It should be noted that the frequency of oscillation ω o is determined by the phase characteristics of the feedback loop. the loop oscillates at the frequency for which the phase is zero The steeper the phase shift as a function of frequency φ(ω) the more stable the frequency of oscillation
Non-linear Amplitude Control Generally difficult to design circuits with Aβ = 1 as circuit parameters vary with temperature, time and component values If Aβ < 1 Oscillation ceases If Aβ > 1 Oscillations grow until circuit saturates Require mechanism to force Aβ = 1 this is accomplished by a nonlinear circuit for gain control Design circuit with Aβ > 1 as voltage of oscillation increases gain control mechanism kicks in and reduces gain to 1 Design circuit with right half plane poles. Gain control pulls the poles back to the imaginary axis Two approaches. The first approach uses a limiter circuit, oscillations are allowed to grow until the level reaches the limiter set value. Once the limiter comes into operation, the amplitude remains constant. The limiter should be designed to minimize nonlinear distortion The second method uses a resistive element in the feedback loop whose resistance can be controlled by the sinusoidal output amplitude. Diodes or JFETs (operating in triode region) are commonly used
A popular limiter circuit for amplitude control can be seen below:
Wien - Bridge Oscillator Opamp - RC Oscillators
The loop gain can be found by multplying the transfer function of the feedback path v a (s) / v o (s) by the amplifier gain L( s) = 1 + R R 2 1 Z p Z p + Z s L( jω) = R 1+ 2 R1 3+ j ωcr 1 ωcr The loop gain will be a real number (i.e. the phase will be zero) at one frequency given by ω ω ο ο CR = = 1 ω 1 CR ο CR To obtain sustained oscillations at this frequency the magnitude of the loop gain should be unity which can be achieved by setting R2 R 1 = 2
To ensure that oscillation starts, one chooses R 2 / R 1 slightly greater than 2 The amplitude of oscillation can be controlled using a non-linear limiter as seen below
Phase Shift Oscillator The basic structure of the phase shift oscillator is shown below. It consists of a negative gain amplifier (- k) with a three section (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 180 o. Only at this frequency will the phase shift around the loop be 0 o (360 o ). Three RC sections are required to produce a 180 o phase shift at a finite frequency The value of k is chosen to be slightly higher than the inverse of the magnitude of the RC network transfer function at the frequency of oscillation
Active Filter Tuned Oscillator
The opamp RC oscillator circuits are useful for operation in the 10 Hz - 1 MHz range due to limitations in passive component size (Low frequency) and opamp slew rate (high frequency) for higher frequencies, circuits that employ transistors together with LC tuned circuits or crystals are commonly used
LC and Crystal Oscillators Oscillators utilizing transistors and LC tuned circuits or crystals are useful for operation in the range from 100 khz to 500 MHz. They exhibit higher Q than RC types (more stable), however, LC oscillators are difficult to tune over wide ranges and crystal oscillators operate at a single frequency. The extremely stable response of crystal oscillators has made them very popular particularly for digital timing signals
t v = L ( L β L ) e τ τ = C R + + substituting v = β L at t = T gives + L 1 β L+ T 1 = τ ln 1 β 1 Similarly for the discharge cycle it can be shown L 1 β + L T 2 = τ ln 1 β If L = L and T = T +T Then T = + 1 2 1+ 2τ ln 1 β β The square wave generator can be made to have variable frequency by adjusting C and/or R The waveform across C can be made almost triangular by using a small value for the parameter β
Phase Lock Loops (PLLs) Useful building block available from numerous vendors as a single IC. Contains a phase detector, low pass filter, amplifier and VCO. Combines digital and analog techniques Applications include tone decoding, demodulating AM and FM signals, frequency multiplication and synthesis and synchronization of signals from noisy sources
Phase Lock Loop Block Diagram
Phase detector compares two frequencies and generates an output which is a measure of their phase difference The phase error is filtered and amplified and causes the VCO frequency to deviate in the direction of f IN If conditions are right the VCO will quickly lock to f IN maintaining a fixed phase relationship with the input signal When locked the PLL control voltage is a DC signal which is a measure of the input frequency (Tone Decoding, FM Decoding) The VCO output is a locally generated signal and thus provides a clean replica of f IN which may itself be noisy If a modulo N counter is inserted between the VCO and phase detector the VCO will generate a multiple of the input frequency (high speed clocks, power line suppression)
