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And

arg ( H ( ) ) = arg ( K ) + arg ( j ωτ z1 + 1 ) + arg ( j ωτ z2 + 1 ) + . . . + arg ( j ωτ zM + 1 ) arg ( j ωτ p1 + 1 ) arg ( j ωτ p2 + 1 ) . . . arg ( j ωτ pN + 1 ) alignl { stack { size 12{"arg" \( H \( jω \) \) ="arg" \( K \) +"arg" \( j ital "ωτ" rSub { size 8{z1} } +1 \) +"arg" \( j ital "ωτ" rSub { size 8{z2} } +1 \) + "." "." "." +"arg" \( j ital "ωτ" rSub { size 8{ ital "zM"} } +1 \) } {} #matrix { matrix {matrix { matrix {{} # {} } {} # {}} {} # {} } {} # {}} - "arg" \( j ital "ωτ" rSub { size 8{p1} } +1 \) - "arg" \( j ital "ωτ" rSub { size 8{p2} } +1 \) - "." "." "." - "arg" \( j ital "ωτ" rSub { size 8{ ital "pN"} } +1 \) {} } } {}

4/ Logarithmic magnitude

Taking twenty times the logarithm of the magnitude yields

20 log 10 H ( ) = 20 log 10 K + 20 log 10 j ωτ z1 + 1 + 20 log 10 j ωτ z2 + 1 + . . . + 20 log 10 j ωτ zM + 1 20 log 10 j ωτ p1 + 1 20 log 10 j ωτ p2 + 1 . . . 20 log 10 j ωτ pN + 1 alignl { stack { size 12{"20""log" rSub { size 8{"10"} } \lline H \( jω \) \lline ="20""log" rSub { size 8{"10"} } \lline K \lline +"20""log" rSub { size 8{"10"} } \lline j ital "ωτ" rSub { size 8{z1} } +1 \lline +"20""log" rSub { size 8{"10"} } \lline j ital "ωτ" rSub { size 8{z2} } +1 \lline + "." "." "." +"20""log" rSub { size 8{"10"} } \lline j ital "ωτ" rSub { size 8{ ital "zM"} } +1 \lline } {} #matrix { matrix {matrix { matrix {{} # {} } {} # {}} {} # {} } {} # {}} - "20""log" rSub { size 8{"10"} } \lline j ital "ωτ" rSub { size 8{p1} } +1 \lline - "20""log" rSub { size 8{"10"} } \lline j ital "ωτ" rSub { size 8{p2} } +1 \lline - "." "." "." - "20""log" rSub { size 8{"10"} } \lline j ital "ωτ" rSub { size 8{ ital "pN"} } +1 \lline {}} } {}

Note than both the logarithmic magnitude and the angle are expressed as sums of terms of the form

± 20 log 10 j ωτ + 1 and ± arg ( j ωτ + 1 ) size 12{ +- "20 log" rSub { size 8{"10"} } \lline j ital "ωτ" +1 \lline " " matrix { {} # {}} " and " matrix { {} # {}} +- " arg" \( j ital "ωτ" +1 \) } {}

Therefore, to plot the frequency response we need to add terms of the above form.

5/ Decibels

It is common to plot frequency responses as Bode diagrams whose magnitude is expressed in decibels. The decibel, denoted by dB, is defined as 20 log 10 Η size 12{"20 log" rSub { size 8{"10"} } \lline Η \lline } {} . The following table gives decibel equivalents for a few quantities.

How many decibels correspond to |H| = 50? Express |H| =100/2. Then

20 log 10 ( 100/2 ) = 20 log 10 100 - 20log 10 2 40 - 6 = 34 dB size 12{"20 log" rSub { size 8{"10"} } \( "100/2" \) =" 20 log" rSub { size 8{"10"} } " 100 - 20log" rSub { size 8{"10"} } " 2" approx " 40" "- 6"" = ""34" "dB"} {}

6/ Asymptotes

To plot the frequency response of a system with real poles and zeros, we need to plot terms of the form

