Non-linear transient behavior of second, third, and fourth order phase-locked loop

Carden, F. F.

Jones, T. J.

Martin, C. R.

Merrill, M. D.

English

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`J THE NON-LINEAR TRANSIENT BEHAVIOR
OF SECOND, THIRD AND FOURTH ORDER
PHASE-LOCKED LOOPS
Frank F. Carden
' Associate Professor
ir
^ M. Don Merrill
Y Assistant Trofesso:+
s
1 =
a
Carl R. Martin
o g Research Assistant
v
^ v Thomas J. Jones
3 1 n Research Assistant
d
40
C Communications Research Group
- Electrical Engineering Department
New Mexico State University
aM rYw
"^"" Las Cruces, New Mexico
Abstract. The non-linear transient behavior
of second, third and fourth order phase -locked
loops is obtained from the projections of the re-
spective state spaces onto the various hyper-planes.
For various driving functions, regions of the
parameter space are specified such that the tran-
sients may be determined by the linear model.
INTRODUCTION
The object of this paper is to determine
parameter regions of linear operation with respect
to certain input signals for second, third, and
fourth order phase-locked loops (PLL's). That is,
certain bounds on the input phase driving fun-
ctions will be established such that the PLL op-
erates in much the manner predicted by linear
analysis when subjected to these inputs.
These bounds are determined by comparing the
peak transient error as calculated from the linear
model with that calculated from the more exact non-
linear differential equation for the various PLL,
configurations. When the peak transient errors so
calculated are within 10% of each other, it is con-
cluded that a linear analysis is valid. Thus, for
a given system the magnitude of a particular phase
input is varied until the above condition just
fails to hold. This sets a bound for a given in-
put upon a given PLL configuration. Thus, the
signal and system parameter region is partitioned
into regions of linear and nonlinear operation.
With this knowledge of the bounds available,
easily predictable, stable operation of given loops
are assured for input signals not exceeding the
specified limits.
The input signals of concern are the fre-
quency step, ramp, and parabola. these being the
aperiodic signals for which it is possible to re-
duce the steady-state error to zero in a second,
third, and fourth order PLL respectively. For the
attainment of zero steady-state errors as just
mentioned it is necessary that the system type at
least equal the system order, and these are the
cases that will be considered. For example, when
the second order system is investigated, it is a
second order loop with two pure integrators.
Therefore, the loop filter will contain an inte-
grator. The other integration being contributed
by the voltage controlled oscillator.
For each order PLL, restricted regions of the
parameters space is studied in order to partition
the parameter space into regions of linear and non-
linear operation. The restricted regions investi-
gated are those neighborhoods containing the points
in the parameter space determined by the two opti-
mization criterianat
CASE I Minimum integral square error
CASE II Minimum error for steady state
sinusoidal input as frequency
approaches zero.l
Each of these optimizations require a system type
number equal to the system order.
STATE VARIABLE MODEL
In order to determine the nonlinear response
of the PLL, the describing nonlinear differential
equation is formulated in terms of state variables
and the projections of the state space onto the
various hyperplanes is determined with the aid of
a digital computer. Plots of state variable ver-
sus time are also obtained from analog simulation.
1. S. Gupta and R. J. Solen, "O ptimum Filters for
Second and Third Order-Phase Locked Loops by an
Error-Function Criterion" IEE Trans. on Space
Electronics and Telemetry, Vol. Set-11 pp. 54-62,
June 1965.
21 D'1
For the nonlinear fourth order loop the differen-
tial equation relating the error and input phase
is
0	 F	 0
0	 0
0 0
Oe 0 + Cwo COS Oa Oe + 3 C wo SIN(0)Oa Oa
W 4 SIN Y	 •0••
-C wo COS Oa Oa	 o	 1	 i J	 (6)
+ b wog SIN(Oe )Oa2 - b wog COS (Oe)Oe where
all 0, a12 - 1, a13 0, a14 0
(1)
a21 - 0, a22 0, a23 1, a24 0
as obtained from the general expression
2
K E) SIN 0
e	 i	 p	 e	 (2)
Similarily, the third and second order equations
may be developed:
.Aa = . Oi + bwo SIN Oa Oat - bwo COS Oa•Oa
- aw2 COS 0 0 -w 3 SIN 0
o	 e e	 o	 e
(3)
a31 0, a32 0, a33 = 0, a34	 1
a41 = 0
842 (Y22 Cwo - awo3) COS Y1 + Y2 bwo2 SIN Y1
a43 Y23 Cwo SIN Y1 - bwo2 COS Y1
a44 = Cwo COS Y1
Similarily, the third and second order equations
ii state space are:
awo COS Oa Oa + wo SIN Oa
(4)
The fourth order loop is formulated in terms of
state variables via the following substitutions:
Y1 e 
Y1 Oa. = Y2
Y2
=Oe=Y3
Y3 =•Oa = Y4
(5)
The resulting state variable expression is:
Y1	 all a12 a13 a14	 Y1
Y2	 a21 a22 a23 a24	 Y2	 +
Y3	 a31 a32 a33 a34	 Y3
LY4J	 La4l a42 a43 a44J L Y4
2. A.J. Viterb: Principles of Coherent Communi-
cations, New York: McGraw Hill, 1966 9 p,45,
Y1	 all	 a12	 a13	 Y1
Y2	 -	 a21	 a22	 a23	 Y2
Y3	 a31	 a32	 a33	 Y3
0	 0
0	 0
SIN Y1
	 •0•
i	 (7)
where
all 0, a12	 1, a13 0
a21 
- 
0, a22 0, a23 1
a31=0
a32 = a COS Y1 + bY2 SIN Y1
a33 = b COS Y 
21 D 2
it
and
Y1
	0	 1	 rY1	 0	 0
-	 I	 t	 +
Y2	0 -2tw COS X Y2	 02 SIN Y1	Al
(8)
Digital computer algorithims applicable to the
solution of n first order differential equations
are utilized to obtain numberical solutions of the
state variable equations for various driving fun-
ctions. The projections of the n-dimensional
state space onto the various hyper-planes are ob-
tained. Projections for the linear and nonlinear
second, third, and fourth order loops are obtained
and compared while conducting a parameter sweep.
The corresponding roots of the linearized charac-
teristic equation for the parameter sweeps are
tabulted below. Figures 1, 2, and 3 depict the
data of the tables. Also shown are the maximum
values of Aw for which the 10% limit on peak over-
shoot holds.
Second order system
TABLE 1
Roots of Linearized Charac- 	 AW for Agreement
teristic Equation	 Within 102
1. Pi -.707 + .707 A1(t) - AW t
(Case I)
P2
 -.707 - j .707 Aw • 1.2
2. Pi -.5 + j .866 A1(t) - AW
(Case II)
P2 -.5 - j .866 AW • 0.8
3. Pi- -1 Ai (t) - Awt
P2 -1 AW - 1.2
Third order system
TABLE 2
Roots of Linearized Charac-	 Aw for Agreement
teristic Equation	 within 10%
Table 2 (Continued)
2. P1
 -
-1.609 01(t) - Awt .01 (t) - 2t2
P2
 -
-.195 + j.764 bw - 2.1 AW - 1.15
P3 - -.195 - j.764
3. P1 - -1.44 01(t) - Awt 01 (t) - 2 t2
P2 - -.280 + j.785 AW - 2.1 AW - 1.15
P3 - .280 - j.785
4. Pl
 -
-1.24 01(t) - Awt 01 (t) - LW t2
P2 - -.381 + j.814 Aw - 2.1 Aw - 1.167
P3 - -.381 - j.814
5. P1 • -1.00 A1(t) - Awt 01(t) - 2 t2
P2 - -.500 + j.866 Aw - 2.178 Aw - 1.217
(Case I)
P3 - -.500 - j.866
rourtn oraer system
TABLE 3
Roots of Linearized Char- Aw for Agreement
acteristic Equation
	
