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A single-phase transformer consists of a core composed of juxtaposed silicon steel sheets. In general, these sheets are cut into some typical shapes, such as E and I or E and E. Normally, this type of cut is used to form shell core. The cuts, type U and I or L, are used to form core type. Note that all of them generate a closed path for the magnetic flux, as we can see in Figure 93-01.

Generally, materials are either hard or soft. Hard materials are those in which magnetization tends to be permanent, while soft materials are used in magnetic circuits of electrical machines and transformers. They are related, although their uses are largely distinct.

Therefore, by adding coils made of copper or aluminum wires to these metal structures, it is possible to build a transformer. In the following items, we will address several characteristics of the transformer.

It is worth noting that, in the study of magnetism, certain variables can be expressed in several units. In some books, some units linked to the old CGS system are still mentioned. Let's look at some relationships.

In the International System, SI, the unit of measurement for magnetic flux is the weber and that for magnetic induction density, B or also known as magnetic flux density, is measured in Tesla, and is equivalent to 1 Wb / m². However, texts and books are still produced where the magnetic flux density is expressed in gauss, G, a unit of the CGS system. The correspondence between T and G is 1 T = 10,000 gauss. On the other hand, the magnetic field intensity, H, has the dimension of ampere-turns per meter in the SI system. And in the CGS system, its unit is the oersteds, Oe. So, one oersted means that this magnetic field intensity is needed to generate a magnetic flux density of one gauss in free space.

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Let's designate the primary inductance as L₁ and the secondary inductance as L₂. Between these two windings there is a coupling determined by the mutual inductance, M ( See Here !!). Thus, these variables are related by the coupling coefficient, according to eq. 78-07 below.

Analyzing this equation, we conclude that, if L₁ = L₂ = L = M, the result is k = 1, something physically impossible. Therefore, the objective is to maximize the value of M.

Two equations already studied will be recalled below.

We are considering that the magnetic medium is air. By joining these two pieces of information, we obtain the following equation.

As we know, the value of μ_o is very small. Its value is μ_o = 4 π x 10^-7   H/m. Considering, in this equation, μ_o and L_n as constants, the variables that we can vary are I and N. Therefore, to obtain a value of B that can be used in transformers, high values ​​of I and N would be necessary. This, in itself, makes the use of materials with low magnetic permeability unfeasible.

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The silicon steel sheets that make up the core of transformers and electrical machines in general have exceptional magnetic properties. They are composed of an iron alloy with the addition of silicon, generally varying between 2% and 4.5% of their composition. As they have a very narrow hysteresis curve, this means that hysteresis losses are reduced, resulting in lower energy losses during magnetization and demagnetization cycles. In addition, their high magnetic permeability facilitates the induction of strong magnetic fields with reduced magnetization currents, further reducing losses.

On the other hand, despite being a ferromagnetic material, silicon steel has low electrical conductivity, which minimizes eddy currents that form in the core. These eddy currents also contribute to energy losses in the transformer.

Due to all these properties, the use of silicon steel in electrical machines makes it possible to build highly efficient devices.

With the aim of further improving the properties of silicon steel, the silicon steel industry is undergoing constant technological advances. Currently, there are three types of silicon steel:

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We can define a transformer as a static machine that transforms AC electrical power at a given voltage and current level to another voltage and current level, using a magnetic field as an intermediate medium. Basically, it is formed by two or more windings, electrically isolated from each other, having a certain electrical resistance and self-inductance. These windings are coupled via a mutual inductance (M), that is, they are mutual magnetic fluxes that must be obligatory variable over time for the phenomenon of self-induction to exist, which, in turn, will produce the electromotive force (EMF) in the windings. mutual inductance was studied in more detail in Chapter 78. The reader can access   ☞     Clicking Here !!

