Transformers

Transformers. An electrical transformer is the name given to any device for producing by means of one electric current another of a different character. The working of such an appliance is, of course, subject to the law of conservation of energy. The resulting current represents less power than the applied current, the difference being represented by the power dissipated in the translating process. Hence an electrical transformer corresponds to a simple machine in mechanics, both transforming power from one form into another with a certain energy-dissipation depending upon frictional losses, or something equivalent to them. Electrical transformers may be divided into several classes, according to the nature of the transformation effected. The first division comprises those which change the form of the power, but keep the type of the current the same; the second those that change the type of the current as well as the form of power. The power given up electrically to any circuit is measured by the product of the effective value of the current, the effective value of the difference of potential between the ends of the circuit and a factor called the power factor. In dealing with periodic currents, the effective value is that called the root-mean-square value (R.M.S.), that is to say, the square root of the mean of the squares of the time equidistant instantaneous values during one complete period (see Electrokinetics). In the case of continuous current, the power factor is unity, and the effective value of the current or voltage is the true mean value. As the electrical measure of a power is always a product involving current and voltage, we may transform the character of the power by increasing or diminishing the current with a corresponding decrease or increase of the voltage. A transformer which raises voltage is generally called a step-up transformer, and one which lowers voltage a step-down transformer.

Again, electric currents may be of various types, such as continuous, single-phase alternating, polyphase alternating, undirectional but pulsating, &c. Accordingly, transformers may be distinguished in another way, in accordance with the type of transformation they effect. (I) An alternating current transformer is an appliance for creating an alternating current of any required magnitude and electromotive force from another of different value and electromotive force, but of the same frequency. An alternating current transformer may be constructed to transform either single-phase or polyphase currents. (2) A continuous current transformer is an appliance which effects a similar transformation for continuous currents, with the difference that some part of the machine must revolve, whereas in the alternating current transformer all parts of the machine are stationary; hence the former is generally called a rotatory transformer, and the latter a static transformer. (3) A rotatory or rotary transformer may consist of one machine, or of two separate machines, adapted for converting a single-phase alternating current into a polyphase current, or a polyphase current into a continuous current, or a continuous current into an alternating current. If the portions receiving and putting out power are separate machines, the combination is called a motor-generator. (4) A transformer adapted for converting a single-phase alternating current into a unidirectional but pulsatory current is called a rectifier, and is much used in connexion with arc lighting in alternating current supply stations. (5) A phase transformer is an arrangementof static transformers for producing a polyphase alternating current from a single-phase alternating current. Alternating current transformers may be furthermore divided into (a) single-phase, (b) polyphase. Transformers of the first class change, an alternating current of single-phase to one of single-phase identical frequency, but different power; and transformers of the second class operate in a similar manner on polyphase currents. (6) The ordinary induction or spark coil may be called an intermittent current transformer, since it transforms an intermittent low-tension primary current into an intermittent or alternating high-tension current.

Alternating Current Transformer

The typical alternating current transformer consists essentially of two insulated electric circuits wound on an iron core constituting the magnetic circuit. They may be divided into (I) open magnetic circuit static transformers, and (2) closed magnetic circuit static transformers, according as the iron core takes the form of a terminated bar or a closed ring. A closed circuit alternating current transformer consists of an iron core built up of thin sheets of iron or steel, insulated from one another, and wound over with two insulated conducting circuits, called the primary and secondary circuits. The core must be laminated or built up of thin sheets of iron to prevent local electric currents, called eddy currents, from being established in it, which would waste energy. In practical construction, the core is either a simple ring, round or rectangular, or a double rectangular ring, that is, a core whose section is like the figure 8. To prepare the core, thin sheets of iron or very mild steel, not thicker than -014 of an inch, are stamped out of special iron (see Electromagnetism) and carefully annealed.

The preparation of the particular sheet steel or iron used for this purpose is now a speciality. It must possess extremely small hysteresis loss (see Magnetism), and various trade names, such as " stalloy," " lohys," are in use to describe certain brands. Barrett, Brown and Hadfield have shown (Journ. Inst. Elec. Eng. Lond., 1902, 31, p. 713) that a silicon iron containing 2.87% of silicon has a hysteresis loss far less than that of the best Swedish soft iron. In any case the hysteresis loss should not exceed 3o watts per kilogram of iron measured at a frequency of 50 - and a flux-density of I o,000 lines per square centimetre. This is now called the " figure of merit " of the iron.

Examples of the shapes in which these stampings - are supplied are shown in fig. 1. The plates when annealed are varnished or covered with thin paper on one side, and then piled up so as to make an iron cdre, being kept together by bolts and nuts or by pressure plates. The designer of a transformer core has in view, first, economy in metal, so that there may be no waste fragments, and second, a mode of construction that facili tates the winding of the wire circuits.

These consist of coils of cotton covered copper wire which are wound on formers and baked after being well saturated with shellac varnish. The primary and secondary circuits are sometimes formed of separate bobbins which are sandwiched in between each other; in other cases they are FIG. I. wound one over the other (fig. 2).