within 10%
1. P1 - -.957 + j 1.23 A1(t)-Awt A1(t)- 6 t3
P2 - -.957 - j 1.23 &0-2.038 &-1.003
P3 • -.043 + j.641
(Case II)
P4 - -.043 -j.641
2. P1 - -.082 + j.657 01(t)-Awt 01(t)- 6 t3
P2 - -.082 - j.657 AW-2.085 AW-1.033
P3 - -1.175 + j.949
P4 - -1.175 - j.949
1. Pi
 -1.755	 81 (t) • Awt 81 (t) • -27t
P2 - 1.23 + j.745
(Cass II)	 Aw • 2.175 Aw - 1.15
P3 - -.123 - j.745
2	
3. P1 - -1.108 + j 1.108 01 (t)-Awt. 01(t)- 6 t3
P2
 - -1.108 - j 1.108 Aw-2.203 AW-1.052
P3 - -.0459 + j.616
P4
 - -.0459 - j.616
21 D 3
Table 3 (Continued)
4. P1 -.901 + j.774 1 Oi (c)-bmt 0 (t)- 6 t3
jW
SD D 
D 1.D
-1.0	 -0.5
{..
P2 - -.901 - j.774 &-2.5 &)-1.226
P3 - -.330 + j.774
P4
 - -.330 - j.774
5. P1 - -.383 + j.924 Oi(t)-bait 0 (t)- 6 t3
P2 - -.383 - j.924 bur2.58 64-1.254
P3 - -.924 + j.383
(Case I)
P4
 - -.924 - j.383
jW
1.0
a
-1.5	 -1.0	 -0.5
0.5
Figure 3. Roots of linearized characteristic
equation for fourth order system. Number by roots
indicates corresponding data from Table 3.
0
	
CONCLUSION
The linearization of equation 1, 2, and 4
for the analysis of the inputs considered yield
valid solutions in the regions of parameter space
specified. That is, with the loop gain constant
and filter parameter adjusted to the region speci-
fied by the roots of the linearized equation,
bounds for which a linear analysis is applicable
are determined.
This work was supported by NASA
grant $NGR-32-003-037
1. 2. 3.	 4.	 S.
Figure 2. Roots of linearized characteristic
equation for third order system. Number by roots
indicates corresponding data from Table 2.
1)
I _ jW 1.5
1.0
0.5
Figure 1. Roots of linearized characteristic
equation for second order system. Number by
roots indicates corresponding data from Table 1.
21 D 4


The non-linear transient behavior of second, third and fourth order phase-locked loops

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