1 Basic Equations of the PLLs

Transcription

1 1 Baic Equation of the PLL 1.1 INTRODUCTION Phae lock loop (PLL) belong to a larger et of regulation ytem. A an independent reearch and deign field it tarted in the 1950 [1] and gained major practical application in cochannel TV. On thi occaion, we find one of the firt fundamental paper [2]. Some 15 year later, we encounter a urveying book by Gardner [3], till mentioned and ued. Since a dozen book were publihed on the topic of PLL problem proper [4] and in connection with frequency ynthei, we would find chapter on PLL in all of the relevant book. Here we hall only mention ome of them [5 8]. But the importance of the topic i tetified by the publication of new book on PLL (e.g., [9 12]) and a wealth of journal article, the important one of which will be cited at the relevant place. A major advantage of modern PLL i the poibility of a widepread ue of off-the-helf IC chip. Their application reult in low-volume, low-weight, and often power-aving device. At the ame time we alo appreciate hort witching time and very high-frequency reolution. We hall find PLL in communication equipment, particularly, in mobile application in low-gigahertz range, in computer, and o on, where we appreciate hort witching time and very high-frequency reolution. However, there are hortcoming too: the limited range for high frequencie (today commercial divider hardly exceed the 5 GHz bound and only laboratory device work in higher range). In the following paragraph we ummarize the baic propertie of PLL with ome deign-leading idea and repeat all the major feature and ue terminology introduced year ago by mechanical engineer [13] and alo ued by Gardner [3] and many other. 1.2 BASIC EQUATIONS OF THE PLL The tak of the PLL i to maintain coherence between the input (reference) ignal frequency, f i, and the repective output frequency, f o, via phae comparion. Another Phae Lock Loop and Frequency Synthei V.F. Kroupa 2003 John Wiley & Son, Ltd ISBN:

2 2 BASIC EQUATIONS OF THE PLL feature of PLL i the filtering property, particularly with repect to the noie where it behavior recall a very narrow low-pa arrangement that i not to be realized by other mean. The theory wa explained in many textbook a we have mentioned in the previou ection. Each PLL ytem i compoed of four baic part: 1. the reference generator (RG) 2. the phae detector (PD) 3. the low-pa filter F L (f ) (in higher-order ytem) 4. the voltage-controlled ocillator (VCO) and work a a feedback ytem hown in Fig Without any lo of generality, we may aume that input and output ignal are harmonic voltage with additional phae modulation v i (t) = V i in[ω i t + φ i (t)] V i in i (t) (1.1) where φ i (t) and φ o (t) are lowly varying quantitie. v o (t) = V o co[ω o t + φ o (t)] V o co o (t) (1.2) Later we hall prove that realization of the phae lock require that input and output voltage mut be in quadrature, that i, mutually hifted by π/2. Phae detector (PD) i a nonlinear element of a different deign and contruction (we hall deal with PD later in Chapter 8, Section 8.4). For the preent dicuion, we aume that the PD i a imple multiplier. In thi cae the correponding output voltage will be v d (t) = K m v i (t) v o (t) (1.3) where K m i the tranfer contant with the dimenion [1/V]. After introduction of eq. (1.1) and (1.2) in the above relation, we get v d (t) = K m V i V o in i (t) co o (t) = 1 2 K mv i V o [in[(ω i ω o )t + φ i (t) φ o (t)] (1.4) + in[(ω i + ω o )t + φ i (t) + φ o (t)]] v i (t) v d (t) v 2 (t) v o (t) RG PD F L VCO Output v o (t) Figure 1.1 Baic feedback network of PLL.

3 BASIC EQUATIONS OF THE PLL 3 In the implet cae we hall aume that the low-pa filter remove the upper ideband with the frequency ω i + ω o but leave the lower ideband ω i ω o without change. Evidently the VCO tuning voltage will be v 2 (t) = K d in[(ω i ω o )t + φ i (t) φ o (t)] K d in e (t) (1.5) where we have introduced the o-called PD gain K d = K m V i V o of dimenion [V/rad]. Note that the phae difference between the input and the output voltage i e (t) = i (t) o (t) (1.6) Voltage v 2 (t) will change the free running frequency ω c of the VCO to o (t) = ω c + K o v 2 (t) (1.7) where the proportionality contant K o i deignated a the ocillator gain with the dimenion [2π Hz/V]. After integration of the above equation and introduction into relation (1.6), we get for the phae difference e (t) e (t) = i (t) ω c t K o v 2 (t) dt (1.8) which can be rearranged a follow: e (t) = ω i t ω c t K o K d in e (t) dt (1.9) and differentiation reveal d e (t) dt = ω K in e (t) (1.10) where we have introduced ω i ω o = ω and K d K o = K. Note that K i indicated a the gain of the PLL with the dimenion [2πHz]. The concluion that follow from the foregoing dicuion i that the phae lock arrangement i decribed with a nonlinear eq. (1.10), the olution of which for arbitrary value ω and K i not known. With certainty we can tate that for ω/k 1 an aperiodic olution doe not exit. Thi concluion tetifie the phae plane arrangement (Fig. 1.2). Without an aperiodic olution, the feedback ytem in Fig. 1.1 cannot reach the phae tability, that i, the output frequency of the VCO, ω o, will never be equal to the reference frequency ω i. However, the DC component in the teering voltage v 2 (t) reduce the original difference between frequencie ω i ω c > ω i ω o (1.11)