In any case the primary and secondary coils must be symmetrically distributed. If they were placed on opposite sides of the iron circuit the result would be considerable magnetic leakage. It is usual to insert sheets or cylinders of micanite between the primary and secondary windings. The transformer is then well baked and placed in a cast-iron case sometimes filled in with heavy insulating oil, the ends of the primary and secondary circuits being brought out through water-tight glands. The most ordinary type of alternating current transformer is one intended to transform a small electric current produced by a large electromotive force (moo to Io,000 volts) into a larger current of low electromotive force (Ioo to 200 volts). Such a step down transformer may be obviCore ously employed in the reverse, direction for raising pressure and. „ Primary reducing current, in which case it L.., _? 'Circuit ' is a step-up transformer. A trans former when manufactured has to Secondary be carefully tested to ascertain, Circuit first, its power of resisting break down, and, second, its energy dissipating qualities. With the first object, the transformer is subjected to a series of pressure tests. If it is intended that the FIG. 2. - Closed Circuit Transprimary shall carry a current former.

AEI A produced by an electromotive force of 2000 volts, an insulation test must be applied with double this voltage between the primary and the secondary, the primary and the case, and the primary and the core, to ascertain whether the insulation is sufficient. To prevent electric discharges from breaking down the machine in ordinary work, this extra pressure ought to be applied for at least a quarter of an hour. In some cases three or four times the working pressure is applied for one minute between the primary and secondary circuits. When such an alternating current transformer has an alternating current passed through its primary circuit, an alternating magnetization is produced in the core, and this again induces an alternating secondary current. The secondary current has a greater or less electromotive force than the primary current according as the number of windings or turns on the secondary circuit is greater or less than those on the primary. Of the power thus imparted to the primary circuit one portion is dissipated by the heat generated in the primary and secondary circuits by the currents, and another portion by the iron core losses due to the energy wasted in the cyclical magnetization of the core; the latter are partly eddy current losses and partly hysteresis losses.

In open magnetic circuit transformers the core takes the form of a laminated iron bar or a bundle of iron wire. An ordinary induction coil is an instrument of this description. It has been shown, however, by careful experiments, that for alternating current transformation there are very few cases in which the closed magnetic circuit transformer has not an advantage. An immense number of designs of closed circuit transformers have been elaborated since the year 1885. The principal modern types are the Ferranti, Kapp, Mordey, Brush, Westinghouse, Berry, Thomson-Houston and Ganz. Diagrammatic representations of the arrangements of the core and circuits in some of these transformers are given in fig. 3.

A B C 3. - Diagrams of (A) Mordey (in section), (B) Kapp and (C) Ganz Transformers.

i, i Primary circuit; 2, 2 Secondary circuit.

Alternating current transformers are classified into (i.) Core and (ii.) Shell transformers, depending upon the arrangements of the iron and copper circuits. If the copper circuits are wound on the outside of what is virtually an iron ring, the transformer is a core transformer; if the iron encloses the copper circuits, it is a shell transformer. Shell transformers have the disadvantage generally of poor ventilaton for the copper circuits. Berry, however, has overcome this difficulty by making the iron circuit in the form of a number of bunches of rectangular frames which are set in radial fashion and the adjacent legs all embraced by the two copper circuits in the form of a pair of concentric cylinders. In this manner he secures good ventilation and a minimum expenditure in copper and iron, as well as the possibility of insulating the two copper circuits well from each other and from the core. An important matter is the cooling of the core. This may be effected either by ordinary radiation, or by a forced draught of air made by a fan or else by immersing the transformer in oil, the oil being kept cool by pipes through which cold water circulates immersed in it. This last method is adopted for large high-tension transformers.

The ratio between the power given out by a transformer and the power taken up by it is called its efficiency, and is best. represented by a curve, of which the ordinate is the efficiency expressed as a percentage, and the corresponding abscissae represent the fractions of the full load as decimal fractions. The output of the transformer is generally reckoned in kilowatts, and the load is conveniently expressed in decimal fractions of the full load taken as unity. The efficiency on one-tenth of full load is generally a fairly good criterion of the economy of the transformer as a transforming agency. In large transformers the one-tenth load efficiency will reach 90% or more, and in small transformers 75 to 80%.

The general form of the efficiency curve for a closed circuit transformer is shown in fig. 4. The horizontal distances represent fractions of full secondary load (represented by unity), and the vertical distances efficiency in percentages. The efficiency curve has a maximum value corresponding to that degree of load at which the copper losses in the transformer are equal to the iron losses.

In the case of modern closed magnetic circuit transformers the copper losses are proportional to the square of the secondary current (I) or to gI, where q=R a -f-R; R being the resistance of the primary and R2 that of the secondary circuit, while a is the ratio of the number of secondary and primary windings of the transformer. Let C stand for the core loss, and V2 for the secondary terminal potential difference (R 4 M.S. value). We can then write as an expression for the efficiency (n) of the transformer (n = 12V2/ (C g12 --12V2). It is easy to show that if C1, V2 and q are constants, but I is variable, the above expression for has a maximum value when C - 0 =0, that is, when the iron core loss C = the total copper losses g122.

The iron core energy-waste, due to the hysteresis and eddy currents, may be stated in watts, or expressed as a fraction of the full load secondary output. In small transformers of i to 3 kilowatts output it may amount to 2 or 3%, and in large transformers of io to 50 kilowatts and upwards it should be i or less than i %. Thus the core loss of a 30-kilowatt transformer (one having a secondary output of 30,000 watts) should not exceed 250 watts.

