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Chapter 15Time Response ofReactive Circuits

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Objectives

• Explain the operation of an RC integrator 

• Analye an RC integrator !ith a single

input pulse

• Analye an RC integrator !ith repetitive

input pulses

• Analye an RC "ifferentiator !ith a single

input pulse

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Objectives

• Analye an RC "ifferentiator !ith repetitive

input pulses

• Analye the operation of an R# integrator 

• Analye the operation of an R#

"ifferentiator 

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The RC $ntegrator 

• %hen a pulse generator is connecte" to the input

of an RC integrator& the capacitor !ill charge an"

"ischarge in response to the pulses

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The RC $ntegrator 

• The rate of charging an" "ischarging "epen"s on

the RC time constant

τ ' RC• (or an i"eal pulse& both e"ges are consi"ere" to be

instantaneous

 )  The capacitor appears as a short to an instantaneous

change in current an" as an open to "c

 )  The voltage across the capacitor cannot change

instantaneously * it can change only exponentially

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Capacitor +oltage

• $n an RC integrator& the output is the capacitor

voltage

• The capacitor charges "uring the time that the pulseis high

• $f the pulse is at its high level long enough& the

capacitor !ill fully charge to the voltage amplitu"e

of the pulse

• The capacitor "ischarges "uring the time that the

 pulse is lo!

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Response of RC $ntegrators to

,ingle*-ulse $nputs• A capacitor !ill fully charge if the pulse

!i"th is e.ual to or greater than 5 time

constants /5τ0

• At the en" of the pulse& the capacitor fully

"ischarges bac through the source

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%hen the -ulse %i"th is E.ual to

or 2reater than 5 Time Constants• As the pulse !i"th is increase"& the shape of the

output pulse approaches that of the input

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%hen the -ulse %i"th is #ess

Than 5 Time Constants• The output voltage !ill not reach the full input

voltage before the en" of the pulse

• The capacitor only partially charges• As the pulse !i"th is re"uce"& the output voltage

 becomes smaller because the capacitor has less time

to charge

• 3o!ever& it taes the capacitor approximately the

same length of time /5τ0 to "ischarge bac to ero

after the pulse is remove"

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Repetitive*-ulse Response of RC

$ntegrators• $f a perio"ic pulse !aveform is applie" to an RC

integrator& the output !aveshape "epen"s on the

relationship of the circuit time constant an" thefre.uency /perio"0 of the input pulses

• $f the pulse !i"th an" the time bet!een pulses are

each e.ual to or greater than five time constants

/T45τ0& the capacitor !ill fully charge an" fully"ischarge "uring each perio" of the input

!aveform

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%hen a Capacitor oes 6ot

(ully Charge an" ischarge• %hen the pulse !i"th an" the time bet!een pulses are

shorter than five time constants& the capacitor !ill not

completely charge or "ischarge

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%hen a Capacitor oes 6ot

(ully Charge an" ischarge• uring the first pulse& the capacitor charges& the

output voltage reaches 7895:

• ;et!een the first an" secon" pulses& the capacitor"ischarges& an" the voltage "ecreases to /1

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,tea"y*,tate Response

• The output voltage !ill buil" up to a stea"y*state

average value in five time constants /5τ0

• $f the time constant is extremely long compare" tothe pulse !i"th& the output voltage approaches a

constant "c voltage9 This value is the average

value of the input

• (or a s.uare !ave& the stea"y*state value is one*

half of the input voltage

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,ingle*-ulse Response of RC

ifferentiators• A series RC circuit in !hich the output voltage is

taen across the resistor is no!n as a

"ifferentiator9 $n terms of fre.uency response& it isa high*pass filter 

• The shape of the "ifferentiator?s resistor voltage is

"etermine" by the charging an" "ischarging action

of the capacitor 

• The next sli"e sho!s the pulse response of a

"ifferentiator for@ t%≥5 τ& an" t%

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Repetitive*-ulse Response of RC

ifferentiators• As the time constant is re"uce" /t%5τ0& both

 positive an" negative portions of the output become

narro!er • The average value of the output is ero

• (or a very long time constant& the output

approaches the shape of the input& but !ith an

average value of ero

• The average value of a !aveform is its "c

component

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Analysis of a Repetitive

%aveform• Consi"er an RC "ifferentiator !ith τ ' t%& an" an

input !aveform of < to B1

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-ulse Response of R# $ntegrators

• The output !aveform is taen across the resistor&

an" un"er e.uivalent con"itions& is the same shape

as that for the RC integrator • %hen a pulse generator is connecte" to the input

of the integrator an" the voltage pulse goes from

its lo! level to its high level& the in"uctor prevents

a su""en change in current

• The in"uctor acts as an open& an" all of the input

voltage is across it at the instant of the rising e"ge

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Response of the $ntegrator to a

,ingle -ulse

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-ulse Response of an R#

ifferentiator • A series R# circuit in !hich the output voltage is

taen across the in"uctor is no!n as a

"ifferentiator • %hen the input pulse goes from its lo! level to its

high level& # loos lie an open& an" all of the

input voltage appears across it at the instant of the

rising e"ge• ,ho!n in the next sli"e are the responses of an R#

"ifferentiator for both time constant con"itions

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-ulse Response of an R#

ifferentiator 

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,ummary

• $n an RC integrating circuit& the output

voltage is taen across the capacitor 

• $n an RC "ifferentiating circuit& the outputvoltage is taen across the resistor 

• $n an R# integrating circuit& the output voltage

is taen across the resistor • $n an R# "ifferentiating circuit& the output

voltage is taen across the in"uctor 

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,ummary

• $n an integrator& !hen the pulse !i"th /t%0 ofthe input is much less than the transient time&

the output voltage approaches a constantlevel e.ual to the average value of the input

• $n an integrator& !hen the pulse !i"th of theinput is much greater than the transient time&the output voltage approaches the shape ofthe input

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,ummary

• $n a "ifferentiator& !hen the pulse !i"th of

the input is much less than the transient

time& the output voltage approaches theshape of the input but !ith an average value

of ero

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,ummary

• $n a "ifferentiator & !hen the pulse !i"th of

the input is much greater than the transient

time& the output voltage consists of narro!& positive*going an" negative*going spies

occurring on the lea"ing an" trailing e"ges

of the input pulses