Title: Hawkins Electrical Guide v. 03 (of 10)
Author: N. Hawkins
Release date: August 24, 2015 [eBook #49769]
Most recently updated: October 24, 2024
Language: English
Credits: Produced by Juliet Sutherland, tallforasmurf and the Online
Distributed Proofreading Team at http://www.pgdp.net
In transcribing this book, the proofreaders found and corrected several minor typographical errors which did not affect the sense of the text. In the caption to Figure 541, the equation for the voltage of a Weston cell at different temperatures was missing a digit "1" and this has been corrected. There is a reference to a Figure 619 but no such figure exists in the original text. There are references to a Figure 119 and a Figure 443; these presumably exist in one of the preceding volumes of the series.
COPYRIGHTED, 1914,
BY
THEO. AUDEL & CO.,
New York.
Printed in the United States.
Action of compass needle--simple galvanometer--difference between galvanoscope and galvanometer--sensibility--action of short and long coil galvanometers--classes of galvanometer--astatic galvanometer--tangent galvanometer--graduation of tangent galvanometer scale--table of galvanometer constants--mechanical explanation of tangent law--sine galvanometer--table of natural sines and tangents--comparison of sine and tangent galvanometers--differential galvanometer--ballistic galvanometer--kick--damping effect--use of mirrors in galvanometers--lamp and scale--damping--D'Arsonval galvanometer: construction, operation; uses--galvanometer constant or figure of merit--shunts.
TESTING AND TESTING APPARATUS 465 to 536
Pressure measurement--Clark cell--Weston cadmium cell--pressure measurement error with ordinary voltmeter--International volt--hydraulic analogy of amperes--coulombs--current measurement--International ampere--voltameters--Ohm's law and the ohm--International ohm--ohm table--practical standards of resistance--various methods of resistance measurement--direct deflection method--method of substitution--resistance box--fall of potential method--differential galvanometer method--drop method--voltmeter method--Wheatstone bridge--usual arrangement of resistances of Wheatstone bridge--ratio coils of Wheatstone bridge--the decade plan--two plug arrangement--"plug out" and "plug in" type of resistance box--testing sets--direct deflection method with Queen Acme set--[typo:ohmeter:ohmmeter]--fall of potential method with Queen Acme set--apparatus for measuring low resistances--how to check a voltmeter--Kelvin wire bridge--internal resistance measurement--Evershed portable ohmmeter set--L and N fault finder--ammeter test--diagram of Queen standard potentiometer--diagrams illustrating loop testing--the Murray loop--the Varley loop--special loop--the potentiometer--location of opens--to pick out faulty wires in a cable--voltage of cell measurement with potentiometer--care of potentiometer--location of faults where the loop is composed of cables of different cross sections.
AMMETERS, VOLTMETERS, AND WATTMETERS 537 to 572
Definition of ammeter--classification of ammeter and voltmeters--moving iron type instrument--Keystone voltmeter--winding in ammeters and volts--connections for series and shunt ammeters--voltmeter connections--Westinghouse ammeter shunts--various types of instrument--plunger type instrument--magnetic vane instrument--inclined coil instrument--Whitney hot wire instruments--principle of electrostatic instruments--multipliers--portable shunts--Siemens electro-dynamometer--station instruments--Thompson watt hour meter--how to read a meter--installation of wattmeters--Westinghouse watt hour meter--Thompson prepayment watt hour meter--how to test a meter--Sangamo watt hour meter--Columbia watt hour meter--Duncan watt hour meter.
OPERATION OF DYNAMOS 573 to 596
Before starting a dynamo--adjusting the brushes--brush position--how to set the brushes--method of soldering cable to carbon brush--brush contact pressure--direction of rotation--method of winding cables with marlin--method of assembling core discs--starting a dynamo--tinning block for electric soldering tool--shunt dynamos in parallel--shunt dynamos on three wire system--how to start a series machine--the term "build up"--how to start a shunt or compound machine--"picking up"--indication of reversed connections--how to correct reversed polarity--finding the reversed coil--loss of residual magnetism--remedy for reversed dynamo--attention while running--lead of brushes--method of taking temperature--lubrication--oils--allowable degree of heating--attention to brushes and brush gear.
