Applied Electronics II

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1 Applied Electronics II Chapter 4: Wave shaping and Waveform Generators School of Electrical and Computer Engineering Addis Ababa Institute of Technology Addis Ababa University Daniel D./Getachew T./Abel G. May 2017 Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

2 Overview 1 Introduction 2 Basic Principles of Sinusoidal Oscillators The Oscillator Feedback Loop The Oscillation Criterion Nonlinear Amplitude Control 3 Op AmpRC Oscillator Circuits The Wien-Bridge Oscillator The Phase-Shift Oscillator 4 Multivibrators Bistable Multivibrators Application of the Bistable Circuit as a Comparator Astable Multivibrator Generation of Square Waveforms Generation of Triangular Waveforms Monostable Multivibrator Generation of a Standardized Pulse Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

3 Introduction Introduction Standard waveforms for example, sinusoidal, square, triangular, or pulse are required in computer, control systems, communication systems, test and measurement systems. A circuit that produces periodic wave forms at its output with out an input is refereed as Oscillator. Oscillator can be classified as 1 Linear Oscillators 1 RC oscillators Wien Bridge Phase-Shift 2 LC oscillators Hartley Colpitts Crystal 2 Non-linear Oscillators 1 Multivibrators bistable monostable astable hapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

4 Basic Principles of Sinusoidal Oscillators The Oscillator Feedback Loop The Oscillator Feedback Loop The basic structure of a sinusoidal oscillator consists of an amplifier and a frequency-selective network connected in a positive-feedback loop. Although no input signal will be present in an actual oscillator circuit, we include an input signal here to help explain the principle of operation. The gain-with-feedback is given by A f (s) = A(s) 1 A(s)β(s) Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

5 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 f 0 the loop gain Aβ is equal to unity L(jω 0 ) = A(jω 0 )β(jω 0 ) = 1 That is, at this frequency the circuit will provide sinusoidal oscillations for zero input signal. At f 0 the phase of the loop gain should be zero and the magnitude of the loop gain should be unity. This is known as the Barkhausen criterion. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

6 Basic Principles of Sinusoidal Oscillators Nonlinear Amplitude Control Nonlinear Amplitude Control The parameters of any physical system cannot be maintained constant for any length of time. As a result, even if Aβ = 1 and ω = ω 0 is achieved then the temperature changes and Aβ becomes slightly less than unity(oscillation will cease) or slightly grater(oscillations will grow in amplitude). It is evident a mechanism is needed to force Aβ remain equal to unity at the desired value of output amplitude. A nonlinear circuit for gain control achieves the task and have the following function. First, to ensure that oscillations will start, designs the circuit such that Aβ is slightly greater than unity. When the amplitude reaches the desired level, the nonlinear network comes into action and causes the loop gain to be reduced to exactly unity. If, for some reason, the loop gain is reduced below unity, the nonlinear network comes into action and causes the loop gain to be increase to exactly unity. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

7 Basic Principles of Sinusoidal Oscillators Nonlinear Amplitude Control The gain control can be implemented using a Limiter Circuit. The figure below is a popular limiter circuit frequently employed for the control of op-amp oscillators. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

8 Basic Principles of Sinusoidal Oscillators Nonlinear Amplitude Control To understand how the circuit operates. Let s consider first the case of a small (close to zero) input signal v I and a small output voltage v O v A is positive and v B is negative. Both diodes D 1 and D 2 will be off. All input current flows through the feed back resistor. v O = (R f /R 1 )v I This is the linear portion of the limiter transfer characteristic in the previous figure. Let us use superposition to find the voltages at nodes A and B. v A = V v B = V R 3 R 2 + R 3 + v O R 2 R 2 + R 3 R 4 R 4 + R 5 + v O R 5 R 4 + R 5 Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

9 Basic Principles of Sinusoidal Oscillators Nonlinear Amplitude Control As v I goes positive v O goes negative v B will become more negative, thus keeping D 2 off. v A becomes less positive. If we continue to increase v I further. A negative value of v O will be reached at which v A becomes -0.7 V or so and diode D 1 conducts. Using the constant voltage-drop model for D 1 and denote the voltage drop V D. The value of v O at which D 1 conducts is the negative limiting level L. L = V R ( 3 V D 1 + R ) 3 R 2 R 2 v I can be found by dividing L by the limiter gain R f /R 1. If v I is increased beyond this value, more current is injected into D 1, and v A remains at approximately V D. Thus R 3 appears in effect in parallel with R f which is ( (R f R 3 )/R 1 ) slope of the transfer function. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

10 Basic Principles of Sinusoidal Oscillators Nonlinear Amplitude Control The transfer characteristic for negative v I can be found in a manner identical to the previous. L + = V R ( 4 + V D 1 + R ) 4 R 5 R 5 The slope of the transfer characteristic in the positive limiting region is (R f R 4 )/R 1. Removing R f altogether results in the transfer characteristic, which is that of a comparator That is, the circuit compares v I with the comparator reference value of 0 V : v I > 0 results in v o L, and v I < 0 yields v o L +. hapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

11 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 + R ] 2 R 1 Z p Z p + Z s Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

12 Where Thus Op AmpRC Oscillator Circuits Z p = R 1 + src L(s) = 1 + R 2/R Zs Z p = The Wien-Bridge Oscillator Z s = 1 + src sc 1 + R 2 /R scr + 1/sCR L(jω) = 1 + R 2 /R j(ωcr 1/ωCR) The phase of the loop gain will be zero at frequency 0 = ω 0 CR 1/ω 0 CR That is ω 0 = 1/CR To obtain sustained oscillations at this frequency, one should set the magnitude of the loop gain to unity. This can be achieved by selecting R 2 /R 1 = 2 To ensure that oscillations will start, one chooses R 2 /R 1 slightly greater than 2 Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

13 Op AmpRC Oscillator Circuits The Wien-Bridge Oscillator Figure: A Wien-bridge oscillator with a limiter used for amplitude control. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

14 Op AmpRC Oscillator Circuits The Phase-Shift Oscillator The Phase-Shift Oscillator The basic structure of the phase-shift oscillator consists of a negative gain amplifier (K) with a three-section (third-order) RC ladder network in the feedback. Figure: A phase-shift oscillator. The circuit will oscillate at the frequency for which the phase shift of the RC network is π. For oscillations to be sustained, the value of K = mag[1/(rcnetwork)] at the oscillation frequency. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

15 Op AmpRC Oscillator Circuits The Phase-Shift Oscillator Figure: practical phase-shift oscillator with a limiter for amplitude stabilization. Diodes D 1 and D 2 and resistors R 1, R 2, R 3, and R 4 for amplitude stabilization. To start oscillations, R f has to be made slightly greater than the minimum required value Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

16 Bistable Multivibrators Bistable Multivibrators Bistable Multivibrators are circuits that has two stable state and move between states when appropriately triggered. Figure: A positive-feedback loop capable of bistable operation. Assume that the electrical noise causes a small positive increment in the voltage v +. The incremental signal will be amplified by A. Much greater signal will appear at the output voltage v O. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

17 Bistable Multivibrators The voltage divider will feed a fraction of the output signal β back to the positive-input terminal of the op amp. If Aβ > 1, as is usually the case, the fed-back signal will be greater than the original increment in v +. This regenerative process continues until op amp saturates at the positive-saturation output level, L +. When this happens, v + becomes L + R 1 /(R 1 + R 2 ). This is one of the two stable states of the circuit. Had we assumed the equally probable situation of a negative increment. The op amp would saturate in the negative direction. v O = L and v + = L R 1 /(R 1 + R 2 ) This is the other stable state. Also note that the circuit cannot exist in the state for which v + = 0 and v O = 0 for any length of time. This is a state of unstable equilibrium(also known as a metastable state). Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

18 Bistable Multivibrators Transfer Characteristics of the Bistable Circuit Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

19 Bistable Multivibrators To derive the transfer characteristics. Assume v O is at L + level. v + = βl +. v I is increased from 0 V. nothing happens until it reaches βl + = V T H When v I > βl + then v O goes negative. The regenerative process takes place until v O = L and v + = βl. Increasing v I further has no effect. Next consider what happens as v I is decreased. Since now v + = βl, the circuit remains in the negative-saturation state until v I = βl. As v I < βl goes below this value the regenerative action takes place v O = L and v + = βl Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

20 Bistable Multivibrators Bistable Circuit as a Comparator The comparator is used for detecting the level of an input signal relative to a preset threshold value. This is noninverting configuration. by using superposition. v + = v I R 2 R 2 + R 1 + v O R 1 R 2 + R 1 Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

21 Bistable Multivibrators Assuming the output voltage at v O = L +. To make a state change v O = L +, v + = 0, v I = V T L. V T L = L + (R 1 /R 2 ) To change from negative state to positive. V T H = L (R 1 /R 2 ) The difference between V T L. and V T H is the Hysteresis. By using limiter circuits to make the output more precise. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

22 Bistable Multivibrators R should be chosen to yield the current required for the proper operation of the zener diodes. L + = V Z1 + V D L = (V Z2 + V D ) thus V T H = L (R 1 /R 2 ) = (V Z2 + V D )(R 1 /R 2 ) thus V T L = L + (R 1 /R 2 ) = (V Z1 + V D )(R 1 /R 2 ) Assuming the zener diodes are identical The hysteresis will be 2(V Z + V D )(R 1 /R 2 ) Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

23 Astable Multivibrator Generation of Square Waveforms A square waveform can be generated by arranging for a bistable multivibrator to switch states periodically. This can be done by connecting the bistable multivibrator with an RC circuit in a feedback loop Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

24 Astable Multivibrator Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

25 Astable Multivibrator 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 a time constant τ = CR. 4 This will continue until v = V T H = βl +. 5 Any further the input seen by the op amp will be negative then v O = L. 6 As a result, v + = βl. 7 The capacitor will then start discharging, and its voltage, v, will decrease exponentially toward L. 8 This new state will prevail until v reaches the negative threshold V T L = βl. 9 Then the bistable multivibrator switches to the positive-output state. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

26 Astable Multivibrator Expression A capacitor C that is charging or discharging through a resistance R toward a final voltage V has a voltage v(t) v(t) = V (V V 0+ )e t/τ where V 0+ 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 T 1 the voltage v, v = L + (L + βl )e t/τ where τ = CR Substituting v = βl + at t = T 1 gives T 1 = τ ln 1 β(l /L + ) 1 β Similarly, during the discharge interval T 2 T 2 = τ ln 1 β(l +/L ) 1 β Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

27 Astable Multivibrator The period T = T 1 + T 2 where L + = L T = 2τ ln 1 + β 1 β Square-wave generator can be made to have variable frequency by switching different capacitors C and by continuously adjusting R. Also, the waveform across C can be made almost triangular by using a small value for the parameter β. However, triangular waveforms of superior linearity can be easily generated using the scheme discussed next. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

28 Astable Multivibrator Generation of Triangular Waveforms The exponential waveforms generated in the astable circuit can be changed to triangular by replacing the low-pass RC circuit with an integrator. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

29 Astable Multivibrator 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 through capacitor C. 3 Causes the output of the integrator to linearly decrease with a slope of L + /CR. 4 This will continue until the integrator output reaches the lower threshold V T L 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 value will become equal to L /R. 7 The integrator output will start to increase linearly with a positive slope equal to L /CR. 8 This will continue until the integrator output voltage reaches V T H. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

30 Astable Multivibrator The period T of the square and triangular waveforms During the interval T 1. V T H V T L = L + T 1 CR from which we obtain During the interval T 2. from which we obtain T 1 = CR V T H V T L L + V T H V T L T 2 = L CR T 2 = CR V T H V T L L Thus to obtain symmetrical square waves we design the bistable circuit to have L + = L. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

31 Monostable Multivibrator Generation of a Standardized Pulse In some applications the need arises for a pulse of known height and width generated in response to a trigger signal. Such a standardized pulse can be generated by the monostable multivibrator. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

32 Monostable Multivibrator The circuit is an augmented form of the astable circuit. A clamping diode D 1 is added across the capacitor C 1. Trigger circuit composed of capacitor C 2, resistor R 4, and diode D 2 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 triggering signal, the output of the op amp is at L +. 2 D 1 is conducting through R 3 and thus clamping the voltage v B to one diode drop above ground. 3 R 4 R 1, so that diode D 2 will be conducting a very small current and the voltage v c (R 1 /(R 1 + R 2 ))L +. 4 The stable state is maintained because βl + is greater than V D1. 5 Now consider the application of a negative-going step at the trigger input. 6 D 2 conducts heavily and pulls node C down. 7 If the trigger signal is of sufficient height to cause v C to go below v B, the op amp will see a net negative input voltage and its output will switch to L. Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

33 Monostable Multivibrator 8 This in turn will cause v C to go negative to βl. 9 D 2 will then cut off, thus isolating the circuit from any further changes at the trigger input terminal. 10 The negative voltage at A causes D 1 to cut off, and C 1 begins to discharge exponentially toward L. 11 The monostable multivibrator is now in its quasi-stable state. 12 When v B goes below the voltage at node C, op-amp output switches back to L + and the voltage at node C goes back to βl Capacitor C 1 then charges toward L + until diode D 1 turns on and the circuit returns to its stable state. The duration T of the output pulse is determined from the exponential waveform of v B, by substituting v B (T ) = βl, v B = L (L V D1 )e t/c 1R 3 βl = L (L V D1 )e T/C 1R 3 Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

34 Monostable Multivibrator Rearranging ( ) VD1 L T = C 1 R 3 ln βl L For V D1 L, this equation can be approximated by ( ) 1 T C 1 R 3 ln 1 β Chapter 4: Wave shaping and Waveform Generators Chapter (AAIT) Three May / 34

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