7. Positive-Feedback Oscillators (continued)

Transcription

1 ecture : Introduction to electronic analog circuit Poitive-Feedback Ocillator (continued) Eugene Paerno, Ocillator for high frequencie: ocillator: Our aim i to develo ocillator with high frequency tability at high frequencie. We have to find a different aroach becaue, a will be hown in the next lecture, the mall-ignal voltage gain decreae with frequency, and, therefore, the frequency tability of the Wien-bridge ocillator would be low at high frequencie. Our aroach will be baed on emloying arallel reonant circuit a feedback network. (At high frequencie inductance are mall and inexenive.) The hae reone of a reonant circuit i real at the frequency of reonance, and the hae condition of the Barkhauen criterion will hold true; hence, we will et the ocillator frequency at the reonant frequency of it feedback network: ω ω. We aw in the reviou lecture that to obtain high frequency tability the loe of the hae reone of the feedback network, or it equivalent quality factor, hould be high. By definition, equivalent quality factor, Q equiv, of a arallel reonant circuit equal it intrinic quality factor: Q equiv Q /(ω ). Hence, to kee Q a high a oible for a given and ω we have to kee a high a oible. Before tarting the develoment of an ocillator, let u firt ee what the hyical meaning of i in a arallel reonant circuit. Fig. how that for a given ω, for examle, ω, i inverely roortional to the reitance of the inductance wire: the lower : the higher, the higher Q Hartley and olitt ocillator According to the oitive-feedback aroach, we have to connect an amlifier to the feedback network and to cloe the feedback loo to obtain an ocillator. In order not to reduce the quality factor of the network, we have to chooe only amlifier with high outut imedance. Otherwie, the outut imedance of the amlifier will hunt in Fig. and reduce Q. Hence, we chooe either E or B amlifier. (Note that it i alo oible to connect an amlifier in erie with a erie reonant circuit. In thi cae, the outut reitance of the amlifier hould be much lower than the erie reitance of the inductor to not to reduce the quality factor of the reonant circuit. A amlifier ha the lowet outut reitance, about 6 Ω at I E ma. However, comared to a lower than Ω erie reitance of the inductor, it i till too high. Of coure, it will be low enough at I E ma, but it i too much of ower conumtion.) In a general cae, the mall-ignal voltage gain of thee amlifier are greater than one. Therefore the tranmiion of the feedback network at the reonant frequency hould be le than unity to atify the amlitude condition of the Barkhauen criterion. Thi tranmiion hould alo be Im[Ζ ] jω jω ϖ + ( + Q ) ϖ + ϖ ϖ ( + / Q ) ϖ e[ζ ] Q ϖ Fig.. A ractical circuit (above), a tranformation of and into and (in the middle), and an electrical equivalent of the hyical circuit: a arallel reonant circuit (below). negative to atify the hae condition of the Barkhauen criterion for a E amlifier and oitive for a B amlifier. Thee requirement can be met if we divide the voltage at the outut of the network a hown in Fig.. Without the reactive voltage divider in Fig. (b), the feedback network tranmiion at ω ω would be equal to unity. Thi i o becaue the ideal reonant circuit (tank) ha an infinite imedance at ω + at ϖ. () Note alo from Fig. that the amlifier inut imedance hould be high in order not to hunt the feedback network. et u now find the Barkhauen condition for the generic model [ee Fig. (b)] for high-frequency ocillator.

