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Extension of the reluctance generated circuit model to an n-layer
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magnetizing inductors can be replaced with a single shunt inductance, as shown in Fig. 2.16. In most cases, the multiple magnetizing inductors in an n-winding transformer can be reduced to a single equivalent without any great error. An exception would be the case where there is an air gap on an outer leg or a magnetic shunt is present. Note that this model performs equally well for transformers with interleaved winding layers. The layers that represent each winding are simply connected in series in order to make the nal model. Even though this model is more complex than the simple Pi model, it has the major advantage of correctly placing the leakage impedances with respect to the windings. This helps to make the simulation of cross-regulation, under varying winding loads, much more accurate in a multiple-winding transformer. Using this modeling process, more and more details from the physical structure can be added to the model. The problem, however, is that the model may become very complex. This makes it more dif cult to use. In general, the simplest possible model that gives acceptable results
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Eliminating multiple magnetizing inductance elements.
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SPICE Modeling of Magnetic Components
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N1i1
N2i2
N3i3
N4i4
Figure 2.17 A four-winding mesh transformer (A), along with its reluctance model (B), and the resulting equivalent circuit (C).
should be used, and complex models should be avoided whenever possible. The need for a complex model depends entirely upon how accurately the small details of the device performance need to be modeled and how willing you are to develop the necessary model. The following examples show more complex applications of reluctance modeling. Figure 2.17 gives an example of a four-winding mesh transformer that might be used in a polyphase power system. The reluctance modeling proceeds as shown previously and results in the model given in Fig. 2.17C. Note how different this model is from an equivalent four-winding junction transformer. Instead of cascaded parallel windings, the windings are in series. This is because mesh and junction transformers are topological duals. Integrated magnetic structures that incorporate transformers and inductors into a common structure are becoming more common. An example of an integrated magnetic forward converter is given in Fig. 2.18a. A sketch of the magnetic structure is given in Fig. 2.18b. The reluctance model and the series of steps required to convert it to a circuit model are shown in Fig. 2.19. Again, the process is exactly as shown earlier; however, it is more complex now. The completed model, which has been inserted back into the circuit simulation, is shown in Fig. 2.20. Using the reluctance modeling procedure, the derivation of an appropriate model is straightforward, although a bit tedious. Without this process, the appropriate model is far from obvious.
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N P2 Lg
N P1
T+L Q1 D
Figure 2.18a.
An integrated magnetic forward converter circuit.
- V + R
+ VP -
N P2
N P1
+ 6 VL 5 iL
Figure 2.18b.
The magnetic structure used in the integrated forward con-
verter.
SPICE Modeling of Magnetic Components
Rc2 Rg Rc FS = iSNS FT Rc
F1 = iPNP1 FR = iRNP2
RCT FL = iLNL
L L+ 2
Rg FL
1 = L + 2
FT = F1+FR Rc = Rc1 = Rc2 Rg >> RCT
T P1
- N Li L
NL i N P1 L
N p1 d c N L dt
2 N P1 Pg
NS i N P1 S
N P 1 PC
L+ 2
+ Pc
N S iS
d T dt
2 N P1 PC
N p1 d 2 N s dt
+ iR N
iT = i P + N N
P2 P1
Ideal T
VP VR N P2
N P1
LC N P1
The reluctance modeling procedure for the transformer used in the forward
converter.
Saturable Core Modeling It would be dif cult to accurately model power circuits without the ability to model magnetics. This section details the SPICE 2 and SPICE 3 methods that are used to simulate various types of magnetic cores including molypermalloy powder (MPP) and ferrite. The presented techniques can be extended to many other types of cores, such as tape wound, amorphous metal, etc.
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Ideal T
+ 4 1
NS Lg N P1
N P2
N P1
N P1
NS LC N P1
C R V0
Vs Dc
Q1 D
The completed forward converter shows how the reluctance derived transformer is integrated into the circuit.
SPICE 2 Compatible Core Model A saturable reactor is a magnetic circuit element consisting of a single coil wound around a magnetic core. The presence of a magnetic core drastically alters the behavior of the coil by increasing the magnetic ux and con ning most of the ux to the core. The magnetic ux density, B, is a function of the applied MMF, which is proportional to ampere turns. The core consists of many tiny magnetic domains that are made up of magnetic dipoles. These domains set up a magnetic ux that adds to or subtracts from the ux that is set up by the magnetizing current. After overcoming initial friction, the domains rotate like small DC motors and become aligned with the applied eld. As the MMF is increased, the domains rotate until they are all in alignment and the core saturates. Eddy currents are induced as the ux changes, thereby causing added loss. A saturable core model that utilizes the PSpice subcircuit feature is available [76]. The saturable core subcircuit is capable of simulating nonlinear transformer behavior including saturation, hysteresis, and eddy current losses. To make the model even more useful, it has been parameterized. This is a technique that allows the characteristics of the core to be determined via the speci cation of a few key parameters. At the time of the simulation, the speci ed parameters are passed into the subcircuit. The equations in the subcircuit (inside the curly braces) are then evaluated and replaced with a value that makes the equationbased subcircuit compatible with PSpice.
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