# Practical Study of Mixed-Core High Frequency Power Transformer

## Abstract

**:**

## 1. Introduction

- The influence of soft magnetic materials of suitable geometry [13] as well as that of copper conductors.

_{cu}depends on the current density of conductors, the impact of proximity effects on the resistance values of primary and secondary conductors, winding configuration and the construction of conductors [14,23]. On the other hand, the core loss P

_{core}depends on the properties of the core material, peak operating flux density B

_{m}, the excitation frequency f

_{s}, the waveform pattern and the core temperature [18,19,24,25]. Thirdly, the design also involves devising a thermal circuit so that the temperature rise in core, copper and insulation are not only within the respective safe operating limit, but there should also be increased uniformity of maximum temperature rise in different parts of windings as well as in the core. Ensuring near-uniform temperature rise is complex because the distribution of power loss in the core is not uniform and so is the case for windings. Moreover, the thermal behavior of each circuit is also different. Ideally, for the design of the thermal circuit for heat removal, the average flux density per cycle is considered to be zero where the core loss is decided by the values of B

_{m}and f

_{s}; it is true when the DC bias in core is absent [17]. The presence of DC bias could adversely affect the performance of the PET in several ways. Primarily, the core loss increases significantly under DC bias; it could work as a hindrance to draw any comparative statement on performance among different PETs. Secondly, depending upon the DC bias capacity of the magnetic circuit, there could be core saturation that affects the performance of the power controller. The DC bias capacity of the magnetic circuit is poor for zero-gap magnetic circuit using high permeability materials (e.g., toroidal core using nanocrystalline materials). The DC bias could be static [26] or dynamic [27,28].

## 2. Power Electronics Transformer for Divergent Load Characteristics

_{cu}could be zero at no load. In the second case, irrespective of the magnitude of the delivered power, the windings always draw the set current. The value of the core loss P

_{core}is negligible at zero power.

_{w}A

_{c}in Equation (1) is used to define the extent of optimization of a PET. At a particular frequency f

_{s}, it suggests a large value of B

_{m}in core as well as the current density J in copper windings, as given below,

_{m}for a square wave input voltage V

_{in}is,

_{w}is the window area, A

_{c}is the core area and n

_{p}is the number of primary turns.

_{core}[18] and copper loss P

_{cu}[23] need to be calculated accurately. For sinusoidal primary voltage, the Steinmetz equation [16] is used to calculate the value of P

_{core}; its parameters are mostly mentioned in core datasheet. However, in high-frequency applications, the primary voltage is rarely sinusoidal. Using the same Steinmetz parameters, the improved generalized Steinmetz equation (iGSE) is used to calculate P

_{core}for any input voltage waveform [17,19]. Using the iGSE, the expression of P

_{core}with square wave excitation is,

_{S}, α and β are Steinmetz parameters, d

_{pwm}is duty cycle of square wave input and W

_{c}is the core weight. For pure square wave input (d

_{pwm}: 1.0), Equation (3) may be modified to,

_{pri}) and secondary (P

_{sec}) windings are,

_{1}and F

_{2}are the ac resistance factors, i

_{p}and i

_{s}are the primary and the secondary current, respectively, and r

_{dc1}and r

_{dc2}are their respective dc resistance values. Both F

_{1}and F

_{2}depend on several factors such as skin and proximity effects where proper choice of copper conductors (litz wire or thin foil) and layout of windings are important [23].

_{in}decides the value of B

_{m}in core; it depends on the power controller and the characteristics of the connected load. Depending upon the value of B

_{m}, the magnetic circuit could face nonlinearity as well as the magnetic saturation. Here, two application types are considered where the dynamics of V

_{in}or B

_{m}are completely different.

#### 2.1. PET for Full-Bridge DC−DC Converter (FBDC)

_{L}and its effective resistance, e.g., battery charging [11], arc welding [33], etc. Even at zero load current the cores could be fully loaded. For a nonlinear load (e.g., welding arc), the dynamic control of DC current I

_{a}would decide the value of V

_{in}or B

_{m}through dynamic change in d

_{pwm}, such as,

_{DC}is the supply voltage, $n={n}_{\mathrm{p}}/{n}_{\mathrm{s}}$ is the ratio of primary (n

_{p}) to secondary (n

_{s}) turns, k

_{1}is constant and the control $u=f\left(e\right)$ is used to ensure zero current error. The transient disturbance in the arc welding process is large [28].