It has been shown that ,00 for the constant p090 tential transformer the iron core loss is constant at all loads, but d17p minishes slightly as the y 60 core temperature rises. so On the other hand, the copper losses due to the resistance of the copper circuits increase 20 about 0 4% per degree 10 C. with rise of tempera0 ture. The current taken in at the primary side of the transformer, when the secondary circuit is unclosed, is called the magnetizing current, and the power then absorbed by the transformer is called the open circuit loss or magnetizing watts. The ratio of the terminal potential difference at the primary and secondary terminals is called the transformation ratio of the transformer. Every transformer is designed to give a certain transformation ratio, corresponding to some particular primary voltage. In some cases transformers are designed to transform, not potential difference, but current in a constant ratio. The product of the root-meansquare (R.M.S.), effective or virtual, values of the primary current, and the primary terminal potential difference, is called the apparent power or apparent watts given to the transformer. The true electrical power may be numerically equal to this product, but it is never greater, and is sometimes less. The ratio of the true power to the apparent power is called the power factor of the transformer. The power factor approaches unity in the case of a closed circuit transformer, which is loaded noninductively on the secondary circuit to any considerable fraction of its full load, but in the case of an open circuit transformer the power factor is always much less than unity at all loads. Power factor curves show the variation of power factor with load. Examples of these curves were first given by J. A. Fleming, who suggested the term itself (see Jour. Inst. Elec. Eng. Lond., 1892, 21, p. 606). A low power factor always implies a magnetic circuit of large reluctance.

The operation of the alternating current is then as follows: the periodic magnetizing force of the primary circuit creates a periodic magnetic flux in the core, and this being linked with the primary circuit creates by its variation what is called the back electromotive force in the primary circuit. The variation of the particular portion Fraction of Full Load. FIG. 4. - Typical Efficiency Curve of Closed Circuit Transformer.

of this periodic flux, linked with the secondary circuit, originates in this last a periodic electromotive force. The whole of the flux linked with the primary circuit is not interlinked with the secondary circuit. The difference is called the magnetic leakage of the transformer. This leakage is increased with the secondary output of the transformer and with any disposition of the primary and secondary coils which tends to separate them. The leakage exhibits itself by increasing the secondary drop. If a transformer is worked at a constant primary potential difference, the secondary terminal potential difference at no load or on open secondary circuit is greater than it is when the secondary is closed and the transformer giving its full output. The difference between these last two differences of potential is called the secondary drop. This secondary drop should not exceed 2% of the open secondary circuit potential difference.

The facts required to be known about an alternating current transformer to appraise its value are (I) its full load secondary output or the numerical value of the power it is will not rise in temperature more than about 60 0 C. above the atmosphere when in normal use; (2) the primary and secondary terminal voltages and currents, accompanied by a statement whether the transformer is intended for producing a constant secondary voltage or a constant secondary current; (3) the efficiency at various fractions on secondary load from one-tenth to full load taken at a stated frequency; (4) the power factor at one-tenth of full load and at full load; (5) the secondary drop between full load and no load; (6) the iron core loss, also the magnetizing current, at the normal frequency; (7) the total copper losses at full load and at one-tenth of full load; (8) the final temperature of the transformer after being left on open secondary circuit but normal primary potential for twenty-four hours, and at full load for three hours.

The matters of most practical importance in connexion with an alternating current transformer are (1) the iron core loss, which affects the efficiency chiefly, and must be considered (a) as to its initial value, and (b) as affected by " ageing " or use; (2) the secondary drop or difference of secondary voltage between full and no load, primary voltage being constant, since this affects the service and power of the transformer to work in parallel with others; and (3) the temperature rise when in normal use, which affects the insulation and life of the transformer. The shellacked cotton, oil and other materials with which the transformer circuits are insulated suffer a deterioration in insulating power if continuously maintained at any temperature much above 80° C. to 100° C. In taking the tests for core loss and drop, the temperature of the transformer should therefore be stated. The iron losses are reduced iq value as temperature rises and the copper losses are increased. The former may be I o to 15% less and the latter 20 °A greater than when the transformer is cold. For the purpose of calculations we require to know the number of turns on the primary and secondary circuits, represented by N 1 and N2; the resistances of the primary and secondary circuits, represented by R 1 and R21 the volume (V) and weight (W) of the iron core; and the mean length (L) and section (S) of the magnetic section. The hysteresis loss of the iron reckoned in watts per lb per too cycles of magnetization per second and at a maximum flux density of 2500 C.G.S. units should also be determined.