± 20 log 10 1 + j ωτ and ± arg ( 1 + j ωτ ) size 12{ +- "20 log" rSub { size 8{"10"} } \lline 1+j ital "ωτ" \lline " " matrix { {} # {}} " and " matrix { {} # {}} " " +- " arg" \( 1+j ital "ωτ" \) } {}

The low and high frequency asymptotes are

7/ Corner frequency

At ω = ω c = 1 / τ size 12{ω "= "ω rSub { size 8{c} } " = 1"/τ} {} , called the corner or cut-off frequency,

  • the low- and high-frequency asymptotes intersect,
  • the magnitude is

± 20 log 10 1+j ωτ = ± 20 log 10 1+j1 = ± 20 log 10 2 1/2 ± 3 dB size 12{ +- "20 log" rSub { size 8{"10"} } \lline "1+j" ital "ωτ" \lline " = " +- "20 log" rSub { size 8{"10"} } \lline "1+j1" \lline " = " +- "20 log" rSub { size 8{"10"} } 2 rSup { size 8{"1/2"} } " " +- "3 dB"} {}

  • the angle is

± arg ( 1+ j ωτ ) = ± arg ( 1+ j ) = ± π / 4 size 12{ +- "arg" \( "1+"j ital "ωτ" \) " = " +- "arg" \( "1+"j \) " = " +- π/4} {}

Example — first-order lowpass system

First-order low pass systems arise in a large variety of physical contexts. For example,

For the parameters M = B = R = C = 1, the frequency responses for the two systems are

H ( ) = V ( ) F ( ) = 1 + 1 and H ( ) = V 0 ( ) V i ( ) = 1 + 1 size 12{H \( jω \) = { {V \( jω \) } over {F \( jω \) } } = { {1} over {jω+1} } matrix { {} # {}} ital "and" matrix { {} # {}} H \( jω \) = { {V rSub { size 8{0} } \( jω \) } over {V rSub { size 8{i} } \( jω \) } } = { {1} over {jω+1} } } {}

Magnitude

For

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

the low-frequency asymptote has a slope of 0 and an intercept of 0 dB and the high-frequency asymptote has a slope of -20 dB/decade and an intercept of 0 dB at the corner frequency.

The corner frequency is 1 rad/sec and the bandwidth is 1 rad/sec. The two asymptotes intersect at ω = 1 where 20 log 10 Η ( ) = -3 dB size 12{"20 log" rSub { size 8{"10"} } \lline Η \( "jω" \) \lline "= ""-3" ital "dB"} {}

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

the low- and high-frequency asymptotes of the angle of the frequency response are 0 and 90 0 size 12{-"90" rSup { size 8{0} } } {} (−π/2 radians), respectively. The angle is 45 0 size 12{-"45" rSup { size 8{0} } } {} at the corner frequency (1 rad/sec).

A line drawn from the low frequency asymptote a decade below the corner frequency to the high frequency asymptote a decade above the corner frequency approximates the angle of the frequency response.

Physical interpretation

With M = B = 1, the frequency response is

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

At low frequencies, |H(jω)| → 1 and arg H(jω) → 0. The inertia of the mass is negligible, and the damping force dominates so that the external force is proportional to velocity.

At high frequencies, |H(jω)| → 1/ω and arg H(jω) → 90 0 size 12{-"90" rSup { size 8{0} } } {} . The inertia of the mass dominates so that the acceleration is proportional to external force and the velocity decreases as frequency increases.

With R = C = 1, the frequency response is

{} H ( ) = V o ( ) V i ( ) = 1 + 1 size 12{H \( jω \) = { {V rSub { size 8{o} } \( jω \) } over {V rSub { size 8{i} } \( jω \) } } = { {1} over {jω+1} } } {}

At low frequencies, |H(jω)| → 1 and arg H(jω) → 0. The impedance of the capacitance is large so that all the input voltage appears at the output.

At high frequencies, |H(jω)| → 1/ω and arg H(jω) → 90 0 size 12{-"90" rSup { size 8{0} } } {} . The impedance of the capacitance is small so that the current is determined by the resistance and the output voltage is determined by the impedance of the capacitance which decreases as frequency increases.

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
I only see partial conversation and what's the question here!
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
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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