PLL Sub System2 Phase-Lock Loop The loop Filter Transfer Function Homogenous solution for Second order differential system Example: RLC: Q factor, critical resistance, Natural frequencies, Damping factor and Ratio PLL: Pull-in range, Pull-in Time, Lock Range, Lock Time Phase Frequency Detector (PFD) Targil4: PLL Design.

The loop Filter The PLL loop filter values will determine: lock time and locking range, pull-in range and pull-in time. proper loop filter values will guarantee non oscillating VCO controlled voltage and respectively will prevent VCO frequency wander, also it will affect the loop immunity to disturbances. Pull-in range: +-  W p = range of input frequencies which the PLL can lock to. e.g.; if Fc=10MHz and  W=1MHz, the pll can lock to input frequencies of 9-11MHz. Pull-in Time: Tp, The time it takes for the loop to lock Lock Range:  W L, once in lock state and protuberated by abrupt changes or variation at input frequencies, we define the range of input frequencies within it the loop will remain locked. If the abrupt frequency changes over interval time step t is smaller than the natural frequency, the pll remain locked.  W l / t < W N 2

XOR PLL loop Filter  =W*t, H(s) closed loop = H(s) open loop/(1+ H(s) open loop) The Phase transfer function of the xor loop pll is: H(s,  ), open loop = H(s,freq)/s, open loop= Kpd*Kf*Kvco /s H(s), closed loop =  (clk)/  (data, clkref) = Kpd*Kf*Kvco/(s+  *Kpd*Kf*Kvco)  =1/N, Kfilter = 1/(1+sRC)  H(s) = Kpd*Kvco*(1/1+sRC)/(s+Kpd*Kvco*1/N *(1+sRC) But.. This equation is of second order system type: H(s)= Kpd*Kvco/RC / (s 2 + s/RC + Kpd*Kvco/NRC) = Fclk/Fref Let’s c more common second order systems, e.g.; we start with un damped spring mass system: Here we will probe damped second order system: Finally we will examine electrical second order system: serial RLC

Equation of Motion : Natural Frequency The following Figure shows a simple un damped spring-mass system, which is assumed to move only along the vertical direction. It has one degree of freedom (DOF), because its motion is described by a single coordinate x. When placed into motion, oscillation will take place at the natural frequency fn which is a property of the system. We now examine some of the basic concepts associated with the free vibration of systems with one degree of freedom.

Equation of Motion : Natural Frequency Newton's second law is the first basis for examining the motion of the system. As shown the deformation of the spring in the static equilibrium position is , and the spring force k  is equal to the gravitational force w acting on mass m By measuring the displacement x from the static equilibrium position, the forces acting on m are k(  +x) and w We now apply Newton's second law of motion to the mass m : But K  = W  By defining the circular frequency Wn by the equation:

Equation of Motion : Natural Frequency and we conclude that the motion is harmonic. The motion Equation is a homogeneous second order linear differential equation, has the following general solution : where A and B are the two necessary constants. These constants are evaluated from initial conditions x(0) The natural period of the oscillation is established from: Wn*T=2pI Note: f, Wn and T, depend only on the mass and stiffness of the system, which are properties of the system

Equation of Motion : Damped Second Order System The homogenous solution of the damped second order system is: Without damping (b=0) the natural frequency is (Wn) the square of the ‘spring constant /mass’, thus c=Wn  and we end up with previous system: Critical damping occur when b=2Wn, thus damping ratio is the ratio of b to critical damping: If  =1, critical damping occurs and the characteristic polynomial is: (s+Wn) 2, In other cases it is (3):

Equation of Motion : Damped Second Order System And the roots are (3) : With real, equal or non real roots depending on |  >=<1 For |  |<1 the complex roots are: Suppose we want to discover Wn and  by measurement (simulation..) then we look at the under damped mass movement (general solution) (2) : Thus the damped resonant frequency will tell us  or Wn, note how it gets lower with it.

