i NAT' AT'L INST. OF STAND & TECH

AlllDb 3HQ115

\

in NBS TECHNICAL NOTE 1093

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Reference

NBS

PUBLICATIONS

Automated Measurement of

Frequency Response of

Frequency-Modulated

Generators Using the Bessel

Null Method

J. R. Major

E. M. Livingston

R.T.Adair

100

.U5753

#1093

1986
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NBS RESEARCH IN7ORMATI0N

100

Automated Measurement of Frequency ^^J^^

Response of Frequency-Modulated ^(f|j

Generators Using the Bessel Null Method ^^

J. R. Major

E. M. Livingston

R.T.Adair

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

Sponsored in part by

U.S. Army CECOM

Ft. Monmouth, New Jersey

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued March 1986
National Bureau of Standards Technical Note 1093

Natl. Bur. Stand. (U.S.), Tech Note 1093, 32 pages (Mar. 1986)

CODEN:NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1986

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402
CONTENTS

Page

1. Introduction <> .c, ^ . «o« o <> o »» »».,<> .<,..<,.. e 1 2. Bessel Null Background » ,,«.»«,.,,,, 3

3. System Description ,»» «,»...«,..... ^ o«.<, o«

.

5 4. Frequency Response Calculations ..,, ,,,.,,,. 15

5. Measurement Accuracy 18

6. Summary o » , o <, . . . . 18

7. Conclusions 19

8. Acknowl edgments 20

9. References , 20

Appendix A Computer Program for Determining Frequency Response of

Frequency -Modulated Generators Automatically 21

Appendix B Computer Program for Curve Fitting 24

m

Automated Measurement of Frequency Response of

Frequency-Modulated Generators Using the Bessel Null Method

J. R. Major

E. M. Livingston

R. T. Adair

This paper describes a Bessel null technique to measure the frequency response of a frequency-modulated rf carrier and a program

to automate frequency response measurements of signal generators

with output frequencies from 0.450 to 2000 MHz. The measurements

obtained using this technique are more precise than those obtained

by a highly trained technician using a manual system.

Automated measurement of this process is desirable since the

manual method is subject to the following problems: (1) excessive

time, (2) error in finding the null, and (3) lack of assurance

that the null is the first Bessel null. Automated measurements

can be performed using a system controller, a spectrum analyzer, a function generator, and a voltmeter (all of which must be compatible and controllable remotely).

The nonlinear relationship between the modulating signal

amplitude and the center frequency amplitude of the carrier is a major obstacle to automated measurement. This problem was solved

by obtaining an approximate formula for this nonlinear curve.

Assurance that the null found is the first Bessel null is

provided by the analysis of the frequency response of the signal

generator under test as displayed on the spectrum analyzer.

Key words: automated; Bessel null; curve fitting; frequency

modulated generators; frequency response; linearization.

1. Introduction

One of the criteria of the performance of a signal generator is its

ability to produce a specified flatness of output amplitude (e.g., ±2 dB)

during frequency modulation over its intended frequency range. The actual

response of a generator to a given range of frequencies of modulating signal

can be determined with the aid of the Bessel null technique as described

herein. When a carrier of radio frequency is frequency modulated by a signal

of lower frequency, variations in the amplitude of the modulating signal are converted to corresponding variations in the carrier frequency [1]. The instantaneous radian frequency co^- varies about the unmodulated carrier frequency

0)^ at the frequency of the modulating signal Wj^ with a maximum frequency

deviation of Aw radians. By definition the ratio of the frequency deviation

1
to the modulating frequency, Aco/oj^j^, is the modulation index and is labeled p.

An increase in the amplitude of the modulating signal corresponds to an increase in the bandwidth occupied by the frequency-modulated carrier, and

hence increases the frequency deviation and consequently the modulation index p.

The average power associated with the frequency-modulated carrier is independent of the modulating signal and is the same as the average power of the

unmodulated carrier.

As the modulation index, p, increases, the energy in the carrier begins

to decrease, thereby transferring more and more energy to frequency sidebands,

which increase in number. For a value of p of lA the carrier amplitude

reaches a minimum or null point, and maximum energy is transferred to the

sidebands. The power series expansion of this power spectrum is expressed in

terms of its coefficients. These are the amplitudes of the carrier frequency

and sidebands. The amplitudes b.ro. defined as Bessel functions of the first

kind (J^).

An accurate measurement of the frequency deviation Aw of the modulated

carrier may be obtained by determining the Bessel null point.

An ampl itude-versus-time display of a frequency-modulated signal is shown

in figure 1 along with the unmodulated carrier signal and the modulating signal.

Figure 1. Frequency modulation, amplitude vs time,
Measurement of deviation from this type of display would be difficult at best,

especially with a wide variation in the frequency of the modulating signal.

Since the measurement of frequency response of the modulated carrier is

dependent on an accurate means of measuring deviation, it is necessary to consider an alternate measurement method.

