Electr Eng (2009) 91:153–160 DOI 10.1007/s00202-009-0127-9

ORIGINAL PAPER

Improving the energy efficiency of induction cooking László Koller · Balázs Novák

Received: 19 January 2009 / Accepted: 18 September 2009 / Published online: 6 October 2009 © Springer-Verlag 2009

Abstract The wide use of induction cooking technology in households is impeded by the price of the hobs, which still has not dropped significantly in spite of the mass production and the availability of high-power (several kW) units. The effective power with conventional technology could be increased only by raising the exciting current flowing in the inductor coil. This means increased loss in the coil, together with lower efficiency caused by the non-linear magnetic behavior of the pot. Using a special type of iron pot, significantly more effective power can be obtained than with conventional ones without increasing the exciting current, improving the energy efficiency of the technology. Calculations were carried out on the 2D finite element model of a given induction hob used with a conventional and a new type of pot. A disadvantage of the special pot is also presented: with conventional hobs the higher efficiency cannot be exploited. Keywords Induction cooking · Novel pot type · Finite element calculations

1 Introduction Induction hobs are the result of a long development process [1–5]. Their general structure is shown in Fig. 1: a stranded (litze) wire inductor coil (2) with ri inner radius and wi width is placed under a (1) glass–ceramic plate. Below this coil a L. Koller · B. Novák (B) Department of Electric Power Engineering, Budapest University of Technology and Economics, Egry József u. 18, 1111 Budapest, Hungary e-mail: [email protected] L. Koller e-mail: [email protected]

segmented ferrite flux guide (3) is located. A resonant inverter (frequency converter power supply unit) supplies the coil with a current of f ≈ 25 kHz frequency. This frequency can slightly vary during operation, mostly with the change of the unit’s power. A pot (4) with r radius and h height is placed on the glass–ceramic surface. The alternating magnetic field produced by the inductor is fully absorbed by the v f thick, ferromagnetic bottom of the pot. The effective power P2 generated in the pot is the sum of the Joule-heat P j and the hysteresis Ph . The thickness of the insulator—namely the glass ceramic plate—between the inductor coil and the bottom of the pot is vk . Besides the improved dynamics of cooking, a power increase in the pot (P2 ) would reduce both the manufacturing and the operational costs of the hobs. With more power the food’s temperature rises faster making the boil-up process shorter, thus reducing both the heat transfer to the environment, and the energy consumption. This can be represented by increasing thermal efficiency ηterm . Pots of the same size but different type have almost the same effective power P2 when placed on the same hob operating at the same current [6]. This effective power influences the electrical efficiency ηIB of the inductor-load system as well, as ηIB =

P2 P2 = , Pin P2 + P1

(1)

where Pin is the active input power measured at the terminals of the inductor coil, and P1 is the power loss of the inductor, mostly the Joule-heat loss of the coil. The resultant electrical efficiency η of the cooking depends also on the inverter’s (ηinv ); therefore, η = ηIB · ηIinv . It would be possible to raise the effective power by increasing the frequency. However, this would reduce the efficiency

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Fig. 1 Induction hob structure with a conventional pot

of the inverter; thus the resultant efficiency would not be improved. Therefore, the power in a conventional pot at a given temperature and at a given exciting frequency can be increased only by increasing the exciting current I1 flowing in the inductor coil comprising N turns. Because of the ferromagnetic behavior of the pot, the power growth rate is decreasing at higher excitations, namely it is not directly proportional with (N · I1 )2 . Besides, the effective power is limited to a maximum by the temperature allowed for the inductor coil and the flux guide, even if a cooling system is used. Although this means that the inductor-load system’s ηIB electrical efficiency is smaller at higher P2 power induced by higher N · I1 excitation, the ηterm thermal efficiency grows more rapidly providing an improved resultant efficiency. The user experiences this improvement as a result of the two effects, but he might not realize that the operational costs could have been further reduced. Because of the significant excitation required, the price of the inductor and the inverter and so of the hob remains high. Consequently, the effective power in conventional pots is limited and can be raised only by using relatively more expensive hobs with less η electrical efficiency. To extend the limit of the maximum power and to improve the η efficiency, we developed a novel type of ferromagnetic pot. The P2 effective power generated in this pot at a particular exciting current is remarkably higher than in a conventional one. Therefore, using this pot will reduce the hob price, raise the maximum power, and save energy. We completed calculations on 2D finite element (FE) models of a given cooktops used with conventional and new pots. The ANSYS FE tool, used for the simulations, provides the ability of solving models with non-linear magnetization curve. During a time-harmonic analysis the software replaces the DC B-H curve with an effective B-H curve, pursuing energy equivalence [7,8]. The magnetization curve of the carbon–steel is shown in Fig. 2a. The ξd penetration, representing the total wave absorption in the steel with ρ =

