the robinson calculator

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The Robinson Calculator

A Simple Computer for Complex Numbers

Introduction

The Electrical Research Board (Australia) was set up in 1945, its general objective was the fostering

of electrical research in universities and it provided extensive funding for a large number of projects.

These led to the establishment of a model power system facility at the University of Sydney in the

1960s.

To assist in the calculation and analysis of complex electrical engineering problems, Mr R.B.

Robinson of the Sydney University invented a calculator that could manipulate 2-D vector quantities.

The Electrical Research Board decided to fund the production of these calculators. The production

models were made by the Amalgamated Wireless (Australasia) Pty. Ltd (AMA). from 1957-1962.

From the research undertaken I have deduced that they were sold at Sydney University in the early

60s for approx. 25 Australian Pounds. It is unknown how many examples where made. But I believe

most where used at the Sydney University on ERA funded projects as the origin of the only known

samples are from that university.

Only a few examples of this calculator remain, I have one and the Sydney Powerhouse Museum hasanother two in their collection.

This paper describes the theory behind the system of calculation and also the production model

construction of the Robinson Calculator.

The Challenge

In the investigation or design of electrical networks the electrical engineer must manipulate 2

dimensional vectors representing voltages, currents, impedances (resistances and capacitive/

inductive reactances), etc. Complex notation is universally employed to perform this manipulation,

aided to some extent by the use of scale vector diagrams when great accuracy is unnecessary.

The processes of addition, subtraction, multiplication, division, and reciprocation of complex

numbers must he performed repeatedly in the investigation of even a simple network. The amount

of labour involved in the performance of the last three of the processes mentioned, when using the

ordinary slide-rule, makes the investigation of a simple ac network a lengthy procedure and sets a

practical limit to the complexity of networks which may be attacked by mathematical analysis.

Numerous means have been devised to facilitate such calculations. The calculator described below

has been developed from the constant need to perform network calculations and to avoid the

shortcomings of alternative methods.


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Development of the Theory

Although the calculator is of circular form, the theory on which it is based is more conveniently

developed in linear form, with subsequent transformation of the result, than developed directly in

circular form.

A complex number will be represented by the notation:

L

Note: the operator j is used as the complex operator in the analysis of ac electrical circuits to avoid

confusing the normal mathematical i complex operator with the symbolfor current.

It follows from elementary concepts that;

(1)

Hence,

.............................. (2)

Taking logarithms to any convenient base,

............... (3)

Equation (3) suggests a means of converting a number from Cartesian to polar form and vice versa.

Consider the slide-rule of Fig. 1.

Fig. 1.

Here, scale C is a logarithmic scale as on an ordinary slide-rule. Slide A carries a scale, marked in

angle , of which the divisions are proportional to log sec . Slide B likewise carries a log cosec

scale.

Thus, if the index lines of scales A and B are set to a and b, respectively, on scale C, a cursor set to

the point at which reads equally on both scales A and B will also indicate r on the C scale.


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Comparison of Fig. 1 with equation (3) should make this clear. The method of conversion from polar

to Cartesian form is obvious and more direct. If a major objection to such a device was to be made, it

would be the difficulty of setting a cursor to a position such that reads equally on two scales,

particularly as the direction of increasing is opposite on the two scales.

Consider now the arrangement shown in Fig. 2. This is a direct development from that of Fig. 1 and

overcomes the difficulty mentioned above. Here, the A and B slides are expanded to carry two

dimensional plots of log sec and log cosec , respectively, against , the scale being uniformly

divided and the axis perpendicular to the direction of the previous scales. The slides are shown in

Fig. 2 with their index lines set to a and b on the C scale, as in Fig. 1, and the position of equal on

scales A and B is now given by the intersection of the curves. A cursor line parallel to the axis will, if

set to this intersection, read r on the C scale.

Fig 2

The arrangement of Fig. 2 forms the basis of the calculator to be described in the next section.

The Calculator Design

General Description :

The calculator consists of four circular slides so constrained as to allow only rotation about the

centre of a circular base. The four slides and base are shown separately in Fig. 3. The base carries atits outer edge the C scale. The first slide, slide D, also carries only a circumferential scale, the outer

diameter of which is equal to the inner diameter of the C scale; the next two slides, A and B, carry

only curves, and their outer diameters, which are equal, coincide with the inner diameter of the D

scale. Small transparent projections on these slides extend over the C and D scales and each carries a

cursor line indicating the relative position of curve and scale. The uppermost of slides A and B is

transparent.


