Let's take a coil with a ferromagnetic core and take out the ohmic resistance of the winding as a separate element as shown in Figure 1.
Figure 1. Inductor with a ferromagnetic core
When an alternating voltage e c is applied in the coil, according to the law of electromagnetic induction, an EMF of self-induction e L arises.
(1) where ψ
- flux linkage, W- the number of turns in the winding, F is the main magnetic flux. We neglect the scattering flux. The voltage applied to the coil and the induced EMF are balanced. According to the second Kirchhoff law for the input circuit, we can write:
e c + e L = i × R exchange, (2)where R obm - active resistance of the winding.
Because the e L >> i × R exchange, then we neglect the voltage drop across the ohmic resistance, then e c ≈ −e L. If the mains voltage is harmonic, e c = E m cosω t, then:
(3)
Let's find the magnetic flux from this formula. To do this, we transfer the number of turns in the winding to the left side, and the magnetic flux Ф to the right:
(4)
Now let's take indefinite integral from the right and left sides:
(5)Since we consider the magnetic circuit to be linear, then only harmonic current flows in the circuit and there is no permanent magnet or a constant component of the magnetic flux, then the integration constant c \u003d 0. Then the fraction in front of the sine is the amplitude of the magnetic flux
(6)whence we express the amplitude of the input EMF
E m = F m × W × ω (7)Its effective value is
(8) (9)Expression (9) is called the basic formula of the transformer EMF, which is valid only for harmonic voltage. With a non-harmonic voltage, it is modified and the so-called shape factor is introduced, equal to the ratio of the effective value to the average:
(10)
Find the shape factor for a harmonic signal, while the average value is found in the interval from 0 to π/2
(11)
Then the form factor is 
and the basic formula of the transformer EMF takes the final form:
If the signal is a sequence rectangular pulses of the same duration (meander), then the amplitude, effective and average values for half the period are equal to each other and its k f = 1. You can find the form factor for other signals. The basic formula for transformer EMF will be valid.
Let's build a vector diagram of a coil with a ferromagnetic core. With a sinusoidal voltage at the coil terminals, its magnetic flux is also sinusoidal and lags the voltage in phase by an angle π / 2, as shown in Figure 2.
We continue our acquaintance with electronic components and in this article we will consider device and principle of operation of the transformer.
Transformers are widely used in radio and electrical engineering and are used for the transmission and distribution of electrical energy in power grids, for powering radio equipment circuits, in converter devices, as welding transformers, etc.
Transformer designed to convert alternating voltage of one magnitude into AC voltage other size.
In most cases, the transformer consists of a closed magnetic circuit (core) with two (windings) located on it, electrically not connected to each other. The magnetic circuit is made of a ferromagnetic material, and the windings are wound with insulated copper wire and placed on the magnetic circuit.
One winding is connected to the source alternating current and called primary(I), voltage is removed from the other winding to power the load and the winding is called secondary(II). A schematic arrangement of a simple transformer with two windings is shown in the figure below.
1. The principle of operation of the transformer.
The principle of operation of the transformer is based on phenomenon of electromagnetic induction.
If an alternating voltage is applied to the primary winding U1, then an alternating current will flow through the turns of the winding io, which around the winding and in the magnetic circuit will create alternating magnetic field. The magnetic field forms a magnetic flux Fo, which, passing through the magnetic circuit, crosses the turns of the primary and secondary windings and induces (induces) variable EMF in them - e1 and e2. And if you connect a voltmeter to the terminals of the secondary winding, it will show the presence of an output voltage U2, which will be approximately equal to the induced emf e2.
When connected to the secondary winding of a load, for example, an incandescent lamp, a current appears in the primary winding I1, which forms an alternating magnetic flux in the magnetic circuit F1 changing at the same frequency as the current I1. Under the influence of an alternating magnetic flux, a current arises in the secondary winding circuit I2, which in turn creates a counteracting magnetic flux according to Lenz's law F2, seeking to demagnetize the magnetic flux that generates it.