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2.7 · 10−7 resistivity at ϑ = 180◦ C, was determined at f = 25 kHz frequency. Although analytical methods [9,10] exist as well, more accurate 1D FE models were used to calculate the current density distribution J . This, normalized to its surface value J (0), is shown in Fig. 2b at different exciting fields (H ) in the direction of wave propagation (x). The values of ξd can be read from these diagrams. It can also be seen that in the range of strong magnetic fields (above Hb belonging to µrmax ) higher field strength, namely lower µr relative permeability causes deeper penetration in agreement with [11]. Besides the eddy current power P j , the hysteresis losses contribute to the heat generation in the ferromagnetic carbon–steel pot as well. We determined this component by using the ph = 1,460 · f · Bm1,6 [W/m3 ]

(2)

empirical formula [12] for calculating the ph hysteresis loss per unit volume. Here, Bm is the peak value of the magnetic flux density within the material. At strong fields, as the penetration, hence the affected volume is increased, the Ph is larger. Although at lower fields the penetration is smaller, lower H means higher Bm (Fig. 2a); consequently, the proportion of Ph to P j is larger in this range, as we will see in Sect. 3.1 Since neither Bm nor the penetration is constant within the pot, ph is also different in different volume elements. Nonetheless, since the magnetic flux density is known in all the tiny elements, the FE method can be used for calculating the hysteresis power of all individual elements by applying formula (2). The total hysteresis loss is the sum of the elements’ losses.

2 The new pot type An expedient option for the new type of pot can be seen in the draft in Fig. 3a. A radially slit, ferromagnetic disc with rb inner radius, wr width, and v thickness, is embedded

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Fig. 2 Magnetization curve of steel (a) and current distributions at different H magnetic fields at f = 25 kHz frequency (b)

Fig. 3 Structure of the new pot type (a), and eddy currents in the disc (b)

in the v f n thick bottom of the pot with rn outer radius. The pot’s height is h. The disc is separated from the bottom by an electrical insulator layer, the thickness of which is vi in the axial and vr in the radial direction. This insulator layer is very thin, merely some hundredths of millimeters, that can be accomplished, e.g., by sticking the disc in the bottom with heat-resisting enamel generally used for covering conventional pots. The basic idea of raising the effective power P2 was to increase the eddy current power P j . The principle of the slit disc is illustrated in Fig. 3b: because of the slit, circular eddy current paths cannot develop in the disc. However, since eddy currents cannot be eliminated by a slit, they find a different route for themselves. Reaching the slit, they flow through the cross-section, and continue their way on the upper side in the opposite direction. Reaching the slit again, they return to the bottom side closing the current path. The full path of a filament can be followed in the bottom of Fig. 3b. The net current at any cross section of the disc (except the slit) is zero, as the currents at both sides are equal, but their direction is opposite. Since current flows also at the upper side, the magnetic field appearing there is absorbed by induced eddy currents in the vned thick bottom of the pot. For better understanding, the disc can be modeled by a v thick infinite plate with a finite slit somewhere at infinity and the pot’s bottom by a semi-infinite conductor at a

vi insulation distance (Fig. 4a). The Joule-heat P j p of this arrangement can be determined and compared with the Jouleheat P j generated in a single semi-infinite conductor without a plate. In this 1D model the parameters of the electromagnetic field can vary only in the direction of the depth. As the net current in the plate is zero, the magnetic field strength between the plate and the semi-infinite conductor is equal with the exciting field, which means that the induced current in the semi-infinite part is the same as it was without the plate. Figure 4b shows the P j p /P j rate in the function of the exciting magnetic field H at different v thicknesses. If the plate is thick enough, namely total wave absorption can occur from both sides, the power generated in it is double that of the semi-infinite conductor. This means that P j p /P j = 300%. If the plate is thin, this ratio is different. For every thickness a maximum can be found at an optimal excitation. Over this excitation the power monotonically decreases and the rate converges to 100%. Using very thin (v < 0.1 mm) plate (disc) is not recommended, as, mostly at higher excitation, no considerable extra power can be gained. In reality the increase in eddy current power is less, as the magnetic field is neither constant at the bottom nor is parallel with it at every point. At low excitations, close to the boundary of the weak and strong magnetic fields (Fig. 2a), the ξd penetration decreases. This reduction in the active, current-conducting cross section can lead to such a high

