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transformer elementary form

Transformers are needed in electronic apparatus to provide the different values of plate, filament, and bias voltage required for proper tube operation, to insulate circuits from each other, to furnish high impedance to alternating but low impedance to direct current, and to maintain or modify wave shape and frequency response at different potentials. The very concept of impedance, so characteristic of electronics, almost necessarily presupposes a means of changing from one impedance level to another, and that means is commonly a transformer.

Impedance levels are usually higher in electronic, as compared with power, equipment. Consider the connected kva on an 11,000-volt power line; it may easily total 1,000,000. Compare this with a large broadcast transmitter operating at the same voltage and drawing 70 kva. The currents in the two cases are 90,000 amp and 6 amp, respectively. For the powder line, the load impedance is 11,000/90,000, or slightly more than 0.1 ohm; for the transmitter it is 11,000/6, or nearly 2,000 ohms. Source impedances are approximately proportional to these load impedances. In low-power electronic circuits the source impedance often exceeds the load impedance and influences the transformer performance even further.

Weight and space are usually at a premium in electronic equipment, and reliability is of paramount importance. Transformers account for a considerable portion of the weight and space, and form a prime component of the reliability.

These and other differences of application render many power transformers unsuitable for electronic circuit use. The design, construction, and testing of electronic transformers have become separate arts, directed toward the most effective use of materials for electronic applications.

3. New Materials. Like all electronic apparatus, transformers are subject to continual change. This is especially so since the introduction of new materials such as

(a) Grain-oriented core steel.

(b) Solventless impregnating varnish.

(c) Inorganic insulating tape.

(d) Improved wire enamel.

(e) Low-loss, powdered iron cores. (/) Ferrite cores.

Through the application of these materials, it has been possible to



(a) Reduce the size of audio and power transformers and reactors.

(b) Increase the usefulness of saturable reactors as magnetic amplifiers.

(c) Reduce the size of high-voltage units.

(d) Design filters and reactors having sharper cut-off and higher Q than previously was thought possible.

(e) Make efficient transformers for the non-sinusoidal wave shapes such as are encountered in pulse, video, and sweep amplifiers,

(/) Extend the upper operating frequency of transformers into the high-frequency r-f range.

Occasionally someone asks why electronic transformers cannot be designed according to curves or charts showing the relation between volts, turns, wire size, and power rating. Such curves are very useful in designing the simpler transformers. However, this idea has not been found universally practicable for the following reasons:

(a) Regulation. This property is rarely negligible in electronic circuits. It often requires care and thought to use the most advantageous winding arrangement in order to obtain the proper IX and IR voltage drops. Sometimes the size is dictated by such considerations.

(£ ) Frequency Range, The low-frequency end of a wideband transformer operating range in a given circuit is determined by the transformer open-circuit inductance. The high-frequency end is governed by the leakage inductance and distributed capacitance. Juggling the various factors, such as core size, number of turns, interleaving, and insulation, in order to obtain the optimum design constitutes a technical problem too complex to solve on charts.

(c) Voltage. It would be exceedingly difficult, if not impossible, to reduce to chart form the use of high voltages in the restricted space of a transformer. Circuit considerations are very important here, and the transformer designer must be thoroughly familiar with the functioning of the transformer to insure reliable operation, low cost, and small dimensions.

{d) Size. Much electronic equipment is cramped for space, and, since transformers often constitute the largest items in the equipment, it is imperative that they, too, be of small size. An open-minded attitude toward this condition and good judgment may make it possible to meet the requirements which otherwise might not be fulfilled. New materials, too, can be instrumental in reducing size, sometimes down to a small fraction of former size.

In succeeding chapters the foregoing considerations will be applied



coil form

primary winding

secondary winding

coil


core laminations

core flux-Fig. 1. Transformer coil and core.

primary winding

secondary winding

A simple transformer coil and core arrangement is shown in Fig. 1. The primary and secondary coils are wound one over the other on an insulating coil tube or form. The core is laminated to reduce losses. Flux flows in the core along the path indicated, so that all the core flux threads through or links both windings. In a circuit diagram

the transformer is represented by the circuit symbol of Fig. 2.

