Applied Electronics II...Applied Electronics II Chapter 4: Wave shaping and Waveform Generators...

Applied Electronics II

Chapter 4: Wave shaping and Waveform Generators

School of Electrical and Computer EngineeringAddis Ababa Institute of Technology

Addis Ababa University

Daniel D./Getachew T./Abel G.

May 2017

Chapter 4: Wave shaping and Waveform Generators (AAIT)Chapter Three May 2017 1 / 34

Overview

1 Introduction

2 Basic Principles of Sinusoidal OscillatorsThe Oscillator Feedback LoopThe Oscillation CriterionNonlinear Amplitude Control

3 Op AmpRC Oscillator CircuitsThe Wien-Bridge OscillatorThe Phase-Shift Oscillator

4 MultivibratorsBistable Multivibrators

Application of the Bistable Circuit as a Comparator

Astable MultivibratorGeneration of Square WaveformsGeneration of Triangular Waveforms

Monostable MultivibratorGeneration of a Standardized Pulse


Introduction

Introduction

Standard waveforms for example, sinusoidal, square, triangular, orpulse are required in computer, control systems, communicationsystems, test and measurement systems.

A circuit that produces periodic wave forms at its output with outan input is refereed as Oscillator.

Oscillator can be classified as

1 Linear Oscillators1 RC oscillators

Wien BridgePhase-Shift

2 LC oscillators

HartleyColpittsCrystal

2 Non-linear Oscillators1 Multivibrators

bistablemonostableastable


Basic Principles of Sinusoidal Oscillators The Oscillator Feedback Loop

The Oscillator Feedback Loop

The basic structure of a sinusoidal oscillator consists of an amplifier anda frequency-selective network connected in a positive-feedback loop.

Although no input signal will be present in an actual oscillatorcircuit, we include an input signal here to help explain theprinciple of operation.

The gain-with-feedback is given by

Af (s) =A(s)

1−A(s)β(s)


Basic Principles of Sinusoidal Oscillators The Oscillation Criterion

The loop gain of the circuit

L(s) = A(s)β(s)

The characteristic equation thus becomes

1− L(s) = 0

If at a specific frequency f0 the loop gain Aβ is equal to unity

L(ω0) = A(ω0)β(ω0) = 1

That is, at this frequency the circuit will provide sinusoidaloscillations for zero input signal.

At f0 the phase of the loop gain should be zero and the magnitudeof the loop gain should be unity. This is known as theBarkhausen criterion.


Basic Principles of Sinusoidal Oscillators Nonlinear Amplitude Control

Nonlinear Amplitude Control

The parameters of any physical system cannot be maintainedconstant for any length of time.As a result, even if Aβ = 1 and ω = ω0 is achieved then thetemperature changes and Aβ becomes slightly less thanunity(oscillation will cease) or slightly grater(oscillations will growin amplitude).It is evident a mechanism is needed to force Aβ remain equal tounity at the desired value of output amplitude.A nonlinear circuit for gain control achieves the task and have thefollowing function.

First, to ensure that oscillations will start, designs the circuit suchthat Aβ is slightly greater than unity.When the amplitude reaches the desired level, the nonlinearnetwork comes into action and causes the loop gain to be reduced toexactly unity.If, for some reason, the loop gain is reduced below unity, thenonlinear network comes into action and causes the loop gain to beincrease to exactly unity.



The gain control can be implemented using a Limiter Circuit. Thefigure below is a popular limiter circuit frequently employed for thecontrol of op-amp oscillators.



To understand how the circuit operates.Let’s consider first the case of a small (close to zero) input signal vIand a small output voltage vO

vA is positive and vB is negative.

Both diodes D1 and D2 will be off.

All input current flows through the feed back resistor.

vO = −(Rf/R1)vI

This is the linear portion of the limiter transfer characteristic inthe previous figure.

Let us use superposition to find the voltages at nodes A and B.

vA = VR3

R2 +R3+ vO

R2

R2 +R3

vB = −V R4

R4 +R5+ vO

R5

R4 +R5



As vI goes positive

vO goes negative

vB will become more negative, thus keeping D2 off.

vA becomes less positive.

If we continue to increase vI further.

A negative value of vO will be reached at which vA becomes -0.7 Vor so and diode D1 conducts.

Using the constant voltage-drop model for D1 and denote the voltagedrop VD. The value of vO at which D1 conducts is the negativelimiting level L−.

L− = −V R3

R2− VD

(1 +

R3

R2

)vI can be found by dividing L− by the limiter gain −Rf/R1.If vI is increased beyond this value, more current is injected into D1,and vA remains at approximately −VD.Thus R3 appears in effect in parallel with Rf which is(−(Rf ‖ R3)/R1) slope of the transfer function.



The transfer characteristic for negative vI can be found in a manneridentical to the previous.

