Thermal Bending Analysis

An Approximate Analytic Model for PWB Warpage

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The basis of the Thermal Bending Analysis

Recall that we decided to use bimaterial beam bending equation to model the warpage the PWB undergoes due to a uniform increase in temperature. The bimaterial beam bending model, as given by the sources we used, is reiterated below:



The following assumptions are made:

• length is much greater than the width, i.e. a beam
• the deflection is small enough to approximate as quadratic

then the following formula holds:

Deflection =

Where:
		L	=	Undeformed Length
		t	=	Undeformed Thickness
			=	Temperature Change
			=	Specific Coefficient of Thermal Bending

Calculating a value for undeformed length

The deflection calculated from the formula given above would then represent the 1D deflection a beam would undergo. How do we calculate deflection for a 2D surface like a PWB? And why does this affect the value used for undeformed length?

First, apply the beam approximation to each edge of the PWB.



When the beams deform due a temperature change, the deflection can be either up or down, depending on which side has the lower effective coefficient of thermal expansion. (For temperature increases, deflection occurs toward the side of lower thermal expansion.) Therefore, there are 4 possible deformed configurations:



Notice that in all of these cases, if we assume that the deformation at the edges completely defines the deformation within the interior of the PWB, the total PWB deflection is the sum of the deflection at each edge. Moreover, for both beams, the only variation is in the effective undeformed length of the beam- the thickness, coefficient of bending, and temperature change are constant. Therefore, these terms can be represented by a constant term k, and we have:









by Pythagorean's theorem.

So, we use the diagonal undeformed length of the PWB because it allows us to capture the 3D behavior of the deformed PWB.

To demonstrate that even with the geometric requirement that the shape of the edges completely determines the shape of the interior and with quadratic deformation curves it is still possibe to realistically model the 3D forms of PWB warpage, we are providing a 3D surface viewer free here.

Determing

is derivable from the material properties stored in the PWB Product model: the relative coefficients of thermal expansion (CTE) of each major material, the relative volumes of each material, and their respective elastic moduli. However, we have adopted the simpler approach of establishing by fitting it to experimental measurements of warpage. As of July 6, 1996, we have only used data from [Yeh et. al., 1993], but we will shortly be adding additional data points from [Stitler, 1996]. Note that since is derived from the materials' CTEs, which vary according to temperature, is also a function of temperature.

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This page last modified on: July 6, 1996