2 ecture : Introduction to electronic analog circuit A " β ( ) + A ε ε v" + v ( o ε ε vε + A + ) + ( + + ( + ) o ). () Amlifier ina a oa Feedback network eactive voltage divider At ω ω, (a) + + Ideal voltage amlifier Tank o oa for j ϖ,, and. () A at ω ϖ + at ω eactive voltage divider the hae condition i met, and () et the amlitude condition A A β ( ) A A ( + ). (4) Ttranmiion at ω /( + ) (b) Fig.. Generic mall-ignal equivalent circuit for ocillator. One can ee from (4) that the amlifier gain A hould be oitive if the and imedance are of different tye, and A hould be negative if the and imedance are of the ame tye. To illutrate thi intuitively recall that at ω the tank ha an infinite imedance, and therefore no current i flowing in o. Thu, the current flowing in the tank doe not leave it: the ame current flow through all the tank imedance. Note however that thi current flow in the ooite direction through the and imedance. Therefore, if thee imedance are of different tye the voltage dro on them are of the ame ign [ee Fig. (b)] and the tranmiion of the tank i oitive. Would and be of the ame tye, the voltage dro on them would be of different ign, and the tranmiion of the tank would be negative. Note from () that for o, the amlitude condition can be atified for any frequency. Therefore, oa hould not equal zero if we want and we do want our circuit to ocillate at a ingle frequency. Will claify the ocillator with two inductance and two caacitor in the tank a the Hartley and olitt ocillator, correondingly.

3 ecture : Introduction to electronic analog circuit Examle circuit V B ϖ ϖ + V B B g r m o. (E) B V E B v o I V T V A + V I E VA V >> VE VT 5mV 4 ina >>X o B v be h ie g m v be ro v o X Fig.. Examle circuit: a olitt ocillator.

4 ecture : Introduction to electronic analog circuit Quartz crytal ocillator Skin effect The tyical value of the quality factor of reonant circuit at high frequencie i. It cannot be increaed ignificantly either by increaing frequency or by increaing the diameter of the wire of the inductor becaue of the kin effect. At frequencie tarting from about khz, the current ditribution within the wire become inhomogeneou (ee Fig. 4): the eddy current decreae the current denity at the wire center and increae it at the edge of the wire cro ection. A a reult, the effective cro-ectional area of the wire decreae with the quare root of frequency and the wire reitance increae. It can be hown that the quality factor of a arallel reonant circuit, Q /(ω ), equal to that of the inductor Q ω / (ee Fig. ). Therefore, increaing ω doe not increae Q becaue ' increae with ω. Having a limited Q, we have limited frequency tability. If we want to imrove it, we have to do omething with the inductor. The effective erie reitance (ES) of caacitor i relatively mall and can be neglected. Fig. 4. Skin effect. D current: no kin effect I dc H dc A current: kin effect I eddy I ac H ac I eddy Piezoelectric effect We will relace the inductor with a grain of and, or which i the ame with a quartz crytal. When a quartz crytal i deformed by an external mechanical tre (ee Fig. 5), electric charge aear on the crytal urface (think of a quartz ga lighter!). Thi i called the direct iezoelectric effect, and the material that exhibit it are called iezoelectric material. onverely, an external electric field caue the train, which change the crytal dimenion. Thi i called the revere iezoelectric effect. Suoe now that we connect for a hort time a dc voltage ource to a quartz crytal having electrode on it ooite face (ee Fig. 6). The alied voltage will tre the crytal, and after removing it, the crytal like a tuning fork after triking it will generate mechanical ocillation at a elfreonant frequency. Due to the alternating mechanical tre within the crytal we will ee inuoidal voltage between the crytal electrode. Thi voltage will decay due to mechanical loe in the crytal. It can be hown that the time t.9-. that it take to the enveloe of the crytal voltage to decreae from the level of 9% to % of it maximum value i roortional, for a given reonant frequency, to the crytal quality factor (ee the equation in Fig. 6). Thinking of a -khz tuning fork, t.9-. can eaily be econd, which correond to a quality factor of thouand. The tyical quality factor of quartz crytal i even much higher; it can aroach 5. Fig. 5. Piezoelectric effect.

