Good day to all! In the last article, I talked about anti-aliasing filter chokes and outlined the principle of their calculation. However, these types of chokes are not used very often in household appliances, since in low-power devices it is often more effective to use capacitive filters. The most common type of chokes used in electronic devices is AC chokes. Their features, operating principles and calculation of the parameters of such chokes will be discussed in this article.
To assemble a radio-electronic device, you can pre-make a DIY KIT kit using the link.
Features of the AC choke
An AC choke, just like any other choke, is an inductor with a ferromagnetic core. This type of inductor is connected in series with the load, similar to a smoothing inductor, but unlike it, the current flowing through the AC inductor does not have a DC bias current. In this regard, the AC inductor is widely used in ballast and current-limiting circuits, powerful antenna and filter devices, as well as in various pulse voltage converters.
Regardless of the use of the inductor in the circuit, its operation is based on the dependence of its reactance XL on the frequency f of the current IH flowing through it and the voltage drop across the inductor UL
AC choke.
Thus, the voltage across the inductor UL is determined by the inductance of the inductor L and the parameters of the current flowing through the inductor: the frequency of the current f and the value of the current in the circuit IH.
Parallel oscillatory circuit.
If you connect an inductor and a capacitor, you get a very interesting element of radio engineering - an oscillatory circuit. If you charge a capacitor or induce E.M.F. in the coil using an electromagnetic field, the following processes will begin to occur in the circuit: The capacitor, when discharged, excites an electromagnetic field in the inductor. When the charge is depleted, the inductor returns the stored energy back to the capacitor, but with the opposite sign, due to the E.M.F. self-induction. This will be repeated again and again - electromagnetic oscillations of a sinusoidal shape will appear in the circuit. The frequency of these oscillations is called the resonant frequency of the circuit, and depends on the values of the capacitor capacitance (C) and the inductance of the coil (L).
A parallel oscillatory circuit has a very high resistance at its resonant frequency. This allows it to be used for frequency selection (selection) in the input circuits of radio equipment and intermediate frequency amplifiers, as well as in various master oscillator circuits.
Effect of non-magnetic gap on the throttle
In previous articles, I talked about the negative effect of core saturation on reducing the magnetic permeability μe and inductor inductance L, which lead to a distortion of the shape of the current flowing through the inductor.
The shape of the current flowing through the inductor: for an unsaturated core (1) and for a saturated core (2).
This figure shows the distortion of the sinusoidal voltage current shape when the inductor operates in the saturated and unsaturated sections of the magnetization curve. The degree of distortion of the voltage waveform also depends on the ratio of the inductor reactance to the active load resistance XL/RH. That is, when the core is saturated, the lower this ratio, the lower the degree of voltage waveform distortion. Thus, the introduction of a non-magnetic gap, in addition to stabilizing the inductance value, within a wide range of current changes, allows alternating current to pass through the inductor without significant changes.
In addition to the factors described above, the introduction of a non-magnetic gap leads to some features that must be taken into account when developing and manufacturing chokes with a gap. The main feature is the broadening of the magnetic flux in the gap.
Broadening of the magnetic flux in the non-magnetic choke gap: the choke rod (left) and its cross section (right). The dotted line indicates the dimensions of the increased cross-section due to bulging of the magnetic flux.
This phenomenon is due to the fact that in a choke with a gap, the magnetic flux goes beyond the space between the two ends of the cut core, so the cross-sectional area in the non-magnetic gap seems to increase.
The dimensions of the cross-sectional broadening depend on the length of the inductor winding lob, the cross-sectional area of the core Se and the length of the non-magnetic gap lз. Broadening the magnetic flux reduces the magnetic resistance of the circuit and, therefore, increases the inductance of the inductor. To take into account the broadening of the magnetic flux and the increase in inductance, a buckling coefficient F is introduced, which takes into account the broadening of the magnetic flux in the non-magnetic gap. Therefore, the value of the inductor inductance will be determined by the following expression
where ω is the number of turns of wire in the winding,
μ0 – magnetic constant, μ0 = 4π*10-7 H/m,
μе – equivalent (relative) magnetic permeability of the core,
Sе – equivalent cross-sectional area of the core,
lе – equivalent length of the magnetic line of the core.
lM – length of the magnetic line in the core.
