«atlOMl Bureau of Standard,

Library, k.u. Bldi>

OCT 8 1964

^ecknlcciL ^v2ote

Reference book not to be

taken from the library

250

A NOMOGRAM FOR COMPUTING

AND

A NOMOGRAM FOR COMPUTING

g + jb

c + jd

g + jb

c + id

1

HOWARD S. BOWMAN

^1

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards is a principal focal point in the Federal Government for assuring

maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. Its responsibilities include development and maintenance of the national stand- ards of measurement, and the provisions of means for making measurements consistent with those standards; determination of physical constants and properties of materials; development of methods

for testing materials, mechanisms, and structures, and making such tests as may be necessary, particularly for government agencies; cooperation in the establishment of standard practices for incorporation in codes and specifications; advisory service to government agencies on scientific and technical problems; invention and development of devices to serve special needs of the Government; assistance

to industry, business, and consumers in the development and acceptance of commercial standards and

simplified trade practice recommendations; administration of programs in cooperation with United

States business groups and standards organizations for the development of international standards of practice; and maintenance of a clearinghouse for the collection and dissemination of scientific, technical, and engineering information. The scope of the Bureau's activities is suggested in the following

listing of its four Institutes and their organizational units. Institute for Basic Standards. Electricity. Metrology. Heat. Radiation Fhysics. Mechanics. Applied Mathematics. Atomic Physics. Physical Chemistry. Laboratory Astrophysics.* Radio Standards Laboratory: Radio Standards Physics; Radio Standards Engineering.** Office of Standard Ref- erence Data. Institute for Materials Research. Analytical Chemistry. Polymers. Metallurgy. Inorganic Materials. Reactor Radiations. Cryogenics.** Office of Standard Reference Materials. Central Radio Propagation Laboratory.** Ionosphere Research and Propagation. Troposphere

and Space Telecommunications. Radio Systems. Upper Atmosphere and Space Physics.

Institute for Applied Technology. Textiles and Apparel Technology Center. Building Research.

Industrial Equipment. Information Technology. Performance Test Development. Instrumentation.

Transport Systems. Office of Technical Services. Office of Weights and Measures. Office of Engineering Standards. Office of Industrial Services.

* NBS Group, Joint Institute for Laboratory Astrophysics at the University of G)lorado. ** Located at Boulder, Colorado.
NATIONAL BUREAU OF STANDARDS

Technical Note 250

ISSUED OCTOBER 2, 1964

A NOMOGRAM FOR COMPUTING

^^

c+jd

AND

A NOMOGRAM FOR COMPUTING

^^

c+jd

Howard S. Bowman

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale bv the Sniieriiiffiident of Documents, U.S. Govenuiiciil I'riiil in;; oilic Washington, D.C., 20402 - Price 15 cents
Contents

Part One. A Nomogram to Compute

a + .jb c + jd

1.

2.

3.

4,

5.

6.

7.

In troduc t ion

Construction Details

Application

Directions for Finding Real and Imaginary Values —

Limitations -

Tables

Drawing of Nomogram to Compute

a + .1b c + jd

Part TwOo A Nomogram to Compute

a + .jb c + jd

1.

2.

3.

4.

Construction Details

Instructions for Finding Absolute ValueJustification

Drawing of Nomogram to Compute ;—rr

c + jdj

Page

- 1

- 2

- 3

- 3

- 4

- 5,6

- 7

References8

8

9

12

13

11
A Nomogram for Computing ^ • ^ ' c + jd

and

A Nomogram for Computing

Howard S. Bovmian

c + jd

This report gives two nomograms designed

and constructed for use in computing complex

ratios of two dimensional vector quantities.

One nomogram can be used to compute the real and

imaginary parts of a vector ratio_, and the other

to compute the absolute value of such a function.

Whenever a large number of vector ratios are required in data reduction^ the use of these

nomograms saves time in manual processing.

Part One. A Nomogram for Computing ,

' •{ . c + jd

1, Inroduction

Repeated solutions of intricate problems are often facilitated by

the construction and use of time saving nomograms or abacs as they are sometimes called. The theory of nomograms is found in many publications

[1^2,3_,4^ 5]^ The problem which initiated this study was that of repeated

computations of the difference between two complex ratios required for

the evaluation of the mechanical impedance of the human forehead and mastoid [6] , The calibration constant^ K, and the mechanical impedance^ Z,

Involved in these determinations are written as follows:

K -

Z =

(1)

where the E's are measured complex quantities.