Figures 2 and 3 show a display of amplitude versus frequency of an un- modulated carrier and a frequency -modulated signal, respectively, as observed

on a spectrum analyzer. These figures provide the observer with a clear

concept of frequency deviation. However, actual measurement of deviation from

measurements shown in figure 3 is difficult due to the wide variations in the

frequency of the modulating signal.

Modulation analyzers can be used to make accurate measurements, providing

of course that the frequency range of the analyzer is adequate. In this case,

the 2.0 GHz upper limit of the frequency range of the signal generator to be

tested for frequency response exceeded the range of available modulation

analyzers.

The Bessel null method has a distinct advantage in obtaining a precise

measurement of deviation. With this method the amplitude of the modulating

signal is increased until the amplitude of the center frequency of the modulated carrier reaches a minimum (see fig. 4). The disadvantage of this method

is the excessive time required to perform the test. This disadvantage can be

overcome by automating this measurement procedure.

2. Bessel Null Background

With a fixed frequency of modulation the carrier amplitude can be reduced

to a minimum by increasing the amplitude of the modulating signal. This null

point is not dependent on the modulating frequency. The Bessel formulas and

functions provide verification. Those shown below are taken from reference

[2]. If a sinusoidal modulating signal of the form A^cosw^t is used and has a maximum frequency deviation Aw then the instantaneous frequency is

w^(t) = 00^ + Aw (cos w^t) (1)
MKR 40 . 000 MHz 12.90 dBm

CENTER 40 . 000 MHz RES BW 1 kHz VBW 3 kHz

SPAN 100 kHz

SWP 300 msec

Figure 2. Frequency spectrum of unmodulated carrier.

CENTER 40

.

000 MHZ

HES BW 1 kHz VBW 3 kHz

SPAN 100 kHz

SWP 300 msec

Figure 3. Frequency spectrum of frequency -modulated carrier
where Aw is independent of the modulating frequency, co^, and is proportional

to the amplitude of the modulating signal. The Bessel function is derived

from spectral analysis as shown by Carlson [3], from the power spectral series

00 X (t) = A y J (p) cos (cD + nw )t, (2)

c ' c ^ n ^^' ^ c m ^ ' n=_oo

where Jn(P) is the nth order Bessel function. Examination of this power

spectral series shows that the Bessel function represents the relative amplitude of the carrier and is a function of the modulation index. When the zero- order Bessel function Jq(P) = 0, p = 2.4, 5.5, etc. (p = —)

.

m

3. System Description

A spectrum analyzer is used to monitor the amplitude of the carrier and

is connected to the unit under test, the modulating signal generator, and the

system controller as shown in figure 5. The system controller must be able to

control the amplitude and frequency of the signal generator using the program

discussed below (see appendix A), Also, the system controller must be able to

control functions of the spectrum analyzer such as center frequency, resolution bandwidth, video bandwidth, attenuation, marker, delta marker, and

marker amplitudes.

The program used with the system controller contains the solution to the

problems of (1) working with a nonlinear relationship between the modulating

signal amplitude and the carrier amplitude, and (2) the requirement that the

system use the first null of the zero order Bessel function rather than any of

the subsequent nulls.

The null as described above cannot always be obtained. Instead, a mini- mum is obtained in this test procedure.

A nonlinear relationship exists between the amplitude of the modulating

frequency and the amplitude of the carrier frequency. The change required to

bring the first tenth of reduction in carrier frequency amplitude is many

times greater than the change required for the last tenth of reduction. This

is a substantial problem for a manual test and a potentially difficult problem

for an automatic test. A conventional approach to the automation of this test
MKR 40 000 MHz

-24 . 00 dBm

CENTER 40.000 MHz

RES BW 1 kHz VBW 3 kHz

SPAN 100 kHz

SWP 300 msec

Figure 4. Frequency spectrum of frequency-modulated signal with

amplitude of modulating signal adjusted to "null" carrier,

UNIT

UNDER

TEST

FM ll INPUT

RF

OUTPUT

OUTPUT

SPECTRUM

RNRLYZER

INPUT

SIGNAL

GENERATOR

Figure 5. Equipment setup for automated measurements of frequency

response of frequency modulated generators using the Bessel

null method.
is to make the increments sufficiently small to provide the accuracy required

by the specifications. For example, with a carrier frequency of 40 MHz the

amplitude of a 100 kHz modulating signal at the null is approximately 500 mV.

The increment in modulating signal at the null point that would produce

changes in the carrier amplitude is less than 0.5 mV. Therefore, over 1000 of

these increments would be required at each modulating frequency. A firstorder approximation of the time required for this test is greater than the

time required for the manual test.

To characterize the nonlinear relationship, one could use a look-up table

of measured values or a mathematical formula derived from a curve fitted to

the measured data. To provide for adequate resolution, a look-up table would

require an excessive number of values. This would create a measurement

problem in that a large amount of time would be required to acquire the data.