4. Transformer Fundamentals. The simple transformer of Fig. 2 has two windings. The left-hand winding is assumed to be connected to a voltage source and is called the primary winding. The right-hand winding is connected to a load and is called the secondary. The transformer merely delivers to the load a voltage similar to that impressed across its primary, except that it may be smaller or greater in amplitude.

In order for a transformer to perform this function, the voltage across it must vary with respect to time. A d-c voltage such as that of a storage battery produces no voltage in the secondary winding or power

Fig. 2. Simple transformer.

to the performance and design of several general types of electronic transformers. The remainder of this chapter is a brief review of fundamental transformer principles. Only iron-core transformers with closed magnetic paths are considered in this introduction. Air-core transformers, with or without slugs of powdered iron, are discussed in a later chapter on high-frequency transformers. Most transformers operate at power frequencies; it is therefore logical to begin with low-frequency principles. These principles are modified for other conditions in later chapters.



62 N2

where ei = primary voltage 62 = secondary voltage iVi = primary turns N2 = secondary turns.

in the load. If both varying and d-c voltages are impressed across the primary, only the varying part is delivered to the load. This comes about because the voltage e in the secondary is induced in that winding by the core flux ф according to the law

e= - lX 10- (1)

This law may be stated in words as follows: The voltage induced in a coil is proportional to the number of turns and to the time rate of change of magnetic flux in the coil. This rate of change of flux may be large or small. For a given voltage, if the rate of change of flux is small, many turns must be used. Conversely, if a small number of turns is used, a large rate of change of flux is necessary to produce a given voltage. The rate of change of flux can be made large in two ways, by increasing the maximum value of flux and by decreasing the period of time over which the flux change takes place. At low frequencies, the flux changes over a relatively large interval of time, and therefore a large number of turns is required for a given voltage, even though moderately large fluxes are used. As the frequency increases, the time interval between voltage changes is decreased, and for a given flux fewer turns are needed to produce a given voltage. And so it is that low-frequency transformers are characterized by the use of a large number of turns, whereas high-frequency transformers have but few turns.

If the flux Ф did not vary with time, the induced voltage would be zero. Equation 1 is thus the fundamental transformer equation. The voltage variation with time may be of any kind: sinusoidal, exponential, sawtooth, or impulse. The essential condition for inducing a voltage in the secondary is that there be a flux variation. Only that part of the flux which links both coils induces a secondary voltage.

In equation 1, if ф denotes maxwells of flux and t time in seconds, e denotes volts induced.

If all the flux links both windings, equation 1 shows that equal volts per turn are induced in the primary and secondary, or

ei Ni



5. Sinusoidal Voltage. If the flux variation is sinusoidal,

Ф = Фтах sin wt

where Фщах is the peak value of flux, со is angular frequency, and t is time. Equation I becomes

e = -ЛГФшахСо cos ot X 10~ (3)

or the induced voltage also is sinusoidal. This voltage has an effective value

E = 0.707 X 27г/ЛГф^ X 10

= 4.44/Л^Ф„ X 10- (4)

where / is the frequency of the sine wave. Equation 4 is the relation between voltage and flux for sinusoidal voltage.

Sufficient current is drawn by the primary winding to produce the flux required to maintain the winding voltage. The primary induced voltage in an unloaded transformer is just enough lower than the impressed voltage to allow this current to flow into the primary winding. If a load is connected across the secondary terminals, the primary induced voltage decreases further, to allow more current to flow into the winding in order that there may be a load current. Thus the primary of a loaded transformer carries both an exciting current and a load current, but only the load part is transformed into secondary load current.