As a result of the demagnetizing action of the flow F2 magnetic flux is established in the magnetic circuit Fo equal to the flow difference F1 and F2 and being part of the flow F1, i.e.
Resulting magnetic flux Fo ensures the transfer of magnetic energy from the primary winding to the secondary and induces an electromotive force in the secondary winding e2, under the influence of which current flows in the secondary circuit I2. Due to the presence of magnetic flux Fo and there is a current I2, which will be the more, the more Fo. But at the same time, the more current I2, the greater the opposing flow F2 and therefore less Fo.
From what has been said, it follows that for certain values of the magnetic flux F1 and resistance secondary winding and loads appropriate EMF values are set e2, current I2 and flow F2, providing the balance of magnetic fluxes in the magnetic circuit, expressed by the formula above.
Thus, the flow difference F1 and F2 cannot be equal to zero, since in this case there would be no main thread Fo, and without it there could not be a stream F2 and current I2. Therefore, the magnetic flux F1, created by the primary current I1, always more magnetic flux F2 generated by the secondary current I2.
The magnitude of the magnetic flux depends on the current that creates it and on the number of turns of the winding through which it passes.
The voltage of the secondary winding depends on the ratio of the number of turns in the windings. With the same number of turns, the voltage on the secondary winding will be approximately equal to the voltage supplied to the primary winding, and such a transformer is called dividing.
If the secondary winding contains more turns than the primary, then the voltage developed in it will be more tension supplied to the primary winding, and such a transformer is called raising.
If the secondary winding contains fewer turns than the primary, then its voltage will be less than the voltage supplied to the primary winding, and such a transformer is called lowering.
Consequently. By selecting the number of turns of the windings, at a given input voltage U1 get what they want output voltage U2. To do this, they use special methods for calculating the parameters of transformers, with the help of which the windings are calculated, the cross section of the wires is selected, the number of turns is determined, as well as the thickness and type of the magnetic circuit.
The transformer can only work in AC circuits. If its primary winding is connected to a source direct current, then a magnetic flux is formed in the magnetic circuit that is constant in time, in magnitude and direction. In this case, no alternating voltage will be induced in the primary and secondary windings, and therefore no electrical energy will be transferred from the primary circuit to the secondary. However, if a pulsating current flows in the primary winding of the transformer, then an alternating voltage will be induced in the secondary winding, the frequency of which will be equal to the frequency of the current ripple in the primary winding.
2. Transformer device.
2.1. Magnetic core. magnetic materials.
Purpose magnetic core is to create a closed path for the magnetic flux, which has a minimum magnetic resistance. Therefore, magnetic circuits for transformers are made of materials with high magnetic permeability in strong alternating magnetic fields. The materials must have low eddy current losses so as not to overheat the magnetic circuit at sufficiently high values of magnetic induction, be cheap enough and not require complex mechanical and thermal processing.
Magnetic materials, used for the manufacture of magnetic cores, are produced in the form of separate sheets, or in the form of long tapes of a certain thickness and width, and are called electrical steels.
Sheet steels (GOST 802-58) are produced by hot and cold rolling, grain-oriented strip steels (GOST 9925-61) are produced only by cold rolling.
Also used are iron-nickel alloys with high magnetic permeability, for example, permalloy, permindur, etc. (GOST 10160-62), and low-frequency magnetically soft ferrites.
For the manufacture of a variety of relatively inexpensive transformers are widely used electrical steels, having a low cost and allowing the transformer to work both with constant magnetization of the magnetic circuit, and without it. The most widely used cold-rolled steels best performance compared to hot rolled steels.
Alloys with high magnetic permeability used for the manufacture of pulse transformers and transformers designed to operate at elevated and high frequencies of 50 - 100 kHz.
The disadvantage of such alloys is their high cost. So, for example, the cost of permalloy is 10-20 times higher than the cost of electrical steel, and permendur is 150 times higher. However, in some cases, their use can significantly reduce the weight, volume, and even the total cost of the transformer.
