Analysis and Design of a Third Order Phase-Lock Loop

Transcription

1 Analyi Deign of a Third Order Phae-Lock Loop DANIEL Y. ABRAMOVITCH Ford Aeropace Corporation 3939 Fabian Way, MS: X- Palo Alto, CA Abtract Typical implementation of a phae-lock loop (PLL) are econd order. In thi paper we examine the advantage diadvantage of a third order phae-lock loop. Among the advantage i more deign freedom which can reult in uperior noie rejection lower teady-tate error than a econd order PLL. Chief among the diadvantage of the third order PLL i the difficulty of analyzing it tability in the region of nonlinear operation. In thi paper we will treat the third order PLL a a nonlinear control ytem: firt examining the mall ignal (linear) operation then extending the analyi to the nonlinear region. A ueful et of tool from the nonlinear control ytem world, the econd method of Lyapunov LaSalle Theorem, will be ued to derive tability condition for the nonlinear model. I. INTRODUCTION θ(t) Σ ψ(t) in( ) K u(t) p K( +b +b 0) + +a φ(t) Fig.. y(t) Σ + + γ(t) x(t) Third Order Phae-Lock Loop Block Diagram Typical implementation of a phae-lock loop (PLL) are econd order [], [], [3], [4]. In thi paper we examine the advantage diadvantage of a third order phaelock loop. Among the advantage i more deign freedom which can reult in uperior noie rejection lower teadytate error than a econd order PLL. Chief among the diadvantage of the third order PLL i the difficulty of analyzing it tability in the region of nonlinear operation. The differential equation for the third order phae-lock loop in Figure are : dφ y(t), () ψ(t) θ(t) φ(t), () u(t) in ψ(t), (3) d [ x + a dx d ] u K + b du + b 0u(t) (4) Equation () repreent the VCO, () define the error function, (3) i the mixer, (4) define the filter. Here θ(t) γ(t) are external input to the ytem where θ(t) i the input phae, γ(t) i the VCO noie, ψ(t) i the phae error, φ(t) i the ytem output. The other variable, u(t), x(t), y(t) are merely internal variable. Define q dy. Then ome tediou, but traightforward algebra lead to the following ytem of differential equation: dψ dy dq y(t)+ q(t) (6) K b 0 in ψ(t) {( K y ) } in ψ(t)+b co ψ(t) y(t) {a + K co ψ(t)} q(t) { d ( ) θ +K co ψ(t) in ψ(t) } + b co ψ(t) + d γ + a dγ. (7) If we aume that the ytem i initially quiecent, i.e., ψ(0 )y(0 )q(0 )x(0 )0, that neither θ(t) nor γ(t) are impulive, then the initial condition can be derived in a tediou but traightforward fahion. Thee are: ψ( ) θ( )θ 0, (8) x( ) K in θ 0, (9) y( ) γ( )+K in θ 0, (0) The original derivation of the differential equation for thi third order loop came from Dr. John Y. Huang, of Ford Aeropace. (5)