COUPLING OF DYNAMOS 597 to 610
Series and parallel connections--coupling series dynamos in series; in parallel--equalizer--shunt dynamos in series; in parallel--switching dynamo into and out of parallel--to cut out a machine--dividing the lead--compound dynamos in series; in parallel--equalizer connection--switching a compound dynamo into and out of parallel--equalizing the load--shunt and compound dynamos in parallel.
DYNAMO FAILS TO EXCITE 611 to 622
Various causes--brushes not properly adjusted--defective contacts--incorrect adjustment of regulators--speed too low--testing for break--insufficient residual magnetism; remedy--open circuits--test for field circuit breakers--probable location of breaks--Watson armature discs--Fort Wayne commutator truing device--short circuits--Watson armature--wrong connections--reversed field magnetism.
Causes--how avoided--various faults--short circuit in individual coils--location of faulty coil--test for break in armature lead--bar to bar test for open or short circuit in coil or between segments--short circuits between adjacent coils--alternate bar test for short circuits between sections--short circuits between sections through frame or core of armature; between sections through binding wires--partial short circuits in armatures--method of testing for breaks--burning of armature coils--Watson field coils--grounds in armatures--method of locating grounded armature coil--magneto test for grounded armatures--method of binding armature winding--breaks in armature circuit.
CARE OF THE COMMUTATOR AND BRUSHES 635 to 652
Conditions for satisfactory operation--oil for commutator--attention to brushes--Bissell brush gear--two kinds of sparking--commutator clamp--causes of sparking--bad adjustment of brushes--rocking--bad condition of brushes--brushes making bad contact--bad condition of commutator--detection of untrue commutator--high segments--"flats"--causes of flats; remedy--method of repairing broken joint between commutator segment and lug--segments loose or knocked in--how to re-turn a commutator--Bissell commutators--overload of dynamo--method of repairing large hole burned in two adjacent bars of a commutator--operating dynamos with metal brushes--indication of excessive voltage--method of smoothing commutator with a stone--causes of excessive voltage--loose connections, terminals, etc.,--breaks in armature circuit--sandpaper holder for commutator--short circuits, in armature circuits; in field--breaks in field--sandpaper block--short circuits in commutator.
Various causes--how detected--procedure--heating, of connections; of brushes, commutator and armature--excessive heating--ventilated commutator--self-oiling bearing--some causes of hot bearing--effect of hot bearings--points relating to hot bearings--operation above rated voltage and below normal speed--forced system of lubrication--heating of field magnets--causes of eddy currents in pole pieces--detection of moisture in field coils--indication of short circuits in field coils.
OPERATION OF MOTORS 663 to 696
Before starting a motor--starting a motor--various starting resistances--starting boxes--speed regulators--Cutler Hammer starter--time required to start motor--how to start--sliding contact starters--series motors on battery circuits--starting a shunt motor--multiple switch starters--effect of reverse voltage--rheostat with no voltage and overload release--failure to start--starting panel--Cutler Hammer starting rheostats--Allen Bradley automatic starter--Monitor starter with relay for push button control--a remote control of shunt motors--regulation of motor speed; various methods--Monitor printing press controller--speed regulation of series motor, by short circuiting sections of the field winding--varying the speed of shunt and compound motors--Cutler Hammer multiple switch starter--regulation by armature resistance--Compound starter--regulation by shunt field resistance--Holzer Cabot instructions for shunt wound motor--Reliance adjustable speed motor--Cutler Hammer reversible starter--combined armature and shunt field control--selection of starters and regulators--Watson commutators--organ blower speed regulator--General Electric controller--speed regulation of traction motors--controller of the Rauch and Lang electric vehicles--two motor regulation--controller connection diagrams--stopping a motor.
If a compass needle be allowed to come to rest in its natural position, and a current of electricity be passed through a wire just over it from north to south, the north seeking end of the needle will be deflected toward the east. If the wire be placed under the needle and the current continued from north to south the needle will be deflected toward the west. Again, if the current be passed from north to south over the needle, and back from south to north under the needle, as shown in fig. 504, the magnetic effect will be doubled, and the needle deflected proportionately. Upon these phenomena depend the working of galvanometers.
Fig. 503.--Effect of neighboring current upon a magnetic needle. Above the needle and parallel to it is a conductor carrying an electric current, the current flowing in the direction indicated by the arrow. This causes the north pole of the needle to turn toward the east. If the conductor be held below the needle, its north pole will turn in the opposite direction or toward the west. These movements are easily determined by Ampere's rule as follows: If a man could swim in the conductor with the current, and turn to face the needle, then the north pole of the needle will be deflected toward his left hand.
