In this paper an airgap length of 0.5 mm has been assumed. At an airgap of 0.5 mm, a maximal error of 5% can be found between the measured and analytical calculated inductances.It should be noticed that rotating the cores with a small airgap in between, requires an accurate assembly of the transformer.
B. Electric model
To complete the electric equivalent circuit, winding resis-tances, Rp, Rs and resonant capacitors, Cp, Cs have been added to the circuit, as shown in Fig. 9.
1) Winding resistance: The voltage applied from the half bridge converter to the rotating transformer has a square-waveform, which gives rise to the AC-losses due to harmonics.An analytical expression for the wire resistance in case of non-sinusoidal waveforms has been derived by [9], based on Dowell’s formula for AC-resistance.
2) Resonant capacitors: On both sides of the transformer,resonant capacitors have been added to overcome the voltage drops across the leakage inductance, by locally boosting the voltage and, thereby, increasing the magnetic flux density.Resonance capacitors can be placed in series or parallel to the winding at either side of the transformer.
On the primary side the resonance capacitor has been placed in series to act as a DC-blocking capacitance and to create a zero crossing resonance voltage. This makes it possible to use zero-current switching, to minimize the switching losses.Placing the primary capacitor in parallel would results in a high current in the resonance loop due to the high frequency input voltage. This current would increase the power losses and should therefore by avoided.
On the secondary side the resonant capacitor has been added to boost the power transfer capability. Figure 10 shows the normalized value of Cp for a changing magnetic coupling for series and parallel resonance on the secondary side [10]. To make the resonance capacitor on the primary side insensitive for magnetic coupling changes, which are for example caused by vibration during rotating, the resonance capacitor on the secondary side is placed in series to the secondary winding.
21
The frequency at which the circuit operates at resonance,fres, can be calculated by
fres?1 （12）
2?LlknCnFurthermore, the resonance circuit acts a filter for higher harmonics and, thereby, decreases the conduction losses.
3) Power losses: Conduction and core losses are the main power losses in the rotating transformer. The conduction losses, Pcond, have been calculated by
22
22Pcond?IprmaRp?IsrmaRs （13）
Where Iprms is the primary rms-current, which consists out of the reflected load current and the magnetizing current. The core losses, Pcore, have been calculated by the Steinmetz equation
（14）
whereCm, C(T), xandyare specified material constants and Vcore is the core volume.Both the core and the conduction losses are dependent on the frequency.
Increasing the frequency under a constant power transfer, boosts the conduction losses because of the rising AC winding resistance and decreases the core losses because of
the lower magnetic flux density. For a specific power transfer,an optimum betweenfres
and the magnetic flux density can be found, resulting in minimal core and conduction losses.
C. Thermal model
The core and conduction losses cause a temperature rise in the transformer. It is
important to investigate the thermal behavior of the transformer during the design, because the relative permeability of the core material as well as the power losses in the
core are temperature dependent. A thermal model allows the estimation of the average
winding and core temperature. behavior of the transformer during the design, because the relative permeability of the core material as well as the power losses in the core are
temperature dependent. A thermal model allows the estimation of the average winding and core temperature.
A thermal equivalent circuit, shown in Fig. 11, is made using a finite-difference modeling technique, where the thermal resistance concept is used for deriving the heat transfer [11].The thermal model is derived by dividing the upper half of the geometry
into six regions, where regionItill V represent the core and regionV Irepresents the transformer winding. Five nodes are defined for each region and the heat transfer
between the nodes is modeled by a thermal resistance. Conduction resistances are used
model heat transfer inside the regions and convection resistances are used to model the heat transfer between the border of the regions and the air. No heat transfer is assumed
R?23
at the left and lower boundary of the model. The power losses in each region are presented by a heat source and inserted in the middle node of each region.
The average temperature at each node has been calculated by determining the heat transfer between the nodes, expressed by
?Rth??T???Q? (3-15)
Where Rth is a matrix which consists of all thermal resistances between the nodes,Tis a vector comprising the temperature at each node andQis a vector with all heat energy flowing into the transformer. The thermal resistances are defined using the heat transfer coefficients for conduction and convection, given in Table II.
1) Verification: To verify the thermal equivalent model, a 2D thermal finite element model has been created, based on the thermal assumptions. The temperature has been obtainedin the center of each region and shown in Table III. The largest error between the analytical and numerical calculated increase compared to the environment temperature of 20?C is 6.9%.
24