The solution to the above nonlinear problem is the determination of an equation that adequately describes the nonlinear relationship. Therefore, a "curve-fit" program was developed to determine an equation that would fit the

data of the relationship between the modulating signal amplitude and the carrier amplitude.

Since the plot of the data of modulating signal amplitude and carrier

amplitude appears to be an exponential function, a simple trial and error

curve fit program was implemented. This program allows the user to select a formula and determine its fit to the data.

The following discussion provides an explanation of the curve-fit program

using three different trial equations:

y = X where B is a constant, (3)

y = C log X where C is a constant, (4)

«(x + C)

y = ^ where A, C X D are constants. (5)

(The values of these constants will be determined during the curve-fitting

process.) The data shown in table 1 were measured using the spectrum analyzer

and are the basis of information from which an equation is to be determined.
Table 1. Amplitude of modulating signal versus carrier frequency amplitude.

Amplitude of 10 kHz Incremental amplitude of 40 MHz carrier

modulating signal (V), 10 kHz modulating signal (V), amplitude (dBm),

X (0.498 - x) y

0.498 13

0.384 0.114 3

0.463 0.035 -7

0.491 0.007 -17

0.498 -22.8 (minimum)

0.515 -0.017 -17

The curve-fit program allows the user first to plot the data of the non- linear relationship from table 1, as shown in figure 6.

The next step in the curve-fit program is to select a trial formula; for

example, eq (3). The program then plots the curve and allows the user to make

changes as needed to find an adequate curve fit. The curve fit of eq (3) is

shown in figure 7.

Figure 8 is a plot of the next trial formula which is:

y = C log X. (4)

Figure 8 contains a set of data points which describe the increments in

modulating signal amplitude to reach minimum carrier amplitude as shown in

table 1 (points marked with a "y"). T^e "null" in table 1, which is a minimum

of carrier amplitude, was achieved at a modulating signal amplitude of

0.498 V. This corresponds to an increment to achieve "null" or zero volts

(see table 1). The incremental data were useful since the program to control

the modulating signal amplitude uses increments of increasingly finer resolution to "zero in" on the "null" of the carrier.

The third trial formula is of the form

a(x + C)
CKJ

hX

E 10

CO

Amplitude,

o

o X

- X

-

0)S-20

o

-

i

X

X

0.1 0.2 0.3

Modulating Signal, Volts

0.4 0.5

Figure 6. Plot of data from table 1 to be used in "curve-fit" program,

20

00

o

-D U 3

1-10

^-20

o

-30

-

- ^^^^-^^

X

—

Y = X^B X

—

X

1 1 \x

0.1 0.2 0.3

Modulating Signal, Volts

0.4 0.5

Figure 7. Curve-fit result of y = x
0.1 0.2 0.3 0.4

A Modulating Signal to Reach Null, Volts

0.5

Figure 8. Curve-fit result of y = C log x.

(where A, C, and D are constants that may be changed to provide a curve fit).

This formula was found to be adequate to describe the nonlinear relationship

of the data in table 1 (see figure 9).

As the frequency of the modulating signal is varied, the curve (of modulation signal amplitude versus carrier amplitude) changes in amplitude. The

data points Z in figure 9 are typical for a modulating signal whose frequency

is less than 10 kHz. Therefore, this requires an equation that can be easily

changed. Changes in the constant A of eq (5) as shown in figure 9 provide the

necessary flexibility. In the computer program, A is changed to a smaller

value in two cases. The first case occurs when a minimum is obtained and

second and third resolutions are performed. The second case occurs when the

frequency of the modulating signal is reduced.

The curve-fitting program (see appendix B for a complete listing) was sufficient and adequate for this test. Additional programs can be obtained to

provide more information on curve fitting. For example, there are programs

10
20

E 10

o

oT34-1 Q.

E

< -10 ^

CD

^.^- o -20

-30

0.1 0.2 0.3 0.4

A Modulating Signal to Reach Null, Volts

0.5

Figure 9. Curve-fit result of y

= /x+C)

"15

which are sufficiently sophisticated to provide a formula that fits the data

precisely. If this type of program is not available, the book "Fitting

Equations to Data" [4] will be useful. The process of linearization is shown

in chapter 3. An example follows which demonstrates the use of this linearization process. The data in table 2 (which was generated from the formula

1 ?

a = X

*

) will be used in the example.

The first step is the selection of a formula type to investigate. A good

choice for the data in table 2 is an exponential function in the form y = x (even if the answer is not known ahead of time).

Table 2. Data to provide a case study for the linearization process

1 ? (generated from a = x

* ).

1 1 2 2.297

3 3.737

4 5.278

5 6.899

6 8.586

7 10.33

8 12.13

9 13.97

15.85

11
If

y = x^ (6)

then

In y = B In x; (7)

Therefore, In y is linear in In x as shown in [4],

Also

^=lnx. (8)

and solving for x gives In y

X = e

^ . (9)

Since a linear plot is sought the data in table 2 are plotted in the form

of e versus x, where the data in table 2 are values of a for corre- sponding X values. The result is the straight line shown in figure 10 where

B = 1.2. Therefore, the correct formula for the data in table 2 is

1 o y = X . If the incorrect formula had been selected, the resulting curve would not have been a straight line and the obvious task would have been to

select another formula. If, however, the incorrect value of a constant such

as B in this example were selected, a few trials would show that the plot

would quickly approach the data points.

