Author: YehLiang Hsu, YuanChan Hsu, MingSho Hsu (20000804)；modified:
YehLiang Hsu (20030207); recommended: YehLiang Hsu (20000804).
Note: This paper is published in Transactions of the ASME, Journal of
Electronic Packaging, Vol. 124, No. 3, September, 2002, p. 178~183.
Shape optimal design of contact
springs of electronic connectors
Abstract
An electronic
connector provides a separable interface between two subsystems of an
electronic system. The contact spring is probably the most critical component
in an electronic connector. Mechanically, the contact spring provides the
contact normal force, which establishes the contact interface as the connector
is mated. However, connector manufacturers have a basic struggle between the
need for high normal contact forces and low insertion forces. Designing
connectors with large numbers of pins that are used with today’s integrated
circuits and printed circuit boards often results in an associated rise in
connector insertion force. It is possible to lower the insertion force of a
connector by redesigning the geometry of the contact spring, but this also
means a decrease in contact normal force.
In this paper,
structural shape optimization techniques are used to find the optimal shape of
the contact springs of an electronic connector. The process of the insertion of
a PCB into the contact springs of a connector is modeled by finite element
analysis. The maximum insertion force and the contact normal force are
calculated. The effects of several design parameters are discussed. The
geometry of the contact springs is then parameterized and optimized. The
required insertion force is minimized while the normal contact force and the
resulting stress are maintained within specified values. In our example, the
insertion force of the final contact spring design is reduced to 68.3% of that
of the original design, while the contact force and the maximum stress are
maintained within specified values.
Keywords: connector; contact spring;
insertion force; shape optimal design.
Introduction
A connector
provides a separable interface between two subsystems of an electronic system.
The main function of a connector is to carry a signal or to distribute power.
There are three types of connectors: circuit board to circuit board, wire to
circuit board, and wire to wire [Mroczkowski, 1994]. Note that here the term
“wire” also includes cables. The contact spring is probably the most critical
component in a connector. Mechanically, the contact spring provides the contact
normal force, which is a very important index for connector design.
All separable electrical contacts require a force to hold together
the male and female halves when mated. The contact normal force provides the
force that establishes the contact interface as the connector is mated, and it
maintains the stability of the contact interface against mechanical
disturbances during the application life. Moreover, the contact resistance of a
connector is dependent on the normal force. To reduce the constriction
resistance, a high contact normal force is desired.
Normal force is
also required to maintain the stability of the contact interface. Sawchyn and
Sproles [1992] examined connectors with different geometry parameters. Their
experimental results also showed that connectors which are designed with
contact geometries that provide higher local pressures, are more effective in
overcoming the interference of the dust contamination. For metal connector
contacts, a normal force of 100g
has been a widely accepted design guideline that provides a comfortable margin
to ensure reliability in general applications.
On the other
hand, designing connectors with large numbers of pins that are used with
today’s integrated circuits and printed circuit boards (PCB) often results in
an associated rise in connector insertion force. Too high an insertion force
could not only cause problems in assembly, but also cause mechanical failure in
other parts of the electronic package. Connector manufacturers have a basic
struggle between the need for high normal contact forces and low insertion
forces. The magnitudes of normal contact force and insertion force of a
connector are closely related. It is possible to lower the insertion force of a
connector by redesigning the geometry of the contact spring, but this also
means a decrease in contact normal force.
Design concepts
other than contact springs havealso been proposed in order to reduce the
insertion force without sacrificing contact forces of an electronic connector.
Zero insertion force (ZIF) connectors were widely discussed in the early ‘90
[Bertoncini, et al., 1991; Chikazawa, et al., 1990; Engel, et al., 1989] and
are commonly seen in electronic systems these days. However, gold plates and
contact springs are still widely used in boardtoboard connectors.
Structural
topology and shape optimization has been a very active research field [Hassani
and Hinton, 1998; Hsu, 1995]. The idea is to combine geometrical modeling,
finite element analysis, and optimization into an integrated structural design
process. The purpose of structural shape optimization is to determine the
optimal shape of the structure that has the best performance, while satisfying
all design requirements. The shape optimization methodology can certainly be
applied to the design of the geometry of the connector contact springs. Sehring
[1990] has suggested using finite element analysis techniques for design
optimization of electronic connectors.
In this paper,
structural shape optimization techniques are used to find the optimal shape of
the contact springs of an electronic connector. The process of the insertion of
a PCB into the contact springs of a connector is modeled by finite element
analysis. The maximum insertion force and the contact normal force are
calculated. The effects of several design parameters are discussed. The
geometry of the contact springs is then parameterized and optimized. The
required insertion force is minimized while the normal contact force and the
resulting stress are maintained within specified values. In our example, the
insertion force of the final contact spring design is reduced to 68.3% of that
of the original design, while the contact force and the maximum stress are
maintained within specified values.
Modeling the contact springs of a connector
Figure 1 shows a
pair of contact springs of a boardtoboard connector made of copper alloy. The