_{1}, F

_{2}and d

_{pwm}. For the arrangement of secondary side rectifier of Figure 4, the primary current i

_{p}and the current i

_{s}in each bifilar secondary could be expressed as,

_{core}because the core in FBDC often faces both the static or dynamic DC bias. The static DC bias could be compensated by a simple approach [26]. The dynamic DC bias [27,28] in core would depend on how the control u (or d

_{pwm}) reacts to ripple in steady state error as well as the load transients. The DC bias is more prominent when the loop gains are large where the ripple in I

_{a}becomes transparent in control input u [28]. Under DC bias conditions, the values of Steinmetz parameters drifts.

#### 2.2. PET for Series Resonant Induction Heating Equipment

_{L}at the frequency decided by the tank circuit parameters L4 and Cr. The coil facilitates the power transfer when a metallic object is taken close to the coil. For a noncontact mode of power transfer, normally, the coil current i

_{L}is kept large. To reduce the stress on primary side components, as shown in Figure 5, the induction heating transformer (IHT) is used to step up the inverter current. The value of P

_{cu}is always at its rated value. The loading of its magnetic circuit and hence the core loss P

_{core}depends on the power P

_{OUT}drawn through L4; it depends on multiple parameters, such as,

_{L}is the coil current, f

_{s}is frequency of i

_{L}and the parameter K

_{c}depends on coupling between the coil and the load. R

_{eq}represents the effective load resistance reflected in the tank circuit.

_{dc}is connected to eliminate any static DC bias present in the core. Zero voltage switching (ZVS) condition of switches Q5–Q8 is inherently achieved because the phase-locked loop (PLL) ensures f

_{s}at slightly higher than the resonant frequency ${f}_{r}=1/2\pi \sqrt{{L}_{4}{C}_{r}}$. Plus, the buck converter controls the input voltage V

_{in}to achieve near-zero current switching of Q5–Q8. Under ZVZCS condition, the inverter input voltage V

_{CH}or the primary voltage of IHT could be approximated as,

_{ac}is the ac resistance of L4 and n is the turns ratio of IHT. At no load, the value of B

_{m}is negligible because the value of ${V}_{\mathrm{in}}\approx 0.787n{r}_{\mathrm{ac}}\left|{i}_{L}\right|$ is small. The change in load of IHT, i.e., the change in R

_{eq}is never abrupt. Its value increases when a job is brought close to the coil (i.e., when more power is drawn through L4), and decreases gradually either near the Curie point or when the job is taken away from the coil head mechanically. They ensure that the dynamic change in B

_{m}is also not abrupt. Moreover, the response time of the buck chopper decides the dynamics of V

_{in}. Therefore, the prospect of dynamic DC bias in IHT would be small. Furthermore, the slow dynamic DC bias in the core, if any, could be effectively tackled in the PLL loop [36].

_{p}and n

_{s}; selecting a core material of suitable geometry to afford optimal values of B

_{m}and J. In ZVZCS conditions, the value of n would be maximum at n

_{max}because the load behaves as resistive,

_{cu}. Large values of B

_{sat}and small values of P

_{c}of nanocrystalline cores would allow optimal choice of n

_{p}as well.

_{r}should be accurate. The value of R

_{eq}would be more at higher values of f

_{r}[37]. When L4 is loaded, its inductance value drifts down to, say, L

_{eq}; then, the corresponding value of f

_{r}is,

_{lk}is leakage inductance of IHT; its large value would be a hindrance to effective power transfer [37]. The value of L

_{lk}is small for high-permeability ungapped toroidal cores.

_{pri}consists of triangular wave magnetizing current with peak at I

_{m}plus the reflected sinusoidal coil current i

_{L}. For minimum phase error between V

_{pri}and i

_{pri}of IHT, the value of I

_{m}should be small; it is expressed as,

_{m}is the mean core length. A large value of µ

_{r}is needed for small value of I

_{m}and n

_{p}.

_{m}and L

_{lk}and also the minimum of number of turns where the value of B

_{m}would be large.

