Third-order extension to present second-order systems extends their Doppler tracking capabilities. It widens receiver pull-in range, decreases pull-in time, lowers voltage-controlled oscillator (VCO) noise (determining when no signal is present), and lessens susceptibility to VCO drift

Crow, R. B.

Tausworthe, R. C.

English

NASA Technical Reports Server

•	 August 1974	
.	 B74-10104 
NASA TECH BRIEF 
•	 NASA Pasadena Office 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, . Springfield, Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Third-Order Phase-Locked Loop Receiver 
. 
.
The problem: 
As the more-difficult deep space missions come into 
being, there is a corresponding stringency of require-
ments placed on the tracking instruments. To date, 
second-order phase-locked receivers, used both in space-
craft and in ground tracking stations, have performed so 
satisfactorily that there has been practically no reason 
to consider a more complicated system. The deep space 
missions, however, produce Doppler rate profiles that 
may cause up to 300 steady-state phase errors in the 
unaided second-order loops. Such errors in the second-
order receivers decrease the efficiency with which 
command or telemetry data are detected (by 1.25 dB 
at 300). In addition, they make acquisition of lock 
difficult and faulty and increase the likelihood of 
cycle slipping and loss of lock.
e = rIr1 
Isolation	
r1 =(R 1 +R 2) C1 Amplifier	
= R 5 C 3 
IT2 =RC2 I r =R3C2 
R4 C1'	 R6 
r'Wv', j k = 
R3 FT R5 fl 
Figure 1. Third-Order Loop Filter Which Is An Extension of 
Second-Order Loop by Addition of Two Integrators. 
The solution: 
A simple third-Order extension to the present second-
order systems extends their Doppler tracking capabilities. 
In addition, it widens the receiver pull-in range, de-
creases pull-in time, lowers the voltage-controlled oscil-
lator (VCO) noise (determining when no signal is 
present), and lessens the susceptabiity to VCO drift.
E	 Isolation I 
>1
 2	
/Amplifie4R+R 
R'	
T € = 
wvr
T1 =(R1+R2)C1 
72 =R2C1 
= R3C4 
6 =R3/R4 
k = T2/T3 
Figure 2. Third-Order Loop Filter Which Is Not An Extension of 
Usual Second-Order Loop. 
How it's done: 
The loop filter under consideration may be synthe-
sized in several ways, three designs of which are shown 
in Figures 1, 2, and 3, respectively. They have the same 
transfer function (see figures for parameter definition): 
1+T2s 
F(s) = - + 
1+7- 1 s (1+r1 s)
 (5+7-3 S) 
With the same loop transfer function, the designs operate 
identically once the loop is locked; they differ in their 
lock-in transient behaviors, however, because of the 
possibly different initial capacitor voltages. If all capac-
itors are shorted at time t = 0, they are again identical, 
within the limits introduced by hardware imperfections. 
The two-sided noise bandwidth in the simplified, but 
usual, case of 
€ = 7-2 /r 1 	 1 and 6 ' 1 is given by 
WL =
r /_r—k+1 
- I 
2T2 \ r—k 
where r = Aki 2 /r1 is set equal to 2, with A being the 
rms signal amplitude and k being the loop gain. The 
parameter k is the time-constant ratio 7-2 /T3 . As com-
pared with the loop bandwidth formula of the second-
order loop, the simplified expression above is only 
(continued overleaf) 
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https://ntrs.nasa.gov/search.jsp?R=19740000104 2020-03-17T14:04:52+00:00Z
L0 W is the loop bandwidth at the design 
Note: r0 is the design point value of r, and
RIR1	 4
r23	 r0 (4r0+3) 
T2 = - r0 = 3.375 = Ak (\T) 7	 WL	 r0-1) 
R 1 R4 
71
 
=RC=RC
(R2+R3) (R5+R6) 
Figure 3. Third-Order Loop Filter With Series Integrator 
(Critically Damped, at k=0.25).
S 
slightly increased in complexity, and of course the two 
merge to the same result as k approaches 0. The 
primary determining factors of wL are still r2 and r, 
just as in the second-order loop. The phase error 
variances of any two loops due to input noise are of 
course the same as long as their loop bandwidths are 
the same. 
The theory developed for computing the pull-in 
range of a second-order loop is easily extended to 
account for effects in the third-order loop; there is an 
enhancement in the acquisition range by approximately 
the square root of the added integrator dc gain. In fact, 
experimental evidence verifies this formula exceedingly 
well, all the way out to the point where IF filtering or 
minute equipment bias imperfections begin to limit the 
loop pull in. 
Due to imperfect integrations, there are steady-state 
phase errors that build up because of instantaneous 
frequency detuning and the rate of change of this 
detuning. Compared with the corresponding expression 
for a second-order loop, the steady-state error due to an 
instantaneous frequency offset is reduced by a factor of 
about 6, and the error caused by a frequency rate is 
diminished by a factor of about +(e/k). Such a 
comparison reflects the desirability not only of making 
e and ö very small but also of keeping k as large as other 
factors will permit. 
To minimize the possibility that acquisition is faulty, 
it is necessary that damping be critical or beyond; to 
minimize steady-state error once lock is achieved, k 
should be as large as possible. These two conditions are 
met in slightly different ways according to the type of 
signals to be tracked. If the design is to be for signals of
a fixed level, then k should be set to kmax=0.33 and r 
should be set to produce critical damping at that level, 
r0=3.0. If the design is to be for signals of various 
intensities, then k should be made equal to k 0 =0.25, and 
r should be set for critical damping at the weakest-
expected signal level to ensure that the roots are never 
underdamped, r0=3 375. 
Note: 
Requests for further information may be directed to: 
Technology Utilization Officer 
NASA Pasadena Office 
4800 Oak Grove Drive 
Pasadena, California 91103 
Reference: TSP74-10 104 
Patent status: 
This invention has been patented by NASA (U.S. 
Patent No. 3,704,671). Inquiries concerning nonex-
clusive or exclusive license for its commercial develop-
ment should be addressed to: 
Patent Counsel 
NASA Pasadena Office 
4800 Oak Grove Drive 
Pasadena, California 91103 
Source: Robert B. Crow and

Robert C. Tausworthe of 
Caltech/JPL
under contract to
NASA Pasadena Office
(NPO-1 1941)
S 
B74-10104	 Category 02


Third-order phase-locked loop receiver

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