The experimental examination of a transformer involves the measurement of the efficiency, the iron core loss, and the. secondary drop; also certain tests as to insulation and heating, and finally an examination of the relative phase position and graphic form of the various periodic quantities, currents and electromotive forces taking place in the transformer. The efficiency is best determined by the employment of a properly constructed wattmeter (see Wattmeter). The transformer T (fig. 5) should be so arranged that, if a constant potential transformer, it is supplied with its normal working pressure at the primary side and with a load which can be varied, and which is obtained either by incandescent lamps, L, or resistances in the secondary circuit. A wattmeter, W, should be placed with its series coil, Se, in the primary circuit of the transformer, and its shunt coil, Sh, either across the primary mains in series, with a suitable non-inductive resistance, or connected to the secondary circuit of another transformer, T', called an auxiliary transformer, having its primary terminals connected to those of the transformer under test. In the latter case one or more incandescent lamps, L, may be connected in series with the shunt coil of the wattmeter so as to regulate the current passing through it. The current through the series coil of the wattmeter is then the same as the current through the primary circuit of the transformer under test, and the current through the shunt coil of the wattmeter is in step with, and proportional to, the primary voltage of the transformer. Hence the wattmeter reading is proportional to the mean power given up to the transformer. The wattmeter can be standardized and its scale reading interpreted by replacing the transformer under test by a non-inductive resistance or series of lamps, the power absorption of which is measured by the product of the amperes and volts supplied to it. In the secondary circuit of the transformer is placed another wattmeter of a similar kind, or, if the load on the secondary circuit is non-inductive, the secondary voltage and the secondary current can be measured with a proper alternating current ammeter, A2, and voltmeter, V2, and the product of these readings taken as a measure of the power given out by the transformer. The ratio of the powers, namely, that given out in the external secondary circuit and that taken in by the primary circuit, is the efficiency of the transformer.

In testing large transformers, when it is inconvenient to load up the secondary circuit to the full load, a close approximation to the power taken up at any assumed secondary load can be obtained by adding to the value. of this secondary load, measured in watts, the iron core loss of the transformer, measured at no load, and the copper losses calculated from the measured copper resistances when the transformer is hot. Thus, if C is the iron core loss in watts, measured on open secondary circuit, that is to say, is the power given to the transformer at normal frequency and primary voltage, and if R1 and R2 are the primary and secondary circuit resistances when the transformer has the temperature it would have after running at full load for two or three hours, then the efficiency can be calculated as follows: Let 0 be the nominal value of the full secondary output of the transformer in watts, V 1 and V2 the terminal voltages on the primary and secondary side, N 1 and N2 the number of turns, and A1 and A2 the currents for the two circuits; then 0/V 2 is the full load secondary current measured in amperes, and N 2 N 1 multiplied by 0/V 2 is to a sufficient approximation the value of the corresponding primary current. Hence 0 2 R 2 /V 2 2 is the watts lost in the secondary circuit due to copper resistance, and 0 2 R 1 N 2 2 /V 2 2 N 1 2 is the corresponding loss in the primary circuit. Hence the total power loss in the transformer (= L) is such that 02 (N 1 / 2 022 L = C {- - R2 + N V2 R1 = C }- (R 2 + Rla2)02/V22. Therefore the power given up to the transformer is 0+L, and the efficiency is the fraction 0/(0+L) expressed as a percentage. In this manner the efficiency can be determined with a considerable degree of accuracy in the case of large transformers without actually loading up the secondary circuit. The secondary drop, however, can only be measured by loading the transformer up to full load, and, while the primary voltage is kept constant, measuring the potential difference of the secondary terminals, and comparing it with the same difference when the transformer is not loaded. Another method of testing large transformers at full load without supplying the actual power is by W. E. Sumpner's differential method, which can be done when two equal transformers are available (see Fleming, Handbook for the Electrical Laboratory and Testing Room, ii. 602).

No test of a transformer is complete which does not comprise some investigation of the " ageing " of the core. The slow changes which take place in the hysteretic quality of iron when heated, in the case of certain brands, give rise to a time-increase in iron core loss. Hence a transformer which has a core loss, say, of 300 watts when new, may, unless the iron is well chosen, have its core loss increased from 50 to 300% by a few months' use. In some cases specifications for transformers include fines and deductions from price for any such increase; but there has in this respect been great improvement in the manufacture of iron for magnetic purposes, and. makers are now able to obtain supplies of good magnetic iron or steel with non-ageing qualities. It is always desirable, however, that in the case of large sub-station transformers tests should be made at intervals to discover whether the core loss designed to transform, on the assumption that it m w FIG. 5. - Arrangement for Testing Transformers.

has increased by ageing. If so, it may mean a very considerable increase in the cost of magnetizing power. Consider the case of a 30-kilowatt transformer connected to the mains all the year round; the normal core loss of such a transformer should be about 300 watts, and therefore, since there are 8760 hours in the year, the total annual energy dissipated in the core should be 2628 kilowatt hours. Reckoning the value of this electric energy at only one penny per unit, the core loss costs £Io, 19s. per annum. If the core loss becomes doubled, it means an additional annual expenditure of nearly Li 1. Since the cost of such a transformer would not exceed Ltoo, it follows that it would be economical to replace it by a new one rather than continue to work it at its enhanced core loss.

