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Thermal resistance is a heat property and a measurement of a temperature difference by which an object or material resists a heat flow. Thermal resistance is the reciprocal of thermal conductance.

• 
  (Absolute) thermal resistance R in K/W is a property of a particular component. For example, a characteristic of a heat sink.


• 
  Specific thermal resistance or thermal resistivity R_λ in (K·m)/W is a material constant.


• 
  Thermal insulance has the units (m²K)/W in SI units or (ft²·°F·hr)/Btu in imperial units. It is the thermal resistance of unit area of a material. In terms of insulation, it is measured by the R-value.

Absolute thermal resistance

Absolute thermal resistance is the temperature difference across a structure when a unit of heat energy flows through it in unit time. It is the reciprocal of thermal conductance. The SI units of thermal resistance are kelvins per watt or the equivalent degrees Celsius per watt (the two are the same since the intervals are equal: Δ1 K = Δ1 °C).

The thermal resistance of materials is of great interest to electronic engineers because most electrical components generate heat and need to be cooled. Electronic components malfunction or fail if they overheat, and some parts routinely need measures taken in the design stage to prevent this.

Analogies

Electrical engineers are familiar with Ohm's law and so often use it as an analogy when doing calculations involving thermal resistance.
Mechanical and Structural engineers are more familiar with Hooke's law and so often use it as an analogy when doing calculations involving thermal resistance.

type	structural analogy[1]	hydraulic analogy	thermal	electrical analogy[2]
quantity	... ...displaystyle ... [...]	volume Vdisplaystyle V [m3]	heat Qdisplaystyle Q [J]	charge qdisplaystyle q [C]
potential	displacement Xdisplaystyle X [m]	pressure Pdisplaystyle P [N/m2]	temperature Tdisplaystyle T [K=J/kBdisplaystyle k_mathrm B ]	potential Vdisplaystyle V [V=J/C]
flux	load or force Fdisplaystyle F [N]	flow rate Qdisplaystyle Q [m3/s]	heat transfer rate Q˙displaystyle dot Q [W=J/s]	current Idisplaystyle I [A=C/s]
flux density	stress σdisplaystyle sigma [Pa = N/m2]	velocity vdisplaystyle mathbf v [m/s]	heat flux qdisplaystyle mathbf q [W/m2]	current density jdisplaystyle mathbf j [C/(m2·s) = A/m2]
resistance	flexibility 1/kdisplaystyle 1/k [...]	fluid resistance Rdisplaystyle R [...]	thermal resistance Rdisplaystyle R [K/W]	electrical resistance Rdisplaystyle R [Ω]
conductivity	stiffness kdisplaystyle k [N/m]		thermal conductivity1/Rdisplaystyle 1/R [W/(K·m)]	electrical conductance 1/Rdisplaystyle 1/R [...]
lumped element linear model	Hooke's law ΔX=F/kdisplaystyle Delta X=F/k	Hagen–Poiseuille equation ΔP=QRdisplaystyle Delta P=QR	Newton's law of cooling ΔT=Q˙Rdisplaystyle Delta T=dot QR	Ohm's law ΔV=IRdisplaystyle Delta V=IR
distributed linear model	... ...displaystyle ...		Fourier's law q=−k∇Tdisplaystyle mathbf q =-kboldsymbol nabla T	Ohm's law J=σEdisplaystyle mathbf J =sigma mathbf E

Explanation from an electronics point of view

Equivalent thermal circuits

The diagram shows an equivalent thermal circuit for a semiconductor device with a heat sink:
Qdisplaystyle Q is the power dissipated by the device.
TJdisplaystyle T_J is the junction temperature in the device.
TCdisplaystyle T_C is the temperature at its case.
THdisplaystyle T_H is the temperature where the heat sink is attached.
TAMBdisplaystyle T_AMB is the ambient air temperature.
RθJCdisplaystyle R_theta JC is the device's absolute thermal resistance from junction to case.
RθCHdisplaystyle R_theta CH is the absolute thermal resistance from the case to the heatsink.
RθHAdisplaystyle R_theta HA is the absolute thermal resistance of the heat sink.