4 4 BASIC EQUATIONS OF THE PLL 3 2 ψ e (t) K 1 w K = x ψ e (t) 2x Figure 1.2 Plot of relation (1.10) in the phae plane for different ratio ω/k. 1.3 SOLUTION OF THE BASIC PLL EQUATION IN THE TIME DOMAIN To arrive at the olution we have to introduce ome implification. Neverthele, we gain more inight into the problem Solution in the Cloed Form In the cae where ω/k 1, the differential eq. (1.10) ha a olution after application and eparation of variable. d e (t) = dt (1.12) ω K in e (t) With the aitance of table [14, p. 804], we get a rather complicated cloed-form olution t t 0 = [ ( 2 ω + K π ( ω)2 K arctan 2 ω K tan 4 ) ] e (1.13) 2 where t 0 i a not-yet-defined integration contant. A long a K> ω, the rh will be imaginary and with the aitance tan( jx) = j tanh(x) (1.14)

5 we arrive at t t 0 = SOLUTION OF THE BASIC PLL EQUATION IN THE TIME DOMAIN 5 [ ( 2 K + ω π K2 ( ω) arctanh 2 K ω tan 4 ) ] e 2 1 = K2 ( ω) ln 1 + (K + ω)/(k ω) tan(π/4 e /2) 2 1 (K + ω)/(k ω) tan(π/4 e /2) (1.15) and after computing tan(π/4 e /2) the ought olution i [ K ω e = 2arctan K + ω 1 exp[ ] K2 ( ω) 2 (t t 0 )] 1 + exp[ K 2 ( ω) 2 (t t 0 )] + π (1.16) 2 For the teady tate, that i, for t, the lh of eq. (1.10) equal zero, with the reult e = arcin ω (1.17) K Linearized Solution From the preceding analyi we conclude that the olution of the repective differential equation, in the cloed form, i very complicated even for a very imple PLL arrangement. Conequently, we may uppoe that for more ophiticated PLL ytem it would be practically impoible. However, the ituation need not be o gloomy after the introduction of implification that are not far from reality. In the firt tep we find that the time-dependent phae difference e (t) at the output of the PD in the cloed PLL i mall and prone to the implification in e (t) e (t) (1.18) Thi aumption i upported with the reality that a lot of PD are linear or nearly linear in the working range (ee dicuion in Chapter 8). In uch a cae, the introduction of (1.18) into (1.10) reult in the following implification: d e (t) = ω K e (t) (1.19) dt Solution of thi differential equation i eay, ( e (t) = e Kt e0 ω ) + ω (1.20) K K where e0 i the integration contant, that i, the phae at the tart for t = 0. Further invetigation reveal that the phae difference in the teady tate compenate the frequency difference (cf. (1.17)). e = ω (1.21) K

6 6 BASIC EQUATIONS OF THE PLL 1.4 SOLUTION OF BASIC PLL EQUATIONS IN THE FREQUENCY DOMAIN By auming the phae difference e (t), in the locked tate, to be alway maller than π/2, the reult i the equality between input and output frequencie ω i = ω o (1.22) In other word the PLL ytem i permanently in the phae equilibrium. The ituation being uch, we can rearrange relation (1.7) to ω o + φ o (t) = ω c + K o v 2o + K d K o in[φ i (t) φ o (t)] (1.23) where the term K o v 2o hift the VCO frequency ω o to be equal to the input frequency ω i (of (1.22)). Evidently, in the teady tate we get the following relation between the VCO free running frequency and the locked frequency Combination with (1.23) reveal where K = K d K o. In the teady tate the difference φ e ω o = ω c + K o v 2o (1.24) φ o (t) = K in[φ i (t) φ o (t)] (1.25) φ e (t) = φ i (t) φ o (t) (1.26) i generally mall. Conequently, we may apply the following linearization φ o (t) = K[φ i (t) φ o (t)] (1.27) and employ advantage of the Laplace tranform (with a tacit aumption of the zero initial condition) o () = K[ i () o ()] (1.28) After rearrangement we arrive at the baic PLL tranfer function o () i () = H() = K + K (1.29) or at i () o () i () = e() i () = 1 H() = + K (1.30) between input and PD output error.

