Summary
Equivalent circuit is an important part of the transformer simulation, this article briefly describes the impact of the relevant parameters on the equivalent circuit.
Keywords: equivalent circuit
Citation
We assume that the transformer is perfect, then we want to get an ideal transformer is assumed:
(1) The core material has a sufficiently high permeability and is assumed to be infinitely large (magnetic permeability μ => ∞)
(2) The core magnetizing current is small enough and is considered to approach 0. (Reluctance R => 0)
(3) The loss in the iron core is negligible.
(4) All fluxes are perfectly coupled between the windings without loss of magnetic flux. (Coupling factor k = 1)
(5) Coil capacitance is small enough to be ignored.
Of course, a true transformer does not have these assumptions. Although a well-designed transformer will be closely related to their rated current and operating frequency. In the following section, we will study a transformer equivalent circuit that includes an ideal transformer The effect of all the actual parameters These non-ideal factors play a major role in determining the actual conversion of a transformer.
Limit the permeability
If the permeability μ is finite, the reluctance R is not zero, and the iron core magnetizing current will flow and maintain the iron core flux. According to the reluctance related formula, we introduce: i1 = φR / N1 + i2N2 / N1 = im + i2 / n.
Current im is the magnetizing current, i1 is the input current, i2 is the output current, φ is the magnetic flux, R is the reluctance, and N1 and N2 are the number of input and output turns. In the current phase of the primary coil, Primary windings in parallel, can represent the additional current in the equivalent circuit shown in Figure 1:
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Core loss
Hysteresis loss
The hysteresis loop B / H curve explains the hysteresis relationship and the hysteresis-related loss of a periodically magnetized core B and H. It is shown by the curve that it is proportional to the area of the seal and that the area of the curve itself is proportional to the frequency However, if it is a constant frequency, the hysteresis loss can be deduced from the STEINMETZ equation: ph = kh × Bmax1.6.
Here, Ph is the hysteresis loss, B is the magnetic flux density, H is the magnetic field strength, and Kh is the material constant.
2. Overcurrent loss
Faraday's law means that an overcurrent loss is generated around the flux path, thus creating a loop current in the core material. The defined resistance of the core results in a loss of power loss with a loss proportional to the square of the frequency. However, f and uniform flux distribution (two maximum approximation) pe = ke × Bmax2, where pe is the overcurrent loss, ke is the material constant.
3 core loss resistance
The combination of hysteresis and overcurrent loss results in an effective approximation of the core losses.
Pe = ke × Bmax2 + kh × Bmax1.6 ≒ α × φmax2 And the magnetic flux φmax is proportional to the voltage V1max, α is a factor. Therefore: Pe => V12max, V1 is the input voltage.
Although this is a fairly rough approximation, it enables us to simulate the core loss as a parallel resistor RC in the primary winding, as shown in Figure 2.
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In order to reduce core losses, we either use high-impedance materials (such as ferrite materials) or use a core construction type that can resist over-current flow.
Coil resistance
The wire that is typically used to wind a transformer coil will be a non-zero value resistor. Ohmic losses will be generated in each winding. This effect, included in a simple equivalent circuit, requires a series resistor to be added to On each coil.
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In order to reduce the coil loss, we risk the use of a large radius wire, or reduce the number of turns. Rp, Rs used to indicate the primary winding resistance.
Leakage flux
Due to the transformer and other coupling coils, the influence of the magnetic field generated by the second or other coils on the primary must also be taken into account. The inductance caused by the effect of the flux coupling between the two coils is called mutual inductance.
We assume that the core around the two sets of coils in the case. Under normal circumstances, the two sides of the coil flux is not exactly the same, as some leakage flux exists, as shown in Figure 4:
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From Ampere's law we come to:
φ12 = a (N1i1 + N2i2) φ11 = b N1i1φ22 = c N2i2, where φ is the magnetic flux, N is the number of turns, and i is the current, where a, b and c are used to represent the actual proportional constants.
From Faraday's law, we learned:
V1 = N1 × d / dt × (φ11 + φ12) and V2 = N2 × d / dt × (φ22 + φ12)
V1 = 'N12 (a + b) × di1 / dt' + N1N2a × di2 / dt and V2 = 'N22 (a + c) × di2 / dt' + N1N2a × di1 / dt
inferred:
The primary inductance of the primary winding can be expressed as: Llp = N12 (a + b) and Lls = N22 (a + c)
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Consider first the primary coil, where a, b, and c denote the actual proportional constants. The aN12 term is considered as the ideal self-inductance of the coil ignoring the leakage flux. BN12 represents the effect of leakage flux (eg, 'leakage inductance Therefore, to include the effect of the leakage inductance in the equivalent circuit, we can add an inductor in series with an ideal coil, which also applies to the secondary winding as shown in Figure 5. Factors that affect the magnitude of the leakage inductance include winding Line skills and core geometry.
Distribution capacitance
According to the structure of the transformer windings, there is distributed capacitance between the layers when the power is on. The size of the capacitor mainly depends on the winding geometry, the dielectric constant of the core material and other packaging materials (such as the epoxy material for the product package or PTFE tape insulated between coils).
The second capacitor effect is due to the number of turns in the coil and the capacitance between adjacent turns, although this effect is small (and thus the full capacitance is subtracted) when the capacitance between the coils in series is smaller than in parallel. To simulate this Distributed winding capacitance allows us to load a lumped capacitance through each set of ideal coils in a transformer equivalent circuit as shown in Figure 6. The CDP and CDS distributions in the figure represent the primary distributed winding capacitance.
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Winding capacitance
Depending on the structure of the transformer, the capacitance between the windings (CWW in Figure 7) is generated adjacent to each other between the two windings.The size of this capacitor depends primarily on the geometry of the winding, the dielectric constant of the transformer core material, and other packaging materials This capacitance tends to be small in comparison to the transformer distributed capacitance, and its effect can only be seen at higher transformers than the higher cut-off frequency (see later explanation of transformer frequency response).
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Conclusion
In conjunction with all of the non-idealities described above, we obtain the general equivalent transformer circuit of Figure 8.
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Symbol Description:
V1, V2 that input and output voltage;
N represents the number of turns;
Cww said the capacitance between the windings;
CDP, CDS represents the primary distribution of winding capacitance;
Rp, Rs represents the primary winding resistance;
Rc represents the parallel resistance in the primary coil;
Lm represents the primary inductance.