F is a coefficient that takes into account the broadening of the magnetic flux in the gap.
Two-wire line inductance
A two-wire line (Fig. 1.16, a) consists of two parallel wires of the same radius r0, having a greater length I compared to the distance d between them. Magnetic permeability of the wire material (g, environment) is µ0. The currents I in the forward and return wires differ only in direction; the origin of coordinates is taken at the center of the cross section of the left wire.
For an individual wire, due to its axial symmetry, neglecting the field distortion at its ends, applying the total current law to a circle of radius gives:
When integrating over a circle lying inside a separate wire, only part LX of the total current flowing inside a circle of radius x is covered, equal to when the current is uniformly distributed over the cross section
In the air between the wires on the line connecting the centers of their sections, the directions of the fields created by both currents according to the right screw rule coincide and the field strengths and induction add up:
The same formulas are also valid for i.e. outside the line, but here they give the field difference.
Inside the left wire of the line, the field strength and induction from both wires will be:
Inside the right wire, respectively,
In Fig. Figure 1.16b shows the distribution of field strength and induction along the x axis for the magnetic permeability of the wire material µ > µ0. In the middle between the wires the field is minimal, but does not go to zero. The field is also not zero on the wire axes.
On the inside of the wires, the field strength and induction are greater than on the outside. Unlike field strength, induction has a discontinuity at the surface of the wires. To calculate the line inductance, it is necessary to find the flux linkage. The elementary flux passing through the Idx pad in the air between the wires is
All flow between wires is external flow
at the same time it is an external flux linkage, since it couples with the circuit once. That's why
and the corresponding external inductance
For most lines, the distance d between the wires significantly exceeds the radius r0 of the wires. In this case
To determine the internal inductance corresponding to the internal flux, for d > r0, the field inside the line wire can be calculated as the field of an isolated wire, since the field created by the second wire inside the first, compared to the field of the first, is negligible. Then the elementary flux inside the wire
Since the flow dФi does not cover the entire current, but only part of it [see. formula (1.3)], elementary flux linkage
All flow between wires is external flow
Accordingly, the internal inductance
Total line inductance
With copper or aluminum wires () in most cases the second term can be neglected compared to the first and then
For steel wires (), the main part of the flux is internal flux and inductance
will practically not depend on the distance between the wires.
Principles for calculating AC chokes
The calculation of the AC choke is carried out similarly to the calculation of the smoothing choke, but taking into account the initial conditions. So for an AC inductor, the determining parameters are: the required inductance L, the applied voltage UL, the AC frequency f, the overheating of the inductor. In addition, it is necessary to determine the material of the inductor core, which will determine the saturation induction BS and the maximum induction in the core Bm, which, to prevent saturation of the core, is selected from the condition
Calculations of an AC inductor are based on expressions for determining the magnitude of the effective voltage dropped across the inductor UL
where f is the frequency of alternating current,
L – inductance of the inductor,
I – effective value of the inductor current.
Then, taking into account the expression for the inductance of a closed-core inductor and the expression for the maximum inductance in the core, the voltage on the inductor will depend on the following parameters
where μе is the equivalent magnetic permeability of the core,
μ0 – magnetic constant, μ0 = 4π•10-7 H/m,
ω – number of turns of the inductor winding,
Se – equivalent cross-section of the inductor core,
le is the equivalent length of the magnetic path of the inductor core,
Bm – maximum value of core magnetic induction,
ka is the current (voltage) amplitude coefficient of the inductor.