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Note the model above. E₁ represents the so-called counter electromotive force, the voltage that opposes the voltage applied to the primary, V₁. If V₁ is constant, then E₁ will also be constant, as will the magnetic flux in the transformer core. This constancy of values ​​must be maintained regardless of the electric current supplied by the secondary to the load. Therefore, the primary winding must absorb from the feeder line, in addition to the magnetizing current, an electric current I'₁. This current is called the primary reaction current.

Furthermore, the flux must lag behind the induced EMF by 90°. A transformer requires less magnetic material if it operates at a higher magnetic flux density. Therefore, from an economic point of view, a transformer is designed to operate close to the saturation region of the magnetic core, although this increases the presence of harmonics in the wave.

When the voltage applied to the primary of a transformer is sinusoidal, the mutual flux established in the core is considered sinusoidal. The no-load current, I_P (exciting current), will be non-sinusoidal due to the hysteresis loop containing the fundamental harmonic and all odd harmonics.

Consider the core hysteresis loop as shown in Figure 93-04. We know that Φ = B A and I = (H L_n ) /N. Thus, the hysteresis loop is plotted in terms of flux Φ and current I rather than B and H, so that the current required to produce a given value of flux can be read directly. The no-load current waveform, I_P, can be found from the sinusoidal flux waveform and the Φ - I (hysteresis loop) characteristics of the magnetic core. The graphical procedure for determining the I_P waveform is shown in Figure 93-04.

We observe that the no-load current waveform presents odd harmonics, such as the third, fifth and other higher orders, which rapidly intensify as the maximum flux approaches saturation. However, the third harmonic is the most relevant. For most applications, harmonics above the third order can be considered insignificant. In the no-load current, the contribution of the third harmonic corresponds approximately 5% to 10% of the fundamental at nominal voltage.

In load situations, the total primary current corresponds to the phasor sum of the load current and the no-load current. Because the magnitude of the load current is significantly greater than that of the no-load current, the primary current appears almost like a sinusoidal wave in practical terms during operation under load. Therefore, when the transformer operates in the saturation region, the magnetizing current has a higher proportion (30 to 40%) of harmonics.

If the values ​​of the current required to generate a given flux are compared with the flux in the core, for various values, we can construct a simple graph of the magnetizing current flowing in the core winding. This graph is shown in the lower part of Figure 93-04.

With respect to the magnetizing current, we should note the following points:

Regarding the loss current in the transformer's magnetic core, it is important to note that:

The total current in the no-load core is called the exciting current of the transformer. It is simply the sum of the magnetizing current and the core loss current.

In a well-designed power transformer, the exciting current is much smaller than the full-load current of the transformer.

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The magnetizing current of a transformer is in the order of 3 to 5% of the nominal current. This occurs when the transformer is operating in the steady state. However, when the transformer is energized for the first time, there may be a sudden transient current circulation in the primary winding, with a high magnitude. The flux may reach a maximum value equivalent to twice the usual flux. This causes the core to saturate, resulting in a high peak in the magnetizing current. This current, called inrush current, can reach values ​​between 10 to 20 times the value of the nominal current of the transformer.

The value of the inrush current depends on the instant at which the transformer is connected to the power supply network. The inrush current will be maximum when the input voltage passes through the value zero. This current is minimized when the input voltage passes through its maximum value, that is, at the peak. However, in practice, it is not possible to select a specific instant in time to activate the switch connecting the transformer to the power supply network. Thus, high values ​​of the inrush current generate mechanical forces of great magnitude on the transformer windings. Therefore, the windings must be firmly fixed around the core to withstand great mechanical forces without moving, which could seriously compromise the transformer's performance or even lead to its destruction.

When a transformer is connected to supply power, in the worst case scenario, the transient inrush current causes a high value of magnetic flux in the core. The flux in the core can reach twice the normal value. This is called the doubling effect of the magnetic flux.

It is important to consider that when a transformer is disconnected from the supply network, it may not be possible to reduce the magnetic flux to zero. For this reason, the transformer core retains some residual flux, which we will call Φ_r.