## 3. Mixed-Core Transformer Configuration

_{core}could be fixed and that of P

_{cu}would be decided by the load. Along with reducing the core and the copper losses, the optimization process involves design of a thermal circuit to ensure near-uniform temperature rise in core and also in copper so that the PET is enabled to deliver more power. Due to multiple loss centers, the thermal circuit of the core and windings are coupled. For the magnetic circuit, the distribution of heat and its removal by thermal convection could be improved if the value of K as well as that of the surface area of core are increased. To have requisite flux A

_{c}B

_{m}, several cores are integrated. Often, for magnetic compatibility, cores of the same material with the same batch code are preferred. It is important to find whether such arrangement is best suitable for efficient heat removal, both from the core assembly as well as from the windings. On the other hand, can some other combinations, such as the hybrid core configuration, manage the heat loss or the thermal issues better?

#### 3.1. Thermal Behavior of Power Electronic Transformer

_{c}

_{ore}and P

_{cu}, respectively, increase exponentially with B

_{m}and with ${J}^{2}$. The design optimization of a PET is complex because the layout of the windings and thermal behavior of the PET often contradict. As shown in Figure 3a,b, two windings are overlaid for better magnetic coupling and also for reduced eddy current loss in core [29]. Such arrangement needs good heat removal features because the bulk of total power loss $\left({P}_{\mathrm{tot}}={P}_{\mathrm{core}}+{P}_{\mathrm{cu}}\right)$ is concentrated around a small core segment where the secondary winding is laid above the primary. Removing the heat loss from the multilayered primary winding is difficult because there exists insulation on either side of each layer and also between the windings. The prospect of creation of hot spot is more in the primary [3]. Furthermore, due to the increased impact of the proximity effect, the value of P

_{cu}would be more in multilayered winding. Though the thermal circuit of PET is coupled in a complex manner, the limiting values of B

_{m}and J would be decided by the effectiveness of the thermal circuit [3,39,40]. With given loss, the safe operating temperature for the core would depend on the soft magnetic materials and that for copper would be on the insulation of litz wire strands as well as that placed between the layers.

_{hs}of the PET; it could be decided by the temperature differential $\Delta T={T}_{\mathrm{hs}}-{T}_{\mathrm{amb}}$, such as,

_{amb}is the ambient temperature. R

_{PET}mostly consists of thermal conduction (R

_{th}) and thermal convection (R

_{conv}). Though it plays certain role, the heat transfer by radiation is ignored here. For compact design of PET, apart from reducing the total loss P

_{tot}, the value of $\Delta T$ should be minimum. Large surface area of the core and the secondary winding are available for heat transfer. The major part of P

_{tot}is removed by thermal convection [21,41,42] where the speed of the moving medium would play a significant role. The expressions of conductive and convective thermal resistances are,

_{cond}and A

_{conv}, respectively, are areas available for conduction and convection and h

_{cond}and h

_{conv}are corresponding heat transfer coefficients. The value of h

_{conv}depends on thermal conductivity of the attached medium and also on its speed where fan cooling improves its value significantly. Backed by practical validation, the finite element method (FEM) was extensively used to establish reasons behind the formation of hot spots in core and copper in high-power transformers [3]. It was realized that the temperature rise in copper was alarming on multilayer winding with constrained heat transfer features, e.g., the primary winding of Figure 3a. In multilayer winding, a significant part of P

_{cu}is concentrated in the internal layer of primary winding closest to the core where the proximity effect is more prominent. The hot spot temperature T

_{hs}is located here; its value needs to be reduced. It could be made possible if a part of P

_{pri}close to I-section, in particular, is channelized to the ambience through the core. Considering the heat conduction is symmetrical around the I-section of Figure 3a, the overall heat conduction circuit of half of the PET (shown in Figure 6a), is represented in Figure 6b. The part P

_{cu1}of total copper loss P

_{cu}that could be channelized through the core is expressed as [41],

_{w}and R

_{wa}, respectively, are the conductive and the convective resistance of the winding, and R

_{F}and R

_{ca}, respectively, are the effective thermal resistance of the coil former and the core. It is clear that more heat loss (i.e., P

_{cu1}) would be channelized through the core if either of R

_{ca}or P

_{core}or together could be reduced through design or selection of suitable soft magnetic material; reduction of R

_{F}would also play certain assisting role.

