Electromagnetic induction energy storage technology, as a clean and highly efficient method of energy conversion, plays a vital role in modern power systems, particularly in the context of renewable energy integration and peak-load shaving. In the system I study, the heating pipe is the core component that converts electromagnetic energy into thermal energy. During operation, a medium-frequency high current flows through the heating pipe, causing variations in its equivalent inductance. This inductance directly influences the system’s power factor, resonant state, and overall energy conversion efficiency. To match the proper capacitor in the controller and make the system operate near the resonant point with maximum efficiency, accurate calculation of the heating pipe equivalent inductance is essential.
In this thesis, I focus on a practical electromagnetic induction energy storage system and develop both an analytical inductance model and a field-circuit coupled simulation approach. The work covers material magnetic property measurement, nonlinear equivalent magnetic network modeling, and joint simulation with control strategies. The main contributions are as follows: first, I established a test platform for measuring the magnetic properties of Q235 ferromagnetic steel; second, I proposed an analytical model for heating pipe inductance that accounts for the nonlinear B–H curve and frequency-dependent skin depth; finally, I introduced a co-simulation method using Magnet and Simulink to emulate the real driver operation and validated the model with experiments on a 25 kW prototype energy storage system.
1. Fundamentals of Electromagnetic Induction Energy Storage
The electromagnetic induction energy storage system consists of a three-phase rectifier, a DC filter, a high-frequency inverter, a resonant capacitor, and the heating pipe with its induction coil. The heating pipe can be regarded as a short-circuited secondary winding of a transformer, where the primary is the multi-turn induction coil. When the inverter supplies a medium-frequency current \(i_1\) to the coil, a rapidly alternating magnetic field is generated. This field induces an electromotive force \(e\) in the conductive steel pipe, which produces eddy currents \(i_2\). Because of the pipe’s effective resistance, these eddy currents generate Joule heat that is transferred to water flowing inside the pipe, thus storing thermal energy.
According to Faraday’s law, the induced electromotive force in the heating pipe is given by
$$
e = -N\frac{d\Phi}{dt}
$$
where \(N\) is the number of coil turns and \(\Phi\) is the magnetic flux. When the flux varies sinusoidally, the RMS value of the induced voltage can be expressed as
$$
E = 4.44 f N \Phi_m
$$
where \(f\) is the operating frequency and \(\Phi_m\) is the peak magnetic flux.
The equivalent circuit of the heating system is shown as a transformer model, where \(L_1\) and \(R_1\) are the primary coil inductance and resistance, while \(L_2\) and \(R_2\) are the equivalent eddy-current inductance and resistance of the steel pipe. The reflected impedance at the primary side can be written as
$$
Z_{eq} = R_{eq} + j\omega L_{eq}
$$
$$
R_{eq} = R_1 + \frac{\omega^2 M^2 R_2}{R_2^2 + (\omega L_2)^2}
$$
$$
L_{eq} = L_1 – \frac{\omega^2 M^2 L_2}{R_2^2 + (\omega L_2)^2}
$$
where \(M = k\sqrt{L_1 L_2}\) is the mutual inductance and \(k\) is the coupling coefficient. The above expressions clearly show that the equivalent inductance of the heating pipe is influenced by the operating frequency, the coupling coefficient, and the secondary-side resistance and inductance, which in turn depend on material properties and temperature.
2. Magnetic Property Testing of Q235 Ferromagnetic Material
To accurately model the heating pipe, I first need reliable magnetic characteristics of the construction material, Q235 steel. I built a magnetic testing platform based on a Brockhaus C 510 system. The measurement followed the standard GB/T 3658-2022 for ring-shaped specimens. A toroidal sample was machined from the same batch of steel as the actual heating pipe. The inner diameter, outer diameter, and thickness were 80 mm, 100 mm, and 10 mm respectively. The primary winding had 130 turns and the secondary had 20 turns. Because the specimen experiences strong eddy-current heating, I used insulating fabric and high-temperature tape between the windings and the core. Details of the sample and winding setup are presented in Table 1.