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Here, Φ_c represents an integration constant. Its value can be determined from the initial conditions for t = 0. Consider that, when disconnected from the supply network, the transformer retained a small residual magnetic flux, Φ = Φ_r, in its core. Thus, for t = 0, we have Φ = Φ_r.

Substituting this value into eq. 93-07, we have:

Note that the portion between the red parentheses represents the transient component, while the other portion represents the steady state. And the magnitude of the transient component is given by eq. 93-09 and is a function of φ, where φ is the instant at which the transformer is connected to the power grid.

So, if the transformer is connected at the instant φ = 0, we have cos φ = 1 and eq. 93-09, becomes:

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When a coil is surrounded by a changing magnetic flux, it induces an EMF in the coil according to Faraday's law. If this coil is wrapped around a magnetic material, an EMF will also be induced in the magnetic material by the same magnetic flux. Since the magnetic material is also a good electrical conductor, the EMF induced within the material causes electric currents to appear within it, called eddy currents or eddy currents. The location and path of this current causes it to surround the magnetic flux that produces it. In fact, there are as many closed paths around the magnetic field within the magnetic material as one can imagine.

On the left side of Figure 93-05, we can see some of the paths for the currents induced in a solid magnetic material when the flux density is increasing with time. As the flux in the magnetic material increases, so does the induced EMF in each circular path. The increase in the induced EMF results in an increase in the eddy current in that path. As a result of this current, energy is converted to heat in the resistance of the path. By adding together the power loss in each closed loop within the magnetic material, we obtain the total power loss in the magnetic material caused by the eddy currents. This power loss is called the eddy current loss.

Eddy currents establish their own magnetic flux, which tends to oppose the original magnetic flux. Therefore, eddy currents not only result in eddy current losses within the magnetic material, but also exert a demagnetizing effect on the core. Consequently, a larger MMF is required to produce the same magnetic flux in the core. Demagnetization intensifies as we approach the core axis. The overall effect of demagnetization is to bunch up the magnetic flux toward the outer surface of the magnetic material. This makes the inner part of the core magnetically ineffective.

Due to all these factors, it is necessary to adopt measures that minimize these effects. One way to minimize the adverse effects of eddy currents is to make the magnetic core highly resistive in the direction in which eddy currents tend to flow.

We can do this in practice by stacking thin pieces of magnetic materials to form a magnetic core, as shown on the right side of Figure 93-05. The thin piece, known as a lamination, is coated with varnish or shellac and is commercially available in thicknesses ranging from 0.36 mm to 0.70 mm. The thin coating thus makes each lamination reasonably well electrically insulated from the others. In addition, the magnetic core constructed of these laminations forces eddy currents to travel long, narrow paths within each lamination. The resulting effect is to reduce eddy currents in the magnetic material.

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In free space (nonmagnetic material), the permeability μ_o is constant. Thus, the relation B - H is linear. However, in the case of ferromagnetic materials used in electrical machines, this is not the case, where the relation B - H is strictly nonlinear in two respects: saturation and hysteresis.

The hysteresis nonlinearity is the double-valued B - H relation exhibited in the variation of the H cycle as shown in Figure 93-06.

Starting at point O and a little beyond this point, the curve presents a non-linearity. In the middle of the curve, it presents an "almost" linearity, ending with a region of saturation. In this region, the B field increases much less quickly than H. Near the point P, the saturation is quite pronounced and the magnetic material behaves like free space.

For economic reasons, electrical machines and transformers are designed to work a little beyond the beginning of the saturation zone.

Hysteresis in electrical machines gives rise to the so-called hysteresis loss. These losses are caused by the variation of the magnetization cycle, that is, the material undergoes magnetization in one direction and, subsequently, the direction is reversed. Thus, energy is lost due to molecular friction in the material, that is, the magnetic domains of the material resist being rotated first in one direction and then in the other. Therefore, the magnetic material requires extra energy to overcome this opposition. We can state that these losses per unit volume of the magnetic material in each magnetization cycle are proportional to the area of ​​the hysteresis cycle. We should also note that, if the temperature of the machine increases, the hysteresis losses also increase.