The heat flow can be modelled by analogy to an electrical circuit where heat flow is represented by current, temperatures are represented by voltages, heat sources are represented by constant current sources, absolute thermal resistances are represented by resistors and thermal capacitances by capacitors.

The diagram shows an equivalent thermal circuit for a semiconductor device with a heat sink.

Example calculation

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Consider a component such as a silicon transistor that is bolted to the metal frame of a piece of equipment. The transistor's manufacturer will specify parameters in the datasheet called the absolute thermal resistance from junction to case (symbol: RθJCdisplaystyle R_theta JC), and the maximum allowable temperature of the semiconductor junction (symbol: TJMAXdisplaystyle T_JMAX). The specification for the design should include a maximum temperature at which the circuit should function correctly. Finally, the designer should consider how the heat from the transistor will escape to the environment: this might be by convection into the air, with or without the aid of a heat sink, or by conduction through the printed circuit board. For simplicity, let us assume that the designer decides to bolt the transistor to a metal surface (or heat sink) that is guaranteed to be less than ΔTHSdisplaystyle Delta T_HS above the ambient temperature. Note: T_HS appears to be undefined.

Given all this information, the designer can construct a model of the heat flow from the semiconductor junction, where the heat is generated, to the outside world. In our example, the heat has to flow from the junction to the case of the transistor, then from the case to the metalwork. We do not need to consider where the heat goes after that, because we are told that the metalwork will conduct heat fast enough to keep the temperature less than ΔTHSdisplaystyle Delta T_HS above ambient: this is all we need to know.

Suppose the engineer wishes to know how much power he can put into the transistor before it overheats. The calculations are as follows.

Total absolute thermal resistance from junction to ambient = RθJC+RθBdisplaystyle R_theta JC+R_theta B

where RθBdisplaystyle R_theta B is the absolute thermal resistance of the bond between the transistor's case and the metalwork. This figure depends on the nature of the bond - for example, a thermal bonding pad or thermal transfer grease might be used to reduce the absolute thermal resistance.

Maximum temperature drop from junction to ambient = TJMAX−(TAMB+ΔTHS)displaystyle T_JMAX-(T_AMB+Delta T_HS).

We use the general principle that the temperature drop ΔTdisplaystyle Delta T across a given absolute thermal resistance Rθdisplaystyle R_theta with a given heat flow Qdisplaystyle Q through it is:

ΔT=Q×Rθdisplaystyle Delta T=Qtimes R_theta ,.

Substituting our own symbols into this formula gives:

TJMAX−(TAMB+ΔTHS)=QMAX×(RθJC+RθB+RθHA)displaystyle T_JMAX-(T_AMB+Delta T_HS)=Q_MAXtimes (R_theta JC+R_theta B+R_theta HA),,

and, rearranging,

QMAX=TJMAX−(TAMB+ΔTHS)RθJC+RθB+RθHAdisplaystyle Q_MAX=T_JMAX-(T_AMB+Delta T_HS) over R_theta JC+R_theta B+R_theta HA

The designer now knows QMAXdisplaystyle Q_MAX, the maximum power that the transistor can be allowed to dissipate, so he can design the circuit to limit the temperature of the transistor to a safe level.

Let us substitute some sample numbers:

TJMAX=125 ∘Cdisplaystyle T_JMAX=125 ^circ mboxC (typical for a silicon transistor)

TAMB=21 ∘Cdisplaystyle T_AMB=21 ^circ mboxC (a typical specification for commercial equipment)

RθJC=1.5 ∘C/Wdisplaystyle R_theta JC=1.5 ^circ mathrm C /mathrm W , (for a typical TO-220 package^{[citation needed]})

RθB=0.1 ∘C/Wdisplaystyle R_theta B=0.1 ^circ mathrm C /mathrm W , (a typical value for an elastomer heat-transfer pad for a TO-220 package^{[citation needed]})

RθHA=4 ∘C/Wdisplaystyle R_theta HA=4 ^circ mathrm C /mathrm W , (a typical value for a heatsink for a TO-220 package^{[citation needed]})

The result is then:

Q=125−(21)1.5+0.1+4=18.6 Wdisplaystyle Q=125-(21) over 1.5+0.1+4=18.6 mathrm W

This means that the transistor can dissipate about 18 watts before it overheats. A cautious designer would operate the transistor at a lower power level to increase its reliability.