| Parameter | Value |
|---|---|
| Inner diameter (mm) | 80 |
| Outer diameter (mm) | 100 |
| Thickness (mm) | 10 |
| Effective magnetic path length (m) | 0.283 |
| Mass (g) | 228.18 |
| Primary resistance (Ω) | 0.120 |
| Secondary resistance (Ω) | 0.737 |
The measurement software controlled the excitation frequency and the maximum flux density. I recorded the hysteresis loop family, the normal magnetization curve, the relative permeability curve, and the specific power loss curve. During the test, the maximum excitation current was limited to 40 A and the maximum voltage to 110 V, which constrained the maximum measurable magnetic field strength. To avoid overheating, I allowed the specimen to cool to room temperature after each continuous sweep.
The measured B–H curve is shown in Figure below, where the magnetic flux density starts to saturate at high field strengths. The corresponding relative permeability, \(\mu_r = B/(\mu_0 H)\), first rises to a maximum and then declines as the material enters the saturation region. The loss density also increases nonlinearly with flux density. These data are essential for the equivalent magnetic network model because the steel permeability changes strongly with the local magnetic flux density.
In addition, I measured the electrical resistivity of Q235 steel as a function of temperature using a PPMS device with the van der Pauw method. The sample was a small square of 5 mm × 5 mm × 2.5 mm. Figure below shows the resistivity increasing almost linearly from about \(17.9 \times 10^{-8} \ \Omega\!\cdot\!\text{m}\) at 50 °C to \(28 \times 10^{-8} \ \Omega\!\cdot\!\text{m}\) at 700 °C. This temperature dependence strongly affects the skin depth inside the steel pipe.
2.1 Validation of the measured material data
To verify the measured material properties, I performed a field-circuit coupled loss analysis on the same toroidal specimen. The three-dimensional finite element model was simplified by representing the primary winding as an equivalent current density layer, because the secondary winding remained open. I first fed a pure sinusoidal current with the measured RMS value obtained from the experiment. The computed losses were compared with the measured core loss, and the errors were unexpectedly large (above 50%). The reason was that the actual excitation current in the magnetic tester is not purely sinusoidal; it contains significant harmonics.
I then analyzed the actual current waveform and extracted its harmonic components. The total harmonic distortion (THD) was about 14–17% for flux densities between 0.9 T and 1.4 T, with the third harmonic being dominant. The harmonic amplitudes are summarized in Table 2.
| Flux density (T) | THD (%) | Fundamental (A) | 3rd harmonic (A) | 5th harmonic (A) |
|---|---|---|---|---|
| 0.9 | 14.37 | 11.94 | 1.70 | 0.01 |
| 1.0 | 15.32 | 14.70 | 2.23 | 0.28 |
| 1.1 | 16.15 | 18.00 | 2.88 | 0.39 |
| 1.4 | 17.26 | 32.00 | 5.43 | 0.96 |
By adding the measured harmonic components into the finite element model, the calculated losses became much closer to the experimental values. Table 3 shows the comparison. The maximum error was reduced to 13.37%, which confirms the reliability of the measured B–H and loss data.
| Flux density (T) | Computed loss (W) | Measured loss (W) | Error (%) |
|---|---|---|---|
| 0.9 | 19.86 | 18.86 | 5.30 |
| 1.0 | 27.70 | 25.62 | 8.11 |
| 1.1 | 37.90 | 34.24 | 10.68 |
| 1.4 | 85.40 | 75.33 | 13.37 |
3. Analytical Inductance Model Based on Equivalent Magnetic Network
3.1 Model construction
The heating pipe of the 25 kW energy storage system is a steel tube with a coil wound on the outside. The structure is axisymmetric, with the coil height and pipe length being much larger than the radial dimensions. I used the equivalent magnetic network (EMN) method to solve the nonlinear magnetic field in the heating pipe. The entire solution domain was divided into flux tubes. Each flux tube was modeled as a reluctance element, and the coil excitations were represented as magnetomotive force (MMF) sources. The network was constructed using the nodal magnetic potential method, which is analogous to the nodal voltage method in electric circuits.