Figures in brackets indicate the literature references at the end of

this paper.
Each complex ratio inside the brackets of (1) above can be represented by an expression

a + .jb ^ 1 c + jd N

where N = -7 • * d c

f-!)-nM;j (^>

i

The above expression (2) reduces the number of operations for the

solution of K or Z in that one abac can be used to compute an expression

such as

d' ~ c'

for either real or imaginary part of the complex ratio regardless of the

signs of the variables. Tables I and II given below were made for use with the nomogram to determine the particular process and the proper sign

for the final answers when one is given the signs of the variables.

Other methods of computing the ratio of complex numbers in the rec- tangular form^ such as conversion to magnitude and phase and then back

again_, by slide rule or tables [7] are time consximing and tedious. The

slide rule computation of equation (2) is fairly rapid but more tedious

than using the nomogram and is not an automatic process.

2, Construction Details

In order to perform the products and ratios in the computation^

three logarithmic scales were constructed. The lines are straight_, vertical^ parallel and equidistant. The left vertical line is a five cycle

ascending logarithmic scale while the right vertical line is a seven

cycle descending logarithmic scale. One cycle of the center ascending

logarithmic scale is half as long as one cycle on either of the two other

logarithmic scales. The scales are aligned so that a line drawn through

10 on each of the outer scales is horizontal and intersects the center

line at 1. If a straight line is drawn so that it intersects the three

logarithmic scales^ the point of intersection of the center scale equals

the ratio of the points of intersection of the first and third scales.

From points corresponding to c/d on the center logarithmic scale^ straight

lines are drawn to the left vertical to points corresponding to computed

values of 1/N for given values of c and d, (To prevent overlapping lines,

capital letters are used in some cases. See nomogram and instructions.)

Three linear scales are used for the addition and subtraction^ subsequently called the svraimation and difference processes. In this con- struction the three vertical straight lines are drawn parallel and equidistant. The right and left linear verticals are divided into twenty

equal divisions using horizontal lines labeled in a descending order of

tenths. The center linear scale has twice as many xinits as either of the

other linear scales and is labeled accordingly. Guide lines connect cor- responding points of the left linear scale to the right logarithmic scale.
On the model nomogram the left logarithmic scale is 12 1/2 inches

long and 8 inches away from the right logarithmic scale which is 17 1/2

inches long. The linear scales are 7 1/2 inches long and 2 inches apart,

The distance between the sets of logarithmic and linear scales is 2 1/2

inches. For convenient usage the drawing is mounted on an 18" x 18"

stiff paper board and covered with a sheet of transparent vinyl acetate

which can be marked and easily erased.

Each scale is graduated carefully and labeled with numbers and/or

letters so that the points can be located with a considerable degree of ease and accuracy. (Figure 1 shows the nomogram in reduced size.)

3. Application

The abac was used for computations using data taken from actual

measurements and also sample data for two-dimensional vectors in all four

quadrants. The results were compared with those obtained by use of the

log log vector slide rule, desk calculator and N.B.S. 704 computer. The

average percentage error was less than 3% for 75 computations made by

three inexperienced users of the nomograms. It must be borne in mind

that the accuracy of the results depends on the care taken in drawing and

using the abac.

4. Directions for Using the Nomogram for Finding the

Real and Imaginary Parts of the Complex Ratio —TT^l

For the Real Part:

1. From (c) on I to (d) on III a straight line intersects II. From

this intersection follow the diagonal line, guidelines, or lettered

notation to the corresponding point on I and label that point X^

|_ o N

2. From (d) on I to (a) on III a straight line intersects II. A

straight line through this intersection and the point X^ intersects

r„ la

III. Label this new intersection X,

.

^1 " N

• d

3. From (c) on I to (b) on III a straight line intersects II. A

straight line through this intersection and the point X^ intersects

III. Label this new intersection X^. X^ = ^ o - (Consult Table I

and determine the process and sign of the real part of the ratio

according to the signs of the variables a, b, c, & d.)

Summation Process:

(Note: If the ratio of X^ to X^ or X^ to X^ is greater than 20,000 then

3
the quantities are beyond the range of the following siammation and difference processes of this nomogram and must be added mentally.)

Follow the diagonal line or guide lines from X- to the corresponding

point on IV. Label that point X'

.

1 N

• d

2, Then follow the diagonal line or guide lines from X„ to the corres- ponding point on VI and label that point X', ^2 - N

• c

3. A straight line from X' to X' intersects V at the result (real or imaginary part of the complex ratio).

Difference Process:

1. Locate the point on V corresponding to the larger J xj on III^ and

mark that point X' or X' respectively.