In Great Britain the sheet steel or iron alloy used for the transformer cores is usually furnished to specifications which state the maximum hysteresis loss to be allowed in it in watts per lb (avoirdupois) at a frequency of 50, and at a maximum flux-density during the cycle of 4000 C.G.S. units. When plates having a thickness t mils are made up into a transformer core, the total energy loss in the core due to hysteresis and eddy current loss when worked at a frequency n and a maximum flux-density during the cycle B is given by the empirical formulae T = 0032nB1.5510-7+(tnB)"IO-ls, or 0-9+I.4(t1nB1)21 where T stands for the loss per cubic centimetre, and T 1 for the same in watts per pound of iron core, B for the maximum fluxdensity in lines per square centimetre, and B 1 for the same in lines per square inch, t for the thickness of the plates in thousandths of an inch (mils), and I, for the same in inches. The hysteresis loss varies as some power near to 1.6 of the maximum flux-density during the cycle as shown by Steinmetz (see Electromagnetism). Since the hysteresis loss varies as the I. 6th power of the maximum flux-density during the cycle (B max.), the advantages of a low flux-density are evident. An excessively low flux-density increases, however, the cost of the core and the copper by increasing the size of the transformer. If the form factor (f) of the primary voltage curve is known, then the maximum value of the flux-density in the core can always be calculated from the formula B =E1/4fnS11 where E is the R.M.S. value of the primary voltage, N l the primary turns, S the section of the core, and n the frequency.

The study of the processes taking place in the core and circuits of a transformer have been greatly facilitated in recent years by the improvements made in methods of observing and recording the variation of periodic currents and electromotive forces. The original method, due to Joubert, was greatly improved and employed by Ryan, Bell, Duncan and Hutchinson, Fleming, Hopkinson and Rosa, Callendar and Lyle; but the most important improvement was the introduction and invention of the oscillograph by Blonde', subsequently improved by Duddell, and also of the ondograph of Hospitalier (see Oscillograph). This instrument enables us, as it were, to look inside a transformer, for which it, in fact, performs the same function that a steam engine indicator does for the steam cylinder.' Delineating in this way the curves of primary and secondary current and primary and secondary electromotive forces, we get the following result: Whatever may be the form of the curve of primary terminal potential difference, or primary voltage, that of the secondary voltage or terminal potential difference is an almost exact copy, but displaced 180° in phase. Hence the alternating current transformer reproduces on its secondary terminals all the variations of potential on the primary, but changed in scale. The curve of primary current when the transformer is an open secondary circuit is different in form and phase, lagging behind the primary voltage curve (fig. 6); but if the transformer is loaded up on its 1 For a useful list of references to published papers on alternating current curve tracing, see a paper by W. D. B. Duddell, read before the British Association, Toronto, 1897; also Electrician (1897), xxxix. 636; also Handbook for the Electrical Laboratory and Testing Room (J. A. Fleming), i. 407.

secondary side, then the primary current curve comes more into step with the primary voltage curve. The secondary current curve, if the secondary load is non-inductive, is in step with the secondary voltage curve (fig. 7). These transformer diagrams yield much information as to the nature of the operations proceeding itt the transformer.

The form of the curve of primary current at no secondary load is a consequence of the hysteresis of the iron, combined with the fact that the form of the core flux-density curves of the transformer is always not far removed from a simple sine curve. If e i is at any moment the electromotive force, i i the current on the primary circuit, and b, is the flux-density in the core, then we have the fundamental relation e,=R 1 i 1 -?-SN l db 1 /dt, where R l is the resistance of the primary, and N 1 the number of turns, and S is the cross-section of the core. In all modern closed circuit transformers the quantity R l i l is very small compared with the quantity SNdb/dt except at one instant during the phase, and in taking the integral of the above equation, viz. in finding the value of feidt, the integral of the first term on the right-hand side may be neglected in comparison with the second. Hence we have approximately b, = (SN1) - 'feldt. In other words, the value of the fluxdensity in the core is obtained by integrating the area of the primary voltage curve. In so doing the integration must be started from the time point through which passes the ordinate bisecting the area of the primary voltage curve. When any curve is formed such that its ordinate y is the integral of the area of another curve, viz. y= fy dx, the first curve is always smoother and more regular in form than the second. Hence the process above described when applied to a complex periodic curve, which can by Fourier's theorem be resolved into a series of simple periodic curves, results in a relative reduction of the magnitude of the higher harmonics compared with the funda mental term, and hence a wiping out of the minor irregularities of the curve. In actual practice the curve of electromotive force of alternators can be quite sufficiently reproduced by employing three terms of the expansion, viz. the first three odd harmonics, and the resulting flux-density curve is always very nearly a simple sine curve.

We have then the following rules for predetermining the form of the current curve of the transformer at no load, assuming that the hysteresis curve of the iron is given, set out in terms of flux-density and ampere-turns per centimetre, and also the form of the curve of primary electromotive force. Let the time base line be divided up into equal small elements. Through any selected point draw a line perpendicular to the base line. Bisect the area enclosed by the curve representing the half wave of primary electromotive force and the base line by another perpendicular. Integrate the area enclosed between the electromotive force curve and these two perpendicular lines and the base. Lastly, set up a length on the last perpendicular equal to the value of this area divided by the product of the cross-section of the core and the number of primary turns. The resulting value will be the core flux-density b at the phase instant corresponding. Look out on the hysteresis loop the same flux-density value, and corresponding to it will be found two values of the magnetizing force in ampere-turns per centimetre, one the value for increasing flux-density and one for decreasing. An inspection of the position of the point of time selected on the time line will at once show which of these to select. Divide that value of the ampere-turns per centimetre by the product of the values of the primary turns and the mean length of the magnetic circuit of the core of the transformer, and the result gives the value of the primary current of the transformer. This can be set up to scale on the perpendicular through the time instant selected. Hence, given the form of the primary electromotive force curve and that of the hysteresis loop of the iron, we can draw the curves representing the changes of flux-density in the core and that of the corresponding primary current, and thus predict the rootmean-square value of the magnetizing current of the transformer. It is therefore possible, when given the primary electromotive force curve and the hysteresis curve of the iron, to predetermine the curves depicting all the other variables of the transformer, provided that the magnetic leakage is negligible.