_{tot}while the major part of its surface area is exposed to the ambience. Therefore, if that particular small section of core volume is replaced by a suitable soft magnetic material, then better heat removal feature by convection and radiation could be realized. The new core material is desired to possess following features:

- Superior thermal conductivity.
- Reduced core loss density.
- Higher maximum operating temperature than the parent core.
- The new material must be magnetically compatible.

#### 3.2. Magnetic Compatibility of Different Types of Magnetic Circuits

_{m}of its magnetic circuit. The reluctance circuits of the core assemblies of commonly used high-power PETs of Figure 3a,b could, respectively, be represented in Figure 7a,b. Traditionally, the value of loss density P

_{c}is considered at the same value everywhere. It means, for one type of cores, the value of B

_{m}should be same everywhere. However, in the proposed idea of using the mixed core configuration, the value of loss density could be different. The expression of flux φ linking the windings is,

_{m1}, the total flux ${\phi}_{1}={\phi}_{2}+{\phi}_{3}$ in the central limb is,

_{m2}is the magnetizing current and n

_{p2}is the number of turns at primary. Neglecting the dimensional tolerance of the cores, the dynamic value of R

_{m}of each parallel path plays an important role in flux distribution. When composite core segments are used, the dynamic behavior of µ

_{r}becomes critical for designing the magnetic circuit of Figure 7b. The value of µ

_{r}could change differently with respect to operating value of B

_{m}, f

_{s}, temperature, etc.

#### Characteristics of Different Soft Magnetic Materials

_{r}is extremely important. For better thermal performance, the heat distribution (Equation (15)) should be proper, i.e., more heat needs to be transferred to the core surface area exposed to the environment. Here, both the core loss density P

_{c}and the thermal conductivity K would play significant roles. The basic parameters of popular soft magnetic materials are listed in Table 1 [20]. For ferrite cores, the parametric variation with respect to temperature [13] is large. Compared to others, its Curie temperature is much lower. The reduction in B

_{sat}value vs. temperature is also sharp. For integration of different core types into a magnetic circuit of PET, the value of µ

_{r}at different flux density and core temperature is important. The influence of temperature on µ

_{r}of different soft magnetic materials is shown in Figure 8 [43] where nanocrystalline cores appear to be parametrically robust. It is also known that, for ferrite cores, the value of µ

_{r}changes significantly with flux density [4]. However, for nanocrystalline cores, as shown in Figure 9, the value of µ

_{r}is mostly constant, and only at large values of B

_{m}is there gradual monotonic drooping in its value. Moreover, compared to other soft magnetic materials, the core loss of nanocrystalline cores at a particular frequency is comparatively less at any flux density (shown in Figure 10). Therefore, due to the parametric robustness, small core loss density, superior thermal conductivity, higher saturation flux density and high Curie temperature, the nanocrystalline cores of proper ribbon thickness are superior candidates for mixed-core configuration. It would not disturb the behavior of the magnetic circuit. If geometry permits, Fe-based nanocrystalline cores could ideally be suitable as flux integrators for medium- to high-frequency PETs. The drooping characteristics of µ

_{r}, as shown in Figure 9, make these cores suitable for integration into a magnetic circuit where the MMF feeds several parallel magnetic circuits (shown in Figure 3b).

#### 3.3. Figure of Merit of Mixed-Core Magnetic Circuit for Series Reluctance Model

_{tot}is concentrated around the central I-section EFGH. The hot spot temperature in the core and the primary winding, in particular, would reside in and around the I-section [3]. In order to properly utilize the large surface area of core exposed to the ambient medium for thermal convection and radiation, there needs to be an improvement in spreading the heat loss in the core by thermal conduction. A higher value of thermal conductivity K of core in the I-section EFGH would help remove the concentrated heat loss to the surface area of the magnetic circuit. As detailed in Equation (15), more power loss could be channelized if,

- The value of P
_{c}of the I-section is small. - The value of K of soft magnetic material used in the I-section is more.
- The value of thermal resistance in coil former is reduced.

_{c}and the value of K. From the magnetic characteristics point of view, the role of I-section is not complicated. The Figure of Merit of the new core material for the I-section could be gauged by a suitability parameter S

_{core}(SR); it is introduced in simple form as,

_{c}and the same or larger value of K would result in superior distribution of heat in core. The permeability µ

_{rn}of new I-section should be around the same value as µ

_{rp}, i.e., of parent material. The value of K and other relevant parameters for different core materials are listed in Table 1.