2. From the smaller |x[ on III, follow the line or guide lines to the

corresponding point on IV and mark that point X' or X' respectively.

3. A straight line through X' and X' intersects VI at the results (real or imaginary part of the complex ratio).

For the Imaginary Part:

1. If the real part was determined, first erase X-, X , X', and X'. If

X has not been determined, follow step one under the real part

solution to locate X

o

2. From (d) on I to (b) on III a straight line intersects II. A

straight line through this intersection and the point X intersects

III. Label this new intersection X-

.

3. From (c) on I to (a) on III a straight line intersects II. A

straight line through this intersection and the point X intersects

III. Label this new intersection X-.

4. Consult Table II and determine the process and sign of the imaginary

part of the ratio and follow directions given under the summation or difference process described above.

5. Limitations

With this model nomogram the range of the variables used is .01 <

a, b, c, d, > 1000 with the following limitations:

O.Ul <-^,-^,f,^,^,f,< 100. d-^c-'b-'c-'d'd-'

By making the linear scales and the projected lines from the logarithmic

scale movable, the range on the results can be extended. In moving the

linear scale in order to cover certain ranges on X^ and X„, the decimal
point on the linear scale is shifted to correspond to the particular

number associated with each projected line. The results of the com- putations can be read accurately to two places with the model nomogram.

Table I. Real Part

Conditions on signs

of the variables Process Sign of result

All four signs the same Summation +

Three signs the same, one different:

a or c of

different sign Difference

. i£|xJ>|X2|

b or d of

different sign Difference + if|xJ>|X2|

- if|xj<|xj

Two signs the samei

Signs alternate in

the order a, b^ c, d. Summation

Two consecutive signs

the same in the order

a, b, c, do Summation
Table II, Imaginary Part

Conditions on signs

of the variables

All four signs the same

Three signs the same^ one different:

a or d of

different sign

b or c of

different sign

Process

Difference

Summation

Summation

Sign of results

+ if|x^l>|x2|

- if|x^l<|X2l

Two signs the same:

Signs alternate in

the order a^ b^ d^ c.

Two consecutive signs

alike in the order a, b^

d_, c.

Difference

Difference

+ if|xJ>lX2|

- if|x^l<|X2|

+ if|x^|<|X2|
100 -

100 - 900

1000HI

A NOMOGRAM TO COMPUTE REAL & IMAGINARY PARTS OF

(.01 i a,b,c,ds 1000)

Fig. I

g + jb c + jd
Part Two. A Nomogram for Computing a + jb c + jd

The nomogram and instructions given

below can be used to compute the absolute

value of the ratio of two vector quantities,

1. Construction Details

This nomogram has three vertical lines parallel and equidistant.

The left vertical is an ascending logarithmic scale while the right

vertical is a descending logarithmic scale of the same cycle and length.

The center vertical line is graduated into an ascending logarithmic

scale having cycles one fourth as long as one cycle of either of the two

outer logarithmic scales. The scales are aligned so that a horizontal

line through the 10 on the outer scales intersects the center at 1. The

diagonal lines are drawn from points corresponding to numbers y on the

center vertical to points corresponding to \/T~+^ on the left vertical

and to points corresponding to vi + 1/y on the right vertical. The

diagonals on the right are linear extensions of the diagonals on the

left.

The range on the variables as well as the accuracy of the results

have limitations only due to the practical size of the drawings.

2. Instructions for Using the Nomogram for Computing — .

,

c + jd

From a point corresponding to (b) on the left vertical scale to

the point corresponding to (a) on the right vertical scale^ a straight

line intersects the center vertical scale. From this intersection

follow the diagonal line or diagonal guide lines to a corresponding

point on the left vertical scale. And from this point to the point

corresponding to (c) on the right vertical scale^ a straight line intersects the center vertical. From this intersection follow the horizontal

line or horizontal guide lines to a corresponding point on the left

vertical scale and circle that point.

From a point corresponding to (d) on the right vertical scale to

the point corresponding to (c) on the left vertical scale^ a straight

line intersects the center vertical scale. From this intersection

follow the diagonal line or diagonal guide lines to a corresponding

point on the right vertical scale. And from this point to the point

corresponding to (a) on the left vertical scale, a straight line intersects the center vertical. From this intersection follow the horizontal

line or horizontal guide lines to a corresponding point on the right

8
vertical scale and circle that point. A straight line through the

circled points intersects the center vertical at the answer.

