RL & RC

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RL & RCIntegrator and Differentiator

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An RCintegrator is a circuit that approximates the mathematicalprocess of integration. Integration is a summing process, and a basicintegrator can produce an output that is a running sum of the inputunder certain conditions.

A basic RCintegrator circuit is

simply a capacitor in series with

a resistor and the source. The

output is taken across the

capacitor.

VS

R

C Vout

The RC Integrator

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When a pulse generator is connected to the input of an RCintegrator,the capacitor will charge and discharge in response to the pulses.

When the input

pulse goes HIGH,the pulse

generator acts like

a battery in series

with a switch and

the capacitorcharges.

Switch closes

The output is an

exponentially

rising curve.

R

C

The RC Integrator

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When the pulse generator goes low, the small internal impedance ofthe generator makes it look like a closed switch has replaced thebattery.

The output is anexponentially

falling curve.

R

C

The pulse

generator now actslike a closed switch

and the capacitor

discharges.

The RC Integrator

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Examples

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sFKRC 100)001.0)(100( 1. Time constant2. Compute the Vout for one time constant

3. Time to finish discharging

s 5005

VVVout

3.610)63.0(

Solution

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Waveforms for the RCintegrator depend on the time constant () of thecircuit. If the time constant is short compared to the period of the inputpulses, the capacitor will fully charge and discharge. For an RCcircuit, = RC.The output will reach 63% of the final value in 1.

R

C

What is if R= 10 kand C= 0.022 F? 220 s

The output willreach steady

state in about 5

The RC Integrator

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If is increased, the waveforms

approach the average dc level as in

the last waveform. The output will

appear triangular but with a

smaller amplitude.

Alternatively, the input frequency

can be increased (Tshorter). Thewaveforms will again approach

the average dc level of the input.

t

t

t

t

Vin

Vout

Vout

Vout

The RC Integrator

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Example

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sFKRC 47)01.0)(7.4( 1. Time constant

2. Calculate the first pulse

3. Calculate the second pulse

mveeVV

t

FC 958)1(5)1( 47

10

mveeVV

t

iC 696)(958)( 47

15

4. Calculate the second pulse

VeVmVeVVVV

t

FiFC 52.1)5696(5)( 47

10

Solution

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Solution

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An RCdifferentiator is a circuit that approximates the mathematicalprocess of differentiation. Differentiation is a process that finds the rateof change, and a basic differentiator can produce an output that is therate of change of the input under certain conditions.

A basic RCdifferentiator circuit

is simply a resistor in series with

a capacitor and the source. The


resistor.

VS R

C

Vout

The RC Differentiator

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When a pulse generator is connected to the input of an RCdifferentiator, the capacitor appears as an instantaneous short to therising edge and passes it to the resistor.

The capacitor lookslike a short to the

rising edge because

voltage across C

cannot change

instantaneously.

During this first

instant, the

output follows

the input.

0

VC= 0


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After the initial edge has passed, the capacitor charges and the outputvoltage decreases exponentially.

The voltage across C

is the traditional

charging waveform.

The output

decreases as the

pulse levels off.


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Example

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sFKRC 8.1)120)(15( 1. Time constant

2. tw is bigger than 5 time constant 90 us

Solution

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The falling edge is a rapid change, so it is passed to the output becausethe capacitor voltage cannot change instantaneously. The type ofresponse shown happens when is much less than the pulse width (

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If is long compared to the pulse width, the output does have timeto return to the original baseline before the pulse ends. The

resulting output looks like a pulse with droop.

Vin

5= tw

5>> tw

tw

When 5= tw, the pulse

has just returned to thebaseline when it repeats.

The output shape isdependent on the ratio

of to tw.


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Like the RCintegrator, an RLintegrator is a circuit that approximatesthe mathematical process of integration. Under equivalent conditions,the waveforms look like the RCintegrator. For an RLcircuit, = L/R.

A basic RLintegrator circuit is aresistor in series with an inductor

and the source. The output is

taken across the resistor.VS R

L

Vout

What is the time constant if R= 22 k

and L= 22 H? 1.0 ms

The RL Integrator

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Example

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nsk

H

R

L2

10

20

1. Time constant

Solution

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When the pulse generator output goes high, a voltage immediatelyappears across the inductor in accordance with Lenzs law. Theinstantaneous current is zero, so the resistor voltage is initially zero.

The output is

initially zero

because there is

no current.

VS

R

L

+

The inducedvoltage across L

opposes the

initial rise of

the pulse.0 V

The RL Integrator

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At the top of the input pulse, the inductor voltage decreasesexponentially and current increases. As a result, the voltage across theresistor increases exponentially. As in the case of the RCintegrator, theoutput will be 63% of the final value in 1.