Primary induced voltage would exactly equal primary impressed voltage if there were no resistance and reactance in the winding. Primary current flowing through the winding causes a voltage drop IR, the product of primary current I and winding resistance R. The winding also presents a reactance X which causes an IX drop. Reactance X is caused by the leakage flux or flux which does not link both primary and secondary windings. There is at least a small percentage of the flux which is not common to both windings. Leakage flux flows in the air spaces adjacent to the windings. Because the primary turns link leakage flux an inductance is thereby introduced into the winding, producing leakage reactance X at the line frequency. The larger the primary current, the greater the leakage flux, and the greater the reactance drop IX. Thus the leakage reactance drop is a series effect, proportional to primary current.

6. Equivalent Circuit and Vector Diagram. For purposes of analysis the transformer may be represented by a 1:1 turns-ratio equivalent circuit. This circuit is based on the following assum tions:



(a) Primary and secondary turns are equal in number. One winding is chosen as the reference winding; the other is the referred winding. The voltage in the referred winding is multiplied by the actual turns ratio after it is computed from the equivalent circuit. The choice between primary and secondary for the reference winding is a matter of convenience.

(b) Core loss may be represented by a resistance across the terminals of the reference winding.

(c) Core flux reactance may be represented by a reactance across the terminals of the reference winding.

{d) Primary and secondary IR and IX voltage drops may be lumped together; the voltage drops in the referred winding are multiplied by a factor derived at the end of this section, to give them the correct equivalent value.

(e) Equivalent reactances and resistances are linear.

As will be shown later, some of these assumptions are approximate, and the analysis based on them is only accurate so far as the assumptions are justified. With proper attention to this fact, practical use can be made of the equivalent circuit.

With many sine-wave electronic transformers, the transformer load is resistive. A tube filament heating load, for example, has 100 per cent power factor. Under this condition the relations between voltages and currents become appreciably simplified in comparison with the same relations for reactive loads. In what follows, the secondary winding will be chosen as the reference winding. At low frequencies such a transformer may be represented by Fig. 3(a). The transformer equivalent circuit is approximated by Fig. 3(6), and its vector diagram for 100 per cent p-f load by Fig. 3(c). Secondary load voltage Ei, and load current I are in phase. Secondary induced voltage Es is greater than El because it must compensate for the winding resistances and leakage reactances. The winding resistance and leakage reactance voltage drops are shown in Fig. 3(c) as IR and IX, which are respectively in phase and in quadrature with II and El. These voltage drops are the sum of secondary and primary winding voltage drops, but the primary values are multiplied by a factor to be derived later. If voltage drops and losses are temporarily forgotten, the same power is delivered to the load as is taken from the line. Let subscripts 1 and 2 denote the respective primary and secondary quantities.

E,I, = E2I2 (5)




Es Xn 1

II El IR

Fig. 3. (a) Transformer with resistive load; (b) equivalent circuit; (c) vector

diagram.

so that the voltages are inversely proportional to the currents. Also, from equation 2, they are directly proportional to their respective turns.

(2a)

Now the transformer may be replaced by an impedance Zi drawing the same current from the line, so that

Likewise

12 - E2/Z2

where Z2 is the secondary load impedance, in this case Rl. If these expressions for current are substituted in equation 6,

2 Ke.) ~ Wo/

Equation 7 is strictly true only for negligible voltage drops and losses. It is approximately true for voltage drops up to about 10 per cent of the winding voltage or for losses less than 20 per cent of the power delivered, but it is not true when the voltage drops approach in value the winding voltage or when the losses constitute most of the primary load.

Not onl does the load impedance bear the relation of equation 7



to the equivalent primary load impedance; the winding reactance and resistance may also be referred from one winding to the other by the same ratio. This can be seen if the secondary winding resistance and reactance are considered part of the load, across which the secondary induced voltage Es appears. Thus the factor by which the primary reactance and resistance are multiplied, to refer them to the secondary for addition to the secondary drops, is {Nz/N). If the primary had been the reference winding, the secondary reactance and resistance would have been multiplied by [Nt/N].

