How to calculate the thermal resistance of a PCB

Thermal management is a crucial aspect of designing and manufacturing electronic devices, as heat generated by the components can lead to performance issues and even device failure. To ensure the reliable and efficient operation of electronic devices it is essential to understand how to effectively dissipate heat away from the heat generating devices. These heat generating components are often mounted on printed circuit boards (PCBs) which in many cases are used as a heat sink to cool the attached component.

Thermal resistance is a measure of how well a material or device transfers heat to its surroundings and is often used to predict the thermal performance of a PCB and to calculate the temperature of the attached component. This technical article will provide a detailed explanation of the formulas and method used to calculate the thermal resistance of a PCB. The calculation methodology will take into account the PCB insulation (FR4)/copper layer thicknesses, percent copper coverage in each layer, location of these layers with respect to the heat generating component and usage of thermal vias.

Layout and structure of the PCB

A PCB consists of alternating layers of dielectric and conductive materials typically made using FR4/resin and copper respectively as shown in figure 1. Layers that are used for ground planes often are 100% conductive material. While the layers with attached components, traces and interconnects will have some mixture of conductive and dielectric materials.

Figure 1. PCB layout

The preceding calculation procedure assumes that the component is intimately attached to the PCB using an exposed metal pad or some similar attachment method that creates an extremely low thermal resistance path between the component and the PCB. The component is located at or close to the geometric center of the PCB. The upper and lower surfaces of the PCB are cooled via natural convection or forced convection and radiation. A heat transfer coefficient h_ext is used to quantify the effectiveness of cooling from these outter surfaces. Reference the articles How to design a flat plate heat sink and Performance of a LED flat plate heat sink in multiple orientations for the formulas used to calculate h_ext for natural convection and radiation.

Calculating the thermal resistance of each PCB layer

To facilitate the preceding calculations the rectangular PCB and attached component are converted to circles of equivalent areas r_o and r_i respectively using equations 1 and 2. The approximation of the rectangular areas as circular areas introduces an error of less than 1% at an aspect ratio (width/length) of 1, increasing to an error 8% at aspect ratios of 2.5 or 0.4.

$\displaystyle r_{o}=\sqrt{\frac{W\times L}{\pi}}$ 1

$\displaystyle r_{i}=\sqrt{\frac{W_{s}\times L_{s}}{\pi}}$ 2

The composite nature of the PCB complicates the thermal analysis and does not allow for the PCB to be treated as a simple flat plate heat sink made from uniform thermal conductivity material. The thermal conductivity through the thickness and in the plane of the PCB are different. Additionally, the position of the dielectric and conducting layers from top to bottom affect how well heat is conducted from the heat generating component through the PCB and then to the surroundings.

The thermal resistance calculation is conducted layer by layer to account for the non-uniform thermal conductivity of the PCB. The section of each layer directly under the component is a disk with radius r_i and thickness t_n. This disk will be referred to as the “inner disk”. The periphery of the disk has an equivalent convection coefficient h_o,n that represents the effective cooling of the outer portion of the layer as shown in figure 2.

The subscript n in t_n, h_o,nand all other variables subsequently defined refers to the layer number where layer number 1 is attached to the component and N is the total number of layers in the PCB.

Figure 2. PCB layers and sections used in the analysis.

The outer portion of each layer is an annulus (referred to as the “outer annulus”) with inner and outer radii r_i and r_o respectively.

The calculation for h_o,n is developed with the approximation that the temperature difference between the layers of the PCB is small. As such the flow of heat is predominantly radial in the outer annulus. The resistance to heat flow in the outer annulus can then be approximated by parallel thermal resistances R_o,n as shown in figure 3.

Figure 3. Outer annulus layers and resistance network

The effective thermal conductivity k_eff,n of each layer is:

$\displaystyle k_{eff,n}=\sigma_{c,n}k_{c,n}+(1-\sigma_{c,n})k_{d,n}$ 3

where:
$\displaystyle \sigma_{c,n}$ is the volume fraction of conductor in layer n
$\displaystyle k_{c,n}$ is the thermal conductivity of the conductor of layer n
$\displaystyle k_{d,n}$ is the thermal conductivity of the dielectric of layer n

The value of σ_c,n is from 0 to 1. It can be determined by estimating the conductor coverage area on each layer and dividing that value by the total PCB area (L x W).

