Measurement of power losses in inductive devices in highly efficient power electronic systems is crucial for choosing appropriate components. A calorimetric method for measuring power losses below 10 W is presented in the following paper to decompose the inductive devices’ power losses. A fully enclosed double-jacked calorimetric chamber is constructed to measure the temperature with digital temperature sensors TMP275, providing a resolution of 0.1 W/K. The finite element method is used with analytical calculation to estimate core and conduction losses to decompose power losses. Power losses are measured for an inductor with a 3C95 ferrite core and four types of windings, where three were made of wire with a different number of strands, whereas the fourth has been made using copper foil. All windings have similar cross-sectional areas for comparison.
With an increasing number of renewable energy systems, global trends are focused on increasing electrical energy conversion efficiency with a simultaneous decrease in cost and dimensions of devices. This is usually achieved by operating at higher frequency ranges above 10 kHz. Ferromagnetic-based components are one of the major elements in power electronic devices where power losses, dimensions, and weight can be reduced. In renewable sources, such as photovoltaic panels or low power, domestic wind turbines power electronic devices are used to convert electrical energy. The ferromagnetic materials are used in inductive elements such as chokes and transformers. The primary consideration is picking the material for given operating conditions depending on the device, like in DC/DC or DC/AC converters [1,2]. The following papers’ main objective is to analyze losses occurring in inductors using the calorimetric method, where cores are made of different materials, like ferrites suited for high-frequency currents. The power losses in these devices occur in both winding and core. However, measuring power losses in such devices is not an easy task since these have high Q-factor due to relatively high inductance and very low resistance. The losses cannot be measured using DC current due to additional effects during AC operation, such as skin and proximity effects and core losses that are not generated in the DC field.
It is expected that power losses in analyzed components are at the level of 10 W. The current waveforms through inductive components in most applications are sinusoidal with a high-frequency ripple as shown in Fig. 1(a) or triangular with DC bias in Fig. 1(b). The former is typical for DC/AC converters, whereas the latter for DC/DC converters like BUCK or BOOST. Both waveforms carry a spectra of higher-order harmonics, which are making power losses inherently difficult to analyze. When current and voltage waveforms can be analyzed using Fourier transform, the phase shift between every major harmonic would be nearly 90 deg. The phase shift would decrease only when the skin and proximity effects become significant, resulting in increased AC resistance, which is unwanted .
State of the Art
The calorimetric method poses an alternative to the electric-based measurement of power losses. Comparison between electrical and calorimetric measurements of inductive components shows higher accuracy of the latter, especially in soft-iron powder cores, especially in cores with relative magnetic permeability . The ferrite material 3C95 is an order of magnitude higher permeable to magnetic field  compared to soft iron powders . So far, various such devices were built for high precision calorimetric measurement [6,7]. The use of a heat exchanges, where dissipated power is measured from liquid flow and temperature difference between inlet and outlet, allows for measurement of power losses of hundreds of watts . Higher precision is obtained by using a double-jacket close type chamber  compared to single jacket chambers, due to confinement of heat in the inner chamber, whereas the outer chamber serves and insulation from the environment. The outer chamber can also serve as a reference when equipped with heaters to stabilize the temperature [9,10]. Calorimetric measurements are usually performed to reach thermal steady-state in the system, although it has been shown that estimation of generated power losses by observing temperatures in transient state results in good accuracy . This can be done by having two identical chambers, one with the measured device and the second with a reference heat source controlled via a controller to match the temperature increase in the former. Performing measurement based on the transient state has two advantages: first, it is much quicker than measurement based on reaching steady-state, although less accurate; second, the measured power losses can be higher than nominal for the chamber, since temperature rise is lower than nominal. Besides power losses in inductive elements, also power losses in capacitors can be measured using the same system  or power electronic converters [1,7] with good precession, although some consideration regarding the operating temperature of other components, i.e., semiconductors, should be taken into account to avoid damage due to overheating.
Calorimetric Measurement System
The setup of the chamber and sensor placement is shown in Fig. 2. Both chambers were made of white Styrofoam with thermal conductivity 0.040 W/mK. The outer chamber has inner dimensions of 500 × 300 × 285 mm and 40 mm wall thickness, whereas the inner chamber has inner dimensions of 150 × 150 × 100 mm with 30 mm wall thickness. The inner chamber was located in the very center of the outer chamber on Styrofoam stands to avoid direct contact with the bottom. The inductor has been placed on fiberglass PCB composite at an even distance to every wall.
Eight temperature sensors, TMP275, measure temperature on walls of the inner and outer chambers. Three sensors were placed on walls perpendicular to each other in both chambers. A higher amount of heat flux is expected to escape through the top of the calorimeter due to natural convection; therefore, sensor no. 6 was placed on the inner top wall of the inner chamber, sensor no. 3 on the outer chambers’ inner top wall and sensor no. 1 on the top outer wall of the outer chamber. Multiple sensors were placed in each chamber to ensure that the temperature field is as homogenous inside as possible. The sensor no. 0 measured ambient temperature. As a heater, the Arcol HS50 15R J power resistor has been used with a low electrical resistance temperature coefficient.
