<< Chapter < Page Chapter >> Page >

III. SIMPLE LINEAR FEEDBACK SYSTEM

1/ Black’s formula

K(s) is called the open loop system function, and H(s) = Y (s)/X(s) is called the closed-loop system function. Note, when β(s) = 0, H(s) = K(s)

We can find H(s) by combining

E ( s ) = X ( s ) β ( s ) Y ( s ) and Y ( s ) = K ( s ) E ( s ) size 12{E \( s \) =X \( s \) - β \( s \) Y \( s \) matrix { {} # {}} ital "and" matrix { {} # {}} Y \( s \) =K \( s \) E \( s \) } {}

to obtain

Y ( s ) = K ( s ) X ( s ) β ( s ) Y ( s ) size 12{Y \( s \) =K \( s \) X \( s \) - β \( s \) Y \( s \) } {}

which can be solved to obtain Black’s formula,

H ( s ) = Y ( s ) X ( s ) = K ( s ) 1 + β ( s ) K ( s ) = forward transmission 1 loop gain size 12{H \( s \) = { {Y \( s \) } over {X \( s \) } } = { {K \( s \) } over {1+β \( s \) K \( s \) } } = { { ital "forward" matrix { {} # {}} ital "transmission"} over {1 matrix { {} # {}} - matrix { {} # {}} ital "loop" matrix { {} # {}} ital "gain"} } } {}

Two-minute miniquiz problem

Problem 8-1 Simple position control system

The objective of the position control system shown below is for the output position Y (s) to track the input signal X(s).

a) Determine the closed-loop system function H(s) = Y (s)/X(s).

b) For x(t) = u(t), a unit step, determine the steady state value of y(t).

Solution

  1. We can use Black’s formula to find H(s) as follows

H ( s ) = K ( s + 1 ) ( s + 100 ) 1 + K ( s + 1 ) ( s + 100 ) = K ( s + 1 ) ( s + 100 ) + K size 12{H \( s \) = { { { {K} over { \( s+1 \) \( s+"100" \) } } } over {1+ { {K} over { \( s+1 \) \( s+"100" \) } } } } = { {K} over { \( s+1 \) \( s+"100" \) +K} } } {}

b) The steady-state response to a unit step is simply the response to the complex exponential x ( t ) = 1 . e 0 . t = 1 size 12{x \( t \) =" 1" "." " e" rSup { size 8{0 "." "t "} } =" 1 "} {} which is y ( t ) = 1 . H ( 0 ) . e 0 . t = K/ ( 100 + K ) size 12{y \( t \) =" 1" "." " H" \( 0 \) "." e rSup { size 8{0 "." "t "} } =" K/" \( "100 "+" K" \) } {} . The position error ε = K / ( 100 + K ) 1 = 100/ ( 100 + Κ ) size 12{ε=K/ \( "100"+K \) - 1" = "-"100/" \( "100 + "Κ \) } {} Hence, this position controller (with proportional feedback) has an error that diminishes as the gain increases. However, the error is never zero no matter how large the gain.

Effect of feedback on system performance

Feedback is used to enhance system performance.

  • Stabilize gain
  • Reduce the effect of an output disturbance
  • Improve dynamic characteristics — increase bandwidth, improve response time
  • Reduce noise
  • Reduce nonlinear distortion

Properties of feedback

  • Increase input impedance
  • Decrease output impedance

2/ Stabilize gain

overall gain = 10 Overall gain = 100 × 10 1 + 0 . 099 × 100 × 10 = 10 size 12{ ital "overall" matrix { {} # {}} ital "gain"="10" matrix { {} # {}} ital "Overall" matrix { {} # {}} ital "gain"= { {"100" times "10"} over {1+0 "." "099" times "100" times "10"} } ="10"} {}

Note that both the open-loop and the same gain which equals 10.

But now suppose that the gain of the power amplifier is reduced to 5.

overall gain = 10 Overall gain = 100 × 5 1 + 0 . 099 × 100 × 5 = 9 . 9 size 12{ ital "overall" matrix { {} # {}} ital "gain"="10" matrix { {} # {}} ital "Overall" matrix { {} # {}} ital "gain"= { {"100" times 5} over {1+0 "." "099" times "100" times 5} } =9 "." 9} {}

Note that a change in gain of the power amplifier of 50% leads to a change in gain of the feedback system of only 1%.

The stabilization of the gain resulting from feedback can be appreciated more generally from Black’s formula.

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

If K(s) is large so that |β(s)K(s)|>>1 then

H ( s ) 1 β ( s ) size 12{H \( s \) approx { {1} over {β \( s \) } } } {}

So if H(s) has a gain then β(s) must have an attenuation. If the attenuation β(s) is determined precisely but the gain K(s) varies (with time, temperature, etc.), then we can make an amplifier whose gain is independent of K(s) and determined almost entirely by β(s).

But how can we make β(s) precise?

Consider the non-inverting amplifier with a non-ideal (finite gain) op-amp — network (left), block diagram (right).

For an op-amp (model 741) the gain is typically K 10 7 size 12{K approx "10" rSup { size 8{7} } } {} . Hence, provided KR 2 / ( R 1 + R 2 ) >>1 size 12{"KR" rSub { size 8{2} } / \( R rSub { size 8{1} } +R rSub { size 8{2} } \) ">>1"} {} , we have

V o V i = 10 7 1 + 10 7 R 2 R 1 + R 2 R 1 + R 2 R 2 size 12{ { {V rSub { size 8{o} } } over {V rSub { size 8{i} } } } = { {"10" rSup { size 8{7} } } over {1+"10" rSup { size 8{7} } { {R rSub { size 8{2} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } } } approx { {R rSub { size 8{1} } +R rSub { size 8{2} } } over {R rSub { size 8{2} } } } } {}

Conclusion — the gain of the feedback amplifier depends primarily on the values of the resistors and not on the gain of the op-amp which depends on parameters of transistors which change with time, temperature, etc.