L+ = VR4

R5+ VD

(1 +

R4

R5

)The slope of the transfer characteristic in the positive limiting region is−(Rf ‖ R4)/R1.Removing Rf altogether results in the transfer characteristic, which isthat of a comparator

That is, the circuit comparesvI with the comparatorreference value of 0 V : vI >0 results in vo ≈ L−, and vI< 0 yields vo ≈ L+.


Op AmpRC Oscillator Circuits The Wien-Bridge Oscillator

The Wien-Bridge Oscillator

A Wien-bridge oscillator without the nonlinear gain-control network.

The Loop Gain.

L(s) = A(s)β(s) =

[1 +

R2

R1

]Zp

Zp + Zs



Where

Zp =R

1 + sRCZs =

1 + sRC

sC

Thus

L(s) =1 +R2/R1

1 + ZsZp

=1 +R2/R1

3 + sCR+ 1/sCR

L(ω) =1 +R2/R1

3 + (ωCR− 1/ωCR)

The phase of the loop gain will be zero at frequency

0 = ω0CR− 1/ω0CR

That is ω0 = 1/CRTo obtain sustained oscillations at this frequency, one should set themagnitude of the loop gain to unity. This can be achieved by selecting

R2/R1 = 2

To ensure that oscillations will start, one chooses R2/R1 slightlygreater than 2



Figure: A Wien-bridge oscillator with a limiter used for amplitude control.


Op AmpRC Oscillator Circuits The Phase-Shift Oscillator

The Phase-Shift Oscillator

The basic structure of the phase-shift oscillator consists of a negative gainamplifier (K) with a three-section (third-order) RC ladder network in thefeedback.

Figure: A phase-shift oscillator.

The circuit will oscillate at the frequency for which the phase shift of theRC network is π.

For oscillations to be sustained, the value of K = mag[1/(RCnetwork)]at the oscillation frequency.


Op AmpRC Oscillator Circuits The Phase-Shift Oscillator

Figure: practical phase-shift oscillator with a limiter for amplitudestabilization.

Diodes D1 and D2 and resistors R1, R2, R3, and R4 for amplitudestabilization.To start oscillations, Rf has to be made slightly greater than theminimum required value


Multivibrators Bistable Multivibrators

Bistable MultivibratorsBistable Multivibrators are circuits that has two stable state and movebetween states when appropriately triggered.

Figure: A positive-feedback loop capable of bistable operation.

Assume that the electrical noise causes a small positive increment in thevoltage v+.

The incremental signal will be amplified by A.

Much greater signal will appear at the output voltage vO.



The voltage divider will feed a fraction of the output signal β backto the positive-input terminal of the op amp.

If Aβ > 1, as is usually the case, the fed-back signal will begreater than the original increment in v+.

This regenerative process continues until op amp saturates at thepositive-saturation output level, L+.

When this happens, v+ becomes L+R1/(R1 +R2).

This is one of the two stable states of the circuit.

Had we assumed the equally probable situation of a negativeincrement.

The op amp would saturate in the negative direction.

vO = L− and v+ = L−R1/(R1 +R2)

This is the other stable state.

Also note that the circuit cannot exist in the state for which v+ = 0and vO = 0 for any length of time. This is a state of unstableequilibrium(also known as a metastable state).



Transfer Characteristics of the Bistable Circuit



To derive the transfer characteristics.

Assume vO is at L+ level.

v+ = βL+.

vI is increased from 0 V. nothing happens until it reaches βL+ =VTH

When vI > βL+ then vO goes negative.

The regenerative process takes place until vO = L− and v+ = βL−.

Increasing vI further has no effect.

Next consider what happens as vI is decreased.

Since now v+ = βL−, the circuit remains in thenegative-saturation state until vI = βL−.

As vI < βL− goes below this value the regenerative action takesplace

vO = L− and v+ = βL−



Bistable Circuit as a ComparatorThe comparator is used for detecting the level of an input signal relative to apreset threshold value.

This is noninverting configuration.by using superposition.

v+ = vIR2

R2 +R1+ vO

R1

R2 +R1



Assuming the output voltage at vO = L+.

To make a state change vO = L+, v+ = 0, vI = VTL.

VTL = −L+(R1/R2)

To change from negative state to positive.

VTH = −L−(R1/R2)

The difference between VTL. and VTH is the Hysteresis.

By using limiter circuits to make the output more precise.



R should be chosen to yield the current required for the properoperation of the zener diodes.

L+ = VZ1 + VD

L− = −(VZ2 + VD)

thus VTH = −L−(R1/R2) = (VZ2 + VD)(R1/R2)

thus VTL = −L+(R1/R2) = −(VZ1 + VD)(R1/R2)

Assuming the zener diodes are identical The hysteresis will be2(VZ + VD)(R1/R2)


Multivibrators Astable Multivibrator

Generation of Square Waveforms

A square waveform can be generated by arranging for a bistable multivibratorto switch states periodically. This can be done by connecting the bistablemultivibrator with an RC circuit in a feedback loop



How the circuit operates

1 let the output of the bistable multivibrator be L+.