The elementary theory of the closed iron circuit transformer may be stated as follows: Let N,, N2 be the turns on the primary and secondary circuits, R I. and R2 the resistances, S the Elementary section of the core, and b l and b 2 the co-instantaneous values of the flux-density just inside the primary and secondary windings. Then, if i t and i 2 and e l and e, are the primary FIG. 6. - Transformer Curves at no load.

e l , Primary voltage curve; il, Primary current curve; e2, Secondary voltage curve.

FIG. 7. - Transformer Curves at full load.

e l , Primary voltage curve; il, Primary current curve; e 2 , Secondary voltage curve; i 2, Secondary current curve.

and secondary currents and potential differences at the same instant, these quantities are connected by the equations ei = Rai l +SN 1 e 2 = SN 2 R212.

Hence, if b i = b 2 , and if Ru 1 is negligible in comparison with SN i db/dt, and i=o, that is, if the secondary circuit is open, then el)e2 = N1/N2, or the transformation ratio is simply the ratio of the windings. This, however, is not the case if b 1 and b 2 have not the same value; in other words, if there is magnetic leakage. If the magnetic leakage can be neglected, then the resultant magnetizing force, and therefore the iron core loss, is constant at all loads. Accordingly, the relation between the primary current (i i), the secondary current (12), and the magnetizing current (i), or primary current at no load, is given by the equation N 1 i 1 - N 2 i 2 = N11. Then, writing b for the instantaneous value of the flux-density in the core, everywhere supposed to be the same, we arrive at the identity elil = e212+ (Rri 1 2 -IR 2 i 2) -?- S db (N l i l - N 2 i 2) .

This equation merely expresses the fact that the power put into the transformer at any instant is equal to the power given out on the secondary side together with the power dissipated by the copper losses and the constant iron core loss.

The efficiency of a transformer at any load is the ratio of the mean value, during the period, of the product ecu to that of the product e 2 i 2. The efficiency of an alternating current transformer is a function of the form of the primary electromotive force curve. Experiment has shown' that if a transformer is .tested for efficiency on various alternators having electromotive force curves of different forms, the efficiency values found at the same secondary load are not identical, those being highest which belong to the alternator with the most peaked curve of electromotive force, that is, the curve having the largest form factor. This is a consequence of the fact that the hysteresis loss in the iron depends upon the manner in which the magnetization (or what, here comes to the same thing, the flux-density in the core) is allowed to change. If the primary electromotive force curve has the form of a high peak, or runs up suddenly to a large maximum value, the flux-density curve will be more square-shouldered than when the voltage curve has a lower form factor. The hysteresis loss in the iron is less when the magnetization changes its sign somewhat suddenly than when it does so more gradually. In other words, a diminution in the form factor of the core flux-density curve implies a diminished hysteresis loss. The variation in core loss in transformers when tested on 'various forms of commercial alternator may amount to as much as to %. Hence, in recording the results of efficiency tests of alternating current transformers, it is always necessary to specify the form of the curve of primary electromotive force. The power factor of the transformer or ratio of the true power absorption at no load, to the product of the R.M.S. values of the primary current and voltage, and also the secondary drop of the transformer, vary with the form factor of the primary voltage curve, being also both increased by increasing the form factor. Hence there is a slight advantage in working alternating current transformers off an alternator giving a rather peaked or high maximum value electromotive force curve. This, however is disadvantageous in other ways, as it puts a greater strain upon the insulation of the transformer and cables. At one time a controversy arose as to the relative merits of closed and open magnetic circuit transformers. It was, however, shown by tests made by Fleming and by Ayrton on Swinburne's " Hedgehog " transformers, having a straight core of iron wires bristling out at each end, that for equal secondary outputs, as regards efficiency, open as compared with closed magnetic circuit transformers had no advantage, whilst, owing to the smaller power factor and consequent large R.M.S. value of the magnetizing current, the former type had many disadvantages (see Fleming, " Experimental Researches on Alternate Current Transformers," Journ. Inst. Elec. Eng., 1892).

The discussion of the theory of the transformer is not quite so simple when magnetic leakage is taken into account. In all cases a certain proportion of the magnetic flux linked with Magnetic . the primary circuit is not linked with the secondary Leakage circuit, and the difference is called the magnetic leakage.

