Ocl94 Lloyd Dixon Summary: A brief tutorial on magnetic fundamentals leads into a discussion of magnetic core properties. A modified version of Intusoft' s magnetic core model is presented. Low1requency hysteresis is added to the model. making it suitable for magnetic amplifier applications. Magnetic Fundamentals: Units commonly used in magnetics design are given in Table I, along with conversion factors from the older cas system to the SI system (sys- feme infernational- rationalized MKS). SI units are Fig 1. -Magnetic Core B-H Characteristic used almost universally throughout the world. Equations used for magnetics design in the SI surface of Fig. 1 represents energy per unit volume. system are much simpler and therefore more intu- The area enclosed by the hysteresis loop is unre- itive than their cas equivalents. Unfortunately, coverable energy (loss). The area between the much of the published magnetics data is in the cas hysteresis loop and the vertical axis is recoverable system, especially in the United States, requiring stored energy: conversion to use the SI equations. W/m3 = JHdB Table I. CONVERSIONFACTORS, CGS to SI In Figure 2, the shape is the same as Fig. I, but the axis labels and values have been changed. Figure 2 SI CGS CGS to SI FLUX DENSITY B Tesia GaUSS 10-4 shows the characteristic of a specific core made FIELD INTENSITY H A-T/m Oersted 1000/41t from the material of Figure I. The flux density axis PERMEABILITY(space) ~o 41t.10.7 1 41t.10.7 PERMEABILITY(relative) J.Iy 1 AREA (Core, Window) A m2 cm2 10.4 LENGTH (Core, Gap) . m cm 10.2 TOTAL FLUX = JBdA ~ Weber Maxwell 10.8 TOTAL FIELD = JHdf F,MMF A-T Gilbert 10/41t RELUCTANCE= F/~ R 109/41t PERMEANCE= 1/R = L/N2 P 41t.10.9 INDUCTANCE= P-N2 L Henry (Henry) 1 ENERGY W Joule Erg 10.7 Figure 1 is the B-H characteristic of a magnetic core material -flux density (Tesla) vs. magnetic field intensity (A-Tlm). The slope of a line on this set of axes is permeability (p = B/H). Area on the Fig 2. -Core FlzlX vs. Magnetic Force RI-I is trnnsfonned into the total flux, <I>,through the electrical characteristic of the magnetic core wound entire cross-sectional area of the core, while field with a specific number of turns, N, as shown in intensity is transfonned into total magnetic force Figure 3. around the entire magnetic path length of the core: I = HIe I!\ = B-A ; F = H.f ~ 't' e e The slope of a line on these axes is permeance Area in Fig. 3 again represents energy. this time (P = ci>/F= pAJfe). Penneance is the inductance of in electrical terms: W = JEIdt. The slope of a line one turn wound on the core. in Fig 3 is inductance. Area in Fig. 2 represents total energy -hyster- dt A L = E- = 11N2~ esis loss or recoverable energy. di r- , e Changing the operating point in Fig. lor Fig. 2 requires a change of energy, therefore it can not change instantaneously. When a winding is coupled Low-Frequency Core Characteristics: to the magnetic core, the electrical to magnetic Ferromagnetic core materials include: Crystalline relationship is governed by Faraday's Law and metal alloys, amorphous metal alloys, and ferrites Ampere's Law. (ferrimagnetic oxides).[!] Figure 4 shows the low frequency electrical Faraday's Law: characteristics of an inductor with an ungapped ~ = -~ ci>= 2- rEdt toroidal core of an idealized ferromagnetic metal dt N N J' alloy. This homogeneous core becomes magnetized at a specific field intensity H (corresponding to Ampere's Law: specific current through the winding I=HI/N). At Nl = fHdl ~ HI this magnetizing current level, all of the magnetic dipoles (domains) within the core gradually align, These laws operate bi-directionally. According to causing the flux to increase toward saturation. The Faraday's Law, flux change is governed by the domains cannot align and the flux cannot change voltage applied to the winding (or voltage induced instantaneously because energy is required. The in the winding is proportional to d<1>/dt).Thus, changes occur at a rate governed by the voltage electrical energy is transfonned into energy lost or applied to the winding, according to Faraday's Law. stored in the magnetic system (or stored magnetic Thus, to magnetize this core, a specific magnetiz- energy is transfonned into electrical energy). ing current, IM, is required, and the time to accom- Applying Faraday's and Ampere's Laws, the plish the flux change is a function of the voltage axes can be trnnsfonned again into the equivalent applied to the winding. These factors combined - Fig 3. -Equivalent Electrical Characteristic Fig 4. -Ideal Magnetic Core RI-2 current, voltage, and time -constitute energy put The Effect or Core Thickness: Fig. 1 applies into the core. The amount of