This method can be generalised to include any number of layers of heat-conducting materials, simply by adding together the absolute thermal resistances of the layers and the temperature drops across the layers.

Derived from Fourier's Law for heat conduction

From Fourier's Law for heat conduction, the following equation can be derived, and is valid as long as all of the parameters (x and k) are constant throughout the sample.

Rθ=xA×kdisplaystyle R_theta =frac xAtimes k

where:

• 
  Rθdisplaystyle R_theta is the absolute thermal resistance (across the length of the material) (K/W)


• 
  x is the length of the material (measured on a path parallel to the heat flow) (m)


• 
  k is the thermal conductivity of the material (W/(K·m))


• 
  A is the cross-sectional area (perpendicular to the path of heat flow) (m²)

Problems with electrical resistance analogy

A 2008 review paper written by Philips researcher Clemens J. M. Lasance notes that: "Although there is an analogy between heat flow by conduction (Fourier’s law) and the flow of an electric current (Ohm’s law), the corresponding physical properties of thermal conductivity and electrical conductivity conspire to make the behavior of heat flow quite unlike the flow of electricity in normal situations. [...] Unfortunately, although the electrical and thermal differential equations are analogous, it is erroneous to conclude that there is any practical analogy between electrical and thermal resistance. This is because a material that is considered an insulator in electrical terms is about 20 orders of magnitude less conductive than a material that is considered a conductor, while, in thermal terms, the difference between an "insulator" and a "conductor" is only about three orders of magnitude. The entire range of thermal conductivity is then equivalent to the difference in electrical conductivity of high-doped and low-doped silicon."^[3]

Measurement standards

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The junction-to-air thermal resistance can vary greatly depending on the ambient conditions.^[4] (A more sophisticated way of expressing the same fact is saying that junction-to-ambient thermal resistance is not Boundary-Condition Independent (BCI).^[3]) JEDEC has a standard (number JESD51-2) for measuring the junction-to-air thermal resistance of electronics packages under natural convection and another standard (number JESD51-6) for measurement under forced convection.

A JEDEC standard for measuring the junction-to-board thermal resistance (relevant for surface-mount technology) has been published as JESD51-8.^[5]

A JEDEC standard for measuring the junction-to-case thermal resistance (JESD51-14) is relatively newcomer, having been published in late 2010; it concerns only packages having a single heat flow and an exposed cooling surface.^[6][7][8]

Resistance in Composite Wall

Parallel thermal resistance

Similarly to electrical circuits, the total thermal resistance for steady state conditions can be calculated as follows.

Parallel Thermal Resistance in composite walls

The total thermal resistance

1Rtot=1RB+1RCdisplaystyle 1 over R_tot=1 over R_B+1 over R_C (1)

Simplifying the equation, we get

Rtot=RBRCRB+RCdisplaystyle R_tot=R_BR_C over R_B+R_C (2)

With terms for the thermal resistance for conduction, we get

Rt,cond=L(kb+kc)Adisplaystyle R_t,cond=L over (k_b+k_c)A (3)

Resistance in series and parallel

It is often suitable to assume one-dimensional conditions, although the heat flow is multidimensional. Now, two different circuits may be used for this case. For case (a) (shown in picture), we presume isothermal surfaces for those normal to the x- direction, whereas for case (b) we presume adiabatic surfaces parallel to the x- direction. We may obtain different results for the total resistance Rtotdisplaystyle R_tot and the actual corresponding values of the heat transfer are bracketed by qdisplaystyle q. When the multidimensional effects becomes more significant, these differences are increased with increasing |kf−kg|displaystyle .^[9]

Equivalent thermal circuits for series-parallel composite wall

Radial Systems

Spherical and cylindrical systems may be treated as one-dimensional, due to the temperature gradients in the radial direction. The standard method can be used for analyzing radial systems under steady state conditions, starting with the appropriate form of the heat equation, or the alternative method, starting with the appropriate form of Fourier's law. For a hollow cylinder in steady state conditions with no heat generation, the appropriate form of heat equation is ^[9]