springs are fixed in a plastic housing, with pins sticking out of the housing.
Note that only a portion of the interface between the springs and the housing
is in interference fit. The left and right springs have different shapes. When
the gold plate of a PCB is inserted between the springs, the forces applied by
the springs are unsymmetrical. However, in a connector, the pairs of springs
are arranged in an alternate pattern, i.e., the left and right springs switch
positions in the neighboring pair. Therefore force balance is maintained for
the whole connector.
Figure 1. The configuration of a pair of contact
springs of a connector.
Yamada and Ueno
[1990] presented an analysis of insertion force in elastic/plastic mating. They
considered each instance of the insertion process as a static equilibrium of
the insertion force and the contact force, including the Coulomb’s friction
force. The board inserted between the springs was considered rigid relative to
the springs. As shown in Figure 2, at the contact point, the contact force
direction is offset from the normal direction by the friction angle _{}. The insertion process is assumed to be quasistatic. In the
example presented in Yamada and Ueno’s work, the analytical expression of
insertion force vs. insertion depth matches well with experimental data, using
the static coefficient of friction _{}.
Figure 2. Analysis of insertion force.
Under similar
assumptions, Ling [1998] also presented a useful analysis for mating mechanics
and stubbing of separable connectors. Also referring to Figure 2, there are two
force components at the contact point: the normal force _{} and the
tangential force _{}, and _{}. The projection of the resultant force in the y direction can be expressed as
_{} (1)
where _{} is also the
required insertion force at this contact point. The projection of the resultant
force in the x direction _{} is
_{} (2)
As shown in
these equations, the insertion force and normal force depend heavily on the
geometry of the springs, and cannot be determined using a simple beam model. A
finite element analysis model is built to calculate the insertion force and
normal force of the contact springs in Figure 1. Twodimensional plane stress
elements are used. Contact elements are added at the interface between the PCB
and the springs. The amount of the gap and interference is prescribed as
designed.
The insertion
force and normal force are calculated under similar assumptions discussed in
the literature. First the contact surface of the springs during the insertion
process is identified and divided into discrete contact points, as shown in
Figure 3. It is assumed that there is only one contact point at a given
position of PCB. The insertion process is assumed to be quasistatic, therefore
the static equilibrium equations such as Eq.(1) and (2) are satisfied at a
contact point.
Figure 3. Discrete contact points on the contact
surface of the spring.
As shown in
Figure 3, to find the value of contact normal force _{} at a contact
point _{} when the PCB is
inserted, first the amount of deflection of the spring in the x direction _{} is calculated
from the thickness of the PCB and the geometry of the spring. Then a
secanttype interpolation procedure is used to find the value of _{} that would cause
this deflection _{} in the x
direction. This process continues in an iterative manner until the values of _{} at all discrete contact
points are obtained. Figure 4 shows “normal force versus insertion length” for
the contact springs in Figure 1. The coefficient of friction is assumed to be 0.175 in this figure. Substituting into
Eq.(2), Figure 5 shows “insertion force vs. insertion length” during the
insertion process.
Figure 4. Normal force vs. insertion length.
Figure 5. Insertion force vs. insertion length.
Figure 5 also
shows the effect of the unsymmetrical shapes of the left and right springs.
There are two peaks in the curve of total insertion force. In this case, the
maximum insertion force 59.8g
occurs at the first peak. Because of the unsymmetrical shapes, the left spring
comes into contact first and the peak insertion force of the left spring occurs
earlier than that of the right spring. At this contact point, the insertion
force of the right spring is still low. Therefore when the insertion force
contributed by the left and right springs are added together, the total peak
insertion force is only slightly higher than that of the left spring. If the
shapes of both springs were identical, the peak insertion force would be twice
the peak insertion force of a single spring.
Figure 6 shows
the stress distribution of the left and right springs calculated by the elastic
finite element model. The maximum stresses, 582MPa for the left spring and 752MPa
for the right spring, occur at the root of the springs. The yield stress of
copper alloy is 635MPa. The right spring is over stressed at the root of the
spring.
Figure 6. Stress distribution of the left and
right springs.
The effect of the coefficient of friction
Figure 7 shows
insertion force versus insertion length for _{}, 0.225, 0.300. The maximum insertion force increases from 56.0g (_{}) to 90.7g
(_{}), or almost 60%. Also note that when _{}, the location of the maximum insertion force switches to the
peak of the right spring. In the mean time, the maximum normal force does not
change significantly as the coefficient of friction increases.
The important
role of the coefficient of friction was noticed and utilized in designing
connectors for reducing the insertion force without sacrificing the contact
normal force. For example, Kartlucke, et al. [1992] suggested the application
of thiolbased coatings to silver surfaces to provide good corrosion protection
and a reduction of the insertion and extraction forces required for electrical
connectors.
Figure 7. Insertion force versus insertion length
for various coefficients of friction.
The insertion forces obtained from
our analysis were compared to experimental data. Three samples of the connector
were measured experimentally. The insertion force of each sample was measured
30 times. In the 90 measurements, the insertion forces of the connector samples
ranged from 59.5g
to 69.4g per pin, and
average insertion force was 62.6g
per pin. Comparing with Figure 5 and Figure 7, the coefficient of friction of
our contact springs is estimated to be 0.18~0.21.
In the