energy put in is the only to a very thin toroidal core with Inner Diame- area between the core characteristic and the vertical ter almost equal to Outer Diameter resulting in a axis. In this case, none of this energy is recoverable single valued magnetic path length (7t"D). Thus the -it is all loss, incurred immediately while the same field intensity exists throughout the core, and current and voltage is being applied. the entire core is magnetized at the same current The vertical slope of the characteristic represents level. an apparent infinite inductance. However, there is A practical toroidal core has an O.D. substan- no real inductance -no recoverable stored energy tially greater than Its I.D., causing magnetic path -the characteristic is actually resistive. (A resistor length to increase aDd field intensity to decrease driven by a square wave, plotted on the same axes, with increasing diameter. The electrical result is has the same vertical slope.) shown in Figure 5. As current increases, the critical When all of the domains have been aligned, the field intensity, H, required to align the domains is material is saturated, at the flux level corresponding achieved first at the inner diameter. According to to complete alignment. A further increase in current Ampere: produces little change in flux, and very little volt- NI NI age can exist across the winding as the operating H= = T nD'" point moves out on the saturation characteristic. The small slope in this region is true inductance - Hold on to your hats!! At fIrst, the domains recoverable energy is being stored. With this ideal realign and the flux changes in the new direction core, inductance Lo is the same as if there were no only at the inner diameter. The entire outer portion core present, as shown by the dash line through the of the core is as yet unaffected, because the field origin. The small amount of stored energy is repre- intensity has not reached the critical level except at sented by a thin triangular area above the saturation the inside diameter. The outer domains remain fully characteristic from the vertical axis to the operating aligned in the old direction and the outer flux point (not shown). density remains saturated in the old direction. In If the current is now interrupted, the flux will fact, the core saturates completely in the new decrease to the residual flux level (point R) on the direction at its inner diameter yet the remainder of vertical axis. The small flux reduction requires the core remains saturated in the old direction. reverse Volt-pseconds to remove the small amount Thus, complete flux reversal always takes place of energy previously stored. (If the current is starting from the core inside diameter and progress- interrupted rapidly, the short voltage spike will be ing toward the outside. quite large in amplitude.) As long as the current In a switching power supply, magnetic devices remains at zero (open circuit). the flux will remain are usually driven at the switching frequency by a -forever -at point R. If a negative magnetizing current is now applied to the winding, the domains start to realign in the opposite direction. The flux decreases at a rate determined by the negative voltage across the winding, causing the operating point to move down the characteristic at the left of the vertical axis. As the operating point moves down. the cumulative area between the characteristic and the vertical axis represents energy lost in this process. When the horizontal axis is reached, the net flux is zero - half the domains are oriented in the old direction, half in the new direction. RI-3 voltage source. A voltage across the winding causes considerable additional skewing of the character- the flux to change at a fixed rate. What actually istic. This slope arises from the inclusion of small happens is the flux change starts at the core inner non-magnetic regions in series with the magnetic diameter and progresses outward, at a rate equal to core material. For example, such regions could be the applied volts/turn, Em. The entire core is the non-magnetic binder that holds the particles always saturated, but the inner portion is saturated together in a metal powder core, or tiny gaps at the in the new direction, while the outer portions imperfect mating surfaces of two core halves. remain saturated in the old direction. (This is the Additional magnetic force is required, proportional lowest energy state -lower than if the core were to the amount of flux, to push the flux across these completely demagnetized.) There is in effect a small gaps. The resulting energy stored in these boundary at the specific diameter where the field gaps is theoretically recoverable.