The resulting expression can often be found under the name of the basic formula of transformer EMF, since it establishes a unique relationship between the EMF at the winding terminals and the number of turns of the winding, for a given value of magnetic induction in the core. Then, with a sinusoidal voltage (amplitude factor ka ≈ 1.414), the expression takes the following form
Let's return to the original expression for the voltage on the inductor UL, in which the parameter - the number of turns - is ambiguous. This parameter, among other things (the magnitude of inductance L and magnetic permeability μe of the core) depends on the dimensions of the magnetic circuit, and more specifically on the area of the window SO, which can be calculated using the following expression
where I is the effective value of the inductor current,
ω – number of turns of the inductor winding,
kI – core window utilization factor,
j is the current density in the winding wire.
The parameters kI and j are chosen in the same way as for the smoothing filter choke, that is, the core window utilization factor kI ≈ 0.3, and the current density j = 5 A/mm2.
Then, expressing from this expression the number of turns of wire ω, we obtain
The resulting expression determines the main calculation expression for determining the standard size of the core - the product of the areas of the SeSO core. After transforming the expression for the effective voltage on the inductor UL, we determine the number of turns of the winding ω and the value of the non-magnetic gap δ
where μе is the equivalent magnetic permeability of the core,
μ0 – magnetic constant, μ0 = 4π•10-7 H/m,
Se – equivalent cross-section of the inductor core,
le is the equivalent length of the magnetic path of the inductor core,
Bm – maximum value of core magnetic induction,
ka is the current (voltage) amplitude coefficient of the inductor.
The calculated number of turns is approximate, since due to the broadening of the magnetic flux, the inductance value turns out to be slightly larger for a given number of turns, which in some cases is undesirable. Therefore, it is necessary to recalculate the turns taking into account the magnetic flux broadening coefficient F
It remains to select the cross-section of the winding wire SP
where SO is the window area of the core used,
kI – core window utilization factor,
ω – number of turns of the inductor winding.
The choice of wire cross-section must be made by rounding the resulting value to the nearest nominal value, while it must be taken into account that at high frequencies power losses in the wire increase. Therefore, at a sufficiently high frequency, it is necessary to use a winding wire consisting of several cores, and the core diameter is selected based on the skin depth δ
where f is the frequency of the alternating current flowing through the inductor,
δ – skin layer thickness,
dп – diameter of the core in the winding wire.
After constructive calculation of the core and winding, it is necessary to check the thermal operating conditions of the inductor - heating and overheating of the inductor.
Mutual inductance and its calculation
For two circuits having w1 and w2 turns with currents I1 and I2 (Fig. 1.17), the flux of the first circuit, determined by the current of this circuit, the self-induction flux Ф1l, can be decomposed into the leakage flux Ф1s, penetrating only this circuit, and the mutual induction flux Ф1m, also penetrating the second circuit:
The flux linkage corresponding to flow F11 (provided that this flow penetrates all turns of the primary circuit is equal to
and leakage flux linkage
Similarly for the second circuit
The flux linkage of the second circuit, determined by the current of the first,
and the flux linkage of the first circuit, determined by the current of the second,
It can be shown that
The value M is called mutual inductance and is determined by the configuration of the circuits, their relative position and the magnetic permeability of the medium. Mutual inductance is also measured in henry (H). The total flow passing through the first circuit is
Total primary circuit flux linkage
and accordingly for the second circuit
In these algebraic sums, the first term is always positive, and the sign before the second term is determined by the direction of the currents in the circuits; a positive sign corresponds to the case when the directions of flows Ф1м and Ф2м coincide (see Fig. 1.17). From the above it is clear that
Thus, mutual inductance and inductances always satisfy the inequality
and the coupling coefficient of two circuits used in technical calculations
Similarly, in a system of many loops, the flux linkage of the loop is determined by the currents of all loops:
where Lq is the inductance of the qth circuit, Mqp = Мрq is the mutual inductance of the qth and pth circuits. The general technique for calculating the mutual inductance of circuits is to find the flux linkage that penetrates the circuit q, but created by the current of the pth circuit, and divide it by this current.
How to calculate the interturn capacitance of the inductor winding?
In the inductor, between the turns, layers and metal objects around the inductor, there is some potential difference that creates an electric field. To assess the influence of this field, the concept of interturn capacitance or the inductor’s own capacitance is introduced, the value of which depends on the size and design features of the inductor.