I divided the steel pipe into two regions: a skin-effect region near the surface where eddy currents are concentrated, and a non-skin-effect region deeper inside. The skin depth \(\Delta\) is given by
$$
\Delta = \sqrt{\frac{\rho}{\pi f \mu_0 \mu_r}}
$$
where \(\rho\) is the electrical resistivity, \(f\) is the frequency, and \(\mu_r\) is the relative permeability. For Q235 steel at 20 °C and a frequency of 1 kHz to 25 kHz, the skin depth ranges from about 0.6 mm to 0.12 mm. Because the skin depth is much smaller than the pipe wall thickness, I refined the mesh in the skin-effect region into four layers, while the non-skin-effect region was coarsely divided into three layers.
The complete equivalent magnetic network is illustrated conceptually as a grid of radial and axial permeances, with MMF sources inserted in the coil region. The air region outside the pipe was also discretized using flux tubes to account for leakage flux. The permeance of a rectangular flux tube in the radial direction is computed as
$$
\Lambda_r = \frac{\mu_0 \mu_r h \varphi}{\ln(1 + w/r)}
$$
where \(h\) is the axial height, \(\varphi\) is the angular width, \(w\) is the radial width, and \(r\) is the inner radius. Similar expressions were used for axial and tangential permeances of the selected element shapes.
3.2 Nonlinear magnetic permeance
For the steel portions, the relative permeability is a function of the local magnetic flux density. I fitted the measured B–H curve with a continuous piecewise function so that the permeability can be updated at every iteration. At the beginning, the initial permeability is estimated from the slope between the first two points of the B–H curve:
$$
\mu_{in} = \frac{B(2)-B(1)}{H(2)-H(1)}
$$
The matrix equation of the magnetic network is
$$
\mathbf{\Lambda}_n \mathbf{F}_n = \mathbf{\Phi}_n
$$
where \(\mathbf{\Lambda}_n\) is the nodal permeance matrix, \(\mathbf{F}_n\) is the vector of unknown nodal magnetic potentials, and \(\mathbf{\Phi}_n\) is the source flux vector, which is built from the winding MMF sources. The nodal permeance matrix is assembled from the branch permeances using the incidence matrix \(\mathbf{A}\).
After solving the linear system at the current permeability values, the flux density in each element is calculated. To improve convergence, I used a relaxation technique combined with the nonlinear iteration. The corrected flux density is
$$
B_{ele}^{(k)} = \alpha_1 B_{ele}^{(k-1)} + \alpha_2 B_{ele}^{(k-2)} + (1-\alpha_1-\alpha_2) B_{ele}^{(det)}
$$
with \(\alpha_1\) and \(\alpha_2\) being relaxation factors. The new permeability \(\mu_{new}\) is then obtained from the fitted B–\(\mu_r\) curve. To avoid oscillation, I applied an additional damping:
$$
\mu^{(k)} = f_c \mu^{(k-1)} + (1-f_c)\mu_{new}
$$
where \(f_c\) is a damping coefficient between 0 and 1. The iteration was stopped when the largest change in flux density between consecutive steps was less than \(10^{-3}\) T, and the maximum iteration count was limited to 100.
3.3 Inductance calculation
Once the magnetic field distribution was converged, the magnetic energy stored in the whole solution domain was calculated by integrating the energy density over the volume. For a two-dimensional axisymmetric model,
$$
W = \int_V \left( \int_0^B H \, dB \right) dV
$$
Discretizing the integral over all flux tubes gives
$$
W = \sum_j \omega_{m,j} \Delta V_j
$$
where \(\omega_{m,j}\) is the magnetic energy density in element \(j\). The equivalent inductance is then obtained from the well-known energy relation
$$
L = \frac{2W}{I^2}
$$
where \(I\) is the RMS value of the coil current.
