A tangent function generator is provided which utilizes input signals proportionate to the sine and cosine of a given angle. The equipment accomplishes a trigonometric manipulation which forms an output signal substantially independent of the input reference voltage but proportional to the tangent of the angle. The device uses a multiplier and operational amplifier

Bregman, B. J.

English

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NATIONAL AERONAUTICS AND SPACE ADMINISTRAfION
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U.S.
REPLY TO
ATTN OF: GP
<NASA-Case-MSC-13907-1) -• D E R I V A T I O N OF A
T A N G E N T FUNCTION USING AM I N T E G R A T E D
CIBCUIT F O O B - Q O A D R A N T MUI-TIPLIER Patent
N73-26230 )
(Martin Marietta Corp.). 4 p CSCL 09C Unclas
00/10 06U71
TO: KSI/Scientific & Technical Information Division
Attention: Miss Winnie M. Morgan
FROM: GP/Office of Assistant General Counsel for
Patent Matters
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In accordance with the procedures agreed upon by Code GP
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Elizabeth A. Carter
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Copy of patent cited above
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United States Patent
Fletcher et al. [45]
3,737,639
June 5,1973
[54] DERIVATION OF A TANGENT
FUNCTION USING AN INTEGRATED
CIRCUIT FOUR-QUADRANT
MULTIPLIER
[76] Inventors: James C. Fletcher, Administrator of
the National Aeronautics and Space
Administration with respect to an
invention by; Bernard J. Bregman,
Littleton, Colo.
[22] Filed: May 17,1972
[21] Appl. No.: 254,177
[52} U.S. Cl. 235/186, 235/194, 235/197
[51 ] Int. Cl ....: G06g 7/16, G06g 7/22
[58] Field of Search 235/197, 186, 189,
235/194, 195, 196, 150.25, 61.5 E
[56] References Cited
UNITED STATES PATENTS
2,953,303 9/1960 Sedgfield 235/150.25
2,948,474 8/1960 Seay 235/197
3,250,904 5/1966 Vasuetal 235/186
3,496,565 2/1970 Jenkins.: 235/186 X
3,601,593 8/1971 Preston 235/197 X
3,646,337 2/1972 Bifulco.Jr 235/186
Primary Examiner—Joseph F. Ruggiero
Attorney— Marvin J. Marnock, Marvin F. Matthews
and John R. Manning et al.
[57] ABSTRACT
A tangent function generator is provided which util-
izes input signals proportionate to the sine and cosine
of a given angle. The equipment accomplishes a
trigonometric manipulation which forms an output
signal substantially independent of the input reference
voltage but proportional to the tangent of the angle.
The device uses a multiplier and operational amplifier.
4 Claims, 1 Drawing Figure
PATENIEbJU" 51973 3,737,639
VR
^10 SINE
 irtU; GENERATOR ^
ANGLE \
TO flF \
MEASURED W \
COSINE
GENERA TOR )
,/
22
A A A A A
18
r
lutl It Tl&f 1 C~D
2^
20
3,737,639
1 2
DERIVATION OF A TANGENT FUNCTION USING perhaps 10 volts RMS which is provided to the sine
AN INTEGRATED CIRCUIT FOUR-QUADRANT and cosine generators 12 and 14 to cause the output
MULTIPLIER . signals to swing between selected levels. Quite obvi-
BACKGROUND OF THE INVENTS -10 ±! ""
In control systems, trigonometric functions are quite The apparatus further incorporates an input resistor
often required. Many trigonometric functions have 16. The resistor 16 provides an input for an operational
been generated in the past through the use of large me- amplifier 18. While many operational amplifiers are
chanical resolvers. However, such equipment is unduly made, a suitable operational amplifier is that manufac-
complicated and being mechanical in nature, is sub- 10 tured by National Semi-Conductor bearing model num-
stantially larger and consumes more power than can be ber LM101A. This is an integrated circuit operational
permitted in many cases. Diode circuits have been used amplifier which is more than suitable for the purposes
in the past to approximate a range of trigonometric herein mentioned. The connection of the operational
functions but they have no versatility inasmuch as the amplifier 18 to the sine generator 12 through the resis-
approximate curve generated is not usually subject to 15
 tor J6 heips form an output signai on a conductor 20
change. Accordingly, many devices known in the prior
 which output signal is the signal proportionate to the
art have only provided less than adequate simulation or tangent of the particular angle generated by the appa-
approximation of a particular trigonometric function.
 ratus JQ as will be described in detail hereinafter.