The interturn capacitance C of the winding, being a parasitic parameter, together with the leakage inductance and the inductor’s own inductance form various types of filters and oscillatory circuits. Although this parameter is of small importance, nevertheless, in certain conditions it has to be taken into account, but accurate calculation is difficult due to the large influence of various design parameters, primarily, the relative position of the turns of the wire among themselves. So, bulk-wound coils have the greatest interturn capacitance, and coils with “Universal” type winding or sectional coils have the smallest.
The interturn capacitance Cc of the inductor can be represented as the sum of the capacitances between the inner layer of the winding and the magnetic circuit C1 and the interlayer capacitance inside the winding C2
The capacitance between the inner layer of the winding and the magnetic core can be determined from the empirical formula
where εа is the absolute dielectric constant of the medium around the conductor, εа = ε0εr,
εr – relative dielectric constant,
ε0 – electrical constant, ε0 = 8.85 * 10-12 F/m,
r – radius of the cross section of the wire,
a is the distance between the magnetic core and the axis of the wire,
n – number of turns in the layer,
р1 – perimeter of the turn of the inner layer of the winding.
The relative dielectric constant is taken for the material of the inductor frame; if the design is frameless, then, accordingly, the permeability of air or conductor insulation, depending on the required accuracy.
The capacitance between the winding layers is also calculated using the empirical formula
where рср is the perimeter of the middle turn of the winding,
b – distance between the axes of turns in adjacent layers,
m – number of layers.
In this case, the dielectric constant is taken for the interlayer insulation material.
In all cases, it is necessary to reduce the interturn capacitance of the winding. For this purpose, various types of windings and materials for frames and interlayer insulation with a low dielectric constant are used.
Mutual inductance of two parallel two-wire lines
Let two parallel two-wire lines be arranged symmetrically as shown in Fig. 1.4. Under the condition d> r0, the internal flux in the wires can be neglected compared to the external one. The magnetic flux passing through the first line and created by the current I2 of the second can be found as the sum of the fluxes created by each of the wires of the second line separately. Then the magnetic flux penetrating the first line
distances from the wire of line 1 to the wires of line 2.
Magnetic flux Ф is at the same time the flux linkage of the first line, since it couples with it once; That's why
and mutual inductance
To reduce the coupling coefficient between communication lines l and transmission 2, a transposition of the communication line is used, which consists in crossing the wires of the communication line at equal distances; then the total flux linkage will be zero.
Markings and symbols
In circuit diagrams and technical documentation, chokes are designated by the Latin letter L, and the conventional graphic designation is in the form of semicircles. Their number is not indicated anywhere, but usually does not exceed three pieces. A bold dot placed at the beginning of the semicircles indicates the beginning of the turns. If the inductance is performed on the frame, a straight line is drawn against the image. To indicate the denominations of an element, a code of letters and numbers or color coding is used.
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The numbers indicate the inductance value, and the letter indicates the tolerance. For example, the code 250 J indicates an inductance equal to 25 μH with an error of five percent. When there is only a number on the marking, this means that the tolerance is 20%. Thus, the first two digits indicate the numerical value in microhenry, and the third is the multiplier. The letter D is placed on high-precision products; their error does not exceed 0.3%.
The color marking, in principle, corresponds to the alphanumeric marking, but is only applied in the form of colored stripes. The first two indicate values in microhenry, the third is the multiplication factor, and the fourth is the tolerance. The inductance of the inductor, which shows two orange stripes, brown and white, is 33 μH with a permitted deviation of 10%.
Power smoothing filter chokes
The main parameters of power smoothing filter chokes are inductance, rated bias current, DC resistance, and permissible AC voltage. In many cases, with given overall dimensions and weight, they strive to obtain the highest possible (or specified) inductance with minimal resistance to direct current. Since the inductance of the inductor depends on the bias current and the amplitude of the alternating voltage, it is measured at the rated bias current and a given alternating voltage. Calculation of rectifier filter chokes can be performed using the methodology used for calculating matching transformers with constant bias.
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