Load Estimation for Induction Heating Cookers Based on Series RLC Natural Resonant Current

Abstract

In domestic induction heating applications, cookware can be considered an equivalent load in a series resistor–inductor–capacitor resonant converter. Therefore, the electrical parameters of an equivalent circuit change according to the cookware material, size and the cookware position on the heating coil. This study proposes an online estimation method for detecting the cookware status, determining the material and estimating the equivalent heating resistance of cookware on an induction heating cooker (IHC) for power control. The proposed method could turn off the circuit in abnormal situations such as low equivalent heating coverage rate or non-ferromagnetic cookware and adjust the power in normal situations. In the method, a half-bridge series resonant converter (HBSRC) generates two test patterns with three resonant voltage pulses to detect cookware every 10 ms, only the current feedback information is needed to avoid the calculation loads and times necessary for complex signal operations in software. To verify the proposed method, a digital signal processor based HBSRC with 1000 W was constructed. The maximum errors between the estimated and measured resistance and inductance were 7.14% and 2.91%, respectively. Moreover, power control in emulated user operation reveal that the proposed method and control system can effectively estimate load online to detect cookware status and determine whether to turn off or vary the heating power for an IHC.

1. Introduction

IHCs do not use an open flame, thus, they can replace gas cooktops in tall buildings and apartment complexes in earthquake-prone countries [1]. Moreover, IHCs heat both more quickly and more efficiently than conventional gas stoves do; their efficiency can exceed 80%, higher than that of gas stoves, which is only approximately 40% [2,3,4]. Therefore, IHCs have become popular for both domestic and commercial cooking [5,6]. In an IHC, a high-frequency magnetic flux is generated by a heating coil. This coil is driven by a half-bridge series resonant converter HBSRC as shown in Figure 1, to induce eddy currents on a thin surface of the cookware bottom, resulting in heating [7]. An HBSRC comprises a half-bridge circuit with two insulated gate bipolar transistors (IGBTs), a resonant capacitor, a heating coil, and the cookware on the heating coil. Therefore, an HBSRC can be modeled as a series resistor–inductor–capacitor (RLC) circuit, and the equivalent electrical parameters [8,9], including resistance R_eq and inductance L_eq, could be affected by the cookware material, operating frequency, and equivalent heating coverage rate (EHCR), increasing the complexity of controlling the heating power [10]. Metal cookware can be classified as ferromagnetic and non-ferromagnetic; non-ferromagnetic cookware has a small R_eq, resulting in a large induced current [11] and allows the control to be more difficult. Although the R_eq of ferromagnetic cookware is relatively large, R_eq and L_eq can still be affected by the excitation current due to the current-dependent relative permeability of such cookware [12,13]. Moreover, the EHCR of IHC, which is determined by the geometrical alignment of the cookware and heating coil and by the bottom area of the cookware, affects R_eq and L_eq. Consequently, cookware-related parameters for an IHC cannot be obtained on a test bench in the factory; instead, R_eq and L_eq must be estimated accurately from online information about IHCs. Accordingly, modern high-end IHCs may have functions for estimating R_eq and L_eq for various pieces of cookware and their placement on the heating coil to achieve power control.

R_eq and L_eq can be expressed in an analytical model [14,15,16,17]. One of these studies [14] proposed a mutual-inductance model of the IHC and derived R_eq by eddy current field. Another of these studies [15], R_eq and L_eq are calculated by eddy current loss and magnetic energy, respectively, and the impedance of IHC was obtained based on a linear interpolation between two impedances with and without considering the ferrite core of heating coil. The R_eq and L_eq were derived from the concept of electromagnetic field [16,17] and the coupling effect between coils also considered in the study [15]. Nevertheless, the correctness of derived formulas [14,16,17] was only verified by LCR meter without considering magnetic saturation on the ferromagnetic cookware. Studies and comparison on the load estimation methods of the IHC are shown in

Table 1. Studies and comparison on the load estimation methods of the IHC. Note: Half bridge (HB), quasi-resonant (QR).

