Download Damped Second Order System

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
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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: +-DWp = range of input frequencies which the PLL
can lock to. e.g.; if Fc=10MHz and DW=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: DWL, 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. DWl / t < WN2
XOR PLL loop Filter
f=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, f), open loop = H(s,freq)/s, open loop= Kpd*Kf*Kvco /s
H(s), closed loop = f(clk)/f(data, clkref) =
Kpd*Kf*Kvco/(s+b*Kpd*Kf*Kvco)
b=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 / (s2 + 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 D , and the spring force kD 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(D+x) and w
We now apply Newton's second law of motion to the mass m :
But KD= 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=Wn2 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 x=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 |x|>=<1
For |x|<1 the complex roots are:
Suppose we want to discover Wn and x by
measurement (simulation..) then we look at
the under damped mass movement (general
solution) (2) :
Thus the damped resonant frequency will tell us x or Wn, note
how it gets lower with it.
Thus the damped resonant frequency will tell us x or Wn, If we
measure the (logarithmic) difference between two successive
minima/maximas D we can derive the other value. The
maximas/minimas occur when the motion equation derivative is zero.
This occurs @t= pi/Wd or if t1 and t2 corresponds to successive maxima
We can also take the respective X1/X2 ratio @t1/t2: 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|0t (i(t)d(t) ) + Vo = E(t),
RI(s) + LsI(s)-Li0 + I(s)/Cs + V0/s = E(s)
I(s) = E(s)/(R+Ls+1/Cs) + (Li0 – Vo/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+LCs2+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+LCs2+1 = 0
Serial RLC: Extracted Parameters
The roots or homogeneous solution of second order differential equation:
LCs2+RCs+1 = 0-- s2+Rs/L+1/LC = 0 -- b= R/L, c= 1/LC
Natural frequency, Wn= 1/sqrt(LC),
Damping, a (2a= b)  a= R/2L
Damping ratio, x= b/2Wn = a/Wn = R*sqrt(LC)/2L=
R*sqrt(C/L)/2= R/Rc
Critical resistance, Rc=2sqrt (L/C) (note: what is Q @R=Rc?)
Q-factor, Q= Wn/(2a)
Resonant damped frequency, Wd = Wn *sqrt(1-x2)
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 (‘a’),
• Damping ratio (‘x’),
• 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
XOR PLL , Loop Filter Parameters
H(s)= Kpd*Kvco/RC / (s2 + s/RC + Kpd*Kvco/NRC)
s2 + s/RC + Kpd*Kvco/NRC = 0
Natural frequency, Wn= sqrt (Kpd*Kvco/NRC)
Damping ration, x = 1/(2RC*Wn) = 0.5* sqrt (N/Kpd*Kvco*RC)
Resonant frequency, Wd = Wn *sqrt(1-x2)
Pull in range, DWp= 0.5Pi * sqrt (2xWn*KpdKvco-Wn2) = (zero-poll..)
Pull in Time, Tp= 4 * DW2,(from center)/ (Pi2*Wn3)
Lock Range, DWL=PixWn=Pi/(2RC)
Lock Time, TL= 2Pi/Wn
 Class Targil 2
Use your designed PLL circuit to sim and extract the following
PLL parameters :
• Natural frequency (Wn)
• Damped frequency (Wd)
• Damping factor (‘a’),
• Damping ratio (‘x’),
• 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