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(a) (b) (c)

(a)The base with the C scale(b)The D slide and scales(c)The A slide and Curves(d)The B Slide and Curves(e)The Cursor carrying the

Main and Interpolating

Angle Scales

(d) (e)

Fig 3

The remaining slide, the cursor, is transparent and extends over all scales. It carries both a radial and

a circumferential scale.

Conversion between Polar and Cartesian Formsthe A, B and C Slides

Considering again the arrangement of Fig. 2, it is obvious that this can be transformed to circular

form. The linear C scale of Fig. 2 becomes the circumferential C scale of Fig. 3, logarithmically divided

as before and, in the case illustrated, having two decades around the circumference. The curve slides

A and B of Fig. 3 carry curves relating log sec and log cosec , respectively, circumferentially to

radially, the scale of being uniformly divided between arbitrary limiting radii representing 0 and

90.

Clearly, the relationships between the linear displacements of the slides of Fig. 2 apply also to theangular displacements of those of Fig. 3.

Conversion of a complex number from Cartesian to polar form is accomplished by setting the base

lines of slides A and B to a and b, respectively, on the C scale and the radial cursor line to the

intersection of the curves. The magnitude r may then be read against the cursor line on the C scale.

To read the angle (or to set in the inverse process), the cursor is provided with two scales.

There is firstly a radial uniformly divided scale of along the cursor line. Since the curves of slides A

and B are plotted to this scale radially, may be read by noting the position of the curve intersection

on this cursor scale when the cursor line is set to the intersection. The calibration of this scale of is


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made to cover all four quadrants by co-relating the signs of a and b with the scale and sign of to be

used, or conversely, as shown in Fig. 3.

It is neither convenient nor necessary to make this scale sufficiently long for to be read with a

probable error of less than one degree. For accurate reading of (and particularly for accurate

setting of when near 0 and 90) a subsidiary scale is also carried by the cursor. The derivation of

this scale follows most readily from consideration of the A and B scales of Fig. 1. These scales are

identical in their divisions, being log sec and log cosec scales, differing only in their angle markings

which are everywhere complementary. They are not uniformly divided, but the divisions are well

spaced apart between 45 and 90 on the A scale and between 45 and 0 on the B scale, 90 on A

and 0 on B being at infinity. It follows, therefore, that the distance from the cursor line to the more

distant base line A or B gives an accurate measure of the angle .

To make use of this fact it is not necessary to provide the equivalent circumferential scales to A and

B of Fig. 1. Referring now to Fig. 3, the single circumferential scale shown on the cursor is all that is

required. This cursor scale is a circumferential log sec (or log cosec ) scale having the cursor line

as base and marked only in its more open half It is not calibrated apart from convenient distinction

of the 5 and 10 points because it is used as an interpolating scale in conjunction with the main

radial angle scale described above. One only of the base lines of curves A and B falls on this

interpolating scale at any time and no difficulty is found in interpreting quickly the angle setting.

At this stage it should be pointed out that all accurate readings and settings are made on

circumferential scales. The radial extension of the curves need only be great enough to provide a

reasonable curve intersection. In the linear form of Fig. 2 the curve intersection is always orthogonal.

In the circular form used, the angle of intersection depends upon the ratio of the limiting radii.

In the model constructed this ratio was four, resulting in an intersection angle always better than

50/130 which has proved adequate.

As illustrated by Fig. 3, the model constructed carries two decades on the C scale. Being of eight

inches overall diameter, each decade is approximately ten inches in length, giving good accuracy.

The use of two decades has the result that the curves and the interpolating angle scale on the cursor

cover the range of from 0.6 to 89.4 and the equivalent in the other quadrants. Stated differently,

one part of a complex number is ignored if less than one per cent of the other. If greater accuracy

were required in this direction, more decades could be used or the interpolating scale simply

overlapped on its beginning in spiral form.

Multiplication, Division and Reciprocation -the D Slide

The D slide mentioned earlier is provided in order to facilitate the processes of multiplication,

division and reciprocation. This slide carries a logarithmic scale identical with that of slide C and also

a reversed reciprocal scale. Scales C and D together form a conventional slide which may be used

directly to multiply together the magnitudes of two vectors. The procedure is as follows, the first

complex number is set up using slides A and B in conjunction with the C scale as described in the

previous section for conversion of form. The cursor line is set to the intersection of the curves as

before, but instead of reading the magnitude on the C scale, the base point of the D scale is set to

the cursor line.


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The second complex number is then set up using the A and B slides, now in conjunction with the D

scale. When the cursor line is set to the curve intersection it indicates the magnitude of the product

on the C scale, the angles of the two vectors must be read as each is set up and their sum, which is

the angle of the product vector, computed manually. If one of the factors of the product is known in

polar form the procedure is simplified accordingly.