Refs.	Estimation Method of Load Impedance	Feedback	Circuit Topology	Online Estimation	Computation Loading	Accuracy
[18]	Discrete-Time Fourier Series coefficients from voltage and current waveform.	Voltage and current	HB	╳	High	High
[19]	Resonant voltage and quality factor (Q).	Voltage	HB	╳	High	High
[20]	1. VCE under constant current control. 2. IC under constant voltage control. 3. comparing times at which VCE is greater than the DC link voltage. 4. measuring the diode turn-on time.	Voltage or current	QR	○	low	N/A
[21]	Voltage, current waveform, and related period under different switching intervals	Voltage and current	QR	○	Middle	Low
[22]	Power factor and the absolute amplitude of the impedance.	Voltage and current	QR	○	Low	Middle
[24]	A phase-sensitive detector (PSD) is used to decouple the feedback voltage and current.	Voltage and current	HB	○	High	High
[25]	Using first-order delta–sigma algorithm to digitize the feedback signals of the resonant voltage and current and then calculated the load impedance by discrete Fourier transform.	Voltage and current	HB	○	Middle	Middle
[26]	Particle swarm optimization.	Voltage and current	HB	○	High	High
[27]	Measured the resonant capacitor voltage and voltage harmonics.	Voltage and current	HB	○	Middle	High
[28]	Deep learning from experimental data.	Power, current and Q	HB	○	High	High
[29]	Similar to [24].	Voltage and current	HB	○	High	High
[30]	Combination with PSD and deep learning.	Voltage and current	HB	○	High	High
Proposed method	Key points of current in natural resonant period.	Current	HB	○	Low	High

. R_eq and L_eq can be estimated either offline or online. In offline calculation methods, R_eq and L_eq can be calculated according to the fundamental frequency impedance [18] or resonant capacitor voltage and quality factor [19]. Several studies have focused on developing online estimation methods for reducing the effects of variations in R_eq and L_eq on power control due to cookware movement or removal from the heating coil during heating. For example, studies [20,21,22] have proposed online estimation methods for quasi-resonant converters. Specifically, one of these studies [20] proposed four estimation methods for measuring the V_CE of an IGBT under constant current control, measuring the turn-on current I_C of the IGBT under constant voltage control, comparing times at which V_CE is greater than the DC link voltage, and measuring the diode turn-on time. Another of these studies [21] used the voltage, current waveform, and related period under different switching intervals to calculate equivalent parameters. [22] detected equivalent parameters by using the power factor and the absolute amplitude of the impedance |Z|. As IHCs based on quasi-resonant converters have the shortage of high-voltage stress on the power transistors [23], many studies [24,25,26,27,28,29,30] have focused on estimation methods for HBSRCs; specifically, these studies have proposed estimation methods based on single values or the average value of feedback electrical signals within a bus voltage cycle [24,25,26,27]. One of these studies [24] used a phase-sensitive detector, the inputs of which are multiplied by sine and cosine signals of the feedback voltage and current, respectively, and calculated R_eq and L_eq by using a low-pass filter. Another of these studies [25] adopted a first-order delta–sigma algorithm to digitize the feedback signals of the resonant voltage and current and then used the discrete Fourier transform (DFT) to calculate R_eq and L_eq. In addition, another of these studies [26] adopted particle swarm optimization to obtain the average equivalent resistance. Finally, [27] measured the resonant capacitor voltage and voltage harmonics for estimation. A study also determined the EHCR of cookware by using deep learning [28]. Two studies [29,30] have considered the influence of bus voltage variations on R_eq and L_eq estimations. Furthermore, another study [31] executed power control by changing the operating frequency; the study used the DFT to calculate the equivalent impedance from the resonant voltage and current in an area with a relatively high DC-link voltage in order to achieve favorable power control. Most of the aforementioned online estimation methods require first collecting substantial voltage and current data and then using a field-programmable gate array to perform complex calculations. The calculation capacity of cost-effective microprocessors is limited. Accordingly, this study proposes a method that requires sampling only several specified current values at periodic times during the natural resonant period to perform online R_eq, and L_eq estimation for cookware in order to execute power control.

