# Optimizing Design with the Hanna Curve

## This article discusses the basic concept of the Hanna Curve and the measurement of the inductance in relation to the Hanna curve.

*Inductance is the most fundamental parameter to define an inductive component. In a practical application, the inductance normally performs with non-linear features thus three terms: amplitude inductance, differential inductance, and energetic equivalent inductance are defined to describe the components during the magnetizing and demagnetizing process, as shown in Figure1 [1].*

Among these three definitions, the energetic equivalent inductance is of great significance for energy conversion applications because it matches well with the basic energy charged and discharged operation in the converters, such as DC choppers in buck converters and coupling inductors in Flyback converters [2].

*Figure 1: Definition of amplitude inductance, differential inductance and energetic equivalent inductance.*

Conventionally, the non-linearity performance of the components is derived from the non-linearity of magnetic materials, where the amplitude permeability, incremental permeability and initial permeability etc. are defined initially [4]. However, the description from materials lacks an important aspect-power loss, for a specific inductive component. Besides, the measurement of the material to distinguish the incremental and reversible permeability is normally insufficient for an inductive component, especially under DC-bias condition and the near-saturation condition.

In order to synthesize the design of an inductor, the Hanna Curve was proposed to connect its energy storage capability to its physical geometry, which has been applied for silicon steel at early 20th-century i [3]. In fact, the Hanna Curve is a powerful tool for gapped magnetic components design based on materials study. It was taken over as well for design with gapped ferrite core with consideration of thermal resistance for standardized shaped cores [1].

In this article, the basic concept of Hanna Curve is discussed in the first place and the measurement of the inductance definition and Hanna curve is demonstrated by Bs&T-pulse technology. Two case studies are also presented to discuss the design guidelines from the practical measurement results, to validate the utility of the Hanna Curve as well as the damped oscillation method around the first current peak of voltage-current decay.

## Demand of High Current Power Choke Design

Currently, higher power density and higher power delivery are the essential expectations for power converters. The recent rapid development of power semiconductors with wide-gap-band materials (GaN and SiC) permits the converters working under higher temperature environment. Correspondingly, the magnetic components are also expected working on different conditions from the prior-art. The current standard IEC62024 edition 2, whose target is to characterize an inductive component, while only specifies the performance up to 22 A. Obviously, it cannot satisfy the requirement for modern inductive components operating under higher current excitation and an extended temperature range. Moreover, the uncertainty caused by higher delivered power (i.e. de-rating), higher current and drastic change of the temperature, may affect the systems` reliability.

An example of the emerging requirement for material evaluation is the cross field inductive heating, as illustrated in Fig. 2 [5]. In this typical application, coils will generate a very strong AC magnetic field and the magnetic materials are in charge of regulating the fields, thus the entire system benefits from a high efficiency and practically no EMC issues during the operation. Nevertheless, the operation condition for the materials is on a harsh one with a large flux bias and a high operation temperature, where the performance evaluation for the materials faces critical challenges. Similarly, the difficulty of the materials evaluation also comes up for high-power filter reactors for DC-link converters and electric traction applications etc. [6] [7].

*Figure 2: Principle of cross field heating – an application of inductive components working with high temperature and high excitation current.*

With the emerging requirement of accurate magnetic measurement, the Bs&T pulse technology offers a unique solution to fulfill the requirement for next-generation magnetic component evaluation. In Bs&T bipolar impulse measurement, the damped oscillation method, which has been described in standard IEEE 389, is adopted and it is able not only to provide the performance evaluation under hundreds even thousands of ampere current excitation, but also to minimize the performance shift from temperature rise during the measurement.

The basic concept for damped oscillation method in BsT- pulse micro is illustrated in Figure3. In the first place, the energy for damped oscillation will be charged in the capacitor C, where the voltage can be controlled from 100 V to as high as 1000V to provide enough energy for the oscillation. Once the capacitor being charged, a damped oscillation will be conducted on the D.U.T., during which the voltage and the current of the D.U.T. will be sampled and processed to achieve the performance evaluation, and will complete inductance analysis for magnetization and demagnetization path. The full reversal current enables the loss quantification. Compared with the conventional magnetic measurement with a continuous sinusoidal excitation, this method is particular optimized for high-current application because the total measurement process only persists from tens of nano-second to microsecond. Thus, the measurement temperature of the D.U.T. can be assumed as constant during the measurement. In contrast, the conventional method will suffer the uncertainty from the temperature shift during the entire measurement process, while the systematic error for the total measurement set-up becomes unpredictable either.