Thus the damped resonant frequency will tell us  or Wn, If we measure the (logarithmic) difference between two successive minima/maximas  we can derive the other value. The maximas/minimas occur when the motion equation derivative is zero. This pi/Wd or if t1 and t2 corresponds to successive maxima We can also take the respective X1/X2 as we know the displacements at times differ by pi/Wd are (eq 2) :

Serial RLC The coil can be considered as a RLC resonant circuit. A very low frequencies, the inductor is a short circuit, and the capacitor open circuits (Figure left). This means that the voltage at node C is equal to A if no load is connected to node C. At very high frequencies, the inductor is an open circuit, the capacitor a short circuit (Figure right). Consequently, the link between C and A tends to an open circuit. At a very specific frequency the LC circuit features a resonance effect. The theoretical formulation of this frequency is given by:.

Serial RLC Ri(t) + Ldi(t)/dt +1/C *( intg| 0 t (i(t)d(t) ) + Vo = E(t), RI(s) + LsI(s)-Li 0 + I(s)/Cs + V 0 /s = E(s) I(s) = E(s)/(R+Ls+1/Cs) + (Li 0 – V o /s)/(R+Ls+1/Cs) The first term is the zero state response and the second is the zero input response. Both characteristics will be derived from: (R+Ls+1/Cs). If we are interested in the voltage Vout (t) then since i(t) = Cdv/dt  I(s)= Cs*V(s) we conclude: V(s) = I(s)/Cs = E(s)/(RCs+LCs 2 +1) for the zero state response component. To determine the inverse transform we need to find and examine its poles which are the roots or homogeneous solution of second order differential equation: RCs+LCs 2 +1 = 0

Serial RLC: Extracted Parameters The roots or homogeneous solution of second order differential equation: LCs 2 +RCs+1 = 0--  s 2 +Rs/L+1/LC = 0 --  b= R/L, c= 1/LC Natural frequency, Wn= 1/sqrt(LC), Damping,  (2  = b)   = R/2L Damping ratio,  = b/2Wn =  /Wn = R*sqrt(LC)/2L= R*sqrt(C/L)/2= R/Rc Critical resistance, Rc=2sqrt (L/C) (note: what is Q-factor, Q= Wn/(2  ) Resonant damped frequency, Wd = Wn *sqrt(1-  2 )

Class Targil 1 Design serial RLC circuit be used in 3GHz resonant frequency: sim, extract and compare with theoretical the following second order system parameters : Natural frequency (Wn) Quality factor (Q) Damped frequency (Wd) Damping factor (‘  ’), Damping ratio (‘  ’), Critical resistance (Rc).  Note that for some parameters u will need to run AC analysis while for the others use stimulus (or initial conditions) and perform Transient analysis  Compare with calculated values

H(s)= Kpd*Kvco/RC / (s 2 + s/RC + Kpd*Kvco/NRC) s 2 + s/RC + Kpd*Kvco/NRC = 0 Natural frequency, Wn= sqrt (Kpd*Kvco/NRC) Damping ration,  = 1/(2RC*Wn) = 0.5* sqrt (N/Kpd*Kvco*RC) Resonant frequency, Wd = Wn *sqrt(1-  2 ) Pull in range,  Wp= 0.5Pi * sqrt (2  Wn*KpdKvco-Wn 2 ) = (zero-poll..) Pull in Time, Tp= 4 *  W 2,(from center)/ (Pi 2 *Wn 3 ) Lock Range,  W L =Pi  Wn=Pi/(2RC) Lock Time, T L = 2Pi/Wn XOR PLL, Loop Filter Parameters

 Class Targil 2 Use your designed PLL circuit to sim and extract the following PLL parameters : Natural frequency (Wn) Damped frequency (Wd) Damping factor (‘  ’), Damping ratio (‘  ’), Pull in range Pull in time Lock range Lock time  Note that for some parameters u will need to run AC analysis while for the others use stimulus (or initial conditions) and perform Transient analysis  Compare with calculated values

More on PLL  Phase Frequency Detector (PFD)  The Charge pump principles and Realizations  The loop filter realizations  Targil3: PLL Design Cont