It is necessary to know that the null found is the first null of the zero order Bessel function in both the manual and the automatic tests using the

Bessel null method. Figure 4 is a plot of frequency vs amplitude of the output of a frequency -modulated signal generator at the first null Bessel

function as displayed on a spectrum analyzer. Figure 11 shows this display

for a second Bessel null.

The basis for the solution of the problem of determining the desired

Bessel null is the difference in the "filtered response" of the first null and

that of the second null. Figures 12 and 13 are, respectively, plots of the

first and second null except that the video bandwidth and resolution bandwidth

of the spectrum analyzer have been reduced and produce a "filtered response."

12
Y

IG

15

14

13

12

1 1 10

9

8

7

6

5

4

3

2

1

/ yy

> X

y-> Y / ^^^

Y = ^X-l .2X _^,y X ^^^

y'•

' y

y^^,^-^^ LN C y^

• Y = e

1 . P 1

^ ^t ^y^

^

123456789 10

X

Figure 10. Curve-fit to data points using linearization.

MKP Aa.eea a mhz

-28 . 30 aBm

CENTER 40 . 000 MHz

RES Bw 1 kHz VBW 3 kHz

SPAN 100 kHz

SWP 300 msec

Figure 11. Frequency spectrum showing frequency-modulated signal at second

null of zero order Bessel function.

13
REF 13 OBm ATTEN 30 aB

MKR ^ 24 . kHz

-4 . 10 as

10 as/ -..^

-

F

^-^

MAPI CER I \ 24 . 1 -4 .

3 kH2

10 dE 1

CENTER 40.000 MHZ

RES BW 30 kHz VBW 1 kHz

SPAN 100 kHz

SMP 50.0 msec

Figure 12. Frequency spectrum showing "filtered" first null of the zero order

Bessel function.

10 dB/

13 . dBm ATTEN 30 dB

MKR ^ 24

-0 . 40

.0 kHz

dB

r*-^

""

MARI CER I :^ 24 .1 -0. .

3 kH:

30 dE !

CENTER 40.000 MHZ RES BW 30 kHz VBW 1 kHzSPAN 100 kHz

SWP 50.0 msec

Figure 13. Frequency spectrum showing "filtered" second null of the zero order Bessel function.

14
A comparison of figure 12 and 13 shows that the "filtered response" of the

second null is much flatter (also true of subsequent nulls) than that of the

first null. Therefore, the measurement of the difference in amplitudes at the

markers indicated in figures 12 and 13 is the information that can be used to

determine if the null found is the first Bessel null. If the difference is

greater than 1.5 dB (a number selected between the 4.1 dB difference in fig.

12 and the 0.4 dB difference in fig. 13) then the null found is the first

Bessel null.

A block diagram of the computer program is shown in figure 14 and a complete listing is given in appendix A.

4. Frequency Response Calculations

Here, we define a response parameter R which for an ideal FM modulator

would equal unity (0 dB). In the process of finding the first null at any

modulating frequency, p is held constant.

Recall that since p = — [1] or B = -?— where Af is the deviation from

0) ^ f m m center frequency and f^^ is the modulating frequency, then Af must decrease as fm decreases. Therefore, to normalize Af (and make relative comparisons of

modulating signals and deviations from center frequency), we multiply the rms voltage of the modulating signal at null (V^od^ ^^ ^^^ ratio of the reference

frequency (fp^f) to the frequency of modulation (f[y,od^*

To normalize to a given chosen reference modulation frequency, e.g., 100

KHz, we divide by the rms voltage required to reach null at the reference frequency (y^^f). The normalized reading N is then:

mod ref

or

V . mod Y—

ref

mod

f . ref

(11)

15
FIRST

MODULHTION FIRST NULL?

FREQUENCY

1

< 1 .5 >

DECREASE R

INITIRL

CONDITIONS NEXT

MODULRTION

FREQUENCY —I 1

1

* 1 FIND

CARRIER

R MPLITUDE^

Y=R-^ ( flMPL+37 ) / 1 5 00

FIND NEW

CRRRIER

RMPLITUDE

-1 COMPRRE

n

(REFINE)

DECRERSE R

THIRD TRY

NO YES

1

nr~(^or-nr-r- \^ u n-r<>c.noc.

Figure H. Flow diagram of program to find the first null of the zero order Bessel function.

16
Expressed in dB the response is

R = 20 logN or

- d\j log -n dij log r

. ref ref

(12)

(13)

Typical measurement results have been recorded in table 3 (using equipment setups as shown in fig. 5 and computer programs as described herein).

The program output is a printout showing the response R using eq (10) for each

modulating frequency.

