Stonemason formula

The Steinmetz formula can be used to calculate core losses of inductive components . It bears the name of its discoverer, the German-American engineer Karl Steinmetz , who was the first to calculate these losses through hysteresis and eddy currents .

Discovery of the formula

Eddy currents, which occur in the iron cores of transformers as well as the iron bobbins of generators or electric motors , caused major problems in the early days of electrical engineering , because they reduced the efficiency of the systems on the one hand and caused the components to heat up on the other. The construction of the coil formers from isolated iron plates instead of a solid iron body reduced the eddy currents, but a lot of electrical energy was converted into heat in the transformers. Because the iron cores of the transformer are magnetized under the influence of the alternating magnetic field of the coils and the magnetization does not completely decrease when the magnetic field changes its direction in time with the alternating current frequency, a residual magnetization remains. The so-called remanence must first be overcome by the reversed magnetic field, from which the additional heat losses arise. Karl Steinmetz found the solution for this in hysteresis.

Hysteresis curve

typical hysteresis curve

The atoms of ferromagnetic materials have a magnetic moment. In the unmagnetized state, the magnetic moments of the atoms are oriented in all spatial directions, with atoms in limited cells ( Weiss cells ) having a preferred direction. The boundaries of these cells are called Bloch walls . If you now apply an external magnetic field, the magnetic moments are aligned along the direction of the magnetic field , in that the Weiss cells with magnetic moment grow in the direction of the field at the expense of neighboring cells. One also speaks of moving Bloch walls. Within certain limits, this is a reversible process. When the field strength increases, the Bloch walls jump from defect to defect, which is no longer reversible. If all cells are aligned, the magnetic moments are rotated from their crystal direction into the field direction when the magnetic field is increased further. This behavior, we speak of turning processes here, is reflected in the hysteresis curve (also called BH curve), the course of which is material-dependent.

Reversible Blochwand displacements predominate in the lower area of the new curve. In the middle area, in which the magnetic flux density increases almost linearly with the magnetic field strength , you can see the irreversible jumps of the Bloch walls. In the saturation area, where the increase in the magnetic flux density takes place much more slowly, the turning processes predominate. To achieve the saturation flux density, a saturation field strength must be applied. When the field strength is reduced, many of the displaced Bloch walls get stuck on imperfections and the magnetic flux density decreases along another curve. There is still magnetic flux even after the field strength has decreased to zero. In order to reset the so-called remanence flux density to zero, a certain negative field strength has to be applied, the coercive field strength . ${\ displaystyle B}$ ${\ displaystyle H}$ ${\ displaystyle H_ {s}}$ ${\ displaystyle B_ {R}}$ ${\ displaystyle H_ {C}}$

Steinmetz formula for sinusoidal excitation

Steinmetz recognized that the area within the hysteresis curve corresponds to the core losses (in mW per cm ³ ) per cycle and converted this fact into the following formula:

{\ displaystyle P_ {core} = k \ cdot f ^ {a} \ cdot B ^ {b}}

Here, the average power dissipation per unit volume, the peak value of induction, the core constant, and the frequency of the sinusoidal measuring voltage. The coefficients and depend on the material. For ferrites, for example, the coefficient is between 1.1 and 1.9 and the coefficient is between 1.6 and 3. ${\ displaystyle P_ {core}}$ ${\ displaystyle B}$ ${\ displaystyle k}$ ${\ displaystyle f}$ ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle a}$ ${\ displaystyle b}$

Using the “simple” Steinmetz formula, it is possible to calculate the core losses of inductors whose core designs are based on industry standards. These standard kernels have the same geometry, which is why their kernel constants are also identical. Only the corresponding material characteristics have to be inserted into the equation. Composite inductances are an exception, because even with the same core size, the geometry parameters of these components vary depending on the inductance value. The reason: In contrast to toroidal core constructions or the so-called E-cores, the copper wire is first wound into an air-core coil with composite inductance. Since each coil has a different diameter and a different height, different geometry parameters apply to each inductance and the core constants must be determined individually.

