108 ABSTRACTS OF PAPERS

CALCULATION OF THE MAGNETIC FIELD IN DYNAMO-ELECTRIC MACHINES BYSOUTHWELL'S RELAXATION METHOD

By H. MOTZ, Dr. Ing., M.A.,* and W. D. WORTHY, B.A.*

(ABSTRACT of a paper which was published in December, 1945, in Part II of the Journal.)

The problem of the calculation of flux distribution betweenstator and rotor of dynamo-electric machines has been solvedby F. W. Carter for certain simplified shapes of pole and armaturewhich are amenable to rigorous solution.

Fig. 1.—Developed view of poles and armature with a rectangular net.

The paper explains a method of approximate solution, basedon Southwell's relaxation method, which can be applied toproblems involving any shape of pole and armature. Themethods developed are capable of wide application. They may

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be found useful for the design of magnets, magnetic circuits en-countered in transformers, etc. Without essential modificationthey can be applied to electrostatic problems of high-voltagetechnique and electronics.

The mathematical problem involved in all these cases is thesolution of the equation

(1)

which connects the two-dimensional distribution of potential Vwith the charge density /o, subject to certain boundary conditions.In electrostatic problems the potential values are constant on metalconductors. In the magnetic problems treated in the paper thesurfaces of constant potential are the iron surfaces. A typicalproblem dealt with is shown in Fig. 1, which gives in cross-section a developed view of a salient-pole machine with smootharmature. This problem is equivalent to the electrostaticproblem of the potential distribution between two metal plates,one smooth, the other slotted, to which a certain e.m.f. is applied,In the magnetic case a m.m.f. is maintained between the slottedpart representing yoke and field poles and the armature by afield winding. The problem can be solved for an arbitrarym.m.f. and the solution can be multipliedbya constant to repre-

Fig. 2.—One-eighth of cross-section of machine with distributed windings and slotted armature.(Equipotential lines have been drawn.)

ABSTRACTS OF PAPERS

sent the distribution for any given m.m.f. Saturation effects areneglected.

The first essential step of the approximation method is theintroduction of a net of small but finite mesh length. In Fig. 1a square net of mesh length a is shown and the potential is onlycalculated at points of this net. In the space between the ironboundaries which, in this magnetic problem, is, of course, free ofcharge, p is zero and thus

1P-Vtoe*

= 0 . . . (2)

By means of Taylor expansions and equation (2), the value ofthe potential at any point P of the net can be related to thepotentials at neighbouring points. If the net is fine enough,terms of an order higher than a can be neglected in the seriesexpansions. The potential at any point P can then be shown tobe the arithmetic mean of the values at the four neighbouringpoints, Q, R, S, T, i.e.

The problem is thus to find values for V at every point such thatthe value at any point is the mean of those at the surroundingpoints and with given constant values at points situated at theiron surfaces.

The calculation is started with an arbitrary trial set of valuesat every mesh point. Equation (2) will, of course, not besatisfied. At every point a residual 8P is defined by

VQ + VR + VS+VT-4VP = SF

The S-values represent in certain units the charges concentratedat mesh points which, together with the applied m.m.f., wouldmaintain the assumed potential distribution. However, sincemagnetic charges do not exist these residuals have now to beremoved. A simple arithmetical technique due to R. V. South-well is described by which the potential values are so changedthat the charges are gradually decreased in a systematic processuntil they are finally removed.

The method can be extended to mesh grids of different shapeto fit given boundaries. Near sharp corners of the boundariesa special treatment is necessary.

Some problems rigorously solved by F. W. Carter with thehelp of conjugate functions are recalculated by the above method,and the agreement between flux distributions at the armaturesurface obtained by the two methods is found to be satisfactory.Examples of problems which could not easily be solved by theclassical methods are given. The gap flux and leakage coefficientsfor a typical salient-pole machine with slotted armature are calcu-lated, and indicated in Fig. 2. In this calculation some accountis also taken of the m.m.f. drop along the field pole.

SOME CONSIDERATIONS IN THE DESIGN OF WIDE-BAND RADIO-FREQUENCY AMPLIFIERS

By J. E. COPE.*

(ABSTRACT of a Radio Section paper which was published in December, 1945, in Part III of the Journal.)

The paper deals with the problems confronting the designerof wide-band amplifiers for the receiving systems of television,radar and similar apparatus. By way of introduction, a shorthistorical survey shows that the impetus to design wide-bandradio-frequency amplifiers did not exist until the introduction ofhigh-definition television round about 1935, so that, until thattime, suitable types of amplifying valves were not available.The design of high-slope valves with low capacitances and lowradio-frequency losses proved difficult, and it is only in recentyears that really suitable types have been made available to thecircuit designer. To-day, however, improved types are comingforth much more rapidly, and it may be that before long a typewill be evolved which will be regarded, for a very considerabletime, as a standard valve for wide-band radio-frequencyamplification.

The remainder of the paper deals mainly with the design of awide-band amplifier for a particular television receiver. It isshown that, in order to make the fullest use of the television-transmitter service area, the highest possible sensitivity is de-sirable. An attempt at defining the sensitivity of a televisionreceiver is made, and the limitations imposed by the inherentreceiver noise level are studied. There follows a discussion as towhether this sensitivity should be achieved wholly in the receiveror by means of an economical low-sensitivity receiver with theaddition of an aerial amplifier as a booster for use at distant

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locations. It is argued that either system can be made to operatesatisfactorily and that the final decision may be determined bythe sales policy of the manufacturer concerned.

The influence of the wide pass-band requirements on thedesign of the intervalve couplings is then examined at somelength. It is shown that stagger-tuned single circuits, over-coupled two-element circuits, or a mixture of both, are the mostuseful types of couplings for this class of amplifier. It is pointedout that the stage gain is inversely proportional to the circuitcapacitance, so that trimming capacitors (which add capacitance)are inadvisable, and adjustable dust-iron cores or copper slugsshould be used to resonate the inductors.

Phase distortion, amplitude linearity and detection are thenconsidered, the author expressing the view that the push-pulldiode detector is probably the best detector for the job. Severalmethods of gain control are discussed, and the design difficultiestogether with the advantages and disadvantages of each aretabulated. The practical difficulties of constructing a high-sensitivity high-frequency amplifier are fully examined. Sug-gestions are made on how to solve the problems connected withthe shielding and decoupling of the various circuits in theamplifier.

In conclusion, it is pointed out that, while there is still plentyof room for improvement in circuit technique, the main advancesin the sphere of wide-band radio-frequency amplifiers are likelyto come from improved thermionic valves.