*Figure 3: Bs&T measurement system with damped oscillation method.*

## Hanna Curve validation for Inductors

It is well known that two concerns, which are from windings and from core shape and materials, are main aspects for an optimized inductor design with lower volume and lower temperature rise. Nevertheless, it is not necessary to depict the detailed winding configuration due to the energy storage and delivery of the inductor normalized typically.

Correspondingly, the non-linearity from the core materials must be taken seriously, especially for inductors with significant DC bias.

*Figure 4: Typical Hanna Curve of a gapped core.*

The previous evaluation of the inductor is originally derived from the performance of magnetic materials, which can be qualified by BsT-Pro measurement. Nevertheless, the non-linearity of the inductive components cannot be directly reflected from the features of the materials. The shape of the magnetic core, the winding configuration and especially from the air-gap of the core (distributed or discrete) because they rearrange the distribution of the magnetic flux in the magnetic cores, compared with the components with a closed flux loop, such as toroid cores or typical transformers, due to demagnetization factor. Under these circumstances, the measurement directly for a gapped core should be made before component design, especially with higher current and high temperature applications. In the magnetic materials` measurement, the detailed performance can be describe by “permeabilities”, such as initial permeability and incremental permeability [4], which have been specified by magnetizing condition and measurement configurations. However, the corresponding definition for inductive components has not been established, thus measurement became the only way to specify the components.

A proper description to specify the DC chokes can be Hanna Curve, which was initially proposed by C. R. Hanna in 1927. In the initial curve description, a curve of LI2/V against NI/l for a specified a/l of a magnetic core is plotted to illustrated the energy-store capability of a magnetic core, as shown in Figure 4, where L is the differential inductance, corresponding to incremental permeability µΔ or the reversible inductance µrev ,depending on the excitation of the alternating signal; I is the direct current; V is the volume of the core; N is the number of turns; a is the length of the air gap and l is the effective length of the magnetic flux. The Hanna Curve can also be plotted by pre-magnetization LI2 vs B [8]. Meanwhile, the flux ripple can also be reflected in the Hanna Curve by define as ∆I/Io, or delta ∆B/Bo no matter if the magnetic core operates in the linear area.

*Figure 5: Measurement results for UI 93 ferrite core with Bs&T pulse*

It has been recognized that with the increase of the ratio between the air-gap length to the total effective magnetic length, the energy storage capability will be enhanced in a specific core, which is also described in the Hanna Curve. In the Figure 4, it is noticed that linear accessible flux-linkage extends with the increase of the a/l, which validates the enhancement of the energy storage capability. Moreover, the Hanna Curve for a specific gapped core, which indicates a unimodal function, need to be highlighted because the peak value of LI2/V indicates a unique optimum point for the energy storage capability can be identified for a specific core, which is an essential design guideline for the inductors. Compared with deriving the optimum operation point from the magnetic materials` features and the winding configuration, the Hanna Curve can directly provide a design guide both for optimum H-bias and optimum AC current loop while the optimum current density range can be obtained from the x-axis (NI/l) in the Hanna Curve, which can be provided by the Bs&T pulse measurement.

*Figure 6: Measurement results for UI 93 ferrite core for leakage inductance with Bs&T pulse.*

In particular, the Bs&T micro pulse is able to rapidly provided Hanna Curve for cores of both standard core shapes and customized core, especially under specified operation temperatures. Once the measurement of cores` Hanna Curves are released, the optimum core shape selection can be achieved by reading the Hanna Curve, as well as integrating the evaluation of thermal resistivity, especially for standard Ferrite cores, with proven thermal resistance to limit the uncertainty because of thermal dissipation during the operation.

*Figure 7: Measurement results for EE65/ D9B material ferrite with Bs&T pulse (a) Differential and amplitude inductance (b) energy equivalent inductance*

## Case study

In this section, two case studies for the measurement with Bs&T pulse for gapped inductor are presented, where the non-linearity of the components are highlighted with different inductance definition and excitation current.

*Figure 8: Hanna Curve of D9B in EE65 core with different gap length (in SI).*

### Case study I

In the first case study, a ferrite core in shape UI 93 and winding with 7 turns are tested by Bs&T pulse measurement for the amplitude inductance and differential inductance, as shown in Figure 5. It can be found that both the amplitude inductance and the differential inductance show a nonlinearity with the different excitation current, which not only come from the original feature of the ferrite materials, but also from the shift from core shape, air gap length and also magnetizing/ de-magnetizing process.