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Fig. 4 1D model of the new pot type with eddy currents (a), and the rate of P j p /P j in the function of H at different v plate thicknesses (b)

Fig. 5 2D models of the inductor-load systems: a conventional, b new type of pot

resistance that the current induced by the magnetic field with perpendicular components is much less than at deeper penetrations. The smaller current generates lower field strength between the disc and the bottom, and consequently the effective power drops in both of them. In case of very low excitations this effect can cause even lower power generation in the new pot than in a conventional one. The comparison of the effective power obtained from 2D models and considering also the hysteresis will be discussed in the following section.

3 Comparison of the new with the conventional pot type The 2D axisymmetric models represent a typical arrangement of the inductor-load system, except the flux guide that could be modeled only as a solid disc instead of separated sections. The material and the main dimensions of the two pot types, namely their outer radius, their height, and the thickness of their bottom were the same: rn = r = 105 mm, h n = h = 80 mm v f n = v f = 2 mm. The models of the inductor-load system are shown in Fig. 5. The inner radius of the slit disc was r p = 30 mm and its width was w p = 60 mm. The thicknesses of the insulator layer were vi = 0.05 mm and

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vr = 0.1 mm in every case. During an optimization process the thickness v of the disc was varied. It was assumed that the pots were used with the same cooking zone of the same hob and that the exciting frequency was constant f = 25 kHz. The distance between the bottom and the inductor coil with ri = 22 mm inner radius and wi = 54 mm width was vk = 7.5 mm. Based on the calculation results we compared the amount of effective power generated in the pots, the power distribution in the bottom, the active and the reactive input power at the terminals of the inductor coil, the electrical efficiency, and the magnetic flux density outside the pot. This latter property was calculated along the Cnt contour line forming a cylinder in the 3D space with R K = r + W K radius. 3.1 Effective power Figure 6a–c shows in the function of the excitation N · I1 at v = 0.1, 0.2, 0.4, and 0.8 mm disc thicknesses the rates of the new pot’s P jn eddy-current power, Phn hysteresis power and P2n total effective power to the P j , Ph , and P2 power of the conventional pot, respectively. It can be noted that in the analyzed range the effective power rates monotonically increase if the disc is v = 0.4 or 0.8 mm thick. According

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Fig. 6 Rates of effective power in the function of N · I1 excitation at v = 0.1, 0.2, 0.4, and 0.8 mm thick discs: P jn. /P j eddy current (a), Phn /Ph hysteresis (b), and P2n /P2 total effective power (c)

means that the inductor-coil can be thinner and can be placed closer to the pot improving ηIB .

3.2 Active input power and electrical efficiency

Fig. 7 Variation of effective power in the function of N · I1 excitation at a v = 0.4 mm thick disc: P jn. and P j. eddy current, Phn and Ph hysteresis, P2n , and P2 total effective power

to the hysteresis power the 0.8 mm thickness seems to be slightly better, but this can be neglected as in the total power there is no noticeable difference between these two v values. Considering production and material costs we selected the v = 0.4 mm thick version for further analysis. For example, at N · I1 = 600 A excitation the following considerable results were obtained: P jn. /P j. = 148%, Phn /Ph = 203% and P2n /P2 = 155%. At low excitation (N ·I1 = 100 A) there is only a minor increase in the total power (P2n /P2 = 108%), which is a result of the even decreasing P jn. /P j. (89%) and the increasing Phn /Ph (134%). In practical applications such a low excitation hardly ever occurs, because most of the induction hobs operating at low power alternately turn on and off a medium power at given time intervals. Therefore, the range of very low excitations is only theoretically important. In Fig. 7 the new pot’s P jn. , Phn , P2n and similarly the conventional pot’s P j. , Ph and P2 power in the function of N · I1 can be seen. It can be noted, that at low excitations the eddy-current and hysteresis power are almost equal, but increasing excitation results in much higher eddy current power. For example at N · I1 = 600 A excitation P j /Ph = 7.29 for the conventional, and P jn /Phn = 5.318 for the new pot type. According to the diagram, the necessary reduction of the excitation to get the same power as with a conventional pot can be determined. For example, P2 = 3,273 W can be reached with a conventional one at N · I1 = 600 A, while with a new pot a N · I1 = 470 A excitation is enough. This

