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Power Electronics

Magnetic components and thermal design

The gapped inductor with inductance proved, the saturation check, Steinmetz losses, skin effect and Litz wire, the thermal circuit and heatsink selection with an example, Arrhenius and thermal cycling.

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In a converter the largest, heaviest and often most expensive components are the inductors and transformers, and the power limit of any device is the heat it can get rid of. The design of magnetic components and thermal design therefore decide size, efficiency and lifetime. In this lesson we look at the gapped inductor with the proof of its inductance, the saturation check, core and winding losses, the equivalent thermal circuit with an example of heatsink selection, and the link between temperature and reliability.

The gapped inductor

A power inductor is a winding of N turns on a ferrite or powdered-metal core, with a small air gap, a space of length \ell_g in the magnetic circuit.

Theorem. If the core permeability is much greater than that of air, the inductance is

    \[ L \approx \frac{\mu_0 N^2 A}{\ell_g} \]

where A is the core cross-section: it depends on the gap, not on the material.

Proof. By Ampère’s law along the magnetic circuit, N\,i = H_c\ell_c + H_g\ell_g, with H_c and \ell_c the field and length in the core, H_g and \ell_g in the gap. The flux \Phi = BA is the same in the core and in the gap, so H_c = B/(\mu_r\mu_0) and H_g = B/\mu_0. If \mu_r is large, H_c\ell_c \ll H_g\ell_g and N\,i \approx B\ell_g/\mu_0, i.e. B \approx \mu_0 N i/\ell_g. The flux linkage is \lambda = N\,B\,A = \mu_0N^2A\,i/\ell_g, and the inductance is L = \lambda/i. ∎

The gap looks like a waste, but it is essential: it makes the inductance stable (independent of the material’s permeability, which varies with temperature and from piece to piece), and almost all the energy \frac{1}{2}Li^2 is stored precisely in the gap’s volume of air, where the field H is large. A core without a gap would saturate with tiny currents.

Saturation

Every magnetic material has a saturation flux density B_{sat}, about 0.3-0.4 T for ferrites and 1-1.5 T for iron powders: beyond it, permeability collapses, the inductance vanishes and the current explodes. The design must check that at peak current the flux density stays below the limit. From the previous proof, \lambda = Li = NBA, so

    \[ B_{pk} = \frac{L\,I_{pk}}{N\,A} \]

For a given inductance and current, the product NA must be large enough: that is why an inductor that must carry more current is bigger. In converters the peak current includes ripple and transients: a buck that goes into short circuit drives the inductor into saturation if the protection is not fast enough.

Losses

  • In the core: hysteresis and eddy currents dissipate energy every cycle. They are estimated with the empirical Steinmetz law, P_v = k\,f^\alpha B^\beta per unit volume, with \alpha between 1 and 2 and \beta between 2 and 3, from the manufacturer’s data. What matters is the amplitude of the field ripple, not its mean value.
  • In the windings: besides the DC resistance, at high frequencies the current concentrates on the conductor’s surface because of the skin effect (Electromagnetic Fields lesson on waves): at 100 kHz the skin depth in copper is about 0.2 mm, and a thick wire is almost useless. Litz wire, made of many thin insulated strands twisted together, or foil and printed windings are used. Between adjacent layers there is also the proximity effect, which can multiply the losses.

The thermal circuit

The heat produced in a semiconductor must cross the package, the interface (thermal paste or pad) and the heatsink before reaching the air. Like a current in a series of resistances, the thermal power P produces temperature drops proportional to the thermal resistances (Technical Physics lesson on conduction).

Equivalent thermal circuit of a power component
The dissipated power P, like a current source, flows in series through the junction-to-case thermal resistance, the case-to-sink one and the sink-to-ambient one; the node temperatures are the junction, the case and the heatsink, and the ambient is a voltage source at temperature T a.

    \[ T_j = T_a + P\,(R_{\theta jc} + R_{\theta cs} + R_{\theta sa}) \]

Example. A MOSFET dissipates 20 W, with R_{\theta jc} = 0.5 K/W and a 0.3 K/W interface. The junction must not exceed 150 °C, and a margin is advisable: aim for 125 °C, with air at 40 °C. The available rise is 85 K, which divided by 20 W gives 4.25 K/W in all; subtracting 0.8 K/W for junction and interface, the heatsink must have R_{\theta sa} \le 3.45 K/W. With forced air a heatsink of the same size has a thermal resistance two to five times lower.

For short pulses the transient thermal impedance matters: the heat first warms the mass of the chip and only later reaches the heatsink, and a device can withstand for a few milliseconds a power much greater than the continuous one. Manufacturers give the curves.

Temperature and reliability

Many failure mechanisms (diffusion, electromigration, electrolyte degradation in capacitors) accelerate with temperature according to the Arrhenius law: as a rule of thumb, every 10 °C more halves the lifetime. In electrolytic capacitors, often the most fragile components of a power supply, a rated life of a few thousand hours at 105 °C becomes tens of thousands at 65 °C. Thermal cycling matters too: repeated expansion between different materials fatigues solder joints and the internal bond wires of power modules, and is the main cause of failure in car and wind-turbine inverters. That is why serious designs leave temperature margins and modern modules use assembly techniques such as silver sintering.

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