Ques. Describe a simple galvanometer.
Ans. It consists essentially of a magnetic needle suspended within a coil of wire, and free to swing over the face of a graduated dial.
Ques. What is a galvanoscope and how does it differ from a galvanometer?
Ans. A galvanoscope, as shown in fig. 504, serves merely to indicate the presence of an electric current without measuring its strength. It is an indicator of currents where the movement of the needle shows the direction of the current, and indicates whether it is a strong or a weak one. When the value of the readings has been determined by experiment or calculation any galvanoscope becomes a galvanometer.
Fig. 504.--Effect upon a magnetic needle of a neighboring current in a loop. In this arrangement the same conductor is simply carried back beneath the needle and hence both the upper and lower portions tend to turn it in the same direction, while the side branch or vertical section is ineffective. In accordance with Ampere's swimming rule, the upper wire causes the N pole of the needle to turn to the left, while if a man can imagine himself swimming in the lower wire in the direction of the current, and facing the needle (that is, swimming on his back), the N pole of the needle will turn to his left--that is to the east. The effect of the loop then has double the effect of the single wire in fig. 503.
Ques. For what use are galvanometers employed?
Ans. They are used for detecting the presence of an electric current, and for determining its direction and strength.
Ques. How is the direction and strength of the current indicated?
Ans. When a galvanometer is connected in a circuit, the direction of the current is indicated by the side towards which the north pole of the needle moves, and the current strength by the extent of the needle's deflection.
Fig. 505.--Effect upon a magnetic needle of a neighboring current in a coil. The coil as shown, is equivalent to several loops, that is, the force tending to deflect the needle is equal to that of a single loop multiplied by the number of turns. Hence, by using a coil with a large number of turns, a galvanometer may be made very sensitive so that the needle will be perceptibly deflected by very feeble currents. An instrument, as shown in the figure is called a galvanoscope. When it is accurately constructed, and supplied with a scale showing how many degrees the needle is deflected it is then called a galvanometer.
Ques. How should a galvanometer be set up before using?
Ans. When no current is flowing, the coil should be parallel to the magnetic needle when at rest.
Ques. What is a "sensitive" galvanometer?
Ans. One which requires a very small current or pressure to produce a stated deflection.
It does not follow that a galvanometer which is sensitive for current measurement will also be sensitive for pressure measurement.
Fig. 506.--Bunnell simple detector galvanometer. It has middle clamps and scale divided into degrees.
Ques. Define the term "sensibility."
Ans. With reference to mirror reflecting galvanometers it may be defined in three ways. First, in megohms, the sensibility being the number of megohms through which one volt will produce a deflection of one millimeter with the scale at one meter distance. Second, in micro-volts, the sensibility being the number of micro-volts which applied directly to the terminals of the galvanometer will produce a deflection of one millimeter on a scale one meter from mirror. The sensibility is best stated in megohms for high resistance galvanometers and in micro-volts for low resistance galvanometers, and is frequently given both for galvanometers for intermediate resistance. Third, in micro-amperes, the sensibility being the number of micro-amperes that will give one millimeter deflection with scale at a distance of one meter.
Ques. Upon what does the sensibility depend?
Ans. 1, Upon the number of times the current circulates around the coil, 2, the distance of the needle from the coil, 3, the weight of the needle, 4, the current strength, and 5, the amount of friction produced by its movement.
Fig. 508.--Bunnell horizontal galvanometer. It has two coils, one of which is of zero resistance and one of fifty ohms resistance adapting it to a variety of test.
The needle is usually quite small, and often a compound one. In very sensitive galvanometers, the coils are wound with thousands of turns of very fine wire, and shunts are generally used in connection with them.
NOTE.--Strong currents must not be passed through very sensitive galvanometers, for even if they be not ruined, the deflections of the needle will be too large to give accurate measurements. In such cases the galvanometer is used with a shunt, or coil of wire arranged so that the greater part of the current will flow through it, and only a small portion through the galvanometer.
Ques. What two kinds of coil are used?
Ans. The short coil and the long coil.
Ques. What is the difference between a short coil and a long coil galvanometer?
