Why Is AC Resistance Higher Than DC Resistance?

When calculating copper losses in an electric motor, we usually start with a simple equation:

For DC resistance this is straightforward: calculate the resistance of the winding and multiply it by the square of the current. However, when a motor operates with AC currents, the effective winding resistance can be higher than its DC resistance. This is mainly due to two effects: skin effect and proximity effect.

In this article I explain why these effects occur, how they affect motor windings, and how AC resistance can be estimated during motor design.

What is skin effect?

When AC current flows through a conductor, the alternating magnetic field associated with the current makes the current distribution non-uniform across the conductor cross-section. At higher frequencies, more current tends to flow near the surface of the conductor — this is the skin effect. The depth over which the current density significantly decreases is called the skin depth:

As frequency increases, skin depth decreases. This means less of the conductor cross-section is effectively used to carry current, which increases the AC resistance. For small conductors operating at low electrical frequencies, skin effect may be relatively small — but it becomes more important when conductor dimensions are large compared with the skin depth.

What is proximity effect?

Skin effect is not the only reason for increased AC resistance. In a motor winding, conductors sit close to one another inside the stator slots, and the magnetic fields produced by neighboring conductors can redistribute the current within each conductor. This is the proximity effect.

Depending on the arrangement of the conductors and the direction of the magnetic field, current can become concentrated in particular regions of the conductor cross-section. This increases the effective resistance and therefore the copper losses. Proximity effect can be particularly important in motor windings because conductors are packed closely together; its magnitude depends on conductor dimensions, slot geometry, winding arrangement, and operating frequency.

In practice, skin effect and proximity effect occur together, and both need to be considered when estimating AC winding resistance.

How does MotorDesignSoftware calculate AC resistance?

In MotorDesignSoftware, AC resistance is estimated analytically using the winding geometry, conductor dimensions, material properties and operating frequency. The calculation accounts for the increase in resistance due to skin effect and proximity effect. The resulting AC resistance can then be used to estimate copper losses:

For currents containing significant harmonics, a more complete calculation considers the resistance at each harmonic frequency:

This matters because the resistance experienced by a high-frequency current component can be different from the resistance at the fundamental electrical frequency. Analytical calculations are useful during the early stages of motor design because they provide a practical way to estimate additional copper losses without explicitly resolving the current distribution inside every conductor using FEA. However, the accuracy depends on the analytical model and its assumptions; complex conductor arrangements and field distributions may require more detailed analysis.

Stranded windings vs hairpin windings

One area where AC resistance becomes particularly interesting is the comparison between stranded and hairpin windings. In a conventional stranded winding, each turn is made from relatively thin conductors, which — depending on their dimensions and arrangement — can help limit skin and proximity effects.

Hairpin windings typically use relatively large rectangular copper conductors. They offer manufacturing and packaging advantages and are common where high power density matters, but their larger conductor dimensions can make AC resistance more significant, particularly at higher electrical frequencies. This does not mean hairpin windings are always less efficient — the result depends on several factors, including:

• conductor dimensions and shape
• number of conductors in each slot
• winding arrangement
• copper fill factor
• operating frequency
• current and temperature

A hairpin winding may offer advantages in manufacturing and copper utilisation, while a stranded winding may offer better AC resistance characteristics under certain operating conditions. The best choice depends on the application and the operating range of the motor.

How should AC resistance ideally be calculated?

Analytical methods are useful for estimating AC resistance, but they rely on assumptions about the current distribution and the magnetic field around the conductors. A more detailed approach is to use electromagnetic FEA to calculate the current distribution within the conductor cross-sections. For example, time-harmonic FEA can investigate how skin and proximity effects change with frequency and conductor arrangement.

The mesh must be fine enough to resolve the skin depth, and the model must represent the relevant conductors and magnetic field correctly. For complex windings, additional factors such as parallel paths, conductor transposition and end-winding losses may also need to be considered. This requires more modelling effort and computation time, so analytical calculations remain useful for quick design iterations, with detailed FEA used when greater accuracy is required.

Conclusion

AC resistance is an important consideration when designing motor windings. Skin effect changes the current distribution within individual conductors, while proximity effect changes it through the magnetic fields produced by neighbouring conductors. Both can increase resistance and copper losses compared with a DC resistance calculation.

These effects also help explain why stranded and hairpin windings can behave differently, particularly as electrical frequency increases. Analytical AC resistance calculations provide a practical way to account for them during motor design, and for more complex winding arrangements, detailed electromagnetic FEA can provide further insight into the current distribution and associated losses. When evaluating a motor winding, it is therefore worth considering not just its DC resistance, but also how its effective resistance changes over the operating range.

About the simulation

AC resistance in MotorDesignSoftware is estimated analytically from the winding geometry, conductor dimensions, material properties and operating frequency, accounting for skin and proximity effects — with the option to investigate the current distribution in more detail using electromagnetic FEA.

• FEMM: femm.info/download
• MotorDesignSoftware: get a license