#### 3.4. Figure of Merit of Mixed-Core for Parallelly Connected Magnetic Circuit

_{core}of Equation (19) is modified to,

_{r}(B

_{m}) is the value of µ

_{r}of core at the operating value of B

_{m}. It takes care of the distribution of flux when the condition $(\mathrm{for}{B}_{\mathrm{m}1}{B}_{\mathrm{m}2},{\mu}_{r1}{\mu}_{r2})$ is met. As shown in Figure 9, nanocrystalline cores support such characteristics. Furthermore, as shown in Figure 8, the value of µ

_{r}for this core is also stable against temperature.

## 4. Experimental Validation of Mixed-Core Transformers

_{m}were sluggish, the ungapped toroidal shaped cores were assembled. In the first case, the prospect of transient DC bias was more. The magnetic circuit needed to be built with large DC bias capacity where, for large power applications, EE, UU or CC cores were suitable. On the other hand, for IHT, the prospect of transient DC bias was small. Therefore, ungapped toroidal cores with large permeability were preferred in the magnetic circuit.

#### 4.1. Validation of Mixed-Core PET Where the Magnetic Circuit Uses UU cores

_{c}, the window area A

_{w}, the mean magnetic length l

_{m}, etc.

_{c}on the thermal performance of PET, as shown in Figure 12, two mixed-core transformers (MCT) were built using different ferrite cores with different values of P

_{c}and K. For core assembly, each MCT combined two different core types A and B procured from separate manufacturers. Relevant parameters of these cores are listed in Table 2.

_{DC}was 560 V. The value of inductor L1 was 100 µH. The winding layout of PETs A and B was similar, each had two secondary bifilar windings. To practically compare the magnetic compatibility at high flux density, the turns-ratio n

_{p}:n

_{s}:n

_{s}of PETs was deliberately chosen at 24:2:2. The value of B

_{m}at maximum value of d

_{pwm}at 80% was 0.325T. Moreover, at the rated output power P

_{L}at 4.5 kW (PWM duty cycle d

_{pwm}≈ 57%), the magnetic circuit would be loaded to the rated value of B

_{m}at around 0.2 T. Each PET used one pair of UU cores, the value of A

_{c}was 8.4 cm

^{2}. At rated load current (I

_{a}: 200 A) with d

_{pwm}at 57%, the current density J (A/mm

^{2}) at primary (strand dia.: 0.1 mm, 450 strands) and secondary litz wire (strand dia.: 0.1 mm, 3780 strands) conductors were 3.56 and 4.72, respectively. Due to higher value of J, the power loss in secondary conductors was more.

_{m1}(both secondary: open) in Figure 14a,b demonstrated that the core A and B were magnetically compatible even in the nonlinear zone of the B–H curve. At d

_{pwm}of 80%, the value of B

_{m}was deliberately kept large at 0.325 T so that the magnetic compatibility in the nonlinear zone was verified. The magnetic circuit had rated value of B

_{m}(≈0.2 T) at designed power output.

_{L}was maintained at 22.5 V. At full load, the calculated value of B

_{m}was 0.207 T. Various waveforms of the FBDC using PET A and PET B are shown in Figure 15a and Figure 15b, respectively. Using the Fluke make 59 Mini IR thermometer, the temperature was recorded in each winding and also in the core segment close to the windings. The measured value of the hot spot temperature in the core and the windings of two PETs are listed in Table 3. It was clear that, compared to PET A, the temperature distribution of PET B was superior. The experimental results made it clear that the core type with lower value of P

_{c}and/or possessing higher value of K would be more suitable for the core segment where the windings are laid. It could be stated that the situation would improve further if the I-section was replaced by a suitable nanocrystalline core material.

_{c}and of larger value of K helped reduce the hot spot temperature of PET B. The results mostly tallied with the findings of the temperature profile obtained through the noncontact type Fluke make 59 Mini IR thermometer.