While some function generators for sine and cosine
 Tne operational amplifier is incorporated in the cir-
functions have met with adequate success, function 20
 cuitrywith a feedback loop. Hence, a feedback resistor
generators for the tangent or cotangent have been
 22 is connected to the input of the operational ampli-
rather deficient. Generators of this sort are quite often
 fier 18 The signa, supplied to the resistor 16 is not the
required in equipment such as inertial guidance sys-
 si , directly from the output line 20 but rather the
terns, gun control radar systems, and various and sun-
 QUtput from the multiplier circuit 24. Again, many mul-
dry control circuits. 23
 t.pjjer circuits are known but a suitabie integrated cir-
SUMMARY OF THE INVENTION cuit multiplier is that manufactured by Motorola and
_ bearing model number MC1595L. Quite clearly, other
The present invention is summarized as providing an
 multi Hers can be used if desired. MoreoVer, a conduc-
•ntegraed circuit tangent function generator. More
 tor 26 supplies the signal from the cosine generator 14
particularly, it utilizes as an input, signals representa- 3U
 t- JT. 1 - 2 4
tive of the sine and cosine of a given angle. The cir- _ .? .. ' . .. ..
 e ..
... . . f .._ ° . . For consideration of the operation of the circuitry,
cuitry incorporates an operational amplifier and a mul- .. . . , , , . , , . . .„
.. ,. . . . . . i.
 f * • , some equations have been developed below which willtipher circuit which together form an output signal pro- . . A . r
portionate to the tangent of the angle. The output sig- ex|'ain °Peratlon- "
nal formed is substantially independent of the* refer- 35 The output of the mult.pher c.rcu.t 24 will be first
ence voltage of the sine and cosine functions. The out- Considered. This output voltage .s gwen below ,n Equa-
put signal has a high degree of accuracy, having an tlDn
error of approximately 10 minutes of arc or less, for an- ytt = (VR Cos 6) ( Vm)
gles between 445° and —45°. _ , , . . . . .40 Equation 1 reflects that the output of the multiplier
BRIEF DESCRIPTION OF THE DRAWING circuit is the product of the two input voltages. One
The single FIGURE is a schematic block diagram of inPut volta«e is from the cosine generator 14 while the
the circuit of the present invention which generates a other is the outPut volta«e from the operational ampli-
tangent function. "er
45 Next, the voltage output by the operational amplifier
DESCRIPTION OF THE PREFERRED should be considered. In the operational amplifier 18,
EMBODIMENT the input impedance is quite high, typically measuring
In the single view, the numeral 10 indicates some in the megohm range or greater. In effect, this means
form of apparatus which generates an angle to be mea- that *« inPut current is so negligible as to be almost
sured. The angle to be measured can be generated ei- 5 nonexistent and indeed, it is so small that it can be ig-
ther mechanically or electronically. It may have the "ored. Accordingly, the current flowing through the
form of a rotation of a particular shaft or other me- resistor 16 must equal the current flowing through the
chanical apparatus. On the other hand, it may achieve resistor 22. The current flowing through the resistor 16
the form of an analog or digital representation of the '« » function of the output signal from the sine genera-
angle to be measured. In any case, the angle source 10 tor 12. The current flowing through the resistor 22 is
is incorporated in the single drawing to point out very a function of the signal from the multiplier 24. Equat-
generally that the angle is generated in some suitable ing these two currents. Equation 2 is thus obtained:
form and is subsequently used by the function genera- ., _. - ,_ _ _., . _
tor comprising the present invention. The apparatus 10
 6Q * ' " "' **
is connected in some suitable manner to a sine genera- Equation 2 can be arranged to a different form by re-
tor 12. Additionally, the angle to be measured is sup- arranging variables and hence, Equation 3 is thus ob-
plied as an input to a cosine generator 14. Both the sine tained:
and cosine generators are provided with the same refer- _ „.
ence voltage inputs which are indicated in the drawing.
