A Colpitts Oscillator

A COLPITTS OSCILLATOR: A New Look S Guha Mallick, S Chatterjee1 and B N Biswas Sir J C Bose School of Engineering, SKFGI, Mankundu Hooghly, INDIA 1 Kanailal Vidyamandir (French Section), Chandernagore, Hooghly, INDIA Abstract In Colpitts oscillator some of the questions relating to its large signal analysis have remained unattended. (1) How do the poles move during the growth period of the oscillation and on what parameters the pole movement depend? (2) How does the feedback delay affect the age-old Berkhausen’s Criteria? (3) How does the frequency of oscillation depend on the amount of feedback delay? (4) Do these poles ultimately reside on the imaginary axis? (5) How does the feedback delay affect the steady state amplitude of oscillation? How does the feedback delay affect the movement of the poles? (6) Simple yet elegant energy balance principle instead of nonlinear analysis has been applied to find the steady state amplitude of oscillation. These are some of the unanswered questions that the present paper deals with. Interesting simplified but accurate theory based on the operation of the oscillator near the resonant frequency of the tank circuit has been proposed. 1. Introduction "Oscillator" can mean anything that exhibits periodically time-varying characteristics observed first by Galileo [1, 2], but we are concerned with the type that provides an electrical signal at a specific frequency when supplied only with DC power. A Colpitts oscillator is one such electronic oscillator. Many years ago when the words "oscillator" and "electronics" were not invented in connection with electrical circuits and systems an "Oscillation Generator" was invented by Edwin Henry Colpitts (1872-1949) possibly sometime in the period 1915-1918 [3]. It is one of a number of designs for LC electronic oscillators that use a combination of inductors (L) and capacitors (C) to produce an oscillation at a certain frequency. The distinguishing feature of the Colpitts oscillator is that the feedback for the active device is taken from a voltage divider made of two capacitors in series across the inductor. Historically, it is important to note that when transistors were expensive, Colpitts oscillator is the topological choice because it uses only one transistor and it uses LC circuit elements as the resonator and the quality factor can be made high. 2. Relevance and Requirement of Oscillator An electronic oscillator is a nonlinear circuit with at least two memory components (charge, flux or hysteresis based). When excited with a dc source an oscillator responds with a steady state signal which may be chaotic in nature in case of more than two memory components [9]. The successful implementation of this idea depends on the purity of the oscillator waveform. The amplitude and the instantaneous phase of the oscillator are susceptible to ambient noise. In most cases the amplitude noise is negligible or unimportant because all practical oscillators incorporate amplitude limiter. Z Thus the key parameter is the phase noise at the output of the 3 C1 Vo R oscillator. That is, the most critical specification for any L C1 oscillator is its spectral purity, usually characterized by phase Vs C2 noise. The single side band noise spectral density is expressed C2 Vs as 2 Fig 2: Colpitts Oscillator and its equivalent 2kT 0 S 10log (1) PQ2 0 Q is the quality factor of the tank circuit of the oscillator. P0 is the oscillator power. So increase of oscillator power overpowers the effect of thermal noise and Q increases the selectivity of the tank circuit. Since many wireless transceivers such as mobile phones are battery-powered, we also strive to minimize power consumption. Beyond which increased power consumption cannot be tolerated by practical devices. 978-1-4673-5225-3/14/$31.00 ©2014 IEEE 3. Colpitts Oscillator A typical Colpitts oscillator and its equivalent circuit is shown in Fig. 2.The Colpitts oscillator is a single-transistor implementation of a sinusoidal oscillator which is widely used in electronic devices and communication systems. The Colpitts oscillator thus remains popular because it requires only a single pin to connect the resonator [4-9]. Notwithstanding this popularity and large number of studies, it appears to the authors, certain questions have remained unanswered. For example, large signal analysis of the oscillator has been left unattended. Moreover, the effect of feedback delay has not been looked into which needs modification of the Berkhausen’s Criteria. The present paper presents these aspects. 