Phase Detection Two types (Type I & Type II). Type I can be driven by analog or digital (square-wave) signal. Type II is driven by digital transitions (edges) Simplest Type I (digital) is an exclusive OR gate. Type I (linear) has similar output but uses a four quadrant analog multiplier instead of an XOR Type II phase detector is sensitive only to the relative timing edges of the input signal and the VCO output Type II detector is independent of duty cycle. Type I like 50% duty cycle Type I phase detectors always generate an output wave. This results in a small amount of residual ripple and periodic phase modulation (i.e. the VCO output is distorted with phase modulation sidebands) Type II phase detector sensitive to noise
VCO Many PLL s contain an internal VCO. External VCO ICs are also available Some VCOs offer multiple output waveforms such as sine, square and triangle waves (8038, 2206)
PLL Design PLLs can be treated as negative feedback amplifiers with one difference. In feedback amplifiers the quantity adjusted is the same quantity measured (commonly). In a phase lock loop we measure phase and adjust frequency. Since phase is an integral of frequency this introduces a 90 o phase shift To avoid oscillation we must insure that further phase shifts in the loop do not cause a 180 o phase at any frequency One solution is to eliminate any further phase shifting components by eliminating the low pass filter. This is referred to as a first order loop First order PLLs are useful, however, they do not allow the VCO to smooth out noise or fluctuations on the input signal A second order PLL has low-pass filtering within the feedback loop carefully designed to prevent instabilities Low-pass filtering provides flywheel memory action which maintains lock even when there are small noise and signal fluctuations Low-pass filtering increases capture time and reduces capture range
Second order loops are almost always used PLL design is based on selecting a loop filter and amp which passes the maximum signal frequency you are capturing and reduces the loop gain to less than 1 at 180 o phase shift The loop gain includes the gain of the phase detector, LP filter, amp and VCO VCO acts as integrator with 20 db / decade rolloff and -90 o phase shift K LOOP = K P K F K VCO
Generation of Triangle Waveforms The exponential waveform generated in the astable circuit can be changed to triangular by replacing the low-pass RC circuit with an integrator The integrator causes linear charging and discharging of the capacitor. Because the integrator is inverting it is necessary to use the non-inverting bistable circuit V V T L+ T C R V = V 1 = C R L TH TL TH TL 1 + V V T L = T C R V V 2 = C R L TH TL TH TL 2 If L = -L then symmetrical waveforms are obtained + -
Monostable Multivibrators In some applications the need arises for a pulse of known height and width generated in response to a trigger signal Because the width of the pulse is predictable its trailing edge can be used for timing purposes, such a pulse can be generated by a monostable multivibrator The monostable multivibrator has one stable state in which it can remain indefinitely. It also has a quasi-stable state in which it remains for a predetermined interval equal to the desired width of the output pulse Once the interval expires, the monostable returns to the stable state and remains there awaiting another triggering signal. The circuit is commonly called a one shot A monostable circuit is shown below which is an augmented form of the astable circuit
The duration T of the output pulse is determined by the exponential waveform at v B B ( ) = ( D1 ) v t L L V e t C R 1 3 by substituting β v ( T) = β L ( D1 ) L = L L V e B T C R 1 3 T = C R 1 3 VD1 L ln β L L For V << L This equation can be approximated by T D1 C R 1 3 1 ln 1 β Note: The monostable should not be retriggered again until C 1 has been recharged to V D1 (Recovery period)
Integrated Circuit Timers Commercially available integrated circuit packages contain the bulk of the circuitry needed to implement monostable and astable multivibrators having precise characteristics. The most popular of such ICs is the 555 timer
vc = V CC 1 e t RC 2 vc = vth = VCC at t = T 3 T = CRln 3 11. CR
The rise in v is given by C ( ) v = V V V e C C C C C T L t C R ( + R ) 2 1 v = V = V at t = T ; V = V 3 3 C T H C C H T L CC ( ) ln 0. 69 ( ) T = C R + R C R + R H A B A B A B The fall in v is given by v = V e C T H C t C R v = V 1 2 = V, V = V 3 3 at t = T T = C R ln 2 = 0. 69C R C T L C C T H C C L L B B ( ) T = T + T = 0. 69 C R + 2 R B H L A B