3.4 Experimental validation
I built an experimental platform consisting of an actual heating pipe and an IM3533 LCR meter, which measures inductance at frequencies from 1 mHz to 200 kHz with a basic accuracy of ±0.05%. The LCR meter applies a small AC signal to the coil and derives the inductance from the measured impedance. The measurement was conducted at six frequencies: 1, 5, 10, 15, 20, and 25 kHz. The experimental inductance values are listed in Table 4.
| Frequency (kHz) | Skin depth (mm) | Measured L (μH) | Calculated L (μH) | Error (%) |
|---|---|---|---|---|
| 1 | 0.604 | 174.35 | 161.2 | 7.54 |
| 5 | 0.270 | 107.78 | 100.3 | 6.94 |
| 10 | 0.191 | 94.19 | 88.3 | 6.25 |
| 15 | 0.156 | 88.57 | 82.73 | 6.59 |
| 20 | 0.135 | 85.41 | 79.64 | 6.76 |
| 25 | 0.121 | 83.22 | 77.58 | 6.78 |
The comparison shows that the errors between the analytical model and the measured values are all within 10%. The decreasing trend of inductance with increasing frequency is correctly captured. The remaining discrepancy can be attributed to the simplified treatment of the nonlinear permeability and the discretization error of the magnetic network. The skin depth in the calculation was evaluated using a constant permeability near saturation, which is a reasonable approximation for the high-current operating condition.
4. Field-Circuit Coupled Simulation of the Energy Storage System
4.1 Finite element model with sinusoidal excitation
Before implementing the more realistic driver-coupled simulation, I first performed a finite element analysis using a pure sinusoidal current excitation in the Magnet software. The three-dimensional geometry of the heating pipe was created from the two-dimensional axisymmetric cross-section by rotating 360°. The coil was modeled as a stranded coil with the same equivalent ampere-turns as the actual winding. The material properties were assigned according to the measured B–H curve and resistivity. The excitation was a sinusoidal current with a peak value of 123 A at a frequency of 16.65 kHz.
The simulation results showed that the voltage across the heating pipe had an amplitude of 1225 V. The computed total loss was 22.01 kW, while the experimental loss of the 25 kW system was 25.8 kW, giving an error of 14.7%. The magnetic field energy was 0.2775 J, which corresponds to an equivalent inductance of 73.33 μH. However, the LCR meter measured the inductance as 96.4 μH at the same frequency. The large error of 23.9% indicates that the sinusoidal excitation model is not sufficiently accurate for the actual inverter-driven system.
4.2 Magnet/Simulink joint simulation
To reflect the actual operating conditions, I developed a co-simulation method that couples the finite element model of the heating pipe with the real inverter controller. The Magnet software provides a plug-in that allows the finite element model to be embedded in a Simulink circuit. I removed the current source and replaced the coil excitation with a voltage-driven port. The plug-in receives the instantaneous voltage from the Simulink circuit and returns the instantaneous current for the next time step.
I built the complete controller in Simulink, including the three-phase rectifier, DC bus capacitor, full-bridge inverter with IGBTs, the resonant capacitor, and the heating pipe model represented by the Magnet plug-in. The control strategy consisted of a phase-locked loop (PLL) and a power control loop. The PLL tracks the phase of the output current to generate the appropriate frequency signal, while the power loop adjusts the phase-shift angle to achieve the desired output power. The block diagram of the controller is presented in Table 5.
| Component | Description |
|---|---|
| Rectifier | Three-phase uncontrolled diode bridge |
| DC link | Parallel capacitor for voltage filtering |
| Inverter | Voltage-source full-bridge with IGBTs |
| Resonant tank | Capacitor in series with the heating coil |
| Heating pipe | Magnet finite element model embedded via plug-in |
Figure below shows the gate pulse waveforms of the four IGBTs generated by the phase-shift control logic. The pulses are shifted by the modulation angle to regulate the output power while maintaining the resonant frequency.
The simulated waveforms of the inverter output voltage and current are presented in Figure below. The inverter output voltage amplitude is 556 V, and the output current amplitude is 116 A. The voltage across the heating pipe is about 1076 V peak. Comparing the sinusoidal excitation case, the joint simulation yields a much higher voltage magnitude for the same current, which is consistent with the actual impedance of the heating pipe at this operating point.