 6J
 K
" ~ *»
/K>« <• y» *m "'
As will be appreciated, the maximum value of the sine Since the same quantity is found on the left hand side
and cosine functions swing between 1 .000 to —1 .000. of both Equation 1 and Equation 3, the right hand sides
The reference voltage is a suitable voltage level such as can be equated to obtain Equation 4:
3,737,639
3 4
i/ iv r< a \ — o ID fir c- formation of a negative or positive output signal.
-rM (VR t-os 9) - KitlRit (VR Sin 0) The foregoing js directed to the preferred embodi-
Rearranging variables to isolate the output voltage of ment but the scope of the present invention is deter-
the operation amplifier on the left, Equation 5 is thus mined by the claims which are as follows:
obtained: 5 We claim:
v --» /» (V <u n aM,v r ™ n \ 1. A tangent function generator for generating a sig-
"a> - KZJ/KW ^R &»n «)/lK f t cos «) na, proportionate to the tangent of a given angle de-
On inspecting Equation 5, it will be noted that the rived from inputs of the sine and cosine of the given
reference voltage drops out and moreover, the trigono- angle wherein the sine and cosine functions are refer-
metric functions can be rewritten to incorporate the 10 enced to a given reference voltage and comprising a
tangent. Thus, Equation 6 is obtained: circuit means for forming an output signal proportional
_ _ to the function Sin 0/Cos 6 where the trigonometric
KZO
 ~ ~
RWRI« tan fl functions of 0 are represented by the input signals to
On inspecting Equation 6, it will be noted that the the generator and said output signal is substantially in-
output signal is completely independent of the refer- is dependent of the given reference voltage and is propor-
ence voltage. Moreover, it is proportional to the two tional to the ratio of the value of at least two fixed cir-
resistors and the tangent of the angle being measured. cuit components, said circuit means including opera-
The foregoing describes the tangent generator of the tional amplifier means coupled to receive said sine
present invention. However, if the sine and cosine gen- function signal as an input and multiplier circuit means
erators 12 and 14 shown in the single drawing are re- 20 having two inputs and forming an output corresponding
versed in position, a cotangent generator is provided, to the multiplication of the inputs, one of said two in-
it being observed that the sine and cosine functions of puts being the cosine function signal and the other
Equation 5 would be inverted. input being provided from the output of the operational
In application, the circuitry shown in the single draw- amplifier.
ing has been implemented and has been used over a 25 2. The function generator of claim 1 wherein said
range of ±45° variation. In such application, the error amplifier receives a second signal as a feed back signal
of the tangent generator has been less than 10 minutes applied to the input of said amplifier which signal is de-
of arc. rived from the output of said multiplier circuit means.
The foregoing is directed to the preferred embodi- 3. A function generator as described in claim 2
ment of the present invention. As will be understood, 30 wherein said multiplier means is connected at its output
other types and sources of supply of integrated circuits through a first fixed resistor and the sine function is
can be utilized. Numerous operational amplifiers are in input through a second fixed resistor and the tangent
the market including those manufactured by Fairchild; output signal is proportional to the ratio of the former
Texas Instruments, and other manufacturers. Addition- to the latter.
ally, other multipliers are also known but the multiplier 35 4. A function generator of the character described in
mentioned for the preferred embodiment is preferable claim 2 wherein said multiplier circuit means is a four-
inasmuch as it has the ability to handle four-quadrant quadrant multiplier for multiplying input signals of like
signals. In other words, the multiplier can handle input or dissimilar polarities,
signals of like or dissimilar polarities. This permits the * * * * *
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Derivation of a tangent function using an integrated circuit four-quadrant multiplier

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