Ans. A short coil galvanometer has a coil consisting of a few turns of heavy wire; a long coil galvanometer is wound with a large number of turns of fine wire.
Fig. 509.--Bunnell galvanometer for measurements of instruments, lines, batteries, wires and any object from 1/100 to 10,000 ohms or more.
Ques. What is the action of short and long coil galvanometers?
Ans. With a given current, the total magnetizing force which deflects the needle is the same, but with a short coil, it is produced by a large current circulating around a few turns, instead of a small current circulating around thousands of turns as in the long coil. The short coil being of low resistance is used to measure the current, and the long coil with high resistance, is suitable for measuring the pressure. Hence, a short coil instrument with its scale directly graduated in amperes is an ammeter, and the long coil type with graduation in volts is a voltmeter.
Classes of Galvanometer.--There are numerous kinds of galvanometer designed to meet the varied requirements. According to construction, galvanometers may be divided into two classes, as those having:
Fig. 510.--Astatic needles. Two magnetic needles of equal moment are mounted in opposition on a light support. The whole system is suspended by a delicate fibre, and when placed in a uniform magnetic field such as that of the earth, there will be no tendency to assume any fixed direction, the only restraining influence on the needles being that due to torsion in the suspension fibre.
Either type may be constructed with short or long coil, and there are several ways in which the deflections are indicated. The principal forms of galvanometer are as follows:
Astatic Galvanometer.--It has been pointed out how a compass needle is affected when a wire carrying a current is held over or under it, the needle being turned in one direction in the first instance, and in the opposite direction for the second position of the wire.
Fig. 511.--Connections of single coil astatic needles. The coil surrounds the lower needle and the direction of the current between the two needles tends to turn them the same way.
The earth's magnetism naturally holds the compass needle north and south. The magnetic field encircling the wire, being at right angles to the needle (when the wire itself is parallel therewith), operates to turn it from its normal position, north and south, so as to set it partially east and west. However, on account of the fact that the earth's magnetism does exert some force tending to hold the needle north and south, it is evident that no matter how strong the current, the latter can never succeed in turning the needle entirely east and west. The accomplishment of this is further prevented by the reason of the points of the needle, where the magnetic effect is greatest, quickly passing out of the reach of the magnetic field, where it is now practically operated on only in a slight degree. Thus it would take quite a powerful current to hold the needle deflected any appreciable distance. The use of a shorter needle is, therefore, more desirable.
It is evident in this style of instrument that the effect of the current cannot be accurately measured, because it acts in opposition to the earth's magnetism, and as this is constantly varying, some method must be employed which will either destroy the earth's magnetism or else neutralize it.
In the astatic galvanometer, the earth's magnetism is neutralized by means of astatic needles. These consist of a combination of two magnetic needles of equal size and strength, connected rigidly together with their poles pointing in opposite and parallel directions, as shown in fig. 510. As the north pole of the earth attracts the south pole of one of the needles, it repels with equal strength the north pole of the other needle, hence, the combination is independent of the earth's magnetism and will remain at rest in any position.
Fig. 512.--Connections of double coil astatic needles. With this arrangement, the direction of current in both coils will tend to turn the system in the same direction, making the needles more sensitive than with a single coil as in fig. 511.
If one of the needles be surrounded by a coil, as shown in fig. 511, the magnetic effect of the current will be correctly indicated by the deflection of the needle.
Sometimes each needle is surrounded by a coil, as in fig. 512, the coils being so connected that the direction of current in each will tend to deflect the needles in the same direction.
Ques. For what use is the astatic galvanometer adapted?
Ans. For the detection of small currents.
It is used in the "nil" or zero methods, in which the current between the points to which the galvanometer is connected is reduced to zero.
Fig. 513.--Queen reflecting astatic galvanometer. It is mounted on a mahogany base with levelling screws. A plain mirror is attached above the upper needle. The entire combination of mirror and needles is suspended by unspun silk from the interior of a brass tube, which also carries a weak controlling magnet. A dial 4 inches in diameter and graduated in degrees, enables the deflections of the needle to be accurately read. The mirror can be used with a reading telescope and scale, or by means of a lantern, the image of a slit may be reflected from the mirror to a screen. Resistance, .5 to 1,000 ohms.
Ques. Upon what does the movement of the needles depend?