Table 3. Calculated values of FM frequency response, R,

for f^gf = 100 KHz and V^^^ = 0.498 volts

EXTERNRL

MODULRTION

FREQUENCY

DEVIRTION

RS INDICRTED

MODULRTING

SIGNRL

CRLCULRTED

RESPONSE

(KHz) BY UUT Vmod (VOLTS) R (DB)

0.4 ¥: 0.00149 -2.52

1 0.01 0.0389 -2.15

3 0.09 0.012 -1 .90

10 0.22 0.052 +0.375

30 0.72 0. 14G -0.200

100 2.4 .498 0.00

^ NO METER MOVEMENT DETECTED

17
5. Measurement Accuracy

A preliminary estimate of the measurement uncertainty using the automatic

technique described is ±5 percent. The signal generator used to provide the

modulating signal has a flatness specification of ±4 percent. Therefore, the

measurement accuracy could be improved markedly by using a signal generator

for modulation with a better flatness specification. Thus, the modulating

signal source is the major contributor to the measurement uncertainty.

The actual method of finding a null as described in this document appears

to be highly precise. A comparison was made between the results of the automatic null measurement technique and the reading of the deviation meter on the

signal generator under test. Readings were taken with a deviation of 2.4 as indicated on the unit under test deviation meter. Errors due to parallax in

reading the meter were minimized by using the meter glass as a mirror and

aligning the reflection of the operator's eye with the meter needle. Each

reading was taken with the meter indicator at a specific mark (2.4) to maximize the accuracy and repeatability of reading the meter. Several readings

were taken with each method. Variations observed with the automatic null

measurement method were ±0.1 dB compared to ±0.8 dB using the deviation meter

on the unit under test. Thus, the resolution of the measurements using the

automatic null technique is better than that of the manual method using the

built-in deviation meter. This verifies that the accuracy of frequency

response measurements using the automatic null measurement technique is con- siderably better than manufacturers' stated accuracy based on conventional

techniques.

6. Summary

A method to determine the frequency response of frequency-modulated signal generators is described. The Bessel null method is used to control

deviation since frequency deviation is independent of the modulating frequency. Automated measurements are necessary to eliminate the tedious manual

"nulling" and possibility of nulling on other than first nulls of the zero order Bessel function.

18
An equation for the nonlinear relationship of modulating-signal amplitude

versus carrier amplitude is supplied by a computer program developed for curve

fitting. This equation is a necessary part of the automated measurements.

This technique provides a unique, faster, and more accurate method of

measuring signal -generator frequency response than those commonly used.

7. Conclusions

The Bessel null method of measuring frequency response is potentially

highly precise due to the lack of dependence of modulating frequency on frequency deviation. The measurements must be made automatically or with a highly

skilled technician. If many measurements are necessary the manual method is a very tedious process.

Automated measurements have problems such as nonlinear relationships and

multiple Bessel nulls. This document describes how these problems can be re- solved in a way that is tractable for automatic application.

Additional considerations for future work include the use of a programmable step attenuator placed between the output of the modulating signal

generator and the external frequency modulation input of the signal generator

under test. Then the sub-millivolt increments in modulation voltage will

remain a yery small part of the total modulation voltage required to obtain a first Bessel nul 1

.

Also, the "loading effects" of a variable input impedance of the external

frequency modulation must be considered. For example, an ac-coupled input

could possibly cause a significant phase shift at lower modulating signal frequencies.

Although this test provided results that were repeatable within the submillivolt increments of the modulating signal, a more thorough data analysis

should be performed to establish more accurate uncertainty limits. Other

areas of investigation include, but are not limited to, "sharpness" of null,

frequency deviation other than the first Bessel null, and distortion of the

modulating signal, as well as distortion of the modulated signal with respect

to finding the Bessel null.

19
8. Acknowledgments

The authors wish to express their sincere thanks and appreciation to Edie

DeWeese and Jessie M Page who did an excellent job of preparing this document,

and to Richard Ehret, John Adams, John Smilley, and Norris Nahman for their

helpful suggestions and comments.

A portion of this work was funded by the U.S. Army CECOM, Ft. Monmouth,

New Jersey.

9. References

[1] Schwartz, Mischa. Information, transmission, modulation, and noise. New

York: McGraw Hill; 1959. p. 115-125.

[2] Ranter, Philip F. Modulation noise and spectral analysis. New York:

McGraw Hill; 1965. p. 251.

[3] Carlson, A. Bruce. Communications systems. New York: McGraw-Hill; 1975.

p. 226.

[4] Daniel, Cuthbert; Wood, Fred S. Fitting equations to data. New York:

Wiley-Interscience; 1971. pp. 19-21.