The output

voltage increases

as current builds

in the circuit.

VS

R

L

+

The inducedvoltage across L

decreases.

The RL Integrator

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When the pulse goes low, a reverse voltage is induced across Lopposingthe change. The inductor voltage initially is a negative voltage that isequal and opposite to the generator; then it exponentially increases.

The output voltage

decreases as the

magnetic field

around L collapses.

VS

R

L

+

The induced

voltage across Linitially

opposes the

change in the

source voltage.

Note that these waveforms

were the same in the RC

integrator.

The RL Integrator

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The RLDifferentiatorAn RLdifferentiator is also a circuit that approximates the mathematicalprocess of differentiation. It can produce an output that is the rate ofchange of the input under certain conditions.

A basic RLdifferentiator circuit

is an inductor in series with a

resistor and the source. The


inductor.

VS L

R

Vout

The RL Differentiator

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When a pulse generator is connected to the input of an RLdifferentiator, the inductor has a voltage induced across it thatopposes the source; initially, no current is in the circuit.

Current is

initially zero, soVR= 0.

During this firstinstant, the

inductor develops

a voltage equal

and opposite to

the source

voltage.

VR= 0

VS

L

R

+


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After the initial edge has passed, current increases in the circuit.Eventually, the current reaches a steady state value given by Ohmslaw.

The voltage across R

increases as current

increases.

The output

decreases as the

pulse levels off.

VS

L

R

+



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Next, the falling edge of the pulse causes a (negative) voltage to beinduced across the inductor that opposes the change. The currentdecreases as the magnetic field collapses.

The voltage across R

decreases ascurrent decreases.

The output

decreases

initially and

then increasesexponentially.

VS

L

R

+


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If is long compared tothe pulse width, theoutput looks like a

pulse with droop.

Vin

5= tw

5>> tw

tw

When 5= tw, the pulse

has just returned to thebaseline when it repeats.

As in the case of the RCdifferentiator, the output shape is dependent onthe ratio of to tw.


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Application

An application of an integrator is to generate a time delay. The voltageat Brises as the capacitor charges until the threshold circuit detects

that the capacitor has reached a predetermined level.

SWcloses

Threshold

Time delay

R VoutVin A B VA

VB

Vout

Threshold

circuitSW

C

Application

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Integrator

Time constant

Transient time

A circuit producing an output that approaches the

mathematical integral of the input.

A fixed time interval, set by Rand C, or Rand Lvalues, that

determines the time response of a circuit.

An interval equal to approximately five time constants.

Selected Key Terms

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Steady state

Differentiator

The equilibrium condition of a circuit that occurs after aninitial transient time.

A circuit producing an output that approaches the

mathematical derivative of the input.

Selected Key Terms

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1. The circuit shown is

a. an integrator.

b. a high-pass filter.

c. both of the above.

d. none of the above.

Quiz

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2. The circuit shown is

a. an integrator.

b. a low-pass filter.

c. both of the above.

d. none of the above.

Quiz

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3. Initially, when the pulse from the generator rises, the voltage across Rwillbe

a. equal to the inductor voltage.

b. one-half of the inductor voltage.

c. equal to VS

d. zero.

VS

Quiz

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4. After an RLintegrator has reached steady state from an input pulse, the

output voltage will be equal to

a. 1/2 VS

b. 0.63 VS

c. VS

d. zero

Quiz

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5. The time constant for an RLintegrator is given by the formula

a. = L/R

b. = 0.35RL

c. = R/L

d. = LR

Quiz

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6. The input and output waveforms for an integrator are shown. From the

waveforms, you can conclude that

a. = tw

b. >> tw

c.

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7. If a 20 kresistor is in series with a 0.1 F capacitor, the time constant is

a. 200 s

b. 0.5 ms

c. 1.0 ms

d. none of the above

Quiz

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8. After a single input transition from 0 to 10 V, the output of a

differentiator will be back to 0 V in

a. less than one time constant.

b. one time constant.

c. approximately five time constants

d. never.

Quiz

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9. An interval equal to approximately five time constants is called

a. transient time.

b. rise time.

c. time delay.

d. charging time.

Quiz

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10. Assume a time delay is set by an RCintegrator. If the threshold is set at 63%of the final pulse height, the time delay will be equal to

a. 1

b. 2

c. 3

d. 5

SW

closes

Threshold

Time delay

VA

VB

Vout

Quiz

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Answers:

1. b

2. c

3. d

4. c

5. a

6. b

7. d8. c

9. a

10. a

Quiz

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