This paper presents a digital signal processor (DSP)-based digitally controlled IHC system with the proposed online load estimation method for detecting whether the cookware has normal or abnormal status and for measuring the R_eq to achieve continuous power control. Moreover, the proposed online load estimation method requires only information about the resonant current i_r, namely the current at the high-side IGBT turn-off transient, the time of the first negative half-cycle, and the peak current during the natural resonant period in an HBSRC, only the resonant current information is needed for the proposed method to reduce the calculation loads and times for complex signal operations in software and achieve accurate estimation results. This study derived formulas for the estimated equivalent resistance R_est and equivalent inductance L_eq and verified them by using the simulation software Simulink. To verify the proposed online load estimation method, a measurement method based on two fundamental components, namely the resonant voltage v_r and resonant current i_r, was used to calculate R_eq and L_eq for comparison. The fundamental frequency components v_r and i_r were obtained using bandpass filters in a dual-channel programmable filter equipment (NF 3627) to reach high accuracy. Through the proposed method, R_est and the amplitude of i_r were estimated as power feedback in a closed loop for power control. R_eq and L_eq estimation for normal cookware including that for abnormal cookware detection, and power control were proposed to achieve more accurate heating power control through online load estimation for IHCs. Finally, to verify the proposed online load estimation method, a DSP-based IHC with a rated power of 1000 W was constructed with the specification shown in

Table 2. Electrical specifications of the proposed IHC.

Parameter	Value
Rated input DC voltage (V)	150
Rated output power (W)	1000
Switching frequency fs (kHz)	20
Resonant capacitor (nF)	970
Intermittent estimation period (ms)	10

and examined using both ferromagnetic and non-ferromagnetic cookware at different EHCRs. The maximum error between the estimated R_est and measured R_eq was only 3.55% for ferromagnetic cookware, and the experimental results regarding power control indicated that the proposed method can effectively estimate both R_eq and L_eq online to detect cookware status on an IHC and control heating power.

2. Characteristics of Cookware on an IHC

As cookware materials are excessively complex for modeling or analysis with simulation software, study [32] proposed the impedance measured method by the voltage and current waveforms across the inductor which consists of R_eq and L_eq. However, when the R_eq is much smaller than the reactance of L_eq, the phase between measured voltage and current is closed to 90° to increase the measured errors of R_eq. This study proposes a measurement method for L_eq and R_eq of cookware on an IHC shown in Figure 2. The proposed method uses the fundamental frequency components of resonant voltage v_r and current i_r to calculate L_eq and R_eq; the L_eq and R_eq formulas can be expressed as follows:

$$L_{eq}=\frac{\frac{V_{rp1}}{I_{rp1}}\sin ({\theta }_{v1}-{\theta }_{i1})+\frac{1}{2\pi f_s{C}_r}}{2\pi f_s}$$

(1)

$$R_{eq}=\frac{V_{rp1}}{I_{rp1}}\cos ({\theta }_{v1}-{\theta }_{i1})$$

(2)

$$v_{r1}(t)=V_{rp1}\sin (2\pi f_{s}t+{\theta }_{v1})$$

(3)

$$i_{r1}(t)=I_{rp1}\sin (2\pi f_{s}t+{\theta }_{i1})$$

(4)

where the variables are defined as follows:

• v_r1 and i_r1: resonant voltage v_r and current i_r, respectively, representing the fundamental frequency components.
• V_rp1 and I_rp1: amplitude of v_r1 and i_r1, respectively.
• θ_v1 and θ_i1: phase angle of v_r1 and i_r1, respectively.

To measure v_r1 and i_r1, this study used bandpass filters with a center frequency at f_s in an NF 3627 system, and the HBSRC was operated at f_s with a 50% duty cycle. As shown in

Table 3. Tested cookware.

Cookware Type	A	B
Photo
Bottom side diameter	180 mm	180 mm
Material	Ferromagnetic	Non-ferromagnetic (copper)

, two types of cookware (ferromagnetic and non-ferromagnetic) were used for testing and to calculate R_eq and L_eq at different operating currents, frequencies, and EHCRs. Figure 3 shows the EHCR as defined in the test. The influence of operating current, frequency, and EHCR on R_eq and L_eq could thus be measured. Moreover, the measured R_eq and L_eq values of the heating coil without cookware were 0.15 Ω and 77.9 μH, respectively.

2.1. Measurement at Different Operating Currents

Figure 4 shows the test results obtained by varying the peak current I_rp1 from 1 to 25 A by adjusting V_in at a fixed f_s value of 20 kHz and 100% EHCR. Non-ferromagnetic cookware does not have hysteresis losses and have a relative permeability level of approximately 1. Accordingly, the R_eq and L_eq values of the non-ferromagnetic cookware were not affected by the applied I_rp1, and the measured R_eq values were considerably smaller than the ferromagnetic cookware.