Their other disadvantage is the strong influence on the magnetic permeability of permanent bias, alternating magnetic fields, as well as low resistance to mechanical stress - shock, pressure, etc.
From magnetically soft low-frequency ferrites with high initial permeability are made pressed magnetic cores, which are used for the manufacture of pulse transformers and transformers operating at high frequencies from 50 - 100 kHz. The advantage of ferrites is their low cost, and the disadvantage is low saturation induction (0.4 - 0.5 T) and strong temperature and amplitude instability of magnetic permeability. Therefore, they are used only in weak fields.
The choice of magnetic materials is made on the basis of electromagnetic characteristics, taking into account the operating conditions and purpose of the transformer.
2.2. Types of magnetic circuits.
The magnetic cores of transformers are divided into laminated(stamped) and tape(twisted), made from sheet materials and pressed from ferrites.
Laminated magnetic cores are assembled from flat stamped plates of the appropriate shape. Moreover, the plates can be made from almost any, even very fragile materials, which is the advantage of these magnetic circuits.
Tape magnetic circuits are made of a thin tape wound in the form of a spiral, the turns of which are firmly connected to each other. The advantage of tape magnetic circuits is the full use of the properties of magnetic materials, which reduces the weight, size and cost of the transformer.
Depending on the type of magnetic circuit, transformers are divided into rod, armored and toroidal. Moreover, each of these types can be both rod and tape.
Rod.
In magnetic circuits rod type winding is located on two rods ( rod called the part of the magnetic circuit on which the windings are placed). This complicates the design of the transformer, but reduces the thickness of the winding, which helps to reduce the leakage inductance, wire consumption and increases the cooling surface.
Rod magnetic circuits are used in output transformers with a low noise level, since they are insensitive to the effects of external low-frequency magnetic fields. This is explained by the fact that under the influence of an external magnetic field, voltages opposite in phase are induced in both coils, which, if the turns of the windings are equal, cancel each other out. As a rule, core transformers are made of large and medium power.
armored.
In the magnetic circuit armored type the winding is located on the central rod. This simplifies the design of the transformer, allows more complete use of the winding window, and also creates some mechanical protection for the winding. Therefore, such magnetic circuits have received the greatest application.
Some disadvantage of armored magnetic circuits is their increased sensitivity to low-frequency magnetic fields, which makes them unsuitable for use as output transformers with a low noise level. Most often, medium-power transformers and microtransformers are made armored.
Toroidal.
Toroidal or ring transformers allow better use of the magnetic properties of the material, have low leakage fluxes and create a very weak external magnetic field, which is especially important in high-frequency and pulse transformers. But due to the complexity of manufacturing windings, they are not widely used. Most often they are made of ferrite.
To reduce eddy current losses, laminated magnetic cores are assembled from stamped plates 0.35 - 0.5 mm thick, which are coated on one side with a layer of varnish 0.01 mm thick or with an oxide film.
Tape for tape magnetic circuits has a thickness from several hundredths to 0.35 mm and is also covered with an electrically insulating and simultaneously adhesive suspension or oxide film. And the thinner the insulation layer, the denser the filling of the cross section of the magnetic circuit with magnetic material, the smaller the overall dimensions of the transformer.
Recently, along with the considered "traditional" types of magnetic cores, new forms have been used, which include "cable" type magnetic cores, "inverted torus", coil, etc.
Let's finish this for now. Let's continue in .
Good luck!
Literature:
1. V. A. Volgov - "Details and components of radio-electronic equipment", Energy, Moscow, 1977
2. V. N. Vanin - "Current Transformers", Energia Publishing House, Moscow 1966 Leningrad.
3. I. I. Belopolsky - "Calculation of transformers and chokes of low power", M-L, Gosenergoizdat, 1963
4. G. N. Petrov - “Transformers. Volume 1. Fundamentals of Theory, State Energy Publishing House, Moscow 1934 Leningrad.
5. V. G. Borisov, - " Young radio amateur”, Moscow, “Radio and communication”, 1992
Let's take a coil with a ferromagnetic core and take out the ohmic resistance of the winding as a separate element as shown in Fig. 2.8.