#### 4.2. Validation of Mixed-Core IHT for Parallelly Connected Core Assembly

_{cu}is desired to be minimized because it is always at its rated value. It was analyzed in Section 2.2 that the ZVZCS topology along with using a low-loss magnetic circuit with a high effective value of µ

_{r}and high saturation flux density B

_{sat}would simultaneously minimize the copper content in IHT and also the value of P

_{cu}. It requires the core to operate at a large value of B

_{m}where cores with low loss density and high values of B

_{sat}would be suitable. When both windings are placed in single layers, the impact of proximity effect on P

_{cu}is minimized. Laterally, it would achieve better features for heat removal because windings are exposed to the ambient medium. For the magnetic circuit, the ungapped toroidal cores using nanocrystalline material would be a preferred choice because:

- The value of magnetizing current would be small.
- The value of leakage inductance would be negligible.
- IHT does not need large DC bias capacity; transient load disturbance is small.
- For large value of B
_{m}(>0.25 T), ferrites are not suitable.

_{c}in core D was less. Naturally, the core D resulted superior value of S

_{core}(PR) (see Equation (20)). The operating parameters of the inverter and cores are listed in Table 5. To study the magnetic compatibility, one single-turn search coil was wound on C and D cores each (shown in Figure 19b). The turns-ratio of each IHT was 8:3. For the tank circuit, the value of L4 was 52 µH and that of Cr was 3.6 µF. The inverter frequency was 12.5 kHz. With set coil current at 250 A, the value of J in primary (strand dia.: 0.1 mm, 2880 strands) and secondary (strand dia.: 0.1 mm, 2 × 3780 strands) conductors were 4.15 A/mm

^{2}and 4.21 A/mm

^{2}, respectively.

- The magnetic circuit operated at maximum value of B
_{m}when the secondary was kept open and the primary was excited with full voltage. The control circuit was disabled. The current at primary was the magnetizing current. Waveforms in Figure 20a validated that the two core types were magnetically compatible. Exactly similar nature of induced voltages in the single-turn search coils V_{src}_{-C}(core C) and V_{src}_{-D}(core D) proved that the flux density was shared appropriately. - The tank circuit was connected, but the power delivered through the coil head was zero. Moreover, there was a DC blocking capacitor C
_{dc}of 100 µF added between primary of IHT II and the inverter output. The primary voltage was small. Here, as well, the two core types were found to be magnetically compatible (shown in Figure 20b) because even at very small flux density the readings in both the search coils were similar dynamically. - Magnetic compatibility of cores of IHT II was also tested when the secondary was loaded. The coil head L4 was loaded at 20 kW. The necessary waveforms are shown in Figure 21a. Here, as well, the voltage waveforms of both search coils appeared similar—in magnitude as well as in waveshape. Similar search coil voltage readings at zero secondary current as well as under loaded condition proved that cores C and D were integrated into the magnetic circuit of IHT II.

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Popular arrangements of PET where magnetic circuit is configured in (

**a**) series reluctance circuit model and (

**b**) parallel reluctance circuit model.

**Figure 6.**(

**a**) Symmetrical half of PET of Figure 3a and (

**b**) equivalent thermal circuit to remove heat loss.

**Figure 8.**Behavior of relative permeability of soft magnetic materials vs. temperature [43].

**Figure 9.**Permeability vs. flux density of competitive nanocrystalline cores have similar drooping characteristics.

**Figure 11.**Mixed-core assembly is for better use of the (

**a**) electrical circuit of PET, i.e., the windings, and (

**b**) magnetic circuit or the core assembly.

**Figure 12.**Mixed-core transformers for FBDC where windings are overlaid on the core of material type (

**a**) A for PET A and (

**b**) B for PET B.

**Figure 14.**At zero secondary current, even in nonlinear operating zone of B-H curve (B

_{m}= 0.325 T), the behavior of the magnetic circuit of (

**a**) PET A was similar to that of (

**b**) PET B.

**Figure 15.**Waveforms of FBDC controller when 200 A was drawn by the load and, in that loading condition, the magnetic circuit was loaded with B

_{m}at 0.207 T, for both (

**a**) PET A and (

**b**) PET B.

**Figure 19.**Two 40 kW, 15 kHz transformers: (

**a**) IHT I with original set of C cores and (

**b**) IHT II with mixed-core configuration (C-D-D-C).

**Figure 20.**Magnetic compatibility of C and D cores was verified through similar voltage readings in search coils wound on the respective cores, when (

**a**) the secondary of IHT II was kept open, and (

**b**) the DC blocking capacitor C

_{dc}of 100 µF was added (shown in Figure 5) but the coil L4 was not loaded.