To calculate the electrical efficiency ηIB of the inductorload system, it is necessary to determine the active input power Pin measured at the terminals of the inductor coil. This input power is the sum of the P2 effective power and the P1 inductor loss that comprises of the Joule-heat loss of the ρ = 2.2727 · 10−8  resistivity coil and that of the flux guide (see formula 1). The hysteresis loss in the flux guide was neglected. These values are the results of the FE calculations. For the new pot type the ηIBn , P2n , Pinn and P1n notation were used. Figure 8a, b shows the variation of active input power (Pinn and Pin ) and the electrical efficiency (ηIBn and ηIB ) in the function of the N · I1 excitation for both the new and the conventional pots, respectively. The slight difference between the diagrams in Figs. 7 and 8a is on account of the inductor losses P1 . For example, at N · I1 = 600 A the Pinn /Pin = 150%, that is 5% less than the ratio P2n /P2 of the effective powers. The smaller growth of input power means better efficiency; however, the electrical efficiency is outstanding, barely less than 100% in both cases. The resultant efficiency η of the cooking is degraded by the substantially worse efficiency of the inverter (ηinv ). A remarkable difference can be noticed in Fig. 8b: While the conventional pot’s efficiency drops by 0.74% in the analyzed range at the highest excitation, the efficiency of the new pot drops only by 0.28% less than at the lowest current. Therefore, the power consumption of the new pot is further improved at high excitations, providing even more energy saving during a boil-up process.

3.3 Reactive input power The case of the conventional pot will be analyzed first. Besides the Pin active input power, the Q in reactive input power must also be considered, as it affects the resonant

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Fig. 8 Active input power Pinn and Pin (a) and electrical efficiency ηIBn and ηIB (b) of the inductor-load systems in the function of the excitation N · I1

inverter’s frequency. The capacitance C and Q c reactive power of the inverter’s capacitor is tuned to Q in as well. In case of a serial resonant circuit [2], assuming a given I1 coil current and f frequency—neglecting the losses of the capacitor—the following circuit parameters must be specified. The inductance: L=

Q in , 2 · π · f · I12

the capacitance: C=

I12 2 · π · f · Qc

(3)

and the equivalent resistance: R=

Pin . I12

Substituting these values into the formula (2 · π · f )2 =

R2 1 − L ·C 4 · L2

determining the resonant frequency of the circuit, the reactive power of the capacitor is Q c = Q in +

Pin2 . 4 · Q in

Substituting this into (3), the capacitance is C=

2· · K, π· f I12

(4)

where K =

Q in . 4 · Q 2in + Pin2

To generate f frequency at a given I1 coil current (in case of N turns at N · I1 excitation) the C capacitance—or if the C is constant, the f frequency—is proportional to the K [1/VA] factor. Similarly, in case of the new pot operating at f n frequency, the necessary Cn capacity—or at a given Cn the f n

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frequency—is proportional to a K n factor: Cn =

2 · I12 2 · I12 Q inn · Kn = · . 2 2 π · fn π · f n 4 · Q inn + Pinn

(5)

Dividing Eq. (5) with (4), after reorder the following relation can be obtained: f n Cn Kn · = . (6) f C K The variation of the reactive input power (Q inn and Q in ), the K n and K factors, and the K n /K ratio in the function of N · I1 excitation are shown in Fig. 9a–c, respectively. It can be seen from Fig. 9a that the reactive input power increases with increasing excitation, and its values are higher with the new pot than with the conventional one. The ratio of the two pot’s reactive power is higher than the ratio of the active power, e.g., at N · I1 = 600 A excitation Q inn /Q in = 185%. According to Fig. 9b the change of f frequency with changing excitation can be determined, if the inverter’s parameters are constant. The degree of frequency drop with increasing current is higher with both pot types at the lower section of the analyzed range. This can be a reason why induction hobs are operated only at medium and high power. Figure 9c provides relevant information about the new pot’s use. Regarding Eq. (6) it can be noted that – The f n = f = 25 kHz frequency can be kept only if the capacitance of the resonant circuit is reduced according to the Cn = C · KKn formula. Therefore, the capacitance adjusted to a conventional pot must be replaced with another one. In this case, if we consider an excitation of N · I1 = 600 A, the replacement capacitor must be Cn = 0.55 · C. – If the capacitance was not changed, namely the same inverter was used as with the conventional pot (Cn = C), the frequency of the resonant circuit would decrease according to the f n = f · KKn formula. For example, in our case at N · I1 = 400 A excitation it would drop to f n = 0.55 · f = 0.54 · 25 = 13.59 kHz, that could even