Steinmetz formula for non-sinusoidal excitation

Composite inductors are often used in DC / DC converters that are not galvanically separated and that do not work with sinusoidal alternating current , but with pulsed direct current . In addition to the fact that the current, which is partly responsible for the core loss, now has a triangular time curve, there is another loss factor: the influence of the operating temperature. Because DC / DC converters are used more and more often at higher ambient temperatures, the inductance not only has to cope with the temperature increase due to internal power losses, but also higher ambient temperatures. This affects the iron core so that iron powder ages faster at higher temperatures, which increases core losses. To minimize the effects of thermal aging, it is recommended to keep the maximum operating temperature of the inductance below +125 ° C. In order to determine the operating conditions at the upper temperature limits of the respective application / circuit, the Steinmetz formula applicable for sinusoidal signals can also be used, taking into account the prevailing temperatures:

{\ displaystyle P_ {core} = k \ cdot f ^ {a} \ cdot B ^ {b} (c_ {t2} T ^ {2} -c_ {t1} T + c_ {t0})}

${\ displaystyle a}$ and , again, the so-called Steinmetz frequency or Steinmetz induction coefficients are specified for operating conditions , , and are material constants, the frequency and the operating temperature. ${\ displaystyle b}$ ${\ displaystyle k}$ ${\ displaystyle c_ {t1}}$ ${\ displaystyle c_ {t2}}$ ${\ displaystyle c_ {t0}}$ ${\ displaystyle f}$ ${\ displaystyle T}$

The following applies to non-sinusoidal signals:

{\ displaystyle P_ {core} = f \ cdot k \ cdot f_ {sin_ {e} q} ^ {a-1} \ cdot B ^ {b} (c_ {t2} T ^ {2} -c_ {t1} T + c_ {t0})}

Relative error

Example for the error of the Steinmetz formula with duty cycle> 50% (f = 100 kHz; MnZn core)

The so-called “volt µsec product” indicates the maximum value up to which storage chokes can be controlled due to their magnetically effective area. In other words: the higher the volt-µsec product, the higher the losses. With increasing switching frequency, the required Vµsec product of the storage choke decreases; however, it increases with increasing input voltage. Since, in addition to the volt-µsec product, the duty cycle and the operating frequency of the circuit are essential for calculating the core losses, they also have an influence on the accuracy of the Steinmetz formula. The accuracy of the Steinmetz formula is already lower with a duty cycle of 50%, with small or large duty cycles errors of over 100% can occur. Neglecting the harmonics or the DC bias also leads to inaccuracies in the calculated core losses. The reasons for this are that a different BH curve is set, that a wrong Vμsec product results and that the temperature dependency is not taken into account.

Loss calculation and software tools

In order to make it easier for developers to choose the right inductance or to narrow down the components in question, some manufacturers of passive components offer calculation tools that can also be used to determine the expected core losses.

Individual evidence

↑ ^a ^b Udo Leuschner: Energy knowledge
↑ ^a ^b Dr. Thomas Brander, A. Gerfer, B. Rall, H. Zenkner: Trilogy of inductive components Chapter: Fundamentals
↑ ^a ^b Nicolas J. Schade: Core Losses in Composite Inductors - Vishay Intertechnology, Inc.
↑ TDK-EPC: Customer magazine Components October 2008 Topic: Increase efficiency

[Udo-Leuschner.de-1] Udo Leuschner: Energy knowledge

[Trilogie-2] Dr. Thomas Brander, A. Gerfer, B. Rall, H. Zenkner: Trilogy of inductive components Chapter: Fundamentals

[Vishay-3] Nicolas J. Schade: Core Losses in Composite Inductors - Vishay Intertechnology, Inc.

[TDK-EPC-4] TDK-EPC: Customer magazine Components October 2008 Topic: Increase efficiency