mechanical specifications of the connector, the maximum insertion force is 95g per pin, and the minimum normal force
is 60g per pin. From our
analysis, normal force at the final contact point is 118g for the left spring and 148g for the right spring for _{}. The maximum insertion force during the insertion process is
61.8g. Both contact
force and insertion force satisfy the specifications. The maximum stress at the
root of the right spring is 708.2MPa, which is higher than the yield strength
of the copper alloy (635MPa). The size of the spring is very small (the width
of the spring is about 0.5mm).
Therefore plastic deformation resulted from the high stress was not visually
observable in the experiments.
Another design
parameter we explored was the amount of interference at the interface between
the spring roots and the housing (as shown in Figure 1), which might affect the
stiffness of the springs. However, we found that varying the length and depth
of interference does not have significant effect on the contact normal force
and insertion force of the contact springs.
Constructing the optimization model
In the current
design, the normal contact forces of both left and right springs are much
higher than that required by the specification. In the mean time, the stress of
the right spring is too high. We can optimize the geometry of the contact
springs to further reduce the maximum insertion force, while the normal force
and the maximum stress are kept within specified values. The shape optimization
problem of the contact spring can be described as follows:
minimize the
maximum insertion force
subject to the
normal force has to be higher than a specified value
the
maximum stress has to be lower than the strength of material
There are also constraints on the geometry of the contact springs
required by the manufacturer. These geometry constraints are embedded in the
definition and restriction of the design variables used in the shape
optimization model. The design variables are the positions of the control
points that describe the geometry of the contact spring. As shown in Figure 8,
a total of 48 control points are used to describe the geometry of the springs.
To avoid generating unreasonable shapes during the optimization process, the
control points are allowed to move only in the normal direction as shown in the
Figures. The amount of movement _{} of the 48 control
points are the design variables in the optimization process.
Figure 8. The control points of the springs.
As shown in
Figure 8, the geometry of the spring is divided into three portions: the
contact surface, the bending arm, and the root area. The y coordinates
of the final contact points of both springs are given as part of the
specification of the springs, and have to remain fixed. Therefore, one control
point is located exactly at the final contact point for each spring, as shown
in Figure 8. This control point can only move in the horizontal direction,
which is also its normal direction. A total of 4 control points are used to
describe a smooth contact surface for the right spring, while 3 control points
are used for the left spring.
As discussed
earlier, the contact normal force depends mostly on the stiffness of the
spring, which is decided by the geometry of the bending arm portion of the
spring. 10 control points are used to describe the geometry of the bending arm
of the right spring, and 12 control points are used for the left spring. The
maximum stress and stress concentration may occur around the root area of the
spring. Therefore the profile of the spring at the root area also has to be
carefully designed. 10 control points are used to describe this profile for
each spring. For installation purposes, a lower bound is imposed on the y
coordinates of these control points.
Finally, this
optimization problem can be written into the following mathematical form:
min. _{}
st. _{}
_{}
_{}
_{} (3)
where d is the vector of design variables _{}, i=1,…48, N
is the specified minimum normal force, and S
is the allowable stress of the material of the spring. In our example, N=100g and S=600MPa. Note that we
demand less contact force than in original design. Function _{} is the total
insertion force. Functions _{} and _{} are the normal
contact forces of the left and right springs. These functions and the maximum
stress of the springs _{} and _{} are evaluated by
finite element analysis, as described in the previous sections.
Shape optimization result of the contact springs
Sequential
linear programming (SLP) [Vanderplatts, 1984] is use to find the solution of
Eq.(3). In this optimization algorithm, the optimization model in Eq.(3) is
linearized at the initial design point _{} to form the
following linear programming subproblem:
_{}
_{}
_{}
_{}
_{}
_{} (4)
The
sensitivities (the first derivatives) of functions _{}, _{}, _{}, and _{} required in
Eq.(4) are calculated by finite differences. This linear programming
subproblem is solved to obtain a new design point. The insertion force, normal
force, and the maximum stress of new design point are calculated to see whether
this new design point satisfies the termination conditions. If not, the
optimization model in Eq.(4) is linearized again on this new design point. This
process continues in an iterative manner until the termination conditions are
satisfied.
The last
constraint in Eq.(4) imposes “move limits” on the design variables. This is
because the linear programming subproblem (Eq.(4)) is a good approximation to
the original nonlinear optimization model (Eq.(3)) only in the neighborhood of
the current design point. In our example, the move limit is 0.02mm initially, and is reduced to half
if the new design point obtained from the linear programming subproblem
becomes highly infeasible.
The SLP algorithm terminates after
5 iterations. Table 1 shows the iteration history. The insertion force of the
final contact spring design is reduced to 42.2g,
68.3% of that of the original design (61.8g).
The maximum stress of the right spring drops from 708MPa to 564MPa. The contact
normal forces of the final design are 100g
for the left spring, and 113g
for the right spring. The SLP algorithm terminates because the design obtained
in the 5^{th} iteration satisfies all constraints, and the improvement
in objective function is less than 1% comparing with the objective value of the
4^{th} iteration.
Table 1. The iteration history