Ans. Upon the combined effect of the magnetic attraction of the current which tends to deflect the needles, and the torsion in the suspension fibre which tends to keep the needle at the zero position.
Ques. Does the astatic galvanometer give correct readings for different values of the current?
Ans. When the deflections are small (that is, less than 10° or 15°), they are very nearly proportional to the strength of the currents that produce them.
Thus, if a current produce a deflection of 6° it is known to be approximately three times as strong as a current which only turns the needle through 2°. But this approximate proportion ceases to be true if the deflection be more than 15° or 20°.
Fig. 514.--Central Scientific Co. tangent galvanometer. A 9 inch brass ring is mounted on a mahogany base which rotates on a tripod provided with levelling screws. The needle has an aluminum pointer and jewelled bearing. The winding consists of 300 turns of magnet wire so connected to the plugs in front that 20, 40, 80, or 160 turns or any combination of these numbers may be used. For heavy currents a band of copper is used by connecting to the extra pair of binding posts in the rear of the instrument.
Ques. Why does the instrument not give accurate readings for large deflections?
Ans. The needles are not so advantageously acted upon by the current, since the poles are no longer within the coils, but protrude at the side. Moreover, the needles being oblique to the force acting on them, part only of the force is turning them against the directive force of the fibre; the other part is uselessly pulling or pushing them along their length.
Fig. 515.--Bunnell tangent galvanometer. This instrument is mounted on a circular hard rubber base, 7-3/8 inches diameter, provided with levelling screws and anchoring points. The galvanometer consists of a magnetized needle 7/8 inch in length, suspended at the center of a rubber ring six inches in diameter, containing the coils. There are five coils of 0, 1, 10, 50 and 150 ohms resistance. The first is a stout copper band of inappreciable resistance; the others are of different sized copper wires, carefully insulated. Five terminals are provided, marked, respectively, 0, 1, 10, 50 and 150. The ends of the coils are so arranged that the plug inserted at the terminal marked 50 puts in circuit all the coils; marked at the terminal 50--all except the 150 ohm coil; and so on, till at the zero terminal only the copper band is in circuit. Fixed to the needle, which is balanced on jewel and point, is an aluminum pointer at right angles, extending across a five inch dial immediately beneath. One side of the dial is divided into degrees; on the other side, the graduations correspond to the tangent of the angles of deflection.
Ques. How may correct readings be obtained?
Ans. The instrument may be calibrated, that is, it may be ascertained by special measurements, or by comparison with a standard instrument, the amounts of deflection corresponding to particular current strengths.
Thus, if it be once known that a deflection of 32° on a particular galvanometer is produced by a current of 1/100 of an ampere, then a current of that strength will always produce on that instrument the same deflection, unless from any accident the torsion force or the intensity of the magnetic field be altered.
Fig. 516.--Tangent galvanometer. It consists of a short magnetic needle suspended at the center of a coil of large diameter and small cross section. In practice, the diameter of the coil is about 17 times the length of the needle. If the instrument be so placed that, when there is no current in the coil, the suspended magnet lies in the plane of the coil, that is, if the plane of the coil be set in the magnetic meridian, then the current passing through the coil is proportional to the tangent of the angle by which the magnet is deflected from the plane of the coil, or zero position--hence the name: "tangent galvanometer."
The Tangent Galvanometer.--It is not possible to construct a galvanometer in which the angle (as measured in degrees of arc) through which the needle is deflected is proportional throughout its whole range to the strength of the current. But it is possible to construct a very simple galvanometer in which the tangent of the angle of deflection shall be accurately proportional to the strength of the current.
Fig. 517.--Horizontal section through middle of tangent galvanometer, showing magnetic whirls around the coil and corresponding deflection of needle.
Fig. 518.--Diagram of forces acting on the needle of a tangent galvanometer.
A simple form of tangent galvanometer is shown in fig. 516. The coil of this instrument consists of a simple circle of stout copper wire from ten to fifteen inches in diameter. At the center is delicately suspended a magnetized steel needle not exceeding one inch in length, and usually furnished with a light index of aluminum. When the galvanometer is in use, the plane of the ring must be vertical and in the magnetic meridian. A horizontal section through the middle of the instrument is shown in fig. 517. For simplicity, the coil is supposed to have but a single turn of wire, the circles surrounding the wire representing the magnetic lines of force. By extending the lines of force until they reach the needle, it will be seen that with a short needle, the deflecting force acts in an east and west direction when the galvanometer is placed with its coil in the magnetic meridian.