Thermal conductivities of 398W/m·K for copper and 0.3W/m·K for FR4 are most often used for the conductor and dielectric materials respectively.

The apparent convection coefficient h_ext,n acting on the outer annulus of each layer is:

$\displaystyle h_{ext,n}=\frac{k_{eff,n} \cdot t_{n}}{k_{eff}\cdot t}h_{ext}$ 4

The overall in-plane effective thermal conductivity of the entire PCB, k_effis:

$\displaystyle k_{eff}=\frac{\sum_{n=1}^{N}k_{eff,n}\cdot t_{n}}{t}$ 5

The resistances for the outer annulus of each layer R_o,n are calculated using the solution for a circular fin [1] expressed as:

$\displaystyle R_{o,n}=\frac{1}{2\pi r_{i} \cdot t_{n} \cdot k_{eff,n}\cdot \alpha_{n}} \cdot\frac{K_{1}(\alpha_{n}\cdot r_{o})I_{o}(\alpha_{n}\cdot r_{i})+I_{1}(\alpha_{n}\cdot r_{o})K_{0}(\alpha_{n}\cdot r_{i})}{I_{1}(\alpha_{n}\cdot r_{o})K_{1}(\alpha_{n}\cdot r_{i})-I_{1}(\alpha_{n}\cdot r_{i})K_{1}(\alpha_{n}\cdot r_{o})}$ 6

where:
$\displaystyle I_{0}$ modified zero order Bessel function of the 1st kind
$\displaystyle K_{0}$ modified zero order Bessel function of the 2nd kind
$\displaystyle I_{1}$ modified first order Bessel function of the 1st kind
$\displaystyle K_{1}$ modified first order Bessel function of the 2nd kind
$\displaystyle k_{eff,n}$ thermal conductivity of each layer

$\displaystyle \alpha_{n}=\sqrt{\frac{2\cdot h_{ext,n}}{k_{eff,n}\cdot t_{n}}}$ 7

The multiplication of h_ext by 2 in equation 7 indicates that both the upper and lower surfaces of the PCB are subjected to convective and or radiative cooling.

When equation 4 is substituted for h_ext,n in equation 7, α_n will be a constant value of:

$\displaystyle \alpha=\sqrt{\frac{2\cdot h_{ext}}{k_{eff}\cdot t}}$ 8

As such α can be substituted for α_n in equation 6.

The use of Bessel functions in equation 6 may seem complex, however Bessel functions are readily available in MS Excel or any commercial mathematics software as a built-in function.

With the value of R_o,n known, the equivalent convection coefficient h_o,n acting on the periphery of the inner disk of radius r_i can be calculated with equation 9.

$\displaystyle h_{o,n}=1/(2\pi r_{i} \cdot R_{o,n}\cdot t_{n})$ 9

Combining the PCB layer thermal resistances

The next step in the calculation is to determine the value of h_b,n, the effective heat transfer coefficient acting on the base of the inner disk of each layer as shown in figure 2. This is done by calculating the equivalent heat transfer coefficient that the inner disk of the n^th layer applies to the lower surface of the inner disk of the n-1 layer. Equation 10 uses the equation for an extended surface with convection heat transfer around the periphery and lower surface [2] to calculate h_b,n-1.

$\displaystyle h_{b,n-1}=\frac{\sqrt{h_{ext,n}\cdot P_{s}\cdot k_{eff,n}\cdot A_{s}}}{A_{s}}\times \frac{sinh(m_{n}t_{n})+\frac{h_{b,n}}{m_{n}\cdot t_{n}}cos(m_{n}t_{n})}{cosh(m_{n}t_{n})+\frac{h_{b,n}}{m_{n}\cdot t_{n}}sin(m_{n}t_{n})}$ 10

A_s and P_s are the cross-sectional area and perimeter respectively of the inner disk.

$\displaystyle A_{s}=\pi r^{2}_{i}$ 11

$\displaystyle P_{s}=2\pi r_{i}$ 12

$\displaystyle m_{n}=\sqrt{\frac{h_{ext,n}\cdot P_{s}}{k_{eff,n} \cdot A_{s}}}$ 13

The calculation is started at the last (N_th) layer with h_b,N=h_ext on the bottom surface of the inner disk for that layer. Using equation 10, h_b,N-1 is calculated. This process is repeated N times for all the layers resulting in a heat transfer coefficient h_pcb. This process is illustrated in figure 4.