20
Appendix A Computer Program for Determining Frequency Response

of Frequency -Modulated Generators Automatically

100

105

1 10

1 15

120

125

130

135

140

145

150

155

160

165

170

175

1S0

135

190

195

200

205

210

215

220

225

2 3

235

240

245

250

255

260

265

270

275

280

285

290

295

300

305

310

315

320

325

330

335

340

RESPONSE USING BE SSEL NULL

PRINTER IS 1 PRINT "FM_RESPONS"

PRINTER IS 701

PRINT " FREQUENCY

PRINT

I THIS PROGRhM finds THE RESPONSE WITH RESPECT

3, 1, HND 0.4KHZ. TO FIND THE RESPONSE

AN HTTENURTOR between THE OUTPUT OF THE

TO THE FM INPUT OF THE UNIT UNDER TEST.

TO 100KHZ OF 30, 10,

R T L U E R PRE Q U E N C I E 'E; U S E OSCILLRTOR RND THE INPUT

THE 2: 1 FRCTOR IN

THE OSCILLRTOR

RLSO TO OBTRIH V RSSIGH ifiRa TO 723

RSSIGN C^Sa TO 713

RSSIGN C-Hfiiib TO 7

CLERR liiHpib

R=l . 141

THE RUDIO RNR OUTP

UTPUT TO THE INPUT

OLTRGE RERDINGS FO

UT CRN BE RVOIDED BY CONNECTING

. THIS CONNECTION IS NECESSRRV

R CRLCULRTIONS IN RESPONSE.

THIS IS R FRCTOR

OF RNPLITUDE VS.

OPERRTING CONDITI

FOR THE FIRST

MODULRTION FREQUE

! CHRNGES IN RE

! FOR SECOND BE

Span =1000

Ndf rq=100

D 1 t f

• r q = 2 4 R b = 3

Vb = 30

G S U B S 1.4 b_g o i u b

V100khz = Rriis P R I \\ T " n D U L R T I H PRE Q U E H C Y =100 K HZ";"

PRUSE

Span=300 ! Mdfrq = 30 ! c:30KHZ;'

Dltfrq=72 ! Rb=100 ! Vb = 3 ! GO SUB Sub ^lOiub

GO SUB C ' PRUSE

Ten_k: ! Spari=100 ! Mdfrq=10 ! <10KZ:;'

Dlt.frq = 24 ! Rb=30 ! Vb=.3 ! R=l. 091

GOSUB Sub_goi.ub

GOSUB Calc

PRUSE

Threi='_k : ! ! Srjan = 30 ! ridfrq = :: ! <:3KHZ>

IN THE NUN

NULL (USED

ONS

LINERR EXPRESS

IN "FINDNULL";'

NCY

S BN

SSEL

<100KHZ>

RND VIDEO

NULL CHECH

BW

ION

VMOD" ; Rrns II I I II . II R = 0DE

21
345

350

355

360

Jb-.i 370

375

3S0

335

390

395

400

405

410

415

420

425

430

435

440

445

450

455

460

465

470

475

430

435

490

495

500

505

510

515

520

525

530

535

540

545

550

555

560

565

570

575

580

535

IKHZ::'

D1 t.frq = 7. 2

Rb=10

Vb=. 1

Fl = 1 . 6 1 G S U B S u b _g o s u b

GO SUB Calc

PRUSE

Sf:i.an= 1 Mdfrq=l

Dl t. frq = 2. 4

Rb = 3

Vb=. 03

R= 1.041

GOSUB Sub_go5.ub

G S U E C a 1 c PRUSE

Ndfrq=. 4

Dl tf rq=. 96

Rb=l ! Vb=.03 ! R= 1.001

GOSUB Sub_gc.i.ub

GOSUB C..=il c GOTO End

Outputs: ! CONMRHUS TO RUDIO RHRLVZER RHD SPECTRUM RHRLVZER

U T P U T I? fl a ; " R U PR"; M d f r q ; " K Z R P O V L L " OUTPUT i?Sa;"LF S2 i::F40MZ SPIMZ RL20riN TS El NTl TS"

U T P U T e S a ; " S P "

; S p an ; " K Z

" OUTPUT i?S.=l;"TS M2 TS El TS E2 TS E4 TS MR"

RETURN ! (FOR Outputs::'

Sub_gosub: ! G S U E F i n d n u 1 1

•'. 400HZ;:i

GOSUB cd b>=-s ck R CHECK TO MRKE SURE THRT THE HULL THRT NRS FOUHD

WRS THE FIRST HULL. :; I F THE VOLTRGE INCREMENTS

TO THE OSCILLRTOR NERE TOO LRRGE, THE FIRST NULL

COULD BE JUMPED OVER RND R NULL FOUND ON OTHER

THRN THE FIRST NULL)

G S U B N t rn d f q_r- p t _f d n 1

! GOTO NEXT MODULRTION FREQUENCY OR IF THE SECOND BESS ELL

! TEST INDICRTES; REPERT FINDNULL WITH R LOUER R (SMRLLER

! INCREMENTS TO RVOID JUMPING OVER FIRST NULL)

RETURN ! (FOR Sub_gosub::'