1rddr(krdTdr)=0displaystyle 1 over rd over drleft(krdT over drright)=0 (4)

Where kdisplaystyle k is treated as a variable. Considering the appropriate form of Fourier's law, the physical significance of treating kdisplaystyle k as a variable becomes evident when the rate at which energy is conducted across a cylindrical surface, this is represented as

qr=−kAdTdr=−k(2πrL)dTdrdisplaystyle q_r=-kAdT over dr=-k(2pi rL)dT over dr (5)

Where A=2πrLdisplaystyle A=2pi rL is the area that is normal to the direction of where the heat transfer occurs. Equation 1 implies that the quantity kr(dT/dr)displaystyle kr(dT/dr) is not dependent of the radius rdisplaystyle r, it follows from equation 5 that the heat transfer rate, qrdisplaystyle q_r is a constant in the radial direction.

Hollow cylinder with convective surface conditions in thermal conduction

In order to determine the temperature distribution in the cylinder, equation 4 can be solved applying the appropriate boundary conditions. With the assumption that kdisplaystyle k is constant

T(r)=C1ln⁡r+C2displaystyle T(r)=C_1ln r+C_2 (6)

Using the following boundary conditions, the constants C1displaystyle C_1 and C2displaystyle C_2 can be computed

T(r1)=Ts,1displaystyle T(r_1)=T_s,1 and T(r2)=Ts,2displaystyle T(r_2)=T_s,2

The general solution gives us

Ts,1=C1ln⁡r1+C2displaystyle T_s,1=C_1ln r_1+C_2 and Ts,2=C1ln⁡r2+C2displaystyle T_s,2=C_1ln r_2+C_2

Solving for C1displaystyle C_1 and C2displaystyle C_2 and substituting into the general solution, we obtain

T(r)=Ts,1−Ts,2ln⁡(r1/r2)ln⁡(rr2)+Ts,2displaystyle T(r)=T_s,1-T_s,2 over ln(r_1/r_2)ln left(r over r_2right)+T_s,2 (7)

The logarithmic distribution of the temperature is sketched in the inset of the thumbnail figure.
Assuming that the temperature distribution, equation 7, is used with Fourier’s law in equation 5, the heat transfer rate can be expressed in the following form

Qr=2πLk(Ts,1−Ts,2)ln⁡(r2/r1)displaystyle Q_r=2pi Lk(T_s,1-T_s,2) over ln(r_2/r_1)

Finally, for radial conduction in a cylindrical wall, the thermal resistance is of the form

Rt,cond=ln⁡(r2/r1)2πLkdisplaystyle R_t,mathrm cond =ln(r_2/r_1) over 2pi Lk

See also

• Thermal engineering


• Thermal design power


• Safe operating area

References

• Michael Lenz, Günther Striedl, Ulrich Fröhler (January 2000) Thermal Resistance, Theory and Practice. Infineon Technologies AG, Munich, Germany.


• Directed Energy, Inc./IXYSRF (March 31, 2003) R Theta And Power Dissipation Technical Note. Ixys RF, Fort Collins, Colorado. Example thermal resistance and power dissipation calculation in semiconductors.

Further reading

There is a large amount of literature on this topic. In general, works using the term "thermal resistance" are more engineering-oriented, whereas works using the term thermal conductivity are more [pure-]physics-oriented. The following books are representative, but may be easily substituted.

• 
  Terry M. Tritt, ed. (2004). Thermal Conductivity: Theory, Properties, and Applications. Springer Science & Business Media. ISBN 978-0-306-48327-1.
  


• 
  Younes Shabany (2011). Heat Transfer: Thermal Management of Electronics. CRC Press. ISBN 978-1-4398-1468-0.
  


• 
  Xingcun Colin Tong (2011). Advanced Materials for Thermal Management of Electronic Packaging. Springer Science & Business Media. ISBN 978-1-4419-7759-5.
  

External links

• Guoping Xu (2006), Thermal Management for Electronic Packaging, Sun Microsystems


• http://www.electronics-cooling.com/2012/09/update-on-jedec-thermal-standards/


• The importance of Soil Thermal Resistivity for power companies