Figure 4. Iterative calculation for h_pcb

The heat transfer coefficient h_pcb is the equivalent heat transfer coefficient of the PCB acting on the heat generating component. The thermal resistance of the PCB is then:

$\displaystyle R_{pcb}=\frac{1}{h_{pcb}A_{s}}$ 14

Thermal via calculations

Thermal vias placed under the heat generating component provide a path to more efficiently transfer heat from the heat generating component to the inner layers and base layer of the PCB. This then allows to the heat to be spread across these layers reducing the thermal resistance of the PCB.

Figure 5. PCB with thermal vias

The analysis of the effect of imbedded thermal vias as shown in figure 5 require the calculation of an equivalent through-plane thermal conductivity, k_thru,n for the inner disk of each layer. This is calculated by evaluating the flow of heat through the inner disk of each layer through three parallel thermal resistances:

R_barrel, the thermal resistance of the barrel of the thermal via.
R_filler, the thermal resistance of the filler material of the via if any.
R_matl,n, the thermal resistance of the layer material surrounding the thermal via in the s x s square region shown in figure 5.

$\displaystyle R_{barrel,n}=\frac{t_{n}}{k_{barrel}A_{barrel}}$ 15

$\displaystyle A_{barrel}=\pi\left[ \left(\frac{d_{via}}{2}\right)^2-\left( \frac{d_{via}}{2}-t_{via} \right)^2 \right]$ 16

$\displaystyle R_{filler}=\frac{t_{n}}{k_{filler\times}\pi\left( \frac{d_{via}}{2}-t_{via} \right)^2}$ 17

$\displaystyle R_{matl,n}=\frac{t_{n}}{k_{eff,n}\left[ s^2-\pi\left( \frac{d_{via}}{2} \right)^2 \right]}$ 18

The thermal resistance of the entire s x s, cell, R_cell,n for each layer is:

$\displaystyle R_{cell,n}={\left[ \frac{1}{R_{barrel,n}}+\frac{1}{R_{filler}}+\frac{1}{R_{matl}} \right]}^{-1}$ 19

If the thermal via is not filled then the 1/R_filler term in equation 19 is omitted.

The equivalent thermal conductivity of the inner disk with thermal vias is then:

$\displaystyle k_{via,n}=\frac{t_{n}}{R_{cell,n}}\left( \frac{s}{L_{s}\cdot W_{s}} \right)^{2}$ 20

In equation 13, k_eff,n is replaced with k_via,n and R_pcb is calculated for a PCB with vias under the heat generating component.

Junction to ambient thermal resistance calculations

R_pcb is the thermal resistance from the base of the component to the atmosphere. To determine the junction to ambient thermal resistance, the junction to board thermal resistance typically listed as Θ_jb or R_jb in the component literature is added to R_pcb. The temperature of the junction can now be determined using equation 21.

$\displaystyle T_{j}=T_{amb}+(R_{pcb}+R_{jb})\cdot Q_{s}$ 21

T_amb is the external ambient temperature and Q_s is the heat generated by the component.

This calculation excludes any heat loss from the top of the component exposed to the atmosphere. Typically, this heat loss is small when compared to the heat dissipation from the PCB. If there is a heat sink attached to the component, then the heat loss through the heat sink is significant and must be considered. Figure 6 shows the thermal resistance network for the PCB with a heat sink of thermal resistance R_hs, attached to the component. R_jc is the junction to case (top) thermal resistance.

Figure 6. Thermal resistance network for the PCB

Calculating for the junction temperature from this thermal resistance network yields:

$\displaystyle T_{j}=Q_{s}\left[ \frac{1}{R_{pcb}+R_{jb}}+\frac{1}{R_{hs}+R_{jc}} \right]^{-1}+T_{amb}$ 22

References

[1] F. Incropera, D. DeWitt, (2011). Fundamentals of Heat and Mass Transfer (7th ed.). Hoboken: Wiley. p124

[2] F. Incropera, D. DeWitt, (2011). Fundamentals of Heat and Mass Transfer (7th ed.). Hoboken: Wiley. p118, table 3.4