4.3 Experimental validation on a 25 kW prototype
I built a 25 kW prototype electromagnetic induction energy storage system to validate the simulation. The experimental setup included the actual heating pipe, an insulated coil, a temperature sensor, a power analyzer (PW6001), and a controller cabinet. The controller was connected to the three-phase grid, and the inverter output was connected to the resonant capacitor and the coil. The power analyzer measured the input power, the inverter output voltage and current, and the voltage across the heating pipe.
Figure below shows the measured waveforms. The heating pipe current amplitude is 109.75 A and the voltage amplitude is 1040 V. The phase difference between the voltage and current is 70.827°. The inverter output voltage RMS is 551 V, and the active power is 25.80 kW. The measured inductance of the heating pipe was calculated from the series impedance as 96.4 μH, using the known capacitance of 2 μF.
Table 6 compares the results from the sinusoidal excitation and the joint simulation with the experimental data.
| Item | Sinusoidal excitation | Magnet/Simulink co-simulation | Experimental |
|---|---|---|---|
| Total loss (kW) | 22.01 | 27.7 | 25.8 |
| Loss error (%) | 14.7 | 7.3 | — |
| Inductance (μH) | 73.33 | 101.99 | 96.4 |
| Inductance error (%) | 23.9 | 5.8 | — |
It is clear that the joint simulation is substantially more accurate than the pure sinusoidal excitation. The joint simulation takes into account the switching harmonics and the actual voltage shape generated by the inverter, which are crucial for the eddy current and hysteresis losses in the steel pipe. The remaining small error can be attributed to the simplified model of the IGBT switches and the absence of thermal effects on the material properties.
4.4 Optimal capacitor matching and power improvement
Using the equivalent magnetic network model described in Section 3, I calculated the heating pipe inductance at 16.65 kHz for the 25 kW system. The model yielded 89.3 μH. Based on this value, the resonant capacitance is
$$
C_{res} = \frac{1}{(2\pi f)^2 L} = \frac{1}{(2\pi \times 16650)^2 \times 89.3 \times 10^{-6}} \approx 1.02\ \mu F
$$
To ensure the system operates in a slightly inductive state, I chose a capacitor of 1.3 μF. I then performed the joint simulation with this optimized capacitor while keeping the same DC bus voltage. The resulting voltage across the heating pipe remained almost the same, but the current increased significantly because the load became more resistive. The active power rose to 104.45 kW, which is about 3.77 times higher than the power achieved with the mismatched capacitor. This result clearly demonstrates the importance of accurate inductance prediction for the design and control of electromagnetic induction energy storage systems.
5. Conclusion
In this thesis, I have systematically studied the inductive parameter computation and field-circuit coupled simulation of an electromagnetic induction energy storage system. The main conclusions are summarized below.
First, I established a complete magnetic property testing procedure for the Q235 steel used in the heating pipe. The measured hysteresis loops, loss curves, and resistivity data were validated through loss simulation with harmonic excitation. The results showed that considering the harmonics in the excitation waveform is essential to reduce the simulation error from more than 50% to below 15%.
Second, I proposed an analytical inductance model based on the equivalent magnetic network method. The model accounts for the nonlinear B–H characteristic of the ferromagnetic material and the frequency-dependent skin depth. By combining the nodal magnetic potential method with nonlinear iterative relaxation, the model converges reliably and predicts the heating pipe inductance within 10% error for frequencies from 1 kHz to 25 kHz. The experimental measurements confirm the accuracy of the method.
Third, I developed a field-circuit coupled simulation platform by integrating Magnet with Simulink. This approach incorporates the actual inverter control strategy into the electromagnetic field analysis. Compared with the traditional sinusoidal excitation finite element method, the joint simulation reduces the loss error from 14.7% to 7.3% and the inductance error from 23.9% to 5.8%. The simulated voltage and current waveforms closely match the experimental results obtained from a 25 kW prototype. Finally, using the calculated inductance to match the resonant capacitor significantly improves the output power, demonstrating the practical value of the proposed modeling approach for optimizing electromagnetic induction energy storage systems.