# Encyclopedia Magnetica

Encyclopedia of magnetics and electromagnetics.

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# Thermal resistance

 Stan Zurek, Thermal resistance, Encyclopedia-Magnetica.com, {accessed 2020-01-24}

by S. Zurek, E. Magnetica, CC-BY-3.0

Thermal resistance (Rth) - a parameter describing steady state temperature rise versus dissipated power within a given device.

Thermal resistance is often denoted as Rth with the units of K/W (kelvin per watt)2) or °C/W (degree Celsius per watt).

Thermal resistance depends mostly on geometry and materials involved, and not on operating conditions (e.g. shape of current waveform, or switching frequency). For ferrite transformers it is found to be independent on operating frequency.3)

For example, if thermal resistance of a given device is Rth = 8°C/W, then dissipation of 5 W will make the device to run at a temperature 8°C/W · 5W = 40°C higher than its ambient temperature.

A similar concept is thermal impedance (Zth), which is used if the power is applied not continuously, but intermittently. The thermal impedance can be also used for defining the thermal properties under dynamic (transient) conditions. Thermal resistance applies only to the state of equilibrium (steady state).4)

## Ways of deriving

Experimental measurement of thermal resistance of an ETD44 transformer. The nominal power loss is applied until the transformer reaches thermal equilibrium. The temperature rise (difference between steady state and the initial temperature) divided by the applied power loss, in this case 34.4°C / 3.38 W gives the thermal resistance of 10.2°C/W for this particular prototype. by S. Zurek, E. Magnetica, CC-BY-3.0

The value of thermal resistance for a given device depends on the ratio of its total surface area to its volume, as well as thermal conductivity of the case, access to coolant (e.g. fresh air), emissivity of the surface, etc.5) Therefore, the only certain way of deriving the value of thermal resistance is to empirically measure it under nominal operating conditions, for instance on a prototype sample.

There are several empirical equations linking the size and shape of a given class of devices. For example for EE, EI, ETD and EC ferrite transformers the following formula could be used:6)

 $R_{th} = 53 · (V_{core})^{-0.54}$ (K/W) or (°C/W)

where: Vcore - core volume in cm3 (i.e. this the volume of the ferrite core itself as specified by the manufacturer, and not volume of the whole transformer).

So for instance for ETD44 the core volumeFerroxcube, Data sheet, ETD44/22/15, ETD cores and accessories, 2008, {accessed 8 Jul 2012} is 17.8 cm3 gives a value of 11.2°C/W. And the values given in various sources of thermal resistance of ferrite cores are between 11-12°C/W, so there is a reasonably good agreement.

An experimental measurement for a particular winding configuration and type of bobbin) give a value of 10.2 K/W (see also the graph), which is also quite close, but shows that experimental verification on a given prototype can differ from the literature values.

Empirical data given by Epcos also suggest that the thermal resistance is roughly proportional to the reciprocal of square root (power of -0.5) of the ferrite core volume, which is in agreement with the previous equation:7)

 $R_{th} = \frac{x}{\sqrt{V_{core}}}$ (K/W) or (°C/W)

where x is the proportionality factor.

## Importance

Thermal resistance is an important parameter used for correct design of most electronic power transformers. For instance, enamelled wire has a maximum rated operating temperature, which should not be exceeded.8) Depending on the cooling conditions and size, each transformer will have a specific value of thermal resistance.9)

Hence, knowing the highest expected ambient temperature (e.g. Tambient = 50°C), the highest allowed operating temperature (e.g. Tmax = 155°C), and the thermal resistance of transformer (e.g. 8 K/W) it is possible to calculate the maximum losses allowed in such transformer, which with the example values given above is 13.1 W. So the total loss (e.g. the sum of copper loss and core loss) must be kept below this value.

### Calculator of maximum power loss from thermal resistance

If thermal resistance is known for given device and cooling conditions then a maximum loss dissipation can be calculated from the following equation.

 $P_{max} = \frac{T_{max} - T_{ambient}}{R_{th}}$ (W)

Tmax (°C) =       Tambient (°C) =       Rth (°C/W) =

Pmax (W) =

Note: the maximum temperature must be greater than the ambient value. 