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Abstract-This paper presents a new simulation model for the saturable reactor, an electromagnetic device widely used as a protective element for the thyristor ...
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 35, NO. 2, MAY 1988

A Simulation Model for the Saturable Reactor ANTONIO BARILI, ANGEL0 BRAMBILLA, GIANNANTONIO COTTAFAVA,

AND

ENRICO DALLAGO

Abstract-This paper presents a new simulation model for the saturable reactor, an electromagnetic device widely used as a protective element for the thyristor circuit. The model has been implemented in an improved version of SPICE, which also contains the model of the SCR, and has proved to be fast and accurate on practical cases. Examples given include an industrial chopper for electric drive; the model of saturable reactor has improved the overall simulation accuracy of the chopper.

I. INTRODUCTION

S

ATURABLE reactors are electric devices manufactured by winding a conducting wire around one or more ferromagnetic cores. Fig. l(a) and (b) shows two typical arrangements for construction of saturable reactors. A wide range of performance requirements may be met from a small set of magnetic cores by varying the number of cores in the reactor and/or the number of windings. As a matter of practice, saturable reactors are often realized by mounting the proper number of cores on connection wires. A major application field of the saturable reactor is power semiconductor protection in switching circuits. Their insertion in series with the device being protected limits the current rate of rise during turn-on, in order to avoid device burn-out due to high current density on the device cross section. Saturation effects cause the reactor to switch off automatically after the first and most critical phase of semiconductor turn-on. Anyway, the insertion of reactors is not free of side effects, since they may be the source of oscillations with RC-snubber networks shunting the semiconductor (RC-snubbers control the rate of rise of voltage during turn-off). The primary motivation for this work was the desire to gain a better insight into these effects. 11. SATURABLE REACTOR CHARACTERISTICS The static magnetization characteristics of a typical ferromagnetic material is shown in Fig. 2. The analytical expression for such a curve is usually written as B=poa+p0i@(H)

where B = H = M (H) = po

=

magnetic induction magnetic field magnetization vacuum permeability

(1)

(in Teslas) (in amperes per meter) (in amperes per meter) (in henrys per meter).

Manuscript received November 10, 1986; revised November 16, 1987. A. Barili and A. Brambilla are freelance consultants on electrical CAD. G. Cottafava is with the Dipartirnento di Inforrnatica e Sistemistica, University of Pavia. E. Dallago is with the Department of Electrical Engineering, University of Pavia. IEEE Log Number 8820068.

(b) Fig. 1. (a) Single-core, single-loop reactor. (b) Multicore single-loop reactor.

Bs- saturation flux Br- residual flux

'

€I,saturation field

H,-

coercive field

Fig. 2. Typical B - H plot for a ferromagnetic material.

Expression (1) places into evidence the existence of a linear contribution poH to the induction B , and of a nonlinear one due to material properties. We will not discuss the physics of these phenomena, the interested reader being referred to [ 11, [2]; for our purpose it is only important to note that saturable reactor cores are built using materials with low BJB, ratio (typically 0.1) since hysteresis is just a source of losses in the application we deal with. The following two equations allow us to derive core properties from material properties:

where

S = core cross section (in square meters) A , = effective core cross section (in square meters)

0278-0O46/88/0500-0301$01.OO 0 1988 IEEE

302

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 35, NO. 2, MAY 1988

TABLE I

and

I=

8

REQUIRED MODEL PARAMETERS

H

ZI=IlH

(3) .

I

de dII I = 0 de L = s dI1 I * .x.

zero current differential inductance

I = s

saturation current

e s = 4(Is 1

saturation flux

n =

number of cores

L =o

where

I I

=

ll

=

=

reactor current (in amperes) magnetic path into the core (in meters) effective magetic path (in meters).

saturation differential inductance

C

Parameters A Iand 1, depend on core shape; they are usually tabulated by core manufacturers together with other electric and mechanic parameters. The flux-current characteristic of the core can then be easily derived from the B - H plot for the material, multiplying the scales of the axis by the core factors A , and ll. A simple expression is finally given to represent the 9 - I characteristic of a multiloop multicore reactor as a function of a single-loop single-core one. That is

number of loops nl =

I/

(4)

V

where

A

@* @

= =

n,

=

nl

=

I

*

multiloop multicore reactor flux single-loop single-core reactor flux number of cores number of wire loops

Our modeling effort is aimed at the simulation of the nonlinear @ - I characteristic of cores, but we will suggest ways to account for hysteresis, eddy current losses, and residual flux (B,) effects after model discussion.

Fig. 3.

CP

-

Z plot for the hyperbolic model.

III. MODELDISCUSSION

4

Parameters’ availability and their significance to the enduser is a must for an application-oriented model like the one -_ we intended to develop, so we had some meetings with system designers to identify an optimal set of core parameters (the word optimal above should be used in a strictly pragmatic way). The results of our preliminar research are summarized in Table I. The meaning of some parameters will be clearer in the following discussion. The classic way to express the 9 - I characteristic of a ferromagnetic core is the hyperbolic expression

@(I) = L,I+ tanh (LOm,L.I) -

.

L

I

(5)

The expression is written with reference to Fig. 3. In spite of its compactness and regularity the model has two important drawbacks. First, it does not make use of the I, parameter (see Table I), thus causing a significant loss of information in some practical cases. Second, as a matter of fact, the reactor is saturated for 99 percent of the operating cycle of the system where it is inserted, and behaves like a linear inductor of constant inductance L,. This property should be conveniently exploited in order to gain computational efficiency. The previous considerations led us to writeup of a piecewise polynomial model, defined by the following equations (see

Fig. 4.

-

I plot for the polynomial model.

Fig. 4):

@ ( I= )

C Y ~ I + C Y ~ I ~ O+ < CY I