2 q( ) (b a )K in θ dγ +K co θ 0 K co θ 0 (γ( )+K in θ 0 ).() II. LINEAR ANALYSIS θ(t) Σ ψ(t) K u(t) p K( +b +b 0) + +a φ(t) Fig.. θ() Σ + y(t) Σ + + γ(t) x(t) Linearized Third Order Phae-Lock Loop Block Diagram ψ() K( +b +b 0) +a } {{ } G Fig. 3. γ() + x() Σ + y() }{{} H PLL a a Linear Control Sytem φ() The only nonlinear equation above i Equation (7). In order to linearize thi a mall angle aumption i made, i.e., ψ(t) mall in ψ(t) ψ(t), () co ψ(t), (3) ( ) dψ ψ(t) lowly varying 0; where (4) ( ) dψ ( y + ) K vy y + ( ). (5) Subtituting thee approximation in (7) yield: dq K b 0 ψ(t) K b y(t) (a + K )q(t) { d } θ +K + b + d γ + a dγ. (6) The block diagram for thi i hown in Figure. The initial condition are computed uing Equation () (4), evaluated at t, in Equation (8) () to yield: ψ( ) θ 0, (7) x( ) K θ 0, (8) y( ) γ( )+K θ 0, (9) q( ) (b a )K θ dγ + { } K (γ( )+K θ 0 ) (0). A. Stability of Linear Sytem With very little effort, Figure can be redrawn in the form of a feedback control ytem a hown in Figure 3. A reult from linear ytem theory i that the ytem hown in Figure 3 will be both internally externally table if the tranfer function H ψθ, H ψγ, H yθ, H yγ are all table[5], where H ψθ () +GH 3 + a 3 + a + K ( + b + b 0 ), () H ψγ H yθ H yγ H (3) +GH ( + a ) 3 + a + K ( + b + b 0 ), (4) G (5) +GH K( + b + b 0 ) 3 + a + K ( + b + b 0 ), (6) +GH H ψθ. (7) Thu, providing there are no untable pole-zero cancellation[6], internal external tability of the linearized PLL depend upon the root of α() 3 + a + K ( + b + b 0 ), (8) having negative real part. Routh tability criterion[7] lead to the following condition on (8): a + K > 0, (9) K b > 0, (30) K b 0 > 0, (3) [(a + K )b b 0 ] K > 0. (3) Typically, the gain are choen o that K > 0, thu reducing the tability condition to: a + K > 0, (33) b > 0, (34) b 0 > 0, (35) (a + K )b > b 0. (36) Condition (34) (35) require that the open loop filter be minimum phae. Condition (33) (36) are eentially condition on the gain parameter,, K,.

3 III. NONLINEAR ANALYSIS The econd method of Lyapunov[8] i commonly ued in tability analyi of nonlinear differential equation becaue it doe not require the olution to the differential equation. A very intuitive dicuion of thi can be found in Ogata, [9]. The econd method of Lyapunov i baed on the generalized energy in the ytem. If an energy function of the ytem tate i found which i contantly decreaing, then the ytem i aymptotically table. A general form of a vector differential equation i: ẋ f(x, t) where x, ẋ R n. (37) An equilibrium tate i any tate uch that f(x e,t)0. (38) Uually, a tranformation i made o that the origin of tate pace i an equilibrium tate, i.e., f(0,t)0. (39) Theorem (Lyapunov Main Stability Theorem): For the ytem defined by Equation 37, uppoe there exit a poitive definite calar function of x, V (x), i.e., V (x) > 0 x 0 V (x) 0 x 0, (40) uch that V (x) i negative definite, i.e., V (x) < 0 x 0 (4) V (x) 0 x 0. Then (37) i globally aymptotically table.[8] Theorem (LaSalle Theorem): For the ytem defined by Equation 37, uppoe there exit a poitive definite calar function of x, V (x), uch that V (x) i negative emi-definite, i.e., V (x) 0 x 0 (4) V (x) 0 x 0. Suppoe alo that the only olution of ẋ f(x, t), V (x) 0 i x(t) 0for all t 0. Then ẋ f(x, t) i globally aymptotically table. A. Stability of Nonlinear Sytem 0 Σ ψ in( ) u + Fig. 4. z K( +b +b 0) +a Cloed-Loop Nonlinear Sytem A generalized energy function i any poitive definite function of the ytem tate which i nonvanihing for any tate 0. φ z Fig. 5. KvK Kb 0 Kb y Σ Drawing out the tate variable Stability analyi i done for the homogeneou (no input) equation, therefore it i convenient to redraw Figure a hown in Figure 4. Furthermore, drawing the map from z to φ a hown in Figure 5 will be ueful. Note that the internal tate variable, y, i defined differently from above becaue of the regrouping of the integrator. In order to prove the tability of the nonlinear model, the econd method of Lyapunov [8] will be ued. A Lyapunov function of the kind decribed by LaSalle Lefchetz [0] will be ued. Finally, LaSalle Theorem will be invoked proving tability for ψ ( π, π). The tate equation correponding to Figure 4 5 are: ż in ψ a z (43) ẏ Kb 0 z (44) φ Kż + Kb z + y (45) K in ψ + K(b a )z + y,(46) ψ φ (47) K in ψ K(b a )z y. (48) Chooe V ψ 0 in(σ)dσ + [ z z y P y where p p P p p i a ymmetric, poitive definite matrix. Then V in(ψ) ψ + [ z y ] [ p p p p φ ][ ż ẏ ], (49) ] (50) (5) in(ψ) ψ + p zż + p yż + p zẏ + p yẏ (5) inψ [ K in ψ K(b a )z + y] +p z [ in ψ a z]+p y [ in ψ a z] +p zkb 0 z + p ykb 0 z (53) V in ψ [ K ]+z [p Kb 0 p a ] +inψ {z (p K(b a )) +y (p )} + yz [p Kb 0 p a ]. (54) 3