The most commonly used ferromagnetic materials in transformers and electrical machines are cast iron, cast steel and the best of them, silicon steel. Each of these materials presents a characteristic curve as shown in Figure 93-07.

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When we connect the primary winding to the mains, the transformer core is subjected to magnetization at the mains frequency. Therefore, a certain amount of energy is wasted in this process.

The hysteresis losses, W_h, as stated in the previous item, depend on the area of ​​the hysteresis loop and the material used in the core. These losses can be expressed by eq. 93-14.

In this equation, the variables involved are:

It is important to consider that, when we add small percentages of silicon to the steel, the hysteresis loop becomes smaller and, consequently, there is a reduction in hysteresis losses.

On the other hand, when variations in the magnetic flux occur, currents are induced in the ferromagnetic material, causing parasitic losses (eddy loss). These losses are directly proportional to the squared of the thickness, t, of the laminations used in the manufacture of the transformer core.

In this equation, the variables involved are:

Thus, the sum of the parasitic losses and the hysteresis loss results in the total iron losses, P_fe, given by eq.93-16.

Remembering what was studied in chapter 91, the electromotive force (EMF) induced in the windings of a transformer is given by eq. 91-12 or eq. 91-13. For better understanding, let's repeat eq. 91-12 below. This equation represents the effective or RMS value of the induced voltage. The reader can access this item   ☞     Magnetic Flux and Induced Voltage

Recalling eq. 75-11a where Φ_m = A B_m and substituting this value into eq. 91-12, we obtain:

In the equation above, setting K = 4.44 A N, we can rewrite it as follows:

Analyzing the equation above, we can easily conclude that the value of B_m is proportional to the ratio V / f. Mathematically, we can write:

Therefore, this equation indicates that B_m will remain constant as long as the ratio between the applied voltage and the frequency of the power grid is also kept constant.

Using eq. 93-04 and those developed earlier, we can write:

Thus, keeping the ratio V / f constant, we obtain B_m constant. Then, it is possible to write the following relation:

In this equation, A and B are constants to be determined. Thus, it is possible to determine the values ​​of these constants by measuring the total losses in the iron, P_fe, at two different frequencies, keeping the ratio V / f constant. In this way, we obtain a system of two equations with two unknowns, whose solution is simple, as will be demonstrated below.

By performing a no-load test on the transformer, we can determine the value of the iron losses, Pfe, by reading the value measured by the wattmeter, W, and subtracting the loss caused by the resistance of the primary winding. Mathematically, we can write:

Therefore, we must perform a no-load test on the transformer at two different frequencies, f₁ and f₂. In both situations, it is necessary to adjust the voltage V to keep the value of B_m constant. Thus, it is possible to separate the two components of the losses: loss due to hysteresis and loss due to eddy currents.

To better understand the application of this technique, access the problems   Problem 93-9   and   Problem 93-10

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The presence of the iron core causes a loss because energy is needed to magnetize the core. To represent this loss in the electrical model of the transformer, we add inductance in parallel with the primary winding, symbolized in the model by a reactance designated as X_m.

In addition, we must provide a resistance in parallel with this reactance to represent the losses that occur due to the presence of hysteresis and eddy currents that exist in the iron core. This resistance is symbolized by R_h in the electrical model of the transformer, as can be seen in Figure 93-09.

Note that in the circuit model we have the voltage E₁ across the magnetizing circuit. The current I_h across the resistance R_h is in phase with E₁. However, the current I_m, which flows through X_m, lags behind 90° in relation to E₁. Thus, we can relate the three currents, that is, I_p, I_h and I_m through the equation eq. 93-20 below.

