which is proportional to current flow. In 
this ideal case, variations in the speed of 
sound do not affect the flow measurement. 
There is, however, a small but finite 
time delay in the electronic circuitry of 
each velocimeter which complicates the expres- 
sion for f,. The individual velocimeter out- 
put frequencies taking the time delay (t) into 
account are -- 
C+v 
f= L = CRY |- Cty Fc 
' \+ trv) 1 fle 
Cc- 
(ee gt Cote 
27 | t@-v) ~ L L 
ig 
Typical values for the velocimeters used are 
(C + v)= 1500 meters per second, L = 0.15 
meter and t = 0.6 microseconds. The valve of 
t(C + v)/L > .006 is small enough that the 
higher order terms in the expansions can be 
ignored. The expression for f,, is now -- 
= ev 
ree 
and the percentave error in f, caused by a 
change in C can be shown to be 0.012 times 
the percentage change in © (see Appendix A). 
This makes the error so small that it can be 
ignored. 
. 200 
nae 
The foregoing discussion does not take 
into account differences in path length and 
delay times in the circuitry between the two 
velocimeters. The effect of having these dif- 
ferences is to limit the resolution of the 
instrument. These problems have been invest= 
ligated at LMSC and it has been found in lab- 
oratory models that they limit the resolution 
to +0.1 foot per second. 
DESIGN 
In order to achieve the necessary sensi- 
tivity for a flowmeter of practical dimensions, 
some multiplication of the individual sing- 
around frequencies, before subtraction, is 
necessarye A flowmeter with the desired char- 
acteristics is shown in the block diagram of 
Fige 1. The function of the balanced modu- 
lator is to take the sum and the difference 
of the frequencies of the two signals being 
191 
injected into it. By using the low-pass and 
high-pass filters it is then possible to obtain 
signals prooortional to current speed and vel- 
ocity of sound, respectively. Such an instru- 
ment, in the form of an initial laboratory 
prototype, has been built at LMSC md has been 
shown to function in the laboratory. The gen= 
eral form of this prototype is shown in Fig. 2. 
APPENDIX A 
Taking the expression for the output fre- 
quency -- 
f. 
= 2v _ 2tc 
Vi eee 
it is possible to find the change in this fre- 
quency caused by a change in the speed of sound 
@ == 
The percentage error in fy caused by a change 
A C in the speed of sound is then shown to be - 
A fy 
x100 = 
ees 
2c AG ZC 
ee Sal eA Ray a oe 
= A [ tS +( tC) ...Ix 100 
For the values of C = 1500 m/sec, L = 0.15 
meter andt = 0.6 microseconds, the value of 
2tC/L is found to be 0.012. Therefore, the per- 
centage error in f, caused by a percentage 
change in the speed of sound AC/C x 100 is 
shown to be 0.012 times the percentage chanze 
in the speed of sound. 
NOMENCLATURE 
C = speed of sound, meters per second 
f = frequency, cycles per second 
f.= outout frequency prooortional to C, cycle 
per second 