If, in fig. 518, ab represent the deflecting force acting on the N end of the needle, the component of this force that acts at a right angle to the needle will be
in which, x is the angle of the deflection.
The controlling force is
and when the needle is in equilibrium, the component ae = H sin x is equal and opposite to ac, hence
from which
Since ab is proportional to the current,
in which k is a constant depending upon the instrument. For any other current C',
hence
This means that the currents passing through the coil of a tangent galvanometer are proportional, not to the angle of deflection, but to the tangent of that angle.
Fig. 519.--Diagram illustrating the tangent law. This is the law of the combined action of two magnetic fields upon a magnetic needle. If two magnetic fields be at right angles in direction as indicated in the figure, the resultant field is obtained by the parallelogram of forces and it makes an angle θ with one of the component fields such that tan θ = M + H where M and H are the strengths of the component fields. In the tangent galvanometer this principle is employed in the measurement of currents. A magnetic needle is pivoted in a field of known strength. The current to be measured is passed round a coil (or coils) which generates a field at right angles to the original field. The needle then lies along the direction of the resultant field, and by finding the tangent of its angle of deflection, and knowing the field strength produced by unit current in the coil, the current strength can be found.
Fig. 520.--Graduation of tangent galvanometer scale with divisions representing tangent values. In the figure let a tangent OT be drawn to the circle, and along this line let any number of equal divisions be set off, beginning at O. From these points draw lines back to the center. The circle will thus be divided into a number of spaces, of which those near O are nearly equal, but which get smaller and smaller as they recede from O. These unequal spaces correspond to equal increments of the tangent. If the scale were divided thus, the readings would be proportional to the tangents.
Ques. Upon what does the sensibility of a tangent galvanometer depend?
Ans. It is directly proportional to the number of turns of the coil and inversely proportional to the diameter of the coil.
Ques. How may the tangent galvanometer be used as an ammeter?
Ans. The strength of the current may be calculated in amperes by the formula given below when the dimensions of the instrument are known.
The needle is supposed to be subject to only the earth's magnetism and to move in a horizontal plane. The current is calculated as follows:
amperes = ((H × r)/N) tan x(1)
The constant H, given in the following table represents the horizontal force of the earth's magnetism for the place where the galvanometer is used. Each value has been multiplied by (2π )/10 so that the formula (1) for amperes is correct as given.
Table of Galvanometer Constants.--Values of H.
| Boston | .699 |
| Chicago | .759 |
| Denver | .919 |
| Jacksonville | 1.094 |
| London | .745 |
| Minneapolis | .681 |
| New York | .744 |
| New Haven | .731 |
| Philadelphia | .783 |
| Portland, Me. | .674 |
| San Francisco | 1.021 |
| St. Louis | .871 |
| Washington | .810 |
Fig. 521.--Mechanical explanation of the tangent law. Construct an apparatus as shown in the figure. The short wooden block, NS, represents the magnetic needle. This piece of wood turns around its center, C, which may be an ordinary nail. It will now be seen that two different forces act upon N; namely, the weight, G (one or two ounces), and the changeable weights which are placed in the scoop, W (made of cardboard). The height of the roll, or wheel, R, is such that the cord, RN, runs horizontally, when NS stands vertically, i.e., when there is no weight in the little scoop. If the wheel, R, be placed sufficiently far from NS, the string RN, will always remain almost horizontal, even if NS be deviated. The thin hand on NS moves over a vertical scale, which is divided into equal parts, as shown. This scale may be made of cardboard. If the hand point to division 1 when one ounce is placed in the scoop, it will point to 2 for two ounces, to 3 for three ounces, etc. At 45° the needle is deviated at its greatest angle, and this is, therefore, the sensitivity angle of the tangent galvanometer. The deviating values are, therefore, proportionate to the scale parts 01, 02, and 03, and so on; and, inasmuch as these themselves are tangents, the tangent law will hold good.
Ques. How is the tangent galvanometer constructed to give direct readings?
Ans. To obviate reference to a table, the circular scale of the instrument is sometimes graduated into tangent values, as in fig. 520, instead of being divided into equal degrees.