To measure the inductors’ power losses, the Yokogawa WT5000 precision power analyzer had been connected using the four-wire method to the inductor in question. The inductors were powered from a custom-built inverter—a full-bridge single phase inverter capable of generating various waveforms with DC bias. The inductors were connected through 630 × 0.1 mm Litz wire to reduce additional losses and minimize potential heat sources during operation. To evenly distribute heat in both chambers, fans are required. A single fan was used for the internal, and two fans in the outer chamber. These fans have to operate constantly to force convection and spread air. The fans used in the setup were AAB Super Silent Fan 8 Pro due to relatively low power consumption. Each fan dissipates an additional 1.2 W of power at 12 V, which has to be taken into account.
In some applications, foil is preferable due to the possibility of obtaining a high fill factor and better thermal conductivity of the winding compared to round wires. The main disadvantage of using foil-type winding is the high vulnerability to additional eddy current losses caused by the magnetic field acting perpendicular to the surface of the foil, i.e., due to the fringing effect of the stray magnetic field crossing the inductors’ window. A copper winding is preferable to achieve high efficiency, but in high-voltage and low-current applications, aluminum can reduce overall costs. Some compromise in reducing power losses and achieving a high fill factor is possible by using Litz wire with a rectangular cross section, but that usually comes with a higher cost than foil winding.
Power Loss Decomposition
Finite Element Method Modeling of Inductors
Results and Discussion
The calorimetric chamber was calibrated for every inductor, since installing it into the chamber might result in the Styrofoam lid misalignment and, therefore, changing its parameters. Later, during measurement, the heater used for calibration served as an additional heat source to speed up the heating process to reach steady-state faster. To avoid heating the chamber multiple times, the heater was powered after performing every measurement of an inductor to observe any temperature rise or decrease due to the change of dissipated power and was adjusted accordingly. The current delivered to the heater was such that power losses measured with the power analyzer were similar to confirm the measurement. The way to calculate error has been presented in detail in Ref. . The result of measurement compared to the calibrated calorimetric chamber is presented in Fig. 5. All four inductors with different winding were measured at three different currents of approximate RMS values of 4.0 A, 3.5 A, and 2.6 A. The 4.0 A RMS triangle waveform resulted in approximately 10 W of total losses in 2 × 1.5 mm winding and heated the chamber to . The limiting factor was temperature sensors, which could measure up to . Based on calibrated values and using Eqs. (5), (6), (8), and (11), the losses were decomposed into core, conduction and fringing losses as presented in Fig. 6.
The calorimetric method for estimating low power losses has been presented for inductors. The losses were measured and calculated for chokes, with four different types of winding at three different loads. Although the inductor with Litz wire 400 × 0.1 mm had the smallest cross-section, it resulted in the smallest amount of conduction losses. Having smaller strands in winding also helps reduce fringing losses, which might consist of a greater portion of total power losses, like with 2 × 1.5 mm winding, that were at around 50%. The diameter of strands in 35 × 0.355 mm winding was at the level of skin depth for 35 kHz, and the power losses were an only bit higher compared to 400 × 0.1 mm winding. This is attributed to the current waveforms’ higher-order odd harmonics (3, 5, 7, …) typical for triangular waveforms. The foil winding performs similarly to 2 × 1.5 mm wire winding, primarily because of the high value of additional losses generated in the copper foil. It can also be seen that core losses were lower for 2 × 1.5 mm winding, and this is due to the increased temperature, which decreases core losses and has a minimum at around . The designer may consider this effect at the designing stage of a particular system to increase overall efficiency.
Publication supported by Own Scholarship Fund of the Silesian University of Technology in year 2019/2020.
Conflict of Interest
There are no conflicts of interest.
- A =
magnetic vector potential (Wb)
- B =
magnetic flux density vector (T)
- J =
current density vector (A/m2)
- V =
volume of ferromagnetic core (m3)
- T =
period of current waveform (s)
ferrite loss thermal coefficient 4.26 · 10−6 for 3C95 (W/m3K2) 
ferrite loss thermal coefficient 731.8 · 10−6 for 3C95 (W/m3K) 
ferrite loss thermal coefficient 122.7 · 10−3 for 3C95 (W/m3) 
AC resistance of winding (Ω)
DC resistance of winding (Ω)
thermal resistance of outer chamber walls (K/W)
thermal resistance of inner chamber walls (K/W)
ambient temperature (K)
temperature in inner chamber (K)
reference temperature 23 ()
RMS value of current (A)
- k(T) =
temperature-dependent Steinmetz loss coefficient (W/m3)
- , =
skin and proximity effect coefficients
- B(t) =
absolute value of magnetic flux density at time t (T)
electric resistivity thermal coefficient 0.0039 (1/K)
- α, β =
Steinmetz loss coefficients α = 1.045, β = 2.44 
- δ =
skin depth (m)
- ΔB =
peak-peak value of magnetic flux density (T)
additional total power loss in winding related to skin, proximity, and fringing effects (W)
total power loss in winding (W)
total power loss in magnetic core (W)
total power loss in inner chamber (inductor and fan) (W)
total power loss of inductor (W)
total power loss in outer chamber (W)
total power loss associated with current conduction (W)
power loss generated by fan (W)
- ΔT =
temperature rise (K)
- ϱ =
electrical resistivity (Ωm)
electrical resistivity at reference temperature (Ωm)
magnetic permeability of vacuum 4 · π · 10−7 (H/m)
- μ =
magnetic permeability of material (H/m)