Note in the model above, E₁ represents the so-called counter electromotive force, a voltage that opposes the voltage applied to the primary, V₁. If V₁ is constant, then E₁ will also be constant, as will the magnetic flux in the transformer core. This constancy of values ​​must be maintained for any electric current that the transformer secondary supplies to the load. Thus, the primary winding must absorb from the feeder line, in addition to the magnetizing current, an electric current I'₁. This current is called the primary reaction current.

The total magnetic flux linked with the primary winding can be divided into two components: the resulting mutual flux, which is confined to the iron core and is the result of the combined effect of the electric currents in the primary and secondary; and the stray flux of the primary and secondary, which is linked with itself. On the other hand, most of the stray flux is in the air. Since air does not undergo saturation, the stray flux and the voltage induced by it vary linearly with the primary current I₁. This effect can be simulated, in the electrical model, by a stray reactance represented by X_m, as shown in Figure 93-09.

We must not forget that copper or aluminum wires are used to make transformer windings, and therefore they present electrical resistance. These resistances cause losses due to the Joule effect and are represented in the electrical model by R1 (primary) and R2 (secondary).

Therefore, it should be clear that the voltage V₁ opposes three phasor voltages: the voltage drop across the primary resistance R₁, the voltage drop across the stray reactance X₁, and the back electromotive force E₁ induced in the primary by the resulting mutual flux.

Based on the circuit shown in Figure 93-09, it is possible to develop a phasor plot of the transformer. See Figure 93-10 for the plot with all the details presented by the circuit.

Resume

In short, in order to study the model of a real transformer we must consider four essential points.

Thus, by performing two tests (studied below) on the transformer, we were able to determine the value of resistances of the primary and secondary.

As for reactances , it follows the same principle and with the two tests mentioned above we can determine their values as well as the values of R_h and X_m.

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There is another approach to solving transformer circuits that eliminates the need for explicit voltage level conversions at each transformer in the system. Instead, the necessary conversions are performed automatically by the method itself, without the user having to worry about impedance transformations. Since these impedance transformations can be avoided, circuits containing many transformers can be solved easily with less possibility of error. This method of calculation is known as the per unit (pu) system of measurement.

Thus, when an electrical machine is designed or analyzed using the actual values ​​of its parameters, it is not immediately obvious how its performance compares with that of another machine of a similar type. However, if we express the parameters of a machine as a per-unit (pu) of a base (or reference) value, we find that the per-unit values ​​of machines of the same type but with widely different ratings fall within a narrow range. This is one of the main advantages of a per-unit system. An electrical system has four quantities of interest: voltage, current, apparent power, and impedance. If we select base values ​​of any two of these, the base values ​​of the remaining two can be calculated.

To implement this system, it is necessary that each electrical quantity be measured as a decimal fraction of some level that serves as a base. In the per unit, pu system, we must choose two quantities as the base values. Usually, the ones chosen are voltage and power (or apparent power). Once these base quantities have been chosen, all other base quantities are determined from them, according to the fundamental electrical laws.

Thus, if S_b is the base of apparent power and V_b is the base of electrical voltage, then the base of electrical current and impedance are:

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Um transformador pode ter múltiplos enrolamentos que podem ser conectados em série para aumentar a tensão nominal ou em paralelo para aumentar a corrente nominal. Antes das conexões serem realizadas, no entanto, é necessário que saibamos a polaridade de cada enrolamento. Por polaridade, queremos dizer a direção relativa da força eletromotriz induzida (FEM) em cada enrolamento.

A polaridade de um enrolamento refere-se à característica que mostra a dependência do sentido da FEM induzida em relação ao fluxo que a gera (normalmente assinalada com uma seta ou um ponto). Assim, dois terminais de dois enrolamentos são da mesma polaridade ou homólogos quando estiverem igualmente situados relativamente ao sentido positivo num e noutro enrolamento. Uma vez identificados os terminais do transformador, a ligação em paralelo é feita interligando-se os terminais igualmente identificados nos dois transformadores.