3/ Reduce the effect of an output disturbance

The transfer functions for the input and the disturbance are

Y ( s ) X ( s ) = K ( s ) 1 + β ( s ) K ( s ) and Y ( s ) D ( s ) = 1 1 + β ( s ) K ( s ) size 12{ { {Y \( s \) } over {X \( s \) } } = { {K \( s \) } over {1+β \( s \) K \( s \) } } matrix { {} # {}} ital "and" matrix { {} # {}} { {Y \( s \) } over {D \( s \) } } = { {1} over {1+β \( s \) K \( s \) } } } {}

Therefore, if β(s)K(s) is made arbitrarily large then

Questions & Answers

short run AC curves?
Jasmin Reply
what is short run AC curves?
Jasmin
what is short run curves?
Jasmin
what is short run curves?
Jasmin
nooo am not from India why!?
Godwin Reply
h
Hamid
Godwin which level of education are you please
Millionaires
millionaires am in SHS 2
Godwin
vnsgu BBA ki first sem ki all subjects ki koi app he ?
Modi Reply
one is business economics app
mudasir
all subjects ki koi nhi he ?
Modi
for accounts u can download accounts complete course app ... step by step Al the topics r mentioned
mudasir
Muje BBA ke first semester ke liye chahiye.... Vnsgu
Modi
there is a app for bba
Muhammad
named as college tutor on play store
Muhammad
isme sab subject ki books aa jayegi ?
Modi
han g
Muhammad
baqi main bad main batata hun abi online paper ho raha ha
Muhammad
okk....
Modi
Apki knsi uni ha
Muhammad
aik aur app ha but wo indian ha
Muhammad
Named as bba books
Muhammad
okk
Modi
collage tutor mein join teacher wala hi aata he or kuch nhi aata he
Modi
Mera Vnsgu he
Modi
asslam alaikum
Hamid
BBA books app ki photo send kijiye naa please
Modi
photo kesy send ho skty ha yahan sy?
Muhammad
I mean iska option kahan ha
Muhammad
haa.. sorry
Modi
collage tutor mein join teacher wala hi aata he or kuch nhi aata he
Modi
ok
Muhammad
Mera Vnsgu he
Modi
Ap play store py ja ky waha search ker lain wahan sy asani sy ye app mil jay ga apko
Muhammad
ab books kaha lau ?
Modi
oh sorry
Muhammad
main apko sham rak batata hun abi paper ho raha ha☺☺
Muhammad
kya search maru ?
Modi
yar ap ki books ka name kia ha?
Muhammad
modi
Hamid
who know accounting
Hamid
Hmare pass koi bhi subject ki book nhi he
Modi
abi 1st semester ha
Muhammad
ha... First semester
Modi
Yes sir
Modi
mujy book pary ha pdf ma accounting economic statistics Kay book ha mery pass
Hamid
pdf ma h
Hamid
to send karo naa
Modi
ya pa kasy kro
Hamid
kya ?
Modi
whatsp number do
Hamid
number bejo whatsp ka
Hamid
jaldi
Hamid
technical error bta rha he
Modi
number send nhi ho paa rha he
Modi
fb I'd name batio
Hamid
mery Hamid Ali shaikh ha
Hamid
mein nai use krti
Modi
fb I'd ka name bejo
Hamid
aap kaha se ho ?
Modi
fr kiya hoga
Hamid
ap kiya Kiya chlty ho batio
Hamid
mtlb ?
Modi
kiya use krty h wo Batio ma koxhes krta h fr
Hamid
ma pakistan sa h
Hamid
fb dowled Kro ap fr kam hiGa ap ka
Hamid
Krishi Modi iss fb id pe bhejo
Modi
Modi
Muhammad
profile kon sy h
Hamid
Modi are you from india
Muhammad
yr privacy ha Aik I'd pa req nhi jaty
Hamid
who was the father of economic ?why?
Mahesh Reply
who was the known as father of economic?why?
Mahesh
Rationing and hoarding
Semiat Reply
how do the size of a country's population affect labour force
Evans Reply
a mixed economic system
Ngong Reply
What are the types of price elasticity of demand
Juliana Reply
what are massures to promote geographical mobility of labor?
Ngong
Is to make sure that a labourer to know more about his salary to earn before going to the direction
shehu
what is trade
PHILIPPE Reply
Trade is a basic economic concept involving the buying and selling of goods and services, with compensation paid by a buyer to a seller, or the exchange of goods or services between parties. Trade can take place within an economy between producers and consumers.
Miss
what is fisical policy?
ha Reply
fisical policy or fiscal policy?
Miss
what are.the characteristics of economic goods
Hamis
what are the importance of labour market?
Rachael
how discrib the rural development and their four stages
Sheikh Reply
ye economics se related ha
Sheikh
1..traditional stage..no science and technology is applied hence poor productionuu.2..the take off stage..some development strategies are initiated eg transport system is improved but the traditional cultural belief still remain .3..the prematurely stage..technological methods of production are appl
President
applied leading to higher GDP..4..stage of mass consumption..
President
What is Easiest Formula For National Income?
Tenzin Reply
national income/ agrrigate net value
Sheikh
what do you mean by the supply of goods
sachin Reply
supply of good refer to the total unit of production which is ready to sell at a given price
Tenzin
what is implicit cost
fuseini Reply
Yeah
MOHAMED
any cost that has already occurred but not necessarily shown or reported as a separate expense.
President
The links don't seem to be working
Scorch Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask