 # Signal Generators and Waveform-Shaping Circuits

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Signal Generators and Waveform-Shaping Circuits
C H A P T E R 12 Signal Generators and Waveform-Shaping Circuits

Introduction In the design of electronic systems the need frequently arises for signals having prescribed standard waveforms. There are 2 distinctly different approaches for the generation of sinusoids, most commonly used for the standard waveforms.  Employing a positive-feedback loop that consists an amplifier and an RC or LC frequency-selective network. It generates sine waves utilizing resonance phenomena, are known as linear oscillators (circuits that generate square, triangular, pulse waveforms are called non-linear oscillators or function generators.)  A sine wave is obtained by appropriate shaping a triangular waveform.

The Oscillator Feedback Loop
A basic structure of a sinusoidal oscillator consists of an amplifier and a frequency-selective network connected in a positive-feedback loop. The condition for the feedback loop to provide sinusoidal oscillations of frequency w0 is Barkhausen Criterion:  At w0 the phase of the loop gain should be zero.  At w0 the magnitude of the loop gain should be unity. Figure 12.1 The basic structure of a sinusoidal oscillator. A positive-feedback loop is formed by an amplifier and a frequency-selective network. In an actual oscillator circuit, no input signal will be present; here an input signal xs is employed to help explain the principle of operation.

Figure 12.2

Nonlinear Amplitude Control
Suppose we make Aβ=1 at w=w0,  If the temperature changes, Aβ <1 and oscillation will cease. If Aβ>1, the oscillations will grow in amplitude. We therefore need a nonlinear circuit for gain control to force Aβ to remain equal at the desired value of output amplitude. The function of gain-control mechanism is as follows:  To ensure that oscillation will start, Aβ is designed greater than unity.  As the oscillation grow in amplitude and reaches the desired level, the nonlinear network reduces the loop gain to exactly unity (poles will be pulled back to jw axis.  If the loop gain is reduced below unity, the amplitude will diminish and the nonlinear network increases the loop gain to exactly unity.

Figure 12. 3 (a) A popular limiter circuit
Figure 12.3 (a) A popular limiter circuit. (b) Transfer characteristic of the limiter circuit; L− and L+ are given by Eqs. (12.8) and (12.9), respectively. (c) When Rf is removed, the limiter turns into a comparator with the characteristic shown.

Limiter Circuit for Amplitude Control
A frequently employed limiter circuit is shown below. For small vI, D1/D2 off  slope= As vI continues to go positive, vO goes negative. D1 on/D2 off  As vI continues to go negative, vO goes positive. D1 off/D2 on 

The Wien-Bridge Oscillator Circuits
One of the simplest oscillator is based on the Wein bridge.  For L=1, w0= (RC)-1. R2/R1=2.  To ensure the oscillations will start, R2/R1=2 + δ Figure 12.4 A Wien-bridge oscillator without amplitude stabilization.

The Wien-Bridge Oscillator with A Limiter
The amplitude of oscillation can be determined and stabilized using a nonlinear control network. The positive output peak can be calculated by setting vb=v1+VD2 and writing the node equation at node b while neglecting the current through D2. Figure 12.5 A Wien-bridge oscillator with a limiter used for amplitude control.

Figure 12.6 A Wien-bridge oscillator with an alternative method for amplitude stabilization.

The Phase Shifter Oscillator
The phase-shifter consists of a negative gain amplifier (-K) with a third order RC ladder network in the feedback. The circuit will oscillate at the frequency for which the phase shift of the RC network is 180o. Only at the frequency will the total phase shift around the loop be 0o or 360o. The minimum number of RC sections is 3 because it is capable of producing a 180o phase shift at a finite frequency.

The Phase Shifter Oscillator with a Limiter
To start oscillations, Rf has to be made slightly greater than the minimum required value. (Practice exercise 12.5 and 12.6) Figure 12.8 A practical phase-shift oscillator with a limiter for amplitude stabilization.

The quadrature oscillator is another type of phase-shift oscillator, but the three RC sections are configured so that each section contributes 90° of phase shift. The outputs are sine and cosine (quadrature) because there is a 90° phase shift between op amp outputs. Rf is made equal to 2R, and thus –Rf cancels 2R. Figure 12.9 (a) A quadrature-oscillator circuit. (b) Equivalent circuit at the input of op amp 2.

Negative Impedance Converter
The negative impedance converter (NIC) is a configuration of an operational amplifier which acts as a negative load. source: excerpted from wikipedia

The Active-Filter-Tuned Oscillator
The block diagram of the active-filter-tuned oscillator is shown below.  Assume the oscillations have already started. The output of the bandpass filter will be a sine wave whose frequency is equal to the center frequency of the filter.  The sine-wave signal is fed to the limiter and then produces a square wave. Figure Block diagram of the active-filter-tuned oscillator. Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

Figure 12.11 A practical implementation of the active-filter-tuned oscillator.
Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

LC and crystal Oscillator
For higher freq. (> 1MHz) (a)容串 (b)感串 sedr42021_1312a.jpg Figure Two commonly used configurations of LC-tuned oscillators: (a) Colpitts and (b) Hartley.

Node equation at C sedr42021_1313.jpg
Equivalent circuit of the Colpitts oscillator Node equation at C sedr42021_1313.jpg Figure Equivalent circuit of the Colpitts oscillator of Fig (a). To simplify the analysis, Cm and rp are neglected. We can consider Cp to be part of C2, and we can include ro in R.

Figure 12.14 Complete circuit for a Colpitts oscillator.

sedr42021_1315a.jpg 通常L很大, Cs小 , Cs <<Cp Figure A piezoelectric crystal. (a) Circuit symbol. (b) Equivalent circuit. (c) Crystal reactance versus frequency [note that, neglecting the small resistance r, Zcrystal = jX(w)].

Crystal oscillator = L 串 Cs
sedr42021_1316.jpg Figure A Pierce crystal oscillator utilizing a CMOS inverter as an amplifier.

Bistable Multivibrator
Another type of waveform generating circuits is the nonlinear oscillators or function generators which uses multivibrators. A bistable multivibrator has 2 stable states. The circuit can remain in either state indefinitely and changes to the other one only when triggered. Metastable state: v+=0 and vO=0. The circuit cannot exist in the mestastable state for any length of time since any disturbance causes it to switch to either stable state. Figure A positive-feedback loop capable of bistable operation. Figure A physical analogy for the operation of the bistable circuit. The ball cannot remain at the top of the hill for any length of time (a state of unstable equilibrium or metastability); the inevitably present disturbance will cause the ball to fall to one side or the other, where it can remain indefinitely (the two stable states).

Microelectronic Circuits, International Sixth Edition

Bistable Circuit with Inverting Transfer Characteristics
Assume that vO is at one of its 2 possible levels, say L+, and thus v+=βL+.  As vI increases from 0 and then exceeds βL+, a negative voltage developes between input terminals of the op amp. This voltage is amplified and vO goes negative. The voltage divider causes v+ to go negative, increasing the net negative input and keeping the regenerative process going.  This process culminates in the op amp saturating, that is, vO=L-. Trigger signal The circuit is said to be inverting.

Bistable Circuit with Noninverting Transfer Characteristics
A bistable circuit implementing the noninverting transfer characteristics. The circuit is said to be noninverting.

Figure (a) Block diagram representation and transfer characteristic for a comparator having a reference, or threshold, voltage VR. (b) Comparator characteristic with hysteresis. Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

Application of the Bistable Circuit as a Comparator
To design a circuit that detects and counts the zero crossings of an arbitrary waveform, a comparator whose threshold is set to 0 can be used. The comparator provides a step change at its output every time zero crossing occurs. Figure Illustrating the use of hysteresis in the comparator characteristics as a means of rejecting interference.

Bistable Circuit with More Precise Output Level
Limiter circuits are used to obtain more precise output levels for the bistable circuit. L+ = VZ1 + VD and L– = –(VZ2 + VD), where VD is the forward diode drop. L+ = VZ + VD1 + VD2 and L– = –(VZ + VD3 + VD4).

Microelectronic Circuits, International Sixth Edition

Operation of the Astable Multivibrator (1)
Connecting a bistable multivibrator with inverting transfer characteristics in a feedback loop with an RC circuit results in a square-wave generator. Figure (a) Connecting a bistable multivibrator with inverting transfer characteristics in a feedback loop with an RC circuit results in a square-wave generator. (b) The circuit obtained when the bistable multivibrator is implemented with the circuit of Fig (a). (c) Waveforms at various nodes of the circuit in (b). This circuit is called an astable multivibrator.

Operation of the Astable Multivibrator (2)
The circuit for astable multivibrator. Figure (a) Connecting a bistable multivibrator with inverting transfer characteristics in a feedback loop with an RC circuit results in a square-wave generator. (b) The circuit obtained when the bistable multivibrator is implemented with the circuit of Fig (a). (c) Waveforms at various nodes of the circuit in (b). This circuit is called an astable multivibrator.

Generation of Triangular Waveforms
Triangular waveforms can be obtained by replacing the low-pass RC circuit with an integrator. Since the integrator is inverting, the inverting characteristics of the bistable circuit is required. Figure A general scheme for generating triangular and square waveforms.

Generation of a Standard Pulse (1)
In the stable state, VA=L+ (why?), VB=VD1, VC=βL+ (D2: ON and R4>>R1). When a negative-going step applies at the trigger input:  D2 conducts heavily and pulls node C down (lower than VB). The output of the op amp switch to L- and cause VC to go toward βL-.  D2 OFF and isolates the circuit from changes at the trigger input. D1 OFF and C1 begins to discharge toward L-.  When VB < VC, the output of the op amp switch to L+. Figure (a) An op-amp monostable circuit. (b) Signal waveforms in the circuit of (a).

Generation of a Standard Pulse (2)
Figure (a) An op-amp monostable circuit. (b) Signal waveforms in the circuit of (a).

The 555 Circuit Commercially available integrated-circuit package such as 555 timer exists that contain the bulk of the circuitry needed to implement monostable and astable multivibrator. 2/3 VCC 1/3 VCC Figure A block diagram representation of the internal circuit of the 555 integrated-circuit timer.

A Monostable Multivibrator Using the 555 IC (1)
In the stable state the flip-flop will be in the reset state (why?). As trigger goes below VTL, output of comparator 2 goes high and  Output of flip-flop goes high and Q1 OFF.  Capacitor C begins to charge toward Vcc.  When VC exceeds VTH, output of flip-flop goes low and Q1 ON.  Q1 rapidly discharges C and VC go to 0V.

A Monostable Multivibrator Using the 555 IC (2)
Figure (a) The 555 timer connected to implement a monostable multivibrator. (b) Waveforms of the circuit in (a).

S-R flip-flop Fig.12.28 說明 R S Qn Qn-1 1 timing t=0 Trig=0 Trig=1 t= T
Qn-1 1 Fig 說明 timing t=0 Trig=0 Trig=1 t= T (Vc=Vth) Q’=1, Vc=0 soon, steady state C1 (R) 1 C2 (S) Q (Vo)

An Astable Multivibrator Using the 555 IC (1)
Assume that initially C is discharged and the flip-flop is set (It is fine to start from reset state). VO is high and Q1 is OFF and then  VC charges through RA and RB toward Vcc. When it exceeds VTH, output of comparator 1 is high and VO is low (Q1 is ON).  VC begins to discharge through RB, when it is below VTL, output of comparator 2 is high and VO is high. Figure (a) The 555 timer connected to implement an astable multivibrator. (b) Waveforms of the circuit in (a).