The accuracy of this nomogram depends on the care in drawing_, and

in using it; also the over all size and the range of the scale on which

the answer is found contribute to the accuracy. The drawing Figure 2

in this report is condensed from the model which yielded two place

accuracy.

3. Justification

Let L represent the horizontal line extending from the point cor- responding to 10 on the left axis to the point corresponding to 10 on

the right axis.

Let the distance from L to the point corresponding to 100 on the

left axis be one unit.

The scales are so constructed that the directed distance (considering the ascending direction as positive) from L to a point cor- responding to a number x on the left axis is

i(x) = log X - 1,

from L to a point corresponding to a number y on the center axis is

4 m(y) = /. ,

and from L to a point corresponding to a number Z on the right axis is

r(Z) = 1 - log Z.

The diagonals on the right are linear extensions of the diagonals on the left since

£(Vl + y) + r(\/l + 1/y) ^ (logVl + y - 1) + 1 - logV/l + 1/y)

2 2

2 ^°\y 1 + l/yj 4 \l + 1/yJ 4 '•°\ y + 1 ^

« r log y = m(y) so that m(y) is the average of i(Vl + y) and r(vA + l/y).

The following Is a proof that the results obtained from the foregoing instructions is the correct one:
A straight line drawn from the point corresponding to (b) on the left

axis to the point corresponding to (a) on the right axis will intersect

the center axis at a point corresponding to some number y whose directed

distance m(y) from L is the average of ^(b) and r(a) so that

loS y = ^/„>, _ ^(b) + r(a) _ (log b-1) + (1-log a)

4 ^^^ 2 2

1 , b = -. log -

b^

Hence y = —^

or log y = 2 log - = log( ~

The diagonal from this point of intersection will meet the left

axis at a point corresponding tovl + y = vl + \>^ I or, whose directed

distance from L will be A Vl + b /a^ j. The straight line from that

point to the point on the right axis corresponding to c will intersect

the center axis at a point P whose directed distance D^ from L will be

the average of if Vl + b^/a^ j and r(c)^ i:»£» ^12" _ A n/i + b^/a^j + r(c) _ jogN/l + b^/a^ - ij + (1 - log c)

2

The point to be circled according to the first paragraph of the

instructions is on the same horizontal line as P and hence will have

the same directed distance D, from L.

Similarly^ it may be shown that the point to be circled according

to the second paragraph of the instructions will have a directed

distance

D^ = 1/2 log( >

' , ) from L.

2 \Vl + (d/c)2

The straight line joining the circled points will intersect the

center axis at a point corresponding to some number m(a) from L will be

the average of D- and D„ so that

1/4 log a = m(a)

10
aVl + (b/a)2

(d/c)

1 1 7 ^°S 1 —^^ ^ cV 1 + (ci/c)2

Henee^

a = ^I H- (b/a)2 ^ V^TT^ ^ I a + ib | c\ll + (d/c)2 Vc^ H- d2 jc + jd

In this effort the writer acknowledges and appreciates the support

of the NBS Sound Section staff and especially Mrs, E.L.Ro Corliss for

the initial idea^ Mrs. P.G. Weissler and Mr. E.L. Smith for helpful

suggestions and final checking^ and Mr, D.R. DeAngelis for the careful

drawings.

11
A NOMOGRAM TO COMPUTE

(lia.b.c.d S200)

g+jb

c + jd

200

100

200
References

[1] A.S. Levens, Nomosraphy. 2nd Edition^ John Wiley & Sons_, Inc.^

New York^ 1959.

[2] H.J. Allcock^ et.al .. The Nomogram, 4th Edition^ Pitman Publishing

Corporation, New York, 1950,

[3] A.S. Levens, Nomography, John Wiley & Sons, Inc., New York, 1948.

[4] L.H. Johnson, Nomography and Empirical Equations, John Wiley &

Sons, Inc., New York, 1952.

[5] L, Ivan Epstein, Nomography, Interscience Publishers, Inc.,

New York, 1958.

[6] E.L.R. Corliss, et.al . , "Hearing by Bone Conduction", (not yet

published, NBS, Soxond Section).

[7] John Todd, Table of Arctangents of Rational Numbers, U.S.

Department of Commerce, National Bureau of Standards, Applied

Series II, 1951<,

13

i, U.S. GOVERNMENT PRINTING OFFICE : 1964 O - 744-384








;.:(;.. >

I, :

'•. I' ^ '. ' .' t.. '1

m

'^M