2.2. Measurement at Different Operating Frequencies

R_eq and L_eq were also measured at operating frequencies f_s ranging from 15 to 30 kHz at an I_rp1 value of 20 A and 100% EHCR; the results are shown in Figure 5. Ferromagnetic cookware has a higher coupling magnetic flux than non-ferromagnetic cookware does. Hence, the R_eq values of the ferromagnetic cookware clearly increased with the operating frequency. However, the magnetic flux caused by increasing eddy currents at the bottom of the ferromagnetic cookware counteracted the magnetic flux generated by the heating coil, possibly reducing L_eq.

2.3. Measurement at Different EHCRs

R_eq and L_eq were measured at EHCRs ranging from 0% to 100% at an I_rp1 value of 20 A and f_s of 20 kHz, and the results are shown in Figure 6. The results revealed that the eddy currents and hysteresis losses in the ferromagnetic cookware clearly increased with the EHCR. Although the coupling magnetic flux may increase with the EHCR, the relative permeability of the ferromagnetic cookware was reduced due to magnetic flux saturation at high EHCRs. Thus, variations in L_eq were comparatively subtle. By contrast, the relative permeability of the non-ferromagnetic cookware was near unity; thus, L_eq decreased as the EHCR increased. Moreover, the load characteristic of non-metal and non-ferromagnetic cookware (e.g., glass) is similar to no load (EHCR = 0%).

On the basis of the R_eq and L_eq values measured for the ferromagnetic and non-ferromagnetic cookware by using the proposed method, the following conclusions could be drawn:

• The non-ferromagnetic cookware had substantially lower R_eq and L_eq values; thus, these results can be used as an index to determine the cookware material. As this study focused on R_eq and L_eq estimation for ferromagnetic cookware, the HBSRC can be turned off if the cookware is non-ferromagnetic;
• For the ferromagnetic cookware, L_eq variations were small as I_rp1, operating frequency, and EHCR increased;
• The R_eq values measured for the ferromagnetic cookware changed substantially as the EHCR increased. Therefore, R_eq can be used to detect when the ferromagnetic cookware is moved during heating. For an IHC, I_rp1 can be adjusted to maintain constant power or to turn off the power when the EHCR is low. The IHC used in this study turned off the power when the EHCR was less than 50%.

3. Proposed Load Estimation Method and Power Control System

Figure 7 shows the equivalent circuits of an HBSRC in different operating periods, and Figure 8 displays the waveforms of the resonant voltage v_r and current i_r of a series RLC circuit. The circuit operation is outlined as follows: First, the DC link voltage V_in is sent to the series RLC circuit when the IGBTs Q_h and Q_l are in on and off states, respectively. Subsequently, the IGBTs Q_h and Q_l enter the off and on states, respectively, and the series RLC circuit is operated at its natural resonant period.

3.1. Estimation Method for Equivalent Impedance

If the dead-time effect is neglected, the resonant current i_r during the natural resonant period can be derived as follows [20,21,33,34,35]:

$$i_r(t)=e^{-\alpha t}[B_1\cos {\omega }_{d}t+B_2\sin {\omega }_{d}t]$$

(5)

The initial conditions of i_r(0) and its derivative $\frac{d}{dt}{i}_r(0)$ during natural resonant period can be expressed as follows:

$$i_r(0)=I_1 , \frac{d{i}_r(0)}{dt}=\frac{-1}{L_{est}}(R_{est}{I}_1+V_{C0})$$

(6)

wherein,

• ${\omega }_d(=\sqrt{{\omega }_o{}^2-{\alpha }^2})$: damping resonant angular frequency.
• ${\omega }_o(=\frac{1}{\sqrt{L_{est}{C}_r}})$: natural resonant angular frequency.
• $\alpha (=\frac{R_{est}}{2L_{est}})$: damping coefficient.
• B₁, B₂: coefficients;
• C_r: resonant capacitor;
• L_est: estimated equivalent inductance;
• R_est: estimated equivalent resistance;
• V_C0: initial value of resonant capacitor voltage at the beginning of natural resonant period;
• I₁: initial value of the resonant current i_r at the beginning of natural resonant period.