7 ORDER AND TYPE OF PLL ORDER AND TYPE OF PLL The PLL ytem decribed with relation (1.29) and (1.30) i indicated a PLL of the firt order ince the polynomial in the denominator i of the firt order in (K being a contant). However, generally PLL are much more complicated. To get better inight into the PLL propertie, we hall implify, without any lo of generality, the block diagram to that hown in Fig. 1.3 and introduce the Laplace tranfer function of the individual building circuit, uitable for invetigation of the mall ignal propertie. Invetigation of the above figure reveal that the input phae ϕ i (t) i compared with the output phae ϕ o (t) in the phae detector (ring modulator, ampling circuit, etc.). At it output we get a voltage, v d (t), proportional to the phae difference of the repective input ignal where v d (t) = [ϕ i (t) ϕ o (t)]k d (1.31) the proportionality factor, K d [V/rad], i called the phae detector gain. Next, v d (t) pae the loop filter, F() (a low-pa filter attenuating carrier with frequencie ω i = ω o, and ideally all undeired ideband). Note that the ueful ignal v 2 (t) i a lowly varying DC component, the output voltage of which i given by the following convolution: v 2 (t) = v d (t) h f (t) (1.32) where h f (t) i the time repone of the loop filter. After applying v 2 (t) on the frequency control element of the VCO, we get the output phae ϕ o (t) = ω o (t) dt = ω c t + K o v 2 (t) dt (1.33) with ω c being the VCO free-running frequency. The proportionality factor, K o [2π Hz/V], i deignated a the ocillator gain. Since, in mot cae, K d and K o are voltage-dependent, the general mathematical model of a PLL i a nonlinear differential equation. It linearization, jutified in mall ignal cae ( teady tate working mode), provide a good inight into the problem. After reverting to the Phae detector (PD) Loop filter (F) Voltage-controlled ocillator (VCO) Input w i ; j i (t) v d () K d f e () v d (t) v 2 (t) Output v 2 () = F()v d () f o () = K o v 2 () w o ; j o (t) f o () Figure 1.3 Simplified block diagram of the PLL with individual tranfer function.

8 8 BASIC EQUATIONS OF THE PLL Input ignal Actuating ignal K o K d F() Output ignal Feedback ignal F M () Figure 1.4 Simplified block diagram of the PLL with a tranfer function in the feedback path. whole feedback ytem (Fig. 1.4), we can write for the relation between the input and the output phae in the Laplace tranform notation [ i () o ()F M ()] K dk o F() = o () (1.34) The ratio, o ()/ i (), the PLL tranfer function, i given by H() = KF()F M () 1 + KF()F M() = G() 1 + G() (1.35) where we have introduced the forward loop gain K = K d K o and the open loop gain G() G() = KF()F M() (1.36) Order of PLL In the implet cae there are no filter in the forward or the feedback path. The PLL tranfer function implifie to H() = K (1.37) + K Thi PLL i deignated a the firt-order loop ince the larget power of in the polynomial of the denominator i of the order one. Generally, the tranfer function of the loop filter F() are given by a ratio of two polynomial in. The conequence i that the denominator in H() i of a higher order in and we peak about PLL of the econd order, third order, and o on, in accordance with the order of the repective polynomial in the denominator of (1.35).

9 ORDER AND TYPE OF PLL Type of PLL In intance in which the teady tate error are of major interet, the number of pole in the tranfer function G(), that i, the number of integrator in the loop, i of importance. In principle, every PLL ha one integrator connected with the VCO (cf. eq. (1.33)). For the phae error at the output of the PD we find e () = i () F M () o () (1.38) where o () = e () KF() (1.39) After elimination of o () from the above relation, we get for the phae error e () 1 e () = i () 1 + G() (1.40) Introducing the gain, G(), which i a ratio of two polynomial G() = A() n B() (1.41) we get for the phae error n B() e () = i () A() + n B() (1.42) and eventually with the aitance of the Laplace limit theorem, we get for the final value of the phae error ϕ e (t) [ lim [φ n+1 ] B() e(t)] = lim i () t 0 A() + n B() (1.43) Note that every PLL contain at leat one integrator, that i, VCO; conequently, n 1 (cf. relation (1.34)) Steady State Error Invetigation of the teady tate error in PLL of different order and type will proceed after introduction of the Laplace tranform of the repective input phae tep, input frequency tep, and input teady frequency change into (1.43). ω i = φ i ; ω i = φ i ; ω = ω i 2 2 = φ i 3 (1.44)