original
design

1^{st}
iteration

2^{nd}
iteration

3^{rd}
iteration

4^{th}
iteration

5^{th}
iteration


left

right

left

right

left

right

left

right

left

right

left

right

_{}(MPa)

541

708

498

618

497

591

465

576

489

562

457

564

_{}(g)

118

148

107

126

103

122

102

113

100

114

100

113

_{}(total, g)

61.8

46.7

44.1

43.0

42.3

42.2

Figure 9 shows
the final designs of the contact springs. The shapes of the springs do not have
drastic change, the maximum movement of the control points is less than 9% of
the width of the spring, though the performance of the springs has been
significantly improved. Since the change is so small that it is difficult to
see the difference if the new geometry is superimposed on the original
geometry. The small arrows in the figure are used to indicate the directions of
movement of the control points.
Figure 10. Final designs of the contact springs.
Conclusions and discussions
This paper describes how the
insertion of a PCB into the contact springs of an electronic connector is
modeled by finite element analysis. The insertion force, normal force, and
maximum stress of the contact springs can be calculated, and the analysis
result matches well with experimental data.
Some design guidelines can be
concluded from this analysis:
(1)
The unsymmetrical shapes of the
left and right springs effectively reduce the peak insertion force.
(2)
The coefficient of friction
between the PCB and the contact springs has significant effect on the insertion
force. Reducing the coefficient of friction can reduce the insertion force
without sacrificing the contact normal force.
(3)
The amount of interference at
the interface between the spring roots and the housing does not have
significant effect on the contact normal force and insertion force of the
contact springs.
Connector
manufacturers have a basic struggle between the need for high normal contact
forces and low insertion forces, both are very sensitive to the shape of the
geometry of the contact springs. In this paper, the analysis procedure is
further combined with structural shape optimization techniques to carefully
design the shape of the contact springs. The insertion force of the connector
can be further reduced while the normal contact force and the maximum stress
are maintained within specified values. In our example, after shape
optimization, the shapes of the springs do not have drastic change, though the
performance of the springs has been significantly improved.
Acknowledgement
This research is
partially supported by National Science Council, Taiwan, ROC, grant number
NSC892212E155004. This support is gratefully acknowledged. The authors
also thank Nextronics Engineering Corporation, Taiwan, especially Ms. Angela Liu,
for providing valuable technical data and professional experience for this
research.
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