Substituting Equation (6) into Equation (5) yields the following:

$$i_r(t)=e^{-\alpha t}[I_1\cos {\omega }_{d}t-\frac{R_{est}{I}_1+2V_{c0}}{2L_{est}{\omega }_d}\sin {\omega }_{d}t]$$

(7)

As ω_o² is much larger than α² in IHC applications, ω_d can be replaced by ω_o. Therefore, Equation (7) can be simplified and expressed as follows:

$$i_r(t)=I_P{e}^{-\alpha t}\sin ({\omega }_{o}t+\theta )$$

(8)

where

$$I_p=\sqrt{I_1{}^2+{(\frac{R_{est}{I}_1+2V_{C0}}{2L_{est}{\omega }_o})}^2}$$

(9)

$$\theta ={\tan }^{-1}(\frac{2L_{est}{\omega }_o{I}_1}{R_{est}{I}_1+2V_{C0}})$$

(10)

If the negative half-period time is T/2 and the first zero-crossing time ∆t, I₁ and the peak resonant current in the negative period I_np can be measured, R_est and L_est can be derived as follows:

$$R_{est}=\frac{2L_{est}}{\Delta t+\frac{T}{4}}\ln (\frac{-I_1}{I_{np}}\frac{1}{\sin (2\pi \frac{\Delta t}{T})})$$

(11)

$$L_{est}={(\frac{1}{2\pi f_o\sqrt{C_r}})}^2$$

(12)

3.2. Simulation Results of the Proposed Estimation Method

To verify the effectiveness of the proposed load estimation method, several simulations of cookware on a heating coil were performed under four operating, which involved varying operating frequencies, duty cycles, equivalent impedances, and equivalent resistances. T/2, ∆t, I₁ and I_np could be measured from the simulated waveforms shown in Figure 9, and R_est and L_est could be derived using Equations (11) and (12). The estimation results derived under the four conditions are listed in

Table 4. Simulated results of the estimated impedance derived under different conditions.

	Condition	1	2	3	4
Preset values	DC link voltage Vin (V)	150
	Frequency (kHz)	20	20	20	40
	Duty (%)	10	50	10	50
	Leq (μH)	80	80	30	80
	Req (Ω)	3	3	1	3
Simulated results	I1 (A)	11.8	16.1	13.3	10.5
	Inp (A)	−7.3	−26.1	−13.0	−11.0
	Δt (μs)	18	4.1	5.2	6.5
	T/2 (μs)	28.0	28.0	17.0	28.0
Estimated results	Lest (μH)	81.9	81.9	30.2	81.9
	Rest (Ω)	3.0	3.0	1.0	2.9
Error	Equivalent inductance EOI (%)	2.4	2.4	0.7	2.4
	Equivalent resistance EOR (%)	0.0	0.0	0.0	3.3

where, $\mathrm{EOI} (\%)=\left|\frac{L_{est}-L_{eq}}{L_{eq}}\right|\times 100\%$, $\mathrm{EOR} (\%)=\left|\frac{R_{est}-R_{eq}}{R_{eq}}\right|\times 100\%$

. The maximum errors between the estimated and preset inductances (error of inductance, EOI) and resistances (error of resistance, EOR) were 2.4% and 3.3%, respectively, indicating the effectiveness of the proposed load estimation method.

3.3. Cookware Estimation and Power Control Procedure

The proposed control system, comprising a power control module and a load estimation module, is shown in Figure 10. The HBSRC generates two test patterns with three resonant voltage pulses to detect cookware every 10 ms; each cycle includes a 1 ms estimation time and 9 ms power control time. The power control module uses the fundamental frequency power P₁, which can be expressed as follows:

$$P_1=\frac{1}{2}{I}_{rp1}^2{R}_{est}$$

(13)

I_rp1 is generated by passing the i_r through a bandpass filter at a center frequency of f_s and is measured with a peak detector circuit (PDC). R_est is obtained using Equation (11). The power regulator comprises a PI-type regulator, which generates a duty command for the HBSRC with fixed f_s to achieve the required heating power.