5 ecture : Introduction to electronic analog circuit To undertand how an inductor with a low Q can be relaced by a quartz crytal with a high Q XTA, let u develo it equivalent electric circuit. Since a quartz crytal i a aive ytem with no internal energy ource, excet thoe that are related to the initial condition (initial tre), we have to relace it with a aive electric circuit. Thi circuit hould be able to roduce decaying ocillation in reone to initial condition and hould alo have, like a quartz crytal, a very high dc reitance. A erie reonant circuit (ee Fig. 6) erfectly uit thee condition. Thi circuit include the erie reonance caacitance, the inductance, and the reitance ' that rereent uch mechanical roertie of the crytal a the elaticity, ma, and energy loe, reectively. The arallel reonance caacitance rereent the crytal electric caacitance (meaured between the crytal electrode). We choe uch,,,, and that rovide u with the ame natural reone of the equivalent circuit a the ecific quartz crytal have. To find qualitatively the frequency deendence of the quartz crytal imedance, we find for it equivalent circuit the frequencie of erie (zero imedance; rove thi!) and arallel (infinite imedance) reonance, neglecting (for a very high Q XTA /jω, and are very mall, uch that, << ): Metal cae VmV v XTA (t) VmV Quartz crytal ~.5 mm v XTA (t) π f t.9. Q. P t ~ mm ϖ XTA ϖ + +. (5) VmV v (t)v XTA (t) Fig. 6. Quartz crytal, it natural reone, and equivalent electric circuit. ϖ + << ϖ XTA (jω) Fig. 7 how the quartz crytal imedance a a function of frequency. One can ee that for ω <<ω <<ω, it i inductive. So we have an equivalent of inductance but with much le loe and a much higher quality factor. It i intereting that trying to build a recie ocillator (clock), we ue a tiny art of the ancient and clock. jω XTA Inductor ω ω ω ω aacitor Fig. 7. Imedance of the quartz crytal a a function of frequency.

6 ecture : Introduction to electronic analog circuit Pierce ocillator et u now build a quartz crytal ocillator. We will ue a generic aroach hown in Fig., chooing the olitt configuration becaue it comrie only a ingle inductor that we have to relace with the quartz crytal. We will alo ue a MOS NOT logic gate intead of an analog amlifier. (We are actually building a clock circuit for an electronic watch, communication device, comuter, etc. Such device are uually built by MOS technology.) You will tudy logic gate in a ecial coure, meanwhile it i enough for u to define the tranfer characteritic of a NOT gate (ee Fig. 8) Fig. 8 how that the NOT gate imly tranlate the low and high voltage (binary zero and binary one) at it inut into the high and low voltage (binary one and binary zero), reectively, at it outut. Note alo that the inut imedance of a NOT gate i very high and it outut imedance i very low. In general, it tranfer characteritic i nonlinear; however, there i a linear tranition region. We will ue thi region with a high negative loe to amlify (tranlate) relatively mall analog ignal. To do thi we have to firt define the oerating oint in the middle art of the tranition region. A reitor f connected between the inut and outut of the NOT gate hel u to do thi eaily: no current flow through thi reitor in the tatic tate, hence, no voltage dro acro the reitor and V IN V O. The mall-ignal voltage gain A of the NOT gate equal the loe of it tranfer characteritic at the oerating oint Q. We now can connect to the NOT gate the olitt tank with the quartz crytal intead of the inductor, not forgetting to connect between the gate outut and the tank the o reitor in order not to let it low outut imedance to hunt the tank and reduce the tank quality factor. Note a well that f hould alo have a high enough value in order not to hunt the tank. We divide f in Fig. 8 by +A to find the imedance een by relative to the ground, becaue the voltage dro acro f i (+A ) V for a V unit voltage dro acro. eflecting f to the ground reduce the voltage dro acro it to V. Therefore, we have to decreae the f value to comenate for the voltage decreae acro it and kee it current the ame. The obtained ocillator i called after George Wahington Pierce who wa the firt to ue a quartz crytal to tabilize more reciely the frequency of ocillator (atented in 9, baed on lam, not on NOT gate!). Self reonant frequency of quartz crytal i high, above khz, thi i why Pierce left a niche for Hewlett. >> X +A v O f V H V O f A V O V IN f A Q V IN I A v in o V in V O o v O jω XTA v o NOT gate in the tatic tate Fig. 8. Pierce ocillator baed on a MOS NOT logic gate. HOME WOK Prove that Q /(ω )Q ω /. EFEENES [] S. Sedra, K.. Smith, Microelectronic ircuit, 4th ed. New York: Oxford Univerity Pre, 998. V in