This magnetic leakage constitutes a wasted flux which is noneffective in producing secondary electromotive force. It increases with the secondary current, and can be delineated by a curve on the transformer diagram in the following manner. The*curves of primary and secondary electromotive force, or terminal potential difference and current, are determined experimentally, and then two curves are plotted on the same diagram which represent the variation of (e 1 - R l i 1)/N i and (e 2 +R 2 i 2)/N 2; these will represent the time differentials of the total magnetic fluxes Sb 1 and Sb 2 linked respectively with the primary and secondary circuits. The above curves are then progressively integrated, starting from the time 1 See Dr G. Roessler, Electrician (1895), xxxvi. 150; Beeton, Taylor and Barr, Journ. Inst. Elec. Eng. xxv. 474; also J. A. Fleming, Electrician (1894), xxxiii. 580.

point through which passes the ordinate bisecting the area of each half wave, and the resulting curves plotted to express by their ordinates Sb 2 and Sb2. A curve is then plotted whose ordinates are the differences Sb 2 --5b 2, and this is the curve of magnetic leakage.

The existence of magnetic leakage can be proved experimentally by a method due to Mordey, by placing a pair of thermometers, one of mercury and the other of alcohol, in the centre of the core aperture. If there is a magnetic leakage, the mercury bulb is heated not only by radiant heat, but by eddy currents set up in the mercury, and its rise is therefore greater than that of the alcohol thermometer. The leakage is also determined by observing the secondary voltage drop between full load and no load, and deducting from it the part due to copper resistance; the remainder is the drop due to leakage. Thus if V2 is the secondary voltage on open circuit, and V2' that when a current A2 is taken out of the transformer, the leakage drop v is given by the equation v = (V 2 - V2') - 1 R2A2 + R 1 A 2(N 2/ N 1) 2 }.

The term in the large bracket expresses the drop in secondary voltage due to the copper resistance of the primary and secondary circuits.

In drawing up a specification for an alternating current transformer, it is necessary to specify that the maximum secondary drop between full and no load to be allowed shall not exceed a certain value, say 2% of the no-load secondary voltage; also that the iron core loss as a percentage of the full secondary output shall not exceed a value, say, of i % after six months' normal work.

In the design of large transformers one of the chief points for attention is the arrangement for dissipating the heat generated in their mass by the copper and iron losses.

For every watt expended in the core and circuit a Y P, Design. surface of 3 to 4 sq. in. must be allowed, so that the heat may be dissipated. In large transformers it is usual to employ some means of producing a current of air through the core to ventilate it. In these, called air-blast transformers, apertures are left in the core by means of which the cooling air can reach the interior portions. This air is driven through the core by a fan actuated by an alternating current motor, which does not, however, take up power to a greater extent than about or of the full output of the transformer, and well repays the outlay.

Material.	Dielectric strength in kilowatts per centimetre.	Material.	Dielectric strength in kilowatts per centimetre.
Glass. ... .	285	Lubricating oil. .	83
Ebonite. .. .	538	Linseed oil .	67
Indiarubber.. .	492	Cotton-seed oil. .	57
Mica. ... .	2000	Air film02 cm.
Micanite. .. .	4000	thick. .. .	27
American linen paper		Air film 1.6 cm.
paraffined.. .	540	thick. .. .	48

In some cases transformers are oil-insulated, that is to say, included in a cast-iron box which is filled in with a heavy insulating oil. For this purpose an oil must be selected free from mineral acids and water: it should be heated to a high temperature before use, and tested for dielectric strength by observing the voltage required to create a spark between metal balls immersed in it at a distance of i millimetre apart. Oils, however, are inferior in dielectric strength or spark-resisting power to solid dielectrics, such as micanite, ebonite, &c., as shown by the above table of dielectric strengths (see T. Gray, Phys. Rev., 1898, p. 199).

Polyphase Transformers are appliances of similar construction to the single-phase transformers already described, but modified so as to enable them to transform two or more phase-related primary alternating currents into similar secondary currents. Thus, a three-phase transformer may be constructed with a core, as shown in fig. 8. Each core leg is surrounded with a primary coil, and these are joined up either in star or delta fashion, and connected to the three or four line wires. The secondary circuits are then connected in a similar fashion to three or four secondary lines. In the case of twophase transmission with two separate pairs of leads, single-phase transformers may be S, S S3 FIG. 8. - Brush Threephase Transformer.

employed in each branch, but with two-phase three-wire supply, twophase transformers must be supplied.

Phase Transformers are arrangements of static or rotary transformers intended to transform single-phase alternating currents into polyphase currents. An important system of phase transformation has been described by C. F. Scott.' It is known that if two alternating electromotive forces differing in phase are connected in series, the resulting electromotive force will in general differ in phase and value from either of the components. Thus, if two alternating electromotive forces differing 90° in phase, and having magnitudes in the ratio of I: 1 13, are connected in series, the resulting electromotive force will have a magnitude represented by 2, and the three can be represented by the sides of a triangle which is half an equilateral triangle. If then a two-phase alternator, D (fig. 9), provides two-phase currents, and if the two circuits are connected, as shown, to a pair of single-phase transformers, T 1 and T2, we can obtain three-phase alternating currents from the arrangement. The primaries of both transformers are the same. The secondary circuit of one transformer, T2, has, say, too turns, and a connexion is made to its middle point 0, and this is connected to the secondary of the other transformer which has 87 (= 5 0 1,I 3) turns. From the points A, B, C we can then tap off three-phase alternating currents. The advantages of the Scott system are that we can transform two-phase alternating currents into three-phase for transmission, and then by a similar arrangement retransform back again into two-phase for use. In this manner an economy of 25% in copper is effected, for instead of four transmission lines we have only three. The system adapts itself for the transmission of currents both for power in driving three-phase motors and for working incandescent lamps. A somewhat similar system has been designed by C. P. Steinmetz for producing threephase currents from single-phase (see Electrician, xliii. 236). When a number of alternating electromotive forces are maintained in a closed circuit, the sum of all must be zero, and may be represented by the sides of a closed polygon. The fundamental principle of Mr Steinmetz's invention consists in so choosing the number of these electromotive forces that the polygon must remain stable. Thus, if three single-phase alternators are driven independently at constant speed and excitation, and if they are joined in series, then three wires led away from the junction points will provide three-phase currents to a system from which lamps and motors may be worked.

