# Design considerations for high-frequency coaxial winding power transformers

Abstract: The use of coaxial windings to create low-loss, low-leakage-reactance, power transformers for use in high-frequency soft-switched DC/DC and resonant converters has been demonstrated by M.H. Kheraluwalest al. (1990). Some of the important loss aspects of the design of coaxial winding transformers are examined, including the influence of the skin effect on winding resistance, the variation of core loss caused by nonuniform core flux density, and the choice of the principle dimensions and aspect ratios for maximum efficiency. Experimental measurements on a 50 kVA, 50 kHz unit are included to confirm portions of the analytical results and suggested design procedures. >

## Summary (3 min read)

### I. INTRODUCTION

- NE OF THE MORE important concerns in high fre-0 quency power conversion is the question of magnetic component design, particularly for higher power levels.
- Further, transformer designs which feature an easily calculated leakage inductance are appealing for use in soft switching circuits in which the leakage inductance is a useful circuit component.
- The design of such coaxial transformers is considerably different from that of conventional transformer structures and needs to be better understood.
- As can be seen, the inner winding is wound completely inside the outer conductor.
- Some of these are explained in the following sections.

### 11. FLUX DISTRIBUTION AND CORE LOSS

- The above discussion and (4) were derived by ignoring core saturation (i.e., constant permeability was assumed).
- One of the concerns this raises is the current distribution in the inner and particularly the outer conductor and the effect on winding losses.

### A. Losses in the Outer Winding with Sinusoidal Excitation

- The current distribution in the outer tube is identical to a coaxial transmission line when the transformer windings have one turn cylindrical symmetry and the net current in the outer and inner windings is equal and opposite.
- If the windings are centered, as in Fig. 1 , the effect on the outer tube current distribution should be small, and the coaxial transmission line approximation should give good results.
- Tdr p where J is the current density (magnitude and phase), and T is the radial distance from the axis of symmetry.
- Hence, with saturation-included in the analysis, the actual core loss will look even more like the core loss approximation which assumes uniform flux.

### and

- If the inner winding is also a solid hollow cylinder, the ac resistance of the inner winding can be found using the same method with different boundary conditions.
- Boundary conditions for the inner tube are that there is no magnetic field at the inner surface because there is no enclosed current, and that the magnetic field at the outer surface of the inner tube is tangential.

### B. Losses in the Outer Winding with Nonsinusoidal Current

- The above analysis examined the ac resistance of the outer tube for sinusoidal excitation which is not all that useful by itself for power electronic applications.
- The usual situation involves a transformer driven by a square wave voltage, with a triangular current waveform resulting from inductive switching.
- In terms of outer winding losses, a Fourier series representation can be used on the current waveform in conjunction with the ac resistance at each harmonic to determine the total loss.
- If the designer uses a tube thickness of 1.55 S to minimize the ac resistance at the fundamental, the remaining harmonic currents will essentially see the dc equivalent resistance for one skin depth at each harmonic.
- The resulting ac resistance of the outer tube with a triangular current waveform is 25% greater than when the current waveform is sinusoidal for the same rms current.

### C. Losses in the Inner Winding(s)

- The ac resistance for this type of winding was introduced above and is described by (9).
- A solid tube inner winding exhibits the same minimum resistance phenomenon as the outer tube winding.
- This is the same optimum thickness as the outer tube winding and the shape of the curve is identical to Fig. 5 . Single Tum Litz Winding: Litz wire is typically used over solid or stranded wire when high frequency and high current capacity is needed in transformers and inductors.
- The ac resistance of litz wire can be found using the manufacture's data sheets and is typically close to the dc equivalent of the individual strands in parallel when the gauge on the individual strands is appropriate for the frequency of interest.
- When there are multitums used as the inner winding there is no longer coaxial symmetry between the inner and outer winding and the proximity effect needs to be looked at for precise loss calculations.

### IV. OPTIMUM EFFICIENCY DESIGNS

- Transformer designs are essentially governed by the amount of core needed to avoid saturation and the amount of winding cross sectional area needed to support the full load current.
- It has been shown [l] that one of the main advantages to the coaxial transformer design is its low leakage reactance which is also precisely controllable and hence a useful circuit component.
- This essentially adds an additional constraint to the design if one intends to use the leakage reactance as an actual circuit component rather than simply trying to minimize it.

### A. Design Procedure

- As an initial design, the desired current capacity and choice of conductor type with appropriate fill factor determines the equivalent outer radius (rin) of the inner winding as shown in Fig. 9 .
- The results in the previous section on winding losses indicate that the outer tube resistance will be minimized when the wall thickness is 1.55 6.
- The remaining constraint of the transformer design is to determine the necessary flux cross-sectional area for the transformer core.
- Core loss will be minimized for geometries which maximize cross sectional area per unit volume.

### B. Efficiency Optimization for a 50 kVA, 50 kHz Unit

- The desired leakage reactance and current rating was used to select the winding geometry in the original design.
- As the transformer length is varied the outer tube radius is determined using (1 1).
- This determines the core inner radius, and the core outer radius is adjusted so that the flux cross-sectional area remains fixed.
- The dashed line in Fig. 11 shows where the experimental transformer lies in comparison to the optimal point.
- The curve shows that the transformer loss could be reduced by 30% if the optimal loss dimensions were used over the initial design dimensions.

### C. Optimum Efficiency Length-to-Width Ratios

- In the above optimization example it was shown that the transformer profile for maximum efficiency was rather long and thin.
- This is atypical since maximum efficiency designs for conventional electric machines and transformers tend to be cubical in shape.
- Hence, for maximum efficiency designs, the transformer profile will favor minimum core volume which implies long-thin toroidal cores to provide the necessary flux cross sectional area at the minimum volume.
- It should be noted that as the core is typically a poor thermal conductor, short transformer designs may benefit from additional heat removal along the primaryhecondary conductors to the outside and to the end windings.
- Transformer performance can be substantially improved if forced convection heat transfer is implemented.

### D. Measured Versus Theoretical Losses, 50 kVA, 50 kHz Unit

- Measured losses for the 50 kVA, 50 kHz transformer are compared to the theoretical predictions.
- The outer tube served as the transformer primary.
- The measurement has an error bound which is determined from the phase and magnitude accuracy of the current probe, digitizing resolution of the oscilloscope, and the accuracy of reading a current value from the scope display.
- In order to compare the theoretical and measured results, the inner winding resistance is referred to the outer winding side and added to the outer winding resistance so that the resistance shown in Fig. 12 is the total resistance looking into the transformer.
- The theoretical core loss is found using the manufacture's core loss data and the core loss results presented earlier.

### V. CONCLUSIONS

- The loss aspects of the coaxial winding transformer have been investigated.
- It has been shown that nonuniform flux distribution in the toroidal core has little effect on the core loss, and that computing the core loss by assuming uniform flux density in the core gives a conservative and good estimate of the core loss.
- Minimum ac resistance at the fundamental frequency component in the outer tube winding is achieved by making the wall thickness 1.55 skin depths thick.
- Optimum efficiency designs for coaxial transformers tend to have nonconventional aspect ratios which are long and thin.
- This long-thin profile is also advantageous for heat transfer due to the increased surface area.

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