To estimate R_eq and L_eq in this study, hardware circuits for the bandpass filter, PDC, and zero cross detector (ZCD) were adopted to avoid the calculation loads and times necessary for complex signal operations in software. Thus, the necessary signals for the proposed estimation method in Equations (11) and (12) could be easily obtained. Some details of the proposed load estimation module are presented as follows:

• I_np can be obtained from the negative peak value of i_r by using the PDC;
• T/2 can be obtained using an embedded high-speed counter, and the capture function of the DSP can be used to calculate the time at which two adjacent zero-crossing signals of i_r are output by the ZCD;
• ∆t can be calculated using the time difference between the falling edge signal G_h and the first signal generated by the ZCD when the first zero-crossing signal of i_r is detected during the natural resonant period. Moreover, the time difference can be obtained using the same mechanism as for T/2;
• I₁ can be captured using the embedded analog-to-digital converter in the DSP; the converter is triggered by the falling edge signal G_h;
• If R_est and L_est indicate that the cookware is both ferromagnetic and on the heating coil, the cookware is heated. Otherwise, the HBSRC is turned off.

Figure 11 shows a flowchart of the proposed control system, which is synchronous with the operating frequency (20 kHz). That is, an interrupt (INT) is generated at each 50 μs period to trigger R_est and L_est estimation and power control. The two flags ENest and ENpower in the control flowchart indicate these operations, and the functions of the key blocks are as follows:

• The variable N is a counter of the INTs to generate a period of 10 ms for R_est and L_est estimation;
• An ENest value of 1 indicates live execution of R_est and L_est estimation;
• An ENest value of 0 and ENpower value of 0 indicate that the cookware is non-ferromagnetic or is not on the heating coil, prompting the heating power to be turned off;
• An ENest value of 0 and ENpower value of 1 indicates that the ferromagnetic cookware is being heated under constant power control using R_est.

To demonstrate an example of the control system, Figure 12 shows a control pattern for v_r under normal and abnormal conditions. In this example, three continuous v_r pulses, with 10% duty and spaced at 50 μs, are first adopted to determine whether the cookware status is normal or abnormal. The duty is set to 10% to prevent the non-ferromagnetic cookware or low EHCR from inducing a larger resonant current that may destroy IGBTs. If the cookware status is determined be normal, three continuous v_r pulses, with 50% duty spaced at 50 μs, are used to derive L_est and R_est. Finally, the heating power control system is implemented according to the heating power, as indicated in Equation (13). The control process for normal cookware is presented in Figure 12a. If the cookware status is determined be abnormal, the heating power control is turned off, as shown in Figure 12b.

4. Experimental Results

The test setup and platform comprised an HBSRC, cookware, a digital scope (R&S RTO1014, Munich, Germany), a recorder (Yokogawa DL850, Tokyo, Japan), and the proposed DSP-based IHC, as shown in Figure 13. The proposed load measurement method shown in Figure 2 was verified by implementing it to measure R_eq and L_eq simultaneously.

4.1. R_eq and L_eq Verification

Several test conditions shown in

Table 5. Test conditions involving different EHCR and cookware materials for various cases.

Test Conditions	Normal Situation		Abnormal Situation
	Case 1	Case 2	Case 3	Case 4
Cookware material	Ferromagnetic	Ferromagnetic	Without cookware	Non-ferromagnetic
Frequency (kHz)	20	20	20	20
EHCR (%)	100	50	0	100
Duty (%)	50	50	10	10

including different EHCRs and cookware materials for various cases were used to verify the effectiveness of the proposed load estimation method. Cases 3 and 4 represented an abnormal condition under which the controller should turn off the heating power. As a low R_eq of the heating coil due to the absence of cookware or non-ferromagnetic cookware may induce a large resonant current i_r, the test duty for Cases 3 and 4 was limited to 10% to prevent a large i_r from damaging the HBSRC IGBTs.

Figure 14 presents the v_r1 and i_r1 results generated by passing v_r and i_r through the NF 3627 bandpass filter shown in Figure 2; the generated results were sufficient for calculating L_eq and R_eq using Equations (1) and (2), respectively.

Figure 15 presents the load estimation results; R_est and L_est were calculated online through Equations (11) and (12) by using values obtained from the digital-to-analog converter in the DSP-based controller. If the proposed estimation method detects that L_est is excessively small, it assumes that the cookware is non-ferromagnetic and does not calculate R_est shown in Figure 15d. The measurement and online estimation results for load impedance are summarized in

Table 6. Measured and estimated results for the test cases in Table 5.