4 In order to invoke LaSalle Theorem, we mut have V (ψ, y, z) 0 with V 0 ψ y z 0 V 0. Auming ψ ( π, π) the condition for V 0 are thoe that guarantee that P i poitive definite: p > 0 (55) p p > (p ), which lead to (56) p > 0. (57) (58) The condition that guarantee V 0 are: K > 0, (59) p Kb 0 p a < 0, (60) p K(b a ) 0, (6) p 0, (6) p Kb 0 p a 0. (63) Condition 6 6 imply p (64) p K (b a ), (65) repectively. From Condition 59 we get K > 0 K Kp uing Condition 55 yield > 0 K > 0, (66) b >a. (67) From Condition 63 Equation 64 we get p a. Kb 0 (68) Condition 60 implie K b 0 K (b a )a < 0 or (69) K [b 0 (b a )a ] < 0. (70) Equation 66, K > 0, (7) implie b 0 (b a ) a < 0. }{{} (7) >0 Condition 57 require that p a > 0, (73) Kb 0 which can be aured uing (66) by auming that b 0 a have the ame ign. It i convenient to chooe both b 0 a > 0. Under the above aumption, Condition 56 i completely equivalent to Condition 60. In ummary, chooe, K,, a, b 0, b > 0. Then chooe b > a (74) b 0 (b a )a < 0. (75) Then the above choice of p p P p p implie (76) V in ψ [ K ] +z Kv K (b 0 (b a )a ) (77) 0. (78) Finally, the only place that V (43) (48) can vanih i for z y φ ψ 0, o uing LaSalle Theorem prove tability. IV. CONCLUSIONS Lyapunov tability technique are adequate for analyzing the tability of a third order phae-lock loop i not ubtantially more difficult than that of a econd order loop. We have developed both a linear a nonlinear model of a third order phae-lock loop have developed tability condition for both. It goe without aying that it hould be poible to apply thee ame technique to all order of phae-lock loop, a will be hown in []. V. ACKNOWLEDGEMENTS The author would like to thank Daniel Witmer of Ford Aeropace for introducing the phae-lock loop to him a a nonlinear feedback ytem. The author would further like to thank Profeor Gene Franklin Stephen Boyd of Stanford Univerity Dr. Michael Workman of IBM Corporation for introducing him to the technique that were ued to analyze thi problem. Alo, the author would like to thank Profeor Thoma Cover of Stanford Univerity for treing the importance of looking for unifying principle in all of one work. REFERENCES [] A. Blanchard, Phae-Locked Loop. New York, NY: John Wiley & Son, 976. [] F. M. Gardner, Phaelock Technique. New York, NY: John Wiley & Son, econd ed., 979. ISBN [3] S. C. Gupta, Phae-locked loop, Proceeding of the IEEE, vol. 63, pp , February 975. [4] W. C. Lindey C. M. Chie, A urvey of digital phae-locked loop, Proceeding of the IEEE, vol. 69, pp , April 98. [5] M. Vidyaagar, Control Sytem Synthei: A Factorization Approach. Cambridge, Maachuett: The MIT Pre, 985. [6] T. Kailath, Linear Sytem. Englewood Cliff, N.J. 0763: Prentice- Hall, 980. [7] G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Sytem. Menlo Park, California: Addion-Weley, firt ed., 986. [8] R. E. Kalman J. E. Bertram, Control ytem analyi deign via the Second Method of Lyapunov, Part : Continuou-Time Sytem, Tranaction of the ASME,

5 [9] K. Ogata, Modern Control Engineering. Prentice-Hall Intrumentation Control Serie, Englewood Cliff, New Jerey: Prentice-Hall, 970. [0] J. LaSalle S. Lefchetz, Stability by Liapunov Direct Method. New York, N.Y.: Academic Pre, 96. [] D. Y. Abramovitch, Lyapunov redeign of analog phae-lock loop, Submitted to The IEEE Tranaction on Communication,