In Fig. 3(c) the IR voltage drop subtracts directly from the terminal voltage across the resistive load, but the IX drop makes virtually no difference. How much the IX drop may be before it becomes appreciable is shown in Fig, 4. If the IX drop is 30 per cent of the induced voltage, 4 per cent reduction in load voltage results; 15 per cent IX drop causes but 1 per cent reduction.

7. Magnetizing Current. In addition to the current entering the primary because of the secondary load, there is the core exciting current Jr which flows in the primary whether the secondary load is connected or not. This current is drawn by the primary core reactance and

equivalent core-loss resistance R and is multiplied by N/N2 when it is referred to the secondary side. It has two components: Im, the magnetizing component which flows 90°

lagging behind induced voltage Es] and Ie, the core-loss current which is in phase with Es. Ordinarily this current is small and produces negligible voltage drop in the winding.

Core-loss current is often divided into two components: eddy current and hysteresis. Eddy-current loss is caused by current circulating in the core laminations. Hysteresis loss is the power required to magnetize the core first in one direction and then in the other on alternating half-cycles. Hysteresis loss and magnetization are intimately connected, as can be seen from Fig. 5. Here induced voltage e is plotted against time, and core flux ф lags e by 90 , in accordance with equation 3. This flux is also plotted against magnetizing current in the loop at the right. This loop has the same shape as the B-H loop


Fig. 4. Relation between reactive voltage drop and load voltage.



ELECTRONIC TRANSFORMERS AND CIRCUITS I.4tE


Fig. 5. Transformer voltage, flux, and exciting current.

for the grade of iron used in the core, but the scales are changed so that

where В H

Ic =

Ф = BAc г = HIc/OAttN

core flux density in gauss core cross-sectional area in cm core magnetizing force in oersteds core flux path length in cm.

Current is projected from the -г loop to obtain the alternating current i at the bottom of Fig. 5. This current contains both the magnetizing and the hysteresis loss components of current. In core-material research it is important to separate these components, for it is mainly through reduction of the B-H loop area (and hence hysteresis loss) that core materials have been improved. Techniques have been developed to separate the exciting current components, but it is evident that these components cannot be separated by current measurement only. It is nevertheless convenient for analysis of measurements to add the loss components and call their sum Ie, and to regard the magnetizing component Im as a separate lagging current, as in Fig. 3. As long as the core reactance is large, the vector sum In of Im and Ib is



small, and the non-sinusoidal shape of 1 does not seriously affect the accuracy of Fig. 3.

Core flux reactance may be found by measuring the magnetizing current, i.e., the current component which lags the applied voltage 90° with the secondary circuit open. Because of the method of measurement, this is often called the open-circuit reactance, and this reactance divided by the angular frequency is called the open-circuit inductance. The secondary and primary winding leakage reactances are found by short-circuiting the secondary winding and measuring the primary voltage with rated current flowing. The component of primary voltage which leads the current by 90° is divided by the current; this is the sum of the leakage reactances, the secondary reactance being multiplied by the (turns ratio) , and is called the short-circuit reactance.

Practical cases sometimes arise where the magnetizing component becomes of the same order of magnitude as II. Because current Im flows only in the primary, a different equivalent circuit and vector diagram are necessary, as shown in Fig. 6. Note that the leakage react-

Re Es

Ч El


I,X,

Fig. 6. (a) Equivalent circuit and ib) vector diagram for transformer with high

magnetizing current.

ance voltage drop has a marked effect upon the load voltage, and this effect is larger as Im increases relative to II. Therefore, the statement that IX voltage drop causes negligible difference between secondary induced and terminal voltages in transformers with resistive loads is true only for small values of exciting current. Moreover, the total primary current Ii has a largely distorted shape, so that treating the IR and IX voltage drops as vectors is a rough approximation. For




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