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Fig. 9 Reactive input power Q in and Q inn (a), the K n and K factors (b), and the K n /K ratio (c) of the inductor-load systems in the function of N · I1 excitation

Fig. 10 Effective power distribution ( p An and p A ) along the radius of the pot’s bottom at P2 = 2,138 W (a). Largest values of outer magnetic flux max and B max ) density (Bcnt−n cnt calculated along a Cnt contour line, in the function of effective power P2 (b)

be heard by the human ear. At this modified frequency the effective power would also decrease and the original P2n /P2 = 143% ratio would fall to P2n /P2 = 81%. This means that instead of improving energy consumption, the new pot would even degrade the efficiency.

3.4 Distribution of effective power and outer magnetic flux density The effective power generated per unit surface area ( p An and p A ) in the bottom along the r radius of the new and the conventional pot types at a particular P2 = 2,138 W power is shown in Fig. 10a. The power density distribution is significantly different for the two pots. In case of the conventional one the middle part is much more heated, and the maximum can be located around the rm = 52.5 mm half radius. In the new pot the maximum is at the inner radius (rb = 30 mm) of the disc, and there is a local, but much smaller maximum at its outer radius (rk = 90 mm). Therefore, the characteristic of the heat distribution can be changed by modifying the disc’s parameters. It is important to know the outer magnetic flux density produced by the inductor-load system, in order to limit the exposure of people to the magnetic field. Regulations [13] give the Blim = 6.25 µT limit at f = 25kHz frequency, which is an average value for the whole exposed human body. It can

be ascertained that during induction cooking only a small section of the body is exposed to higher fields, occurring at a height between the pot’s bottom and the inductor coil. The max and B max ) along largest values of magnetic induction (Bcnt−n cnt a Cnt contour line at Wk = 90mm from the outer edge of the pot in the function of the P2 effective power are presented in Fig. 10b. It can be noted that at equal power this magnetic induction is significantly higher with the new pot. For exammax /B max = 170%. However, further ple, at P2 = 3 kW, BCnt−n Cnt from the inductor-load system the field decreases rapidly. For example, with the new pot at P2 = 3,000 W power it is only max = 4.15 µT calculated at W K = 300 mm, which is BCnt−n smaller than the limit above.

4 Conclusions With a relatively simple solution—by embedding a ferromagnetic disc slit in the cross section into the bottom of a pot—the efficiency of the induction cooking can be significantly improved. According to our calculations 1. The effective power generated in the pot can be raised significantly with the new ferromagnetic pot type. It is possible to gain 55% more power at the same excitation. 2. By using the new pot type

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– –

–

–

–

–

Induction hobs with significantly higher limit power can be produced at relatively lower price. The efficiency of the cooking becomes higher as a result of better electrical efficiency and the substantial increase of thermal efficiency during the boilup time. This saves energy and reduces operational costs. Even more convenience is provided as the dynamics of the hob increases, namely the boil-up time becomes shorter. More favorable distribution of effective power can be accomplished in the bottom of the pot as more heat is generated closer to the pot’s center. The characteristic of power distribution can be changed by modifying the pot’s dimensions. The exposure of the human body to the magnetic field is considerably higher at the same effective max /B max can be even 170%), but this power (BCnt−n Cnt does not exceed the acceptable limits. The inductor-load system’s input reactive power is significantly higher, Q inn /Q in can reach even 185%.

3. The advantages of the new pot type cannot be exploited with induction hobs adjusted to conventional pots, because of reduction in frequency. The frequency drop can be avoided by changing the capacitance of the supplying inverter’s resonant circuit. Practically, producing this type of pot entails the development of new hob types as well. The increased costs of the pot’s production will be more than offset by cost reductions resulting from cheaper hobs, and by energy savings. Besides, our results can help to understand the theory of induction cooking and its adoption in practical applications.

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Acknowledgments We would like to express our sincere thanks to the H-TEC Kft, Hungarian subsidiary of HUNDAI Heavy Industries Co. Ltd. for providing us the FE simulation tool.

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