Ntrndf q_rf:it_f dnl : ! R p t _n q f 1 : ! IF De 1 t .=i< -1.5 THEN GOTO Rt n_nqf 1 R=R-. 02

GOSUB Fl ndnul

1

GOSUB S €• c d_b ^ s_.: k GOTO Rpt nqfl

22
59t3

595

600

605

610

615

620

625

6 3

635

640

645

650

655

€GQ

665

670

675

6 S

6S5

690

695

700

705

710

715

720

725

730

735

740

745

750

755

760

765

770

775

780

yo=:

! < F R N t m d f q r p t f d n 1 > Rt.n_riqf 1 : ! RETURN

F i n d n u 1 1 : ! B=l

GOSUB Outputs

Loopl: ! 3UBL0QP OF FIHDNULL

OUTPUT i?S.a; "TS MR"

ENTER CiSa; Rmpl

OUTPUT H-Ra; "Ml " ENTER C-RajRriis Y = R • ( R m p 1+37 :: •• 1500 ! R P P R X I M R T E F R M U L R OF BE S S E L \\ U L

L

U T P U T I? R a ; "

fl N "

; Y ; " V L U P

" NRIT .3

OUTPUT il-Sa; "TS MR"

ENTER eSa; Newarnp

IF H e w arn p > R rn pi THEN GOT R e f i n € GOTO Loopl

R t- f 1 n e : ! 3 U B L P OF F I N D N U L L (2 N D R N D 3 R D C H E C K F N U L L WITH

E = B+1 ! SUCCESSIVELY SMRLLER RMPLITUDE INCREMENTS::'

IF B>3 THEN GOTO End loopl

R=R-.01

Z = Y*2

OUTPUT @Ra; "RN";Z; "VL DN"

WRIT .3

GOTO Loopl

E n d 1 o o pi: ! RETURN ! CFOR FINDNULL)

3e-cd_b€s_ck: ! OUTPUT liiSa; "RE";Rb; "KZ VB";Vb;"KZ TS M3 "

; D 1 t f rq ; " KZ TS MR"

ENTER C-Sa; Delta

R E T U R H ! C F R S e c d_b e- s_c k > Calc: ! CRLCULRTION OF FREQUENCY RESPONSE

R = 2 * L G T ( R m s • V 1 k h z > - 2 * L G T <: M d t ' r q

••• 1 > P R I N T " M D U L R T I N PRE Q U E N C Y

=

P R I N T M d f r q

; " K HZ"; " V M D "

; R m s ; " V "

; " R

=

"

; R

; " D B

RETURN ! CFOR Calc)

End: ! LOCRL C^Hpib

RBORT C^iNpib

END

23
Appendix B Computer Program for Curve Fitting

190

185

lie

1 15

129

125

131-3 135

140

145

159

155

169

165

179

175

139

135

199

195

299

295

219

215

229

225

239

235

249

245

259

255

269

265

279

275

2 3 9

285

299

295

3 9 9

395

319

315

320

325

339

335

349

! RNRLIT

PRINTER IS 1 DIM Tit. l€f[39]

PRINT i:HR$'::12::'

PR I NT TREXV (1,1 9 > , " RNRL I T

PRINT TREKV( 1

, 18) , "< TITLE ? NRXIMUN 33 CHRRRCTERS

L INPUT "123456739 123456739 123456739 12345673 ", T i t 1 e:l IF Tit. U*="" THEN

Tit U$="RNRLIT"

ELSE

END IF

G S U B G f j h_3- € t u p ! c o m rn an d -a- I o g e t i n t, o g r a. p h i c s rn o d 8 an d

G S U B I n c r €• m t. 1 _c u r '.• e G S U B C I.J r u e t, o f i % G S U B P 1 o t _a_t. o t h e_x

PI ot_next.a_x

PI ot_nex_a_x

Plot._log

PI ot._x_t. ot he_b

Y scale

X scale

=• e t. u p g r i d

GO SUE

GOSUB

! GOSUB

! GOSUB

GOSUB

GOSUB _

GOSUB Title

GOTO End

Gph_setup: ! G I N I T GRRPHICS ON

RLPHR OFF

NINDON -. 19, . 55, -79, 39

CLIP 9, . 5, -69, 15

GRID .95,5,9,9

CLIP OFF

RETURN

I nc remt 1 _c urye : ! DRTR . 493, 13

LORG 5

FOR 1=1 TO 5

RERD X RERD Y MOVE X,Y

LRBEL "y"

NEXT I RETURN

Curiv'e_t o_f it: ! DRTR 9, 13, . 334, 3, . 463, -; LORG 5

' FOR 1=1 TO 5

RERD X RERD Y HOVE X,Y

114,3, . 035, -7, . 997, -17,9, -22,

491 , -17, . 493, -22.