Reference must be made to the continuous current transformer. The conversion of a continuous current supplied, say, at Ioo volts, into one having an electromotive force of 10 volts, can of course be achieved by coupling together on the Current same bedplate a suitable electric motor and a dynamo.

The combination is called a motor-dynamo set, and each machine preserves its own identity and peculiarity. The same result may, however, be accomplished by winding two separate armature circuits on one iron core, and furnishing each with its own commutator. The two circuits are interlaced or wound on together. An arrangement of this, kind constitutes a rotatory or rotary transformer, or continuous current transformer. It has the advantage of greater cheapness and efficiency, because one field magnet serves for both armature windings, and there is only one armature core and one pair of bearings; moreover, no shift or lead of the brushes is required at various loads. The armature reactions of the two circuits annul each other. Machines of this description are self-starting, and can be constructed to take in primary current at high pressures, say moo to 2000 volts, and yield another larger current of much lower voltage, say 100 or 150 volts, for use with electric lamps. They are used in connexion with public electric supply by continuous current in many places.

Another important class of rotatory transformer is that also called a rotatory converter, by means of which continuous current is translated into alternating current of one-, twoor three-phase, or vice versa. The action of such an appliance may best be understood by considering the simple case of a Gramme ring armature 1 Proceedings of the National Electric Light Association (Washington, U.S.A., 1894); also Electrician (1894), xxxii. 640.

(see Dynamo) having, in addition to its commutator, a pair of insulated rings on its shaft connected with opposite ends of the armature winding (fig. to). If such a ring is placed in a bipole field magnet, and if a pair of brushes make contact with the commutator C and another pair with the two rings called slip rings, S i S 2, and if continuous current at a constant voltage is supplied to the commutator side, then the armature will begin to revolve in the field, and from the brushes in contact with the slip rings we can draw off an alternating current. This reaches its maximum value when the points of contact of the rings with the armature circuit pass the axis of commutation, or line at right angles to the direction of the magnetic field, for it has at this moment a value which is double the steady value of the continuous current being poured into the armature. The maximum value of the electromotive force creating this alternating current is nearly equal to the electromotive force on the continuous current side. Hence if A is the maximum value of the continuous current put into the armature and V is the value of the brush potential difference on the continuous current side, then 2A is the maximum value of the outcoming alternating current and V is the maximum value of its voltage. Hence 2AV/2 = AV is the maximum value of the outcoming alternating current power, and if we neglect the loss in the armature for the moment, the power given out is equal to the power put in. Hence, assuming a simple harmonic law of variation, the effective value of the alternating current voltage is V/ 1 12, and that of the alternating current is 2Af2. This conclusion follows at once from the fact that the mean value of the square of a sine function is half its maximum value, and hence the R.M.S. value is 1/ 1 12 times the maximum value. The outcoming alternating current has its zero value at the instant when the ends of the diameter of the axis to which the rings are connected are in the direction of the magnetic field of the transformer. Hence the power output on the alternating current side varies from a maximum value AV to zero. The rotatory transformer thus absorbs continuous current power and emits it in a periodic form; accordingly, there is a continual storage and emission of energy by the armature, and therefore its kinetic energy is periodically varying during the phase. The armature is also creating a back-electromotive force which acts at some instants against the voltage driving the current into the armature and at others is creating an electromotive force that assists the external impressed voltage in driving a current through the alternating current side. If we put on another pair of insulated rings and connect them to points of the insulated diameter at right angles to the points of connexion of the first pair of rings, we can draw off another alternating current, the phase of which differs 90° from that of the first. Similarly, if we provide three rings connected to points removed 120° apart on the armature circuit, we can tap off a three-phase alternating current.

Returning to the case of the single-phase rotatory transformer, we may notice that at the instant when the outcoming alternating current is zero the armature is wholly engaged in absorbing power and is acting entirely as a motor. When the alternating current is a maximum, the armature on the other hand is acting as a generator and adds current to the current put into it. The ratio between the potential difference of the brushes on the continuous current side and the root-mean-square or effective value of the voltage between any pair of rings on the alternating current side is called the transformation ratio of the converter.

	Angle			Effective voltage on
Number	between		Voltag	alternating
of slip	connexi ns	Type enerated.	o.	as percentage
rings.	to armatures.			of voltage on continuous current side.
2	180°	Single-phase	11 2 :1	70.71
3	120°	Three-phase	2112:11 3	61.23
4	90°	Two-phase	112:1	70.71
4	90°	Four-phase	2:1	50
6	60°	Three-phase	31/3:1/3	61.23
6	60°	Six-phase	2,/2:1	35'35