An Astable Multivibrator Using the 555 IC (2)
Figure (a) The 555 timer connected to implement an astable multivibrator. (b) Waveforms of the circuit in (a).

Fig.12.29 說明 timing Vc= VTL Vc up Vc=VTH Q’=1, Vc=0 soon, TL Charging
1 Q (Vo) C1 (R) C2 (S)

Figure Using a nonlinear (sinusoidal) transfer characteristic to shape a triangular waveform into a sinusoid. Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

Nonlinear Waveform-Shaping Circuits The Breakpoint Method
Diodes or transistors can be combined with resistors to synthesize network having arbitrary nonlinear transfer characteristics which can be used in waveform shaping. The breakpoint method. Figure (a) A three-segment sine-wave shaper. (b) The input triangular waveform and the output approximately sinusoidal waveform.

Nonlinear Waveform-Shaping Circuits The Nonlinear-Amplification Method
This nonlinear amplification method is based on feeding the triangular wave to the input of an amplifier having a nonlinear transfer characteristics that approximates the sine function. Figure A differential pair with an emitter degeneration resistance used to implement a triangular- wave to sine-wave converter. Operation of the circuit can be graphically described by Fig

Precision Half-Wave Rectifier Superdiode
For instrumentation applications, if the signal to be rectified is 0.1V, it is impossible to employ conventional rectifier circuit. Superdiode shown below could solve this problem. Disadvantages of the superdioide  When vI goes negative and vO=0, the entire magnitude of vI appears in the op input terminal. The op will be damaged if it is not equipped with over-voltage protection.  When vI is negative, the op will be saturated and slows its operation. Figure 12.33

Precision Half-Wave Rectifier An Alternative Circuit
An improved version of the precision half-wave rectifier. The circuit operates in the following manner: vI>0, D2: ON (virtual ground appears) and output of op amp is clamped at one diode drop below ground, D1: OFF  vO=0. vI<0, D1: ON (virtual ground appears), D2: OFF  vO= -vI (R2/R1).  The feedback loop around the op amp remains closed at all times. Hence the op amp remains in its linear operating region (no saturation occurs). Figure (a) An improved version of the precision half-wave rectifier: Diode D2 is included to keep the feedback loop closed around the op amp during the off times of the rectifier diode D1, thus preventing the op amp from saturating. (b) The transfer characteristic for R2 = R1.

Measuring AC Voltages A circuit that could measure the AC voltage is shown below. It consists of a half-wave rectifier and a first-order low pass filter. The DC component of V1 is (VP/π)(R2/R1). The corner frequency of the low-pass filter should be chosen much smaller than the lowest expected frequency wmin of the input sine wave  (CR4)-1 << wmin.  V2 = - (VP/π)(R2/R1)(R4/R3). Figure A simple ac voltmeter consisting of a precision half-wave rectifier followed by a first-order low-pass filter.

Figure E12.28 Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

Precision Full-Wave Rectifier (1)
The principle of full-wave rectifier. Figure Principle of full-wave rectification.

Precision Full-Wave Rectifier (2)
Precision full-wave rectifier based on the above conceptual circuit. VI>0, D2:ON, VI=VO, no current through R1/R2, V- of A1= VI, D1:OFF.  VI<0, V- of A1 is negative, D1 ON. VO= -VI (R2/R1). D2:OFF. Figure (a) Precision full-wave rectifier based on the conceptual circuit of Fig (b) Transfer characteristic of the circuit in (a).

Figure 12.38 Use of the diode bridge in the design of an ac voltmeter.

Precision Peak Rectifier
Including the diode of the peak rectifier inside the negative-feedback loop of an op amp results in a precision peak rectifier. Figure A precision peak rectifier obtained by placing the diode in the feedback loop of an op amp. A buffered precision peak detector shown in the right circuit can hold the value of the peak for a long time. Figure A buffered precision peak rectifier.

Figure 12.41 A precision clamping circuit.
Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.