Figure 2.8 - To the derivation of the transformer EMF formula
When an alternating voltage e c is turned on in the coil, according to the law of electromagnetic induction, an EMF of self-induction e L arises.
(2.8)
where ψ is the flux linkage,
W is the number of turns in the winding,
Ф is the main magnetic flux.
We neglect the scattering flux. The voltage applied to the coil and the induced EMF are balanced. According to the second Kirchhoff law for the input circuit, we can write:
e c + e L = i * R exchange, (2.9)
where R obm is the active resistance of the winding.
Since e L >> i * R exchange, we neglect the voltage drop across the ohmic resistance, then e c ≈ – . If the mains voltage is harmonic е с = E m cos ωt, then E m cos ωt = , whence 
. Let's find the magnetic flux. To do this, we take the indefinite integral of the right and left sides. We get
, (2.10)
but since we consider the magnetic circuit to be linear, only harmonic current flows in the circuit and there is no permanent magnet or constant component, then the integration constant c \u003d 0. Then the fraction in front of the harmonic factor is the amplitude of the magnetic flux, from which we express E m \u003d Ф m * W * ω. Its effective value is
Or we get
where s is the cross section of the magnetic circuit (core, steel).
Expression (2.11) is called the basic formula of the transformer EMF, which is valid only for harmonic voltage. Usually it is modified and the so-called shape factor is introduced, equal to the ratio of the effective value to the average:
. (2.12)
Let's find it for a harmonic signal, but we find the average value on the interval
Then the form factor is 
and the basic formula of the transformer EMF takes the final form:
(2.13)
If the signal is a meander, then the amplitude, effective and average values for half the period are equal to each other and its. You can find the form factor for other signals. The basic formula for transformer EMF will be valid.
Let's build a vector diagram of a coil with a ferromagnetic core. With a sinusoidal voltage at the coil terminals, its magnetic flux is also sinusoidal and lags behind the voltage in phase by an angle π / 2, as shown in Fig. 2.9a.
Figure 2.9 - Vector diagram of a coil with a ferromagnetic
core a) no loss; b) with losses
In a lossless coil, the magnetizing current - reactive current (I p) coincides in phase with the magnetic flux Ф m. If there are losses in the core (), then the angle is the angle of losses for remagnetization of the core. The active component of the current I a characterizes the losses in the magnetic circuit.
Question 7. EMF of transformer windings.
The principle of operation of the transformer is based on the phenomenon of electromagnetic induction (mutual induction). Mutual induction consists in inducing an EMF in an inductive coil when the current changes in the other coil.
Under the influence of alternating current in the primary winding, an alternating magnetic flux is created in the magnetic circuit
which penetrates the primary and secondary windings and induces an emf in them
where are the amplitude values of the EMF.
The effective value of the EMF in the windings is
;
.
The ratio of the EMF of the windings is called the transformation ratio
If , then the secondary EMF is less than the primary one and the transformer is called a step-down transformer, with a step-up transformer.
Question 8. Vector diagram of an ideal transformer idling.
Since we are considering an ideal transformer, i.e. without dissipation and power loss, then the current x.x. is purely magnetizing - , i.e. it creates a magnetizing force that creates a flux, where is the magnetic resistance of the core, consisting of the resistance of the steel and the resistance at the joints of the core. Both the amplitude and the shape of the current curve depend on the degree of saturation of the magnetic system. If the flow changes sinusoidally, then with unsaturated steel, the no-load current curve is almost also sinusoidal. But when the steel is saturated, the current curve is more and more different from the sinusoid (Fig. 2.7.) The current curve x.x. can be decomposed into harmonics. Since the curve is symmetrical about the x-axis, the series contains only harmonics of an odd order. First harmonic current i ( 01) is in phase with the main stream. Of the higher harmonics, the third harmonic of the current is most pronounced i ( 03) .
