082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

436 Chapter 13 Integration
C
R
i
R 1 i 1 i
y f
1
R 2 i y2
2
–
R
i A
y 3 i
X 3 3
+
y i
y R
C
y o
y o
Figure13.3
Simple integrator.
Figure13.4
Summingintegrator usingan operational
amplifier(op amp).
where f =frequencyofthesignal(Hz).ItcanbeseenthatX c
decreaseswithincreasing
frequency, f. For frequencies whereX c
is small compared withR, most of the voltage
droptakesplaceacrosstheresistor.Inotherwords, v o
issmallcomparedwith v R
.ExaminingEquation(13.1)forthecasewhenX
c
isverymuchlessthanR(writtenasX c
≪R),
and v o
≪ v R
, itcan be seen thatEquation (13.4) simplifies to
v i
=RC dv o
dt
(13.5)
Thisequationisonlyvalidfortherangeoffrequencies forwhichX c
≪R.Rearranging
Equation (13.5) yields
dv o
dt
= v i
RC
v o
= 1
RC
∫
v i
dt
The output voltage from the circuit is an integrated version of the input voltage with a
1
scaling factor
RC .
Anelectronicintegratorwhichperformsbettercanbemadefromanoperationalamplifier.
The circuit for an operational amplifier integrator is given in Figure 13.4. The
main advantage of this circuit is the low output impedance and high input impedance,
makingitusefulforelectroniccontrolapplicationsandanaloguesignalprocessing.The
function of the operational amplifier is to amplify the potential difference between the
invertingandnon-invertinginputs.Thesearelabelled −and +respectivelyinthecircuit
diagram.Usuallythegainoftheamplifierisextremelyhigh,soevenasmallvoltagedifferencebetweenthetwoinputswillgiveaverylargeoutputvoltage,whichislimitedby
the voltage supply attached to the device. Notice in this circuit that there is a capacitor
connectedfromtheoutputbacktotheinput.Thiscapacitorprovidesnegativefeedback.
This means that a proportion of the output voltage is fed back to the input and this in
turnservestoreducetheoutput.Asaconsequenceofthistheoverallgainofthecircuit
is limited and the amplifier reaches an equilibrium state where the voltage at point X is
the same as at the non-inverting input, which is connected to earth. For this reason the
point Xinthe circuitis sometimes referredtoas avirtual earth.
Assuming point Xisatzero volts, and using Ohm’s law, gives
i 1
= v 1
R 1
i 2
= v 2
R 2
i 3
= v 3
R 3
(13.6)