10 10 BASIC EQUATIONS OF THE PLL Phae tep After introducing the Laplace tranform of phae tep, φ/, into (1.43), we find out that the final value i zero in all PLL Frequency tep For the frequency tep, ω/, we get lim t φ e2(t) = ω i [ B(0) A(0) ] n=1 = ω i KF(0)F M (0) = ω i K v (1.45) Evidently in all PLL of the econd order, a frequency tep reult in a teady tate phae error inverely proportional to the o-called velocity error contant K v,in agreement with the terminology ued in the feedback control ytem (cf. [13]). In PLL of type 2, with two integrator in the loop, the DC gain F(0) i very large, o K v and conequently the teady tate error i negligible Frequency ramp However, the teady frequency change, ω/ 2, reult in the o-called acceleration or dynamic tracking error K a [ ] B(0) lim φ e3(t) = ω i = ω i (1.46) t A(0) n=2 K a PLL of type 3 can eliminate even the teady tate error ϕ e3 (t) for t to zero. However, PLL of thi type are encountered exceptionally, for example, in time ervice [15], in pace and atellite device [3], and o on. Note that the frequency locked loop may be conidered a 0 type PLL. 1.6 BLOCK DIAGRAM ALGEBRA Actual PLL are often much more complicated than block diagram in Fig. 1.3 or 1.4. For arriving at tranfer function, H() and 1 H(), we can apply the rule of block diagram algebra [13]. Two or more block in erie can be combined into one after multiplication of their Laplace tranform ymbol (ee Fig. 1.5a). A typical example i the addition of independent ection to the fundamental low-pa filter. In the cae where two block are in parallel, the final combination i provided with a mere addition (ee Fig. 1.5b). Invetigation of the relation (1.35) reveal that the feedback block can be put outide of the baic loop [5] H () = 1 H() (1.47) N

11 BLOCK DIAGRAM ALGEBRA 11 F 1 () F 2 () F 1 () F 2 () (a) F 1 () + + F 1 () + F 2 () F 2 () (b) + KF() + KF() N N N (c) + KF() + KF()M M M (d) f i + KF() f o f i KF() f o + M + N N f o Mf i N f o N f o Mf i + f i M Mf i (e) Figure 1.5 Simplification of the block diagram of PLL: (a) erie connection; (b) parallel connection; (c) and (d) feedback arrangement; (e) more complicated ytem. (Reproduced from Fig in C.J. Savant Jr., Baic Feedback Control Sytem Deign. New York, Toronto, London: McGraw-Hill, 1958 by permiion of McGraw Hill, 2002).

12 12 BASIC EQUATIONS OF THE PLL or H () = MH () (1.48) In thi way we arrive at the effective tranfer function, H () and 1 H (), which contain information about the PLL filtering propertie, which will be dicued later. We appreciate thi approach in intance in which a imple frequency divider or frequency multiplier i in the feedback path of the PLL. The rearrangement i reproduced in Fig. 1.5(c) and 1.5(d). Finally, we hall conider the ytem containing a mixer in the feedback path. Relation between output and input phae i [ o () = i () ] o() M i () KF() (1.49) N and rearrangement lead to the implification in accordance with Fig. 1.5(e). REFERENCES [1] W.J. Gruen, Theory of AFC ynchronization, Proc. IRE, 41, , [2] D. Richman, APC Color Sync for Televiion Synchronization, 1953, IRE Conv. Rec., Part 4. [3] M. Gardner, Phae-Lock Technique. New York: Wiley, 1966, [4] W.C. Lindey and C.M. Chie, ed Phae-Locked Loop. New York: IEEE Pre, [5] V.F. Kroupa, Frequency Synthei: Theory, Deign and Application. London: CH. Griffin, [6] V. Manaewtch, Frequency Syntheizer: Theory and Deign. 1 t ed. New York: John Wiley & Son, 1975, lat ed [7] W.F. Egan, Frequency Synthei by Phae Lock. 1981, New York: John Wiley, [8] U.L. Rohde, Digital PLL Frequency Syntheizer. Englewood Cliff, NJ: Prentice Hall, [9] J.A. Crawford, Frequency Syntheizer Deign Handbook. Boton and London: Artech Houe, [10] B.B. Razavi, ed. Monolithic Phae-Locked Loop and Clock Recovery Circuit: Theory and Deign. IEEE Pre, [11] W.F. Egan, Phae-Lock Baic. John Wiley & Son, [12] R. Bet, Phae-Locked Loop: Deign, Simulation, and Application.New York:McGraw-Hill, [13] C.J. Savant Jr., Baic Feedback Control Sytem Deign. New York, Toronto, London: McGraw-Hill, [14] G.A. Korn and T.M. Korn, Mathematical Handbook. New York: McGraw-Hill, [15] J. Tolman, The Czecholovak National Standard of Frequency and Time. Yearbook of the Academy of Science 1967, Praha: Academia, 1969, pp