Should the product be required in Cartesian form it is only necessary to reset slides A and B to

perform the conversion, the result of the multiplication being given essentially in polar form.

To make division of two complex numbers a straightforward process, reciprocal scales and curves

are provided. The reversed scale of slide D has already been mentioned. Reversed curves are marked

on slides A and B (shown dotted in Fig. 3) and a reversed interpolating angle scale provided on the

cursor (not shown in Fig, 3). All of these reversed scales and curves are marked in red, allowing no

confusion with the forward scales, and curves which are marked in black.

The procedure for division of two complex numbers is then exactly the same as for multiplicationexcept that the divisor is set up on the reversed scale D using the reversed curves A and B and the

reversed angle interpolating scale. In other words, the red scales and curves are used in place of the

black ones in setting up the divisor, the procedure being exactly as for multiplication. Of course, the

angles of the two vectors must be subtracted rather than added in order to find the angle of the

quotient

Reciprocation of a complex number is often necessary, as, for example, in converting impedance to

admittance. This is a special case of division and is a simple process on the calculator.

Prototype Model

A circular calculator of this type was made experimentally with hand-drawn scales and curves, and

was constructed in less than one man-week according to Mr Robinson.

Subsequently, the device was used for extensive calculations involving the usual processes and

including Star Delta transformations, etc. Check calculations were performed and it was found

that the error in any operation was consistently less than one per cent and limited at that only by

the hand divided scales.

It was noted that time saving was considerable as compared with alternative means and thedirectness of the method was deemed responsible for the complete absence of errors in reading or

setting in several weeks of intensive use. Parallax error in reading apparently gave no trouble. The

production model was to follow.

The Production Model

The production model as previously stated was made to the order of the Electrical Research Board

by AMA between 1957 and 1962. The finished model is depicted in the following figure;


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Figure 4 THE ROBINSON CALCULATOR

Figure 5 Side elevation


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The calculator was made from 24 distinct parts. 3 metal discs, 3 Perspex discs, 3 plastic discs and

sundry other parts. And measured 220 mm in diameter and approximately 55mm high.

It is easiest to describe the construction of the calculator by the disassembly of the unit. Thefollowing figures describe this.

The rear view of the calculator

showing the hand grip

Removal of the grip required

unfastening two steel nuts, a spring

washer, disc washer and the grip

itself

The removal of the grip uncovers a

felt washer and a disc insert spacer.


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Removal of the Felt washer and

spacer reveals another washer. As

the indent on the felt washer

shows it is sandwiched between

the two washers.

The rear of the calculator after

removal of the washers.

The bolt and metal spacers and

washer from the front of the device

The front Perspex disc with the

metal turning studs attached


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A clear plastic sliding disc

The Imaginary cursor disc (refer (d)

in Fig 3)

Note the metal insert spacer

Another sliding plastic disc


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The real cursor disc (refer (c) in fig

3). Also again note the metal insert

spacer

Another sliding plastic disc

The inner disc containing the C

scale


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The body of the calculator

containing the D scale together

with the DI scale see (b) in fig 3.

Note, contrary to the construction

inferred in fig 3 the D and DI scale

disc are the outermost scales in the

production model

The C scale disc and the D scale disc

assembled

Conclusion.

The calculator described in the paper has the following properties.-

(a) It required only a few operations to perform a given process. For example, to convert a complex

number from Cartesian to Polar form three operations are required, a scale setting for each partof the number and a cursor setting, the magnitude and angle then being read directly. This is the

least practicable number of operations for this process.

(b) Numbers may be directly used in either Cartesian or polar form and results obtained in either

form.

(c) Provision was made for the multiplication, division, and reciprocation of the vector magnitudes

while the associated addition and subtraction of angles is performed manually.

(d) It is of circular form, which results in a compact construction and makes manipulation a

straightforward process by eliminating the off-scale " difficulties inherent in a linear

arrangement.


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(e) It is of simple construction, small and portable, and capable of adequate accuracy for

engineering purposes.

Finally, it should be pointed out that, although designed primarily for use in the manipulation of

complex numbers, the basic property of the instrument is its ability to solve the right angled triangle

and it is applicable to any problem involving such solution.

Acknowledgements.

I would like to thank Professor Vic Gosbell of the Wollongong University for firstly rescuing this

calculator before it was consigned to scrap and secondly donating it to me and describing its

function and method of usage.

the robinson calculator

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