In Figure 6, a secondary winding, which also consists of 7 turns, was added on the same tested core to simulate the core working as a transformer and the secondary winding is short when measured to depict the performance of the leakage inductance, by definition. In this case, the excitation current was pushed to higher than 400 A and due to the flux in air dominates the operation, the linear range of the leakage inductance is found from 50 A to 200 A and then with higher excitation current, the inductance start to show nonlinearity, which should be considered in the component design.

*Figure 9: Hanna Curve of D9B in EE65 core with different temperature and magnetizing/demagnetizing process (in SI).*

### Case study II

In this case study, the measurement was conducted on D9B ferrite material in EE 65 core shape. Firstly, the differential inductance, amplitude inductance and energy equivalent inductance are depicted in Figure 7 with different excitation current. From the energy equivalent inductance measurement, the Hanna Curve of the specified core can be derived as in Figure 8, from which the energy storage capability can be observed directly. In Figure 8, the original form proposed by C.R. Hannah is adopted, where the y-axis represents the energy density stored in the structure while the x-axis represents the flux linkage in the flux path. It is natural that with the increase of the air gap length, the energy will be enhanced correspondingly. From the Hanna Curve, the designer is able to define the energy charging and dis-charging range for the component and also the voltage-second operation range can be defined.

It is well acknowledged that the temperature has a significant influence for the inductors with ferrite core due to lower Curie temperature compared to metal alloyed design and the nonlinearity of the components also shifts between the magnetizing and demagnetizing process, because energy loss is generated during each half cycle. In Figure 9, the Hanna Curve is depicted with different testing temperature and magnetizing/ demagnetizing process. With the damped oscillation method with Bs&T measurement system, the performance shift can be clear observed: under the room temperature, the magnetizing and demagnetizing process own similar energy storage capability; while at higher operational temperature, the two curve shows separate trend but generally the energy density will be higher compared with that with room temperature, which indicates a potential optimal operation point for this magnetic core.

## Conclusion

With the emerging requirement for high-performance magnetic components with higher current and higher temperature working conditions, the damped oscillation method enables proper validation for the magnetic cores, especially for DC chokes application. The Hanna Curve is one of the choices for estimating the performance for a gapped core as well as optimizing the design even without specifying the winding configurations. With the two case studies, the complete inductance analysis, as well as Hanna Curve are depicted based on the measurement data by Bs&T pulse. The measurement validates its energy storage capability not only for evaluating the performance of the components, but also for optimizing the operation point for inductive components, which work as an essential reference for component, converter and system designers. Hanna curve can then be reloaded with BsT-pulse for each magnetic component designer, completely, correctly and easily. It provides the designers and users of magnetic components with limited values for specification.

#### About the Author

**JC Sun** is the founder of Bs&T Frankfurt am Main GmbH, a company located in the north of the metropolis Frankfurt am Main and specializes in the development and manufacture of integrated hyster loop measuring systems. He worked for two decades as a development engineer in power electronics; developing various soft magnetic materials and was a project manager for various companies.

**Yi Dou** is a Ph.D. student working on Power Electronics and has a strong background in high-frequency DC-DC converters, resonant converters, and magnetic design. He is currently under his Doctorate Degree in Power Electronics at DTU - Technical University of Denmark. He also holds a Master's Degree in Electrical and Electronics Engineering at DTU and also holds a Bachelor's Degree in Electrical and Electronics Engineering t Xi'an Jiaotong University.

#### References

- Snelling, Eric Charles. "Soft ferrites" Edition 2 (1988).
- Valchev, Vencislav Cekov, and Alex Van den Bossche. Inductors and transformers for power electronics. CRC press, 2018.
- Hanna, C. R. "Design of reactances and transformers: Which carry direct current." Journal of the AIEE 46.2 (1927): 128-131.
- JC Sun, K Seitenbecher Proper specification of magnetic components BodoPower December edition 2019.
- private communication with Dr. Rainer Menge
- J. Wang, "Challenges Facing Emerging Megawatt Applications: New Technology and Solutions Needed for High-Power Applications," in IEEE Power Electronics Magazine, vol. 6, no. 4, pp. 38-40, Dec. 2019.
- Johann W. Kolar, Jonas E. Huber, "Fundamentals and Application-Oriented Evaluation of Solid-State Transformer Concepts".
- Hilzinger, R., & Rodewald, W. (2013). Magnetic materials: fundamentals, products, properties, applications. Vacuumschmelze.

This article originally appeared in the Bodo’s Power Systems magazine.

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