**Figure 21.**(

**a**) The magnetic compatibility of IHT II was also verified when the coil head was loaded. (

**b**) Waveforms of different variables of the inverter when the IHT was loaded at 35 kW while delivering power to a section of pipe through the coil head L4.

**Figure 23.**Compared to IHT I, the steady state temperature in different parts of IHT II was not only reduced, there was increased uniformity in maximum temperature in various parts of the mixed-core transformer.

Core Material | Si-Steel | Ferrite | Amorphous | Nanocrystalline |
---|---|---|---|---|

Thermal conductivity K, °C/mK | 18.6 | ≤5 | 10 | 10 |

Loss density (W/kg) at 0.1 T, 20 kHz | 20.0 | 1.9 | 4.7 | 0.9 |

Loss density (kW/m^{3}) at 0.1 T, 20 kHz | 150 | 9.1 | 34 | 6.6 |

Relative permeability µ_{r} | 900 | 2200 | 15,000 | 15,000 |

Maximum operating temperature, °C | 150 | 100 | 150 | 120–150 |

Saturation flux density B_{sat} at 25 °C | 1.8 | 4.9 | 1.56 | 1.23 |

Curie Temperature, °C | 770 | 230 | 415 | 570 |

Core Material | Ferrite Core A | Ferrite Core B | ||
---|---|---|---|---|

At temperature | 25 °C | 100 °C | 25 °C | 100 °C |

Initial permeability | 2500 | 2200 | ||

B_{sat}, T | 0.47 | 0.38 | 0.49 | 0.39 |

Curie temp., °C | >230 | >210 | ||

P_{c} @ 0.2 T, 25 kHz, kW/m^{3} | 130 | 65 | 130 | 57 |

P_{c} @ 0.1 T, 100 kHz, kW/m^{3} | 135 | 65 | 140 | 50 |

Thermal conductivity K, W/mK | ≤4.3 | 5.0 |

Transformer | Primary | Secondary | Temperature of Core (°C) |
---|---|---|---|

PET A | 86.9 | 92.8 | Core A: 61.7 |

PET B | 83.6 | 89.3 | Core B: 57.3 |

Parameter | Core C | Core D |
---|---|---|

B_{sat} at 25 °C (130 °C), T | 1.2 (1.18) | 1.25 (1.21) |

Initial permeability μ_{r} | >300 k | >300 k |

K_{S1}, α, β | 0.94, 1.4364, 1.638 | 0.22, 1.608, 1.681 |

Value of μ_{r} at B_{m} = 1 T | >12 k | >12 k |

Ribbon thickness, μm | 30 | 22 |

Area of each core, cm^{2} | 5.25 | 5.62 |

Mean length of core, cm | 29.8 | 29.8 |

P_{c} @0.2 T, 20 kHz, W/kg | 4.98 | 1.82 |

P_{c} @0.6 T, 10 kHz, W/kg | 11.2 | 3.78 |

Th. conductivity K, W/mK | 10 | 10 |

IHT I | IHT II | |
---|---|---|

Delivered power, kW | 35 | |

Input voltage to inverter at full load, V | 450 | |

Inverter frequency at full load, kHz | 12.5 | |

Core area, cm^{2} | 21 | 21.74 |

Value of B_{m} in C, T | 0.535 | 0.517 |

Value of B_{m} in D, T | - | 0.517 |

Core loss density in each of C, W/kg | 12.7 | 12.0 |

Core loss density in each of D, W/kg | - | 4.21 |

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**MDPI and ACS Style**

Paul, A.K.
Practical Study of Mixed-Core High Frequency Power Transformer. *Magnetism* **2022**, *2*, 306-327.
https://doi.org/10.3390/magnetism2030022

**AMA Style**

Paul AK.
Practical Study of Mixed-Core High Frequency Power Transformer. *Magnetism*. 2022; 2(3):306-327.
https://doi.org/10.3390/magnetism2030022

**Chicago/Turabian Style**

Paul, Arun Kumar.
2022. "Practical Study of Mixed-Core High Frequency Power Transformer" *Magnetism* 2, no. 3: 306-327.
https://doi.org/10.3390/magnetism2030022