Fig 2.7 X.X current curve
Effective value of no-load current:
.
(2.22)
Here I 1 m , I 3 m , I 5 m- amplitudes of the first, third and fifth harmonics of the no-load current.
Since the no-load current lags behind the voltage by 90 , the active power consumed by an ideal transformer from the network is also zero, i.e. An ideal transformer draws purely reactive power and magnetizing current from the network.
The vector diagram of an ideal transformer is shown in fig. 2.8.
Rice. 2.8. Vector diagram of an ideal transformer
Question 9 Vector diagram of the idling of a real transformer.
In a real transformer, there is dissipation and losses in steel and copper. These losses are covered by the power R 0 entering the transformer from the network.
where I 0a - effective value of the active component of the no-load current.
Therefore, the no-load current of a real transformer has two leaving: magnetizing - creating the main flow F and coinciding with it in phase, and active:
The vector diagram of a real transformer is shown in fig. 2.9.
Usually, therefore, this component has little effect on the value of the no-load current, but more affects the shape of the current curve and its phase. The no-load current curve is clearly non-sinusoidal, and is shifted in time relative to the flux curve by an angle called the magnetic lag angle.
By replacing the actual no-load current curve with an equivalent sinusoid, the voltage equation can be written in complex form, where all quantities vary sinusoidally:
Considering that the EMF of scattering,
Rice. 2.9. Vector diagram of a real transformer
Rice. 2.11. Transformer voltage vector diagram, no-load mode
In 1876 P.I. Yablochkov suggested using a transformer to power the candles. In the future, the design of transformers was developed by another Russian inventor, a mechanic I.F. Usagin, who suggested using transformers to power not only Yablochkov candles, but also other consumers of electrical energy.
The transformer is electrical apparatus, based on the phenomenon of mutual induction and designed to convert alternating current of one voltage into alternating current of a different voltage, but of the same frequency. The simplest transformer has a steel core and two windings insulated both from the core and from each other.
The winding of a transformer that is connected to a voltage source is called primary winding, and the winding to which consumers are connected or transmission lines leading to consumers is called secondary winding.
An alternating current, passing through the primary winding, creates an alternating magnetic flux, which interlocks with the turns of the secondary winding and induces an emf in them.
Since the magnetic flux is variable, the induced EMF in the secondary winding of the transformer is also variable and its frequency is equal to the frequency of the current in the primary winding.
The variable magnetic flux passing through the core of the transformer crosses not only the secondary winding, but also the primary winding of the transformer. Therefore, an EMF will also be induced in the primary winding.
The magnitude of the EMF induced in the windings of the transformer depends on the frequency of the alternating current, the number of turns of each winding and the magnitude of the magnetic flux in the core. At a certain frequency and a constant magnetic flux, the value of the EMF of each winding depends only on the number of turns of this winding. This relationship between the EMF values and the number of turns of the transformer windings can be expressed by the formula: ?1 / ?2 = N1 / N2, where? 1 and?
The difference between EMF and voltage is so small that the relationship between voltages and the number of turns of both windings can be expressed by the formula: U1 /U2==N1/N2. The difference between EMF and voltage in the primary winding of the transformer becomes especially small when the secondary winding is open and the current in it is zero (idle), and only a small current flows in the primary winding, called the no-load current. In this case, the voltage at the terminals of the secondary winding is equal to the EMF induced in it.
The number showing how many times the voltage in the primary winding is greater (or less) than the voltage in the secondary winding is called the transformation ratio and is denoted by the letter k. k = U1 / U2 ? N1 / N2.
The rated voltage of the high and low voltage windings, indicated on the nameplate of the transformer, refers to the idling mode.
Transformers that serve to increase the voltage are called step-up; their transformation ratio is less than one. Step-down transformers step down the voltage; their transformation ratio is greater than one.
The mode in which the secondary winding of the transformer is open, and an alternating voltage is applied to the terminals of the primary winding, is called idle or idle operation of the transformer.