3.1. Small Signal Analysis Let us refer to the ac equivalent circuit of the Colpitts oscillator as shown in Fig.3. Let the output be denoted as ‘V ’. dv v dv gV C C s 0 (2) v ms 12dt R dt V L gVms sCV12 sCV s 0 R C1 R 3 2 GmVs a0s a1s a2s a3 0 (3) C2 Vs Where aLCC012 , aLCG12 , aCC212, a3 gm G Fig. 3 Small signal equivalent circuit This is a third order characteristic equation of a dynamical system for a self-excited oscillator. It initially starts operating with an insignificantly small value but finally arrives at the steady state behavior. That the system is unstable but finally it is a stable one. The start-up condition requites that the Routh Harwitz condition should satisfy the following G gm ;..,ie gm C12 GC (4) CC12 Again for spontaneous oscillation to build up it is necessary that there should be one real root and one complex conjugate roots placed on the right half plane. That is, denoting the roots as s1 , s2 and s3 , when s1 j , s2 j and s3 is real and they should satisfy a1 G ss123 s (5) aC01 aCC212 2 ss12 s 23 s ss 31 0 (6) aLCC012 agGgG3 mm 2 ss123 s 0 (7) aLCCCC01212 Simplifying gG 2 s m 0 (8) 3 22 CC12 That is, the differential equation is dV g G m (9) dt C12 C It is to be noted that for large signal, gm has to be replaced by Gm. As a result of the decay of V will depend on the amplitude itself. g CGC 2() v m 12 (10) CC11 C 2 t VVe00cos tVt ( )cos t (11) 22 0 (12) it starts growing with time with a finite initial value. But noting the variation of V0 (t) . This indicates the value of the real part of the complex poles on the right half of the s-plane. 3.2 Large Signal Analysis In order to simplify the analysis without affecting the physical behaviour v of the oscillator we assume that the oscillator is operating near the resonance frequency of the tank circuit. We construct the large signal C2 LL1 equivalent circuit as shown in Fig.4. The purpose of this simplification is CC12 to avoid the possibility of chaotic behaviour of the oscillator. We replace C1 R L1 , the series combination of L and C by L where G 2 1 eq V 1/ jC s 2 C VjLjC1/ LL 1 . 02 1 1 C CC12 1 12 LCC Let us denote the large signal trans-conductance by G (V ) one can write 02 2 m s Fig. 4 Large signal equivalent circuit GVeq 00 Yj().0 V (13) 1 Where Yj() G jC 1 jL GGjCeq 2 10 The normalized output voltage variation can be find out as dx0 2()()Ixt10 xt 0 GC2 x0 (14) dt I00 x()() t x 0 t gm C 1 4. Berkhausen’s Criteria in Colpitts Oscillator A typical transistorized Colpitts oscillator is shown in Fig.5 one can easily write VGVZV ms(). s (15) Writing Vs in terms of V one finds that GVm ().. Z 1 (16) Where, Gm is the large signal trans-conductance and Z is the equivalent collector load. The gain of the amplifier without Z3 feedback is given by Vo Z1 R Aj GZjm ()exp() j (17) Feedback Vf According to Barkhausen the oscillation will persist if Z2 Delay Vs Aj.1 (18) Vs Fig.5: Barkhausen Model with Feedback Delay Thus, 0 tan( ) (19) 0 2Q This indicates that the oscillation frequency depends on the feedback delay. Let the total time delay be dQ 2 0 T , QQe (20) d0 2 Although the addition of extra delay changes the frequency of oscillation, yet improves the quality factor of the tank circuit. 5. Pole Movement Case-I: No Feedback Delay: An approximate value of (t) can be determined by (21). This means that the poles j also start moving with time (i.e. pole movement). Therefore it is possible to find the pole movement from a knowledge of the variation of (t) , it is thus possible to appreciate the movement of the frequency of oscillation. Variation of (t) with time is shown in Fig.6 which is computed from the following relation 1 dx ()t 0 (21) x0 dt Case-II: With Feedback Delay Following a similar argument it is possible to calculate the movement of (t) using equation (14). dx 2()()Ixt xt GC 0 10 0 2 x (22) dt I x()() t x t g C 0 00 0m 1 The nature of pole movement is shown in Fig.7. From the figure for the case without delay it can be appreciated that before attaining the steady state oscillation the amplitude and frequency goes on changing. Thus it takes a finite time to settle down to the final value. Now referring to the figure with the delay we can deserve that the real part of the pole takes a spiral motion before it settles down to its steady value. This reflected the transient motion of the oscillator amplitude. The frequency of oscillation also goes on changing before it settles down to the final value.

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