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Reluctance model for the inductor in Fig. 2.8.
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Two
L C1
3 4*
V*
1* 1 2
L L1*
2* 3 2*
L2 * C2
L1*=C1 V *=I
L2 *=C2 C *=L
R *=1/R
Review of the duality transform process.
intersect each of the branches in the original network, are connected between each node. In each of the intersecting branches, current and voltage are interchanged. The result is a new network that is the topological and electrical dual of the original network. A listing of dual quantities is given in Table 2.2.
TABLE 2.2
Duality Relationships Dual element I V q R = G =1/R G = R =1/G L C Short circuit Open circuit D =1 D Current generator Voltage generator Node Mesh
Quantity V I q R G C L Open circuit Short circuit D Voltage generator Current generator Mesh Node
SPICE Modeling of Magnetic Components
The conversion from a reluctance model to a circuit model requires the following steps: Draw the reluctance (R ) model from the device structure and an estimate of the ux paths. Using duality, convert the R model to a permeance (P) model. Scale the P model for the winding turns by multiplying P by N. Scale this model for the winding voltage by multiplying again by N. Replace the scaled permeances with inductors. For multiple windings, use ideal transformers in order to provide the correct voltages. A simple example shows how this process works. Keep in mind that the objective is to convert the physical model, which is in terms of magnetic quantities associated with the actual structure, to an electrical model, which is in terms of lumped inductances, ideal transformers, and winding voltages and currents. This is the model we want to use in the simulation. In Fig. 2.11A, the reluctance network has been simpli ed by combining the material reluctances into one element and the air gap reluctances into another. Figure 2.11B is the dual network in which reluctances have become permeances, the magnetic current ( )
Rg=2R1+4R2 =2R1+4R2 NI 2Rg NI
(A (A)
(C (C)
2NPg
V=N
N Pe
2N Pg
Le = N 2 R1 + 4 R 2 N Rg
Usually not considered for Le >> 2Lg
Le (E) (E
Lg = L g >>
Le (F)
b + 2 (e c)
Reluctance modeling example.
Two
R12* N1 R11 R12 R12 N2 R12* Leakage flux path
Leakage flux path
A two-winding transformer.
has become a magnetic voltage, the magnetic voltage source has become a magnetic current source, and series branches have become parallel branches. The next step, Fig. 2.11C, is to scale the network in order to remove N from the current source, thereby leaving only the winding current, I. must be kept constant; the multiplication of the current source by 1/N implies that each of the permeances must be multiplied by N. The winding voltages are introduced by invoking Faraday s law, V = N . Each element in the network is now multiplied by N, as shown in Fig. 2.11D. The resulting network is now in terms of the winding voltage and the permeances scaled by N 2 . From Eq. (2.8), we know that L = N 2 P, so that the scaled permeances can be replaced by inductances (as shown in Fig. 2.11E and F). We can now apply this process to a two-winding transformer like that shown in Fig. 2.12. The reluctance model, which is shown in Fig. 2.13, includes a voltage source for each winding (N1 and N2 ), a reluctance for the common ux path (R12 ), and reluctances for the
R12 R11 N1i1 R22 N2i2
(N1/N2)2 L22
V1 (C)
L11 L22
L12 (D)
V2 N2
Reluctance model for a two-winding transformer.
SPICE Modeling of Magnetic Components
N1 R3 R2
R3 L1
L2 L3 N1 N2
N1i1
N2i2
A realistic transformer model with multiple layers on the center leg of an E-E core.
leakage ux associated with each winding (R11 and R22 ). The reluctance model is transformed into a permeance model in Fig. 2.13B. This model is then scaled using N1 as the reference winding, and inductances are inserted as shown in Fig. 2.13C. The transformer turns ratio is maintained via the use of an ideal transformer. This is the well-known Pi model. As shown in Fig. 2.13D, L22 can be moved to the secondary by scaling by the square of the turns ratio (N22 /N12 ). The transformer shown in Fig. 2.12 is easy to understand but re ects a physical structure that is rarely used. A much more common transformer structure takes the form of multiple layers on a common bobbin, on the center leg of an E-E core. A cross section of such a transformer is shown in Fig. 2.14A, along with reluctances that represent the core (R1 and R3 ) and the leakage ux between the windings (R2 ). The corresponding reluctance model and the nal circuit model are shown in Fig. 2.14B and C. Note that this model is different from the previous one (Fig. 2.13C). In the case of two windings, the two models can be shown to be equivalent using a delta-wye transform. When four or more windings are present, however, the model does not typically reduce to the Pi model. In fact, the Pi model is not valid for transformers with more than three windings. The extension of Fig. 2.14 to an n-layer transformer is shown in Fig. 2.15. In the typical case, where the magnetizing inductances are large compared with the leakage inductances, the numerous shunt
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