2 The voltage at the positive input terminal will be v+ = βL+.

3 The voltage across C,v−, will rise exponentially toward L+ with atime constant τ = CR.

4 This will continue until v− = VTH = βL+.

5 Any further the input seen by the op amp will be negative thenvO = L−.

6 As a result, v+ = βL−.

7 The capacitor will then start discharging, and its voltage, v−, willdecrease exponentially toward L−.

8 This new state will prevail until v− reaches the negative thresholdVTL = βL−.

9 Then the bistable multivibrator switches to the positive-outputstate.



Expression

A capacitor C that is charging or discharging through a resistance Rtoward a final voltage V∞ has a voltage v(t)

v(t) = V∞ − (V∞ − V0+)e−t/τ

where V0+ is the voltage at t = 0+ and τ = CR is the time constant.

The period T of the square wave can be found as follows.During the charging interval T1 the voltage v− ,

v− = L+ − (L+ − βL−)e−t/τ

where τ = CR Substituting v− = βL+ at t = T1 gives

T1 = τ ln1− β(L−/L+)

1− βSimilarly, during the discharge interval T2

T2 = τ ln1− β(L+/L−)

1− βChapter 4: Wave shaping and Waveform Generators (AAIT)Chapter Three May 2017 26 / 34


The period T = T1 + T2 where L+ = L−

T = 2τ ln1 + β

1− β

Square-wave generator can be made to have variable frequency byswitching different capacitors C and by continuously adjusting R.

Also, the waveform across C can be made almost triangular byusing a small value for the parameter β.

However, triangular waveforms of superior linearity can be easilygenerated using the scheme discussed next.



Generation of Triangular WaveformsThe exponential waveforms generated in the astable circuit can be changed totriangular by replacing the low-pass RC circuit with an integrator.



The integrator causes linear charging and discharging of the capacitor,thus providing a triangular waveform.How the circuit operates.

1 Let the output of the bistable circuit be at L+.

2 A current equal L+/R to will flow into the resistor R and throughcapacitor C.

3 Causes the output of the integrator to linearly decrease with aslope of −L+/CR.

4 This will continue until the integrator output reaches the lowerthreshold VTL of the bistable circuit.

5 The output becomes negative and equal to L−.

6 The current through R and C will reverse direction, and its valuewill become equal to |L−|/R.

7 The integrator output will start to increase linearly with a positiveslope equal to |L−|/CR.

8 This will continue until the integrator output voltage reaches VTH .



The period T of the square and triangular waveformsDuring the interval T1.

VTH − VTL

T1=L+

CR

from which we obtain

T1 = CRVTH − VTL

L+

During the interval T2.VTH − VTL

T2=−L−

CR

from which we obtain

T2 = CRVTH − VTL

−L−

Thus to obtain symmetrical square waves we design the bistable circuit to

have L+ = −L−.


Multivibrators Monostable Multivibrator

Generation of a Standardized PulseIn some applications the need arises for a pulse of known height and widthgenerated in response to a trigger signal. Such a standardized pulse can begenerated by the monostable multivibrator.



The circuit is an augmented form of the astable circuit.A clamping diode D1 is added across the capacitor C1.Trigger circuit composed of capacitor C2, resistor R4, and diodeD2 is connected to the noninverting input terminal of the op amp.

How the circuit operates.

1 In the stable state, which prevails in the absence of the triggeringsignal, the output of the op amp is at L+.

2 D1 is conducting through R3 and thus clamping the voltage vB toone diode drop above ground.

3 R4 � R1, so that diode D2 will be conducting a very small currentand the voltage vc ≈ (R1/(R1 +R2))L+.

4 The stable state is maintained because βL+ is greater than VD1.5 Now consider the application of a negative-going step at the

trigger input.6 D2 conducts heavily and pulls node C down.7 If the trigger signal is of sufficient height to cause vC to go belowvB, the op amp will see a net negative input voltage and its outputwill switch to L−.



8 This in turn will cause vC to go negative to βL−.

9 D2 will then cut off, thus isolating the circuit from any furtherchanges at the trigger input terminal.

10 The negative voltage at A causes D1 to cut off, and C1 begins todischarge exponentially toward L−.

11 The monostable multivibrator is now in its quasi-stable state.

12 When vB goes below the voltage at node C, op-amp outputswitches back to L+ and the voltage at node C goes back to βL+.

13 Capacitor C1 then charges toward L+ until diode D1 turns on andthe circuit returns to its stable state.

The duration T of the output pulse is determined from the exponentialwaveform of vB,

vB = L− − (L− − VD1)e−t/C1R3

by substituting vB(T ) = βL−,

βL− = L− − (L− − VD1)e−T/C1R3



Rearranging

T = C1R3 ln

(VD1 − L−βL− − L−

)For VD1 � |L−|, this equation can be approximated by

T ≈ C1R3 ln

(1

1− β

)


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