Experimental Method	Experimental Results	Normal Situation		Abnormal Situation
		Case 1	Case 2	Case 3	Case 4
Measurement	Leq (μH)	78.8	83.4	77.9	35.9
	Req (Ω)	3.38	1.66	0.14	0.23
Estimation	Lest (μH)	81.1	82.1	78.1	34.9
	Rest (Ω)	3.26	1.69	0.15	N/A
Error	EOI (%)	2.91	1.56	0.26	2.78
	EOR (%)	3.55	1.81	7.14	N/A

. The maximum errors between the measured and estimated inductance and resistance values for the normal situation were 2.91% and 3.55%, respectively, signifying the effectiveness of the proposed load estimation method. Moreover, the required time for online calculations was less than 100 μs after the falling edge of the last tested pulse-width modulation cycle.

4.2. Varying EHCRs during Heating

To test the ability of the proposed method to detect whether the ferromagnetic cookware is placed on the IHC, the tested cookware was moved to the center of the heating coil and then gradually moved away to change the EHCR during the test. As shown in Figure 16, the calculated R_est and L_est changed in accordance with the cookware position. NF was set to 1 if R_est was higher than 1.7 Ω, indicating that the EHCR was nearly 50% and the cookware was ferromagnetic.

4.3. Estimation Pattern in Abnormal and Normal Situation

Figure 17 shows the controlled resonant voltage and current for different cookware situations, including non-ferromagnetic cookware, ferromagnetic cookware with an EHCR of 0%, and ferromagnetic cookware with an EHCR of 100%. On the basis of the calculations (conducted at 10 ms intervals) of the proposed method, heating occurred in only the normal situation that involved ferromagnetic cookware with an EHCR of 100%.

4.4. Constant Power Control

Figure 18 and Figure 19 illustrate the experimental results for power control at 500 and 1000 W, respectively. The step response time of the power control system was less than 2 ms, and the steady-state command tracking error was approximately 0, indicating that the proposed load estimation method and power control system exhibited excellent performance.

Figure 20 shows the experimental results for power control for emulated user operation. The starting power was set to 1000 W (i.e., 100% rated output), followed by 750 and 500 W in succession. The following performance results were obtained for the proposed control system:

• Favorable constant power control was achieved by adjusting the duty value despite the EHCR changing between 100% and 70%. In Figure 20, the green block represents the waveform for P₁ derived after R_est estimations at 10 ms intervals. The peak envelope power P₁ was close to the desired power P₁*;
• According to the NF signal, the power was turned off or on when R_est was lower or higher, respectively, than 1.7 Ω.

5. Conclusions

This study proposes a DSP-based digitally controlled IHC system with an online load estimation method for detecting normal or abnormal cookware status and for obtaining the equivalent heating resistance R_eq to achieve constant power control. The online load estimation method requires only an i_r feedback signal and simple calculations; the heating power control system is based on duty modulation at a fixed operating frequency. An IHC with a rated power of 1000 W was implemented to verify the proposed online load estimation method by using both ferromagnetic and non-ferromagnetic cookware at different EHCRs; the estimation results were compared with measured results. The key contributions of this study are outlined as follows:

1.

The study presents an online R_eq and L_eq estimation method for cookware. The proposed method requires only information about the resonant current i_r, namely the current at the high-side IGBT turn-off transient, the time in the first negative half-cycle, and the peak current during the natural resonant period in an HBSRC. To verify the proposed load estimation method, a measurement method based on the resonant voltage v_r and current i_r, representing fundamental frequency components, was also used to calculate R_eq and L_eq. The maximum error between the estimated R_est and measured R_eq for ferromagnetic cookware in normal situations was only 3.55%; thus, the proposed method can effectively and accurately control heating power. Moreover, the maximum errors between the estimated and measured resistance and inductance for cookware in abnormal situations were 7.14% and 2.78%, respectively. Thus, the method can effectively detect when no cookware is present on the heating coil or if non-ferromagnetic cookware is used and can turn off the heating power.

2.

The proposed power control system is based on fundamental power $P_1(=\frac{1}{2}{I}_{rp1}^2{R}_{est}),$ which can be calculated using R_est and the amplitude of the resonant current I_rp1. Measurements revealed that the control system had a step response time of only 2 ms from no load to 1000 W and had accurate control in the steady state.

3.

The experimental results for power control in emulated user operation reveal that the proposed method and control system in the IHC system can effectively estimate R_eq and L_eq online to detect cookware situations and determine whether to turn off or vary the heating power.