8

24
345 LABEL "x"

350 NEXT I 355 RETURN

3 66 PI o t _a_t. o t h e_x : ! 365 MOVE 0,0

370 FOR .J = -25 TO 15 STEP .2

375 R=l . 141

330 X = R

•

:: J + 3 7 >

- 1500

335 PLOT X,..T 390 IF X>.5 THEN GOTO Rtt

395 NEXT J 400 Ret: ! 405 RETURN

410 PI ot._riexta_x: ! 415 MOVE 0,0

420 FOR J=-25 TO 15

425 R= 1.131

4 3 X = R

•• ( J + 37 >

'' 1 5

435 PLOT X,J

440 NEXT J

445 RETURN

450 P 1 o t _n e x_ =i_x : ! 455 MOVE 0,0

460 FOR J=-25 TO 15

465 fl= 1.101

470 X = R

••••

•:: J + 3 7 :> / 1500

475 PLOT X,J

480 NEXT J 485 RETURN

490 PI ot_l og:

!

495 MOVE 0,0

500 FOR J=.001 TO .5 STEP .01

505 Y='::LOG(.J::'*(20>+55>x3

510 PLOT .j,y

515 NEXT J 520 RETURN

525 P 1 o t _x_t. o t h e_b : ! 530 MOVE 0,0

535 FOR J=.001 TO .5 STEP .01

5 4 m Y

= - ( c: .J * 1 >

•• 2 - >:: 1 3 X 1 . 5 :;i > * ( 1 . 5 > 545 PLOT .J,V

550 NEXT J

555 RETURN

560 Y_scale: ! 565 FOR I =-60 TO 10 STEP 10

570 MOVE -.03,

I

575 LRBEL I 530 NEXT I 585 RETURN

590 X scale: !

25
595

6 m 665

616

615

626

625

6 3 6

6 3 5

646

645

656

655

666

665

676

675

686

63 5

6 9 6

695

766

765

716

715

726

725

736

735

746

745

756

755

F

n

L

h^R

Ti t.

f1L

R

R i rii

6

G

End

6

6

U6

Spi

i:_dP

6

G End

G

P

E

OR

V

RB

EX

ET

1 e OV

RE

ET

O '.•' FF

NOT

N

N

N

H

N

HH

H

H

Hn

:

I.J rri RI

UT

OT

i t RR

RI

HD

= 6 TO

1,19

I

IH

I

;TEP . 1

IJRH

T 1 t. 1 ^

I EY

D GOTO End

S p i n

!

i' 6 L

( 1 L

ci I

(• 4 G

T' 5 L

i' 6 L

t" 7 L

y 8 L

(• 9 L

REEL

REEL

T

T

T REEL

REEL

REEL

REEL

REEL

n U n P G R R P H I C S "

, 3 G T G _d u n i p END", 3 GOTO End it

1 n

i ri

i n R E N V E T H I S "

, 3 G T R e r.i o '... >. S F T K E Y D I S P "

, 3 G T R e ni o '..' e-

:: P R ESS R H Y N U M "

, 3 G T R e- m o '.,.'

.

OR LETTER K E

Y

"

, 3 G T R i m

o

m . T R E G R I H >

"

, 3 G T R e rn o '...' e GOTO Spin

ER IS

IT 2;"

S p i n

761

N"

ICS OFF

CHR:f (12>

26
NBS-n4A (REV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN-1093

2. Performing Organ. Report No. 3. Publ ication Date

March 1986

4. TITLE AND SUBTITLE

Automatic Frequency Response of Frequency- Modulated Generators Using the

Bessel Null Method

5. AUTHOR(S)

E. J. Major, E. M. Livingston, R. T. Adair

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Sponsored in part by

U.S. Army CECOM

Ft. Monmouth, New Jersey

10. SUPPLEMENTARY NOTES

I^J Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most si gnificant information. If document includes a significant

bibliography or literature survey, mention It here)

This paper describes a Bessel null technique to measure the frequency response of a frequency-modulated rf carrier and a program

to automate frequency response measurements of signal generators

with output frequencies from 0.450 to 2000 MHz. The measurements

obtained using this technique are more precise than those obtained

by a highly trained technician using a manual system.

Automated measurement of this process is desirable since the

manual method is subject to the following problems: (1) excessive

time, (2) error in finding the null, and (3) lack of assurance

that the null is the first Bessel null. Automated measurements

can be performed using a system controller, a spectrum analyzer, a function generator, and a voltmeter (all of which must be compatible and controllable remotely).

The nonlinear relationship between the modulating signal

amplitude and the center frequency amplitude of the carrier is a major obstacle to automated measurement. This problem was solved

by obtaining an approximate formula for this nonlinear curve.

Assurance that the null found is the first Bessel null is provided by the analysis of the frequency response of the signal

generator under test as displayed on the spectrum analyzer.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

automated; Bessel null; curve fitting; frequency-modulated generators; frequency

response; linearization

13. AVAILABILITY

031 Unlimited

I I For Official Distribution. Do Not Release to NTIS

[X] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402. Q^ Order From National Technical Information Service (NTIS), Springfield, VA. 22 16 I

14. NO. OF

PRINTED PAGES

32

15. Price

USCOMM-DC 6043-P80

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