LC Tuned Oscillator Two commonly used configurations are the Colpitts and Hartley oscillators. The basic circuit structures without biasing can be seen below
Both circuits utilize a parallel LC circuit connected between the collector and base with a fraction of the tuned circuit voltage fed to the emitter The resistor R models the losses of the inductor, the load resistance of the oscillator and the output resistance of the transistor If the frequency of operation is sufficiently low that we can neglect the transistor capacitances, the frequency of oscillation is determined by the resonant frequency of the parallel tuned circuit (also known as a tank circuit) for the Colpitts oscillator ω ο = 1 1 2 L C C C + C 1 2 For the Hartley oscillator ω ο = 1 ( + ) L L C 1 2 The ratio L 1 / L 2 or C 1 / C 2 determines the feedback factor and thus must be adjusted in conjunction with the transistor gain to ensure that oscillations will start
To determine the oscillation condition for the Colpitts oscillator we replace the transistor with its equivalent circuit To simplify the analysis we neglect the transistor capacitance C µ. Capacitance C π can be considered part of C 2. r π is neglected assuming that r π >> 1 / ω o C 2
A node equation at the transistor collector (c) yields sc v 2 ( 2 ) 1 + gmv + + sc 1+ s LC v = 0 R 2 π π 1 π since v π 0 (oscillations have started) it can be eliminated ( ) ( ) ( ) 2 / 1 2 m 3 2 s LC1C 2 + s LC R + s C + C + g + 1 R = 0 g m 2 1 ω LC2 [ 3 + + j ω ( C C ) LC C ] 1 + 2 ω 1 2 = 0 R R For oscillations to start both the real and imaginary parts must be zero. Setting the imaginary part to zero gives ω ο = 1 1 2 L C C C + C 1 2 which is the resonant frequency of the tank circuit setting the real part to zero gives C2 C = g R 1 m
For sustained oscillation the magnitude of the gain from the base to collector (g m R) must be equal to the inverse of the voltage ratio provided by the capacitive divider vbe C v = ce 2 C1 for the oscillation to start the loop gain must be greater than unity which is equivalent to g R C C m > 2 1 As oscillations grow in amplitude, the transistor s non-linear characteristics reduce the loop gain to unity, thus sustaining oscillations An example of a complete Colpitts oscillator is shown on the next page
The radio frequency choke (RFC) provides a high reactance at ω o but a low DC resistance Unlike the opamp oscillators that incorporate special amplitude control circuitry, LC tuned oscillators utilize the non-linear i C - v BE characteristics of the BJT (or i D versus v GS for FETs) for amplitude control. As the oscillations grow the effective gain of the transistor is reduced below its small-signal value. Thus LC tuned oscillators are known as self-limiting oscillators Reliance on the non-linear characteristics of the BJT (or the FET) implies that the collector (Drain) current waveform will be nonlinearly distorted. Nevertheless, the output voltage swing will still be sinusoidal of high purity because of the filtering action of the LC tuned circuit
Crystal Oscillators A piezoelectric crystal, such as quartz, exhibits electromechanical - resonant characteristics that are very stable (with time and temperature) and highly selective (having very high Q factors) The circuit symbol of a crystal is shown below:
The resonant properties are characterized by a large inductance L (as high as hundreds of henrys), a very small series capacitance C S (as small as 0.0005 pf), a series resistance, r, representing a Q factor ωol / r that can be as high as a few hundred thousand and a parallel capacitance C P (a few picofarads) Capacitor C P represents the electrostatic capacitance between the two parallel plates of the crystal (C P >> C S ) Since the Q factor is so high we can neglect the resistance r and express the crystal impedance as Z ( s) = s C P 1 + s L 1 + 1 s C which can be manipulated to the form S 2 s + 1 1 LCS Z ( s) = scp 2 ( C C ) s + P + S LCS CP we see that the crystal has two resonant frequencies
ω S = 1 ω L C P = 1 S S P L C C C + C series resonance parallel resonance S P for s = jω Z ( jω) = 1 j ω C P ω ω 2 2 ω s 2 2 ω P It can be seen that ω P > ω S, however, since C P >> C S the two resonant frequencies are very close We observe that the crystal reactance is inductive over the very narrow frequency band between ω S and ω P We may use the crystal to replace the inductor in the Colpitts oscillator
The resulting circuit will oscillate at the resonant frequency of the crystal inductance L with the series equivalent of C S and C C CP + 1 2 C + C 1 2 since C S is much smaller than the other capacitances it will dominate and ω ο = 1 = ω LCS S