • ω can be expressed as ω = 2πf.
  • f is the frequency in units of cycles per second which is called hertz (Hz).
  • The frequency response H(j2πf) is often plotted versus f.

4/ Measurement of H(jω)

To measure H(jω) of a test system we can use the system shown below.

However, when the sinusoid is turned on the response contains both a transient and a steady-state component.

For example, suppose

H ( s ) = 5s ( s + 1 ) 2 + 10 2 size 12{H \( s \) = { {5s} over { \( s+1 \) rSup { size 8{2} } +"10" rSup { size 8{2} } } } } {}

The response of this system, y(t), to the input x(t) = cos(2πt) u(t) is shown below.

Note the transient at the onset which damps out after a few cycles of the sinusoid so that the response approaches the particular solution, i.e., this is the steady-state response.

Therefore, to measure H(jω) we turn on the oscillator and wait till steady state is established and measure the input and output sinusoid. At each frequency, the ratio of the magnitude of the output to the magnitude of the input sinusoid defines the magnitude of the frequency response. The angle of the output minus that of the input defines the angle of the frequency response.

More elaborate systems are available for measuring the frequency response rapidly and automatically.

5/ Relation of time waveforms, vector diagrams, and frequency response

Consider and LTI system with system function

H ( s ) = 1 s + 1 size 12{H \( s \) = { {1} over {s+1} } } {}

which has the frequency response

H ( ) = 1 + 1 size 12{H \( jω \) = { {1} over {jω+1} } } {}

The input is

x ( t ) cos ( ωt ) size 12{x \( t \) "cos" \( ωt \) } {}

and the response is

y ( t ) = H ( ) cos ( ωt + arg H ( ) ) y ( t ) = 1 ( ω 2 + 1 ) 1 / 2 cos ( ωt tan 1 ω ) alignl { stack { size 12{y \( t \) = \lline H \( jω \) \lline "cos" \( ωt+"arg"H \( jω \) \) } {} #size 12{y \( t \) = { {1} over { \( ω rSup { size 8{2} } +1 \) rSup { size 8{1/2} } } } "cos" \( ωt - "tan" rSup { size 8{ - 1} } ω \) } {} } } {}

Demo of relation of pole-zero diagram, time waveforms, vector diagrams, and frequency response.

6/ Different ways of plotting the frequency response

The frequency response H(jω) = 1/(jω +1), is plotted in linear coordinates (left) and in doubly logarithmic coordinates called a Bode diagram (right).

IV. BODE DIAGRAMS

1/ Definition and rationale

Frequency responses are commonly plotted as Bode diagrams which are plots of

20 log 10 H ( ) size 12{"20""log" rSub { size 8{"10"} } \lline H \( jω \) \lline } {} plotted versus log 10 ω size 12{"log" rSub { size 8{"10"} } ω} {}

arg H ( ) size 12{"arg"H \( jω \) } {} plotted versus log 10 ω size 12{"log" rSub { size 8{"10"} } ω} {} .

The reasons are:

  • Logarithmic coordinates are useful when the range of |H(jω)| and/or ω is large.
  • Asymptotes to the frequency response are easily plotted.
  • When the poles and zeros are on the real axis, the asymptotes are excellent approximations to the frequency response.

2/ Pole-zero and time-constant form

Consider an LTI system whose system function has poles and zeros on the negative, real axis. It can be displayed in pole-zero form as follows

H ( s ) = C ( s + z 1 ) ( s + z 2 ) . . . ( s + z M ) ( s + p 1 ) ( s + p 2 ) . . . ( s + p N ) size 12{H \( s \) =C { { \( s+z rSub { size 8{1} } \) \( s+z rSub { size 8{2} } \) "." "." "." \( s+z rSub { size 8{M} } \) } over { \( s+p rSub { size 8{1} } \) \( s+p rSub { size 8{2} } \) "." "." "." \( s+p rSub { size 8{N} } \) } } } {}

This system function can be evaluated along the jω axis to yield

H ( s ) = C ( + z 1 ) ( + z 2 ) . . . ( + z M ) ( + p 1 ) ( + p 2 ) . . . ( + p N ) size 12{H \( s \) =C { { \( jω+z rSub { size 8{1} } \) \( jω+z rSub { size 8{2} } \) "." "." "." \( jω+z rSub { size 8{M} } \) } over { \( jω+p rSub { size 8{1} } \) \( jω+p rSub { size 8{2} } \) "." "." "." \( jω+p rSub { size 8{N} } \) } } } {}

By dividing the numerator by each zero and the denominator by each pole, the frequency response can be put in time-constant form as follows

H ( ) = K ( j ωτ z1 + 1 ) ( j ωτ z2 + 1 ) . . . ( j ωτ zM + 1 ) ( j ωτ p1 + 1 ) ( j ωτ p2 + 1 ) . . . ( j ωτ pN + 1 ) size 12{H \( jω \) =K { { \( j ital "ωτ" rSub { size 8{z1} } +1 \) \( j ital "ωτ" rSub { size 8{z2} } +1 \) "." "." "." \( j ital "ωτ" rSub { size 8{ ital "zM"} } +1 \) } over { \( j ital "ωτ" rSub { size 8{p1} } +1 \) \( j ital "ωτ" rSub { size 8{p2} } +1 \) "." "." "." \( j ital "ωτ" rSub { size 8{ ital "pN"} } +1 \) } } } {}

3/ Magnitude and angle

The magnitude and angle can be expressed as

H ( ) = K j ωτ z1 + 1 j ωτ z2 + 1 . . . j ωτ zM + 1 j ωτ p1 + 1 j ωτ p2 + 1 . . . j ωτ pN + 1 size 12{ \lline H \( jω \) \lline = \lline K \lline { { \lline j ital "ωτ" rSub { size 8{z1} } +1 \lline \lline j ital "ωτ" rSub { size 8{z2} } +1 \lline "." "." "." \lline j ital "ωτ" rSub { size 8{ ital "zM"} } +1 \lline } over { \lline j ital "ωτ" rSub { size 8{p1} } +1 \lline \lline j ital "ωτ" rSub { size 8{p2} } +1 \lline "." "." "." \lline j ital "ωτ" rSub { size 8{ ital "pN"} } +1 \lline } } } {}

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
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