//Logo Image
**Author: Yeh-Liang Hsu, Yuan-Chan Hsu, Ming-Sho Hsu
(**05-13-2000 )**；****approved:
Yeh-Liang Hsu (08-09-2000).**

**Note: This paper was presented in “2000 PCB Manufacturing Technology
Conference,” Yuan Ze University, **Chung-Li ,
Taiwan , June
24, 2000.

# Shape optimal design of the
contact springs of a connector

## Abstract

## Introduction

## Modeling the contact springs of a connector

## The effect of the coefficient of friction

## Shape optimization of the contact spring.

## Conclusion and discussions

## Acknowledgement

## Reference

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. The contact spring is probably the most critical component in a connector. Mechanically, the contact spring provides the contact normal force, which establishes the contact interface as the connector is mated. The contact normal force also maintains the stability of the contact interface against mechanical disturbances during the application life.

However, the magnitudes of normal contact force and insertion force of a connector are closely related. Connector manufacturers have a basic struggle between the need for high normal contact forces and low insertion forces. In particular, designing connectors with large number of pins that are used with today’s integrated circuits and printed circuit boards often faces the difficulty caused by the by the 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.

This paper proposes how to find the optimal shape of the contact spring of a 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 the parameters are discussed. The shape of the contact springs is then parameterized and optimized. The required insertion force is minimized while the normal contact force is maintained at a specified value.

** Keywords:** connector; contact spring;
contact normal force; insertion force; shape optimal design.

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 the most important parameter 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 minimum contact resistance of a connector is dependent on the normal force. When two surfaces are brought together, the contact area will be small and the current across the interface will be constricted to flow through this restriction area, which will result in an increase in resistance. This constriction resistance can be expressed as [Wager, 1971].

_{} (1)

where _{} is the
constriction resistance, _{} is the resistance of the wire, *H* is the hardness of the material, and _{} is the normal
force. From this equation, it is obvious that 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
experiment 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

On the other hand, designing connectors with large number of pins that are used with today’s integrated circuits and printed circuit boards often faces the difficulty caused by the by the associated rise in connector insertion force. Too high an insertion force could cause problems in assembly and 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.

Other concepts are also proposed in order to reduce the insertion force without sacrificing contact forces of a connector. Various designs of zero insertion force (ZIF) connectors are discussed [Bertoncini, et al., 1991; Chikazawa, et al., 1990; Engel, et al., 1989]. However, gold plates and contact springs are still used in most board to board connectors.

Structural topology and shape optimization has been a very active research field. The idea is to combine geometrical modeling, finite element analysis, and optimization into an integrated structure 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.

This paper proposes how to find the optimal shape of the contact spring of a 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 the parameters are discussed. The shape of the contact springs is then parameterized and optimized. The required insertion force is minimized while the normal contact force is maintained at a specified value.

Figure 1 shows a pair of contact springs of a board-to board 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 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 is considered rigid, compare to
the deflection of 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 _{}. In the example presented in Yamada and Ueno’s work, the
analytical expression of insertion force vs. insertion depth matches well with
experiment data, using the 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. As shown in 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

_{} (2)

_{} is also the
required insertion force at this contact point. The projection of the resultant
force in the *x* direction _{} is

_{} (3)

A finite element analysis model is built to calculate the insertion force and normal force of the contact springs in Figure 1. Two-dimensional 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.

The insertion force and normal force are calculated under similar assumptions discussed above. First the contact surface of the springs during the insertion process is identified and divided into discrete contact points, as shown in Figure 3. The insertion process is assumed to be quasi-static, therefore the static equilibrium equations such as Eq.(2) and (3) are satisfied at a contact point.

Figure 3. Discrete contact points on the contact surface of the spring.

As shown in
Figure 3, first the amount of deflection of the spring in the *x* direction _{} at a contact
point _{} is calculated
from the thickness of the PCB and the geometry of the spring. Then the value of
_{} that would cause
the deflection _{} is obtained using
a secant-type interpolation procedure. This process continues in an iterative
manner until the values of _{} at all contact
points are obtained. Figure 4 shows normal force vs. insertion length for the
contact springs in Figure 1. The coefficient of friction is assumed to be
0.175. Substituting into Eq.(2), Figure 5 shows insertion force vs. insertion
length.

Figure 4. Normal force vs. insertion length

Figure 5. Insertion force vs. insertion length

In the
mechanical specifications of the connector, the maximum insertion force is

Figure 5 also shows the effect of the unsymmetrical shape of the left and right springs. There are two peaks of insertion force. The left spring comes into contact first, and the peak insertion force of the left spring occurs earlier than that of the right spring. Therefore when the insertion force contributed by the left and right springs are added together, the peak insertion force during the process 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. The maximum stress occurs at the root of the springs. The tensile strength of copper alloy is 635 Mpa. The right spring is over stressed at the root of the spring.

Figure 6. Stress distribution of the left and right springs

Figure 7 shows
insertion force versus insertion length for _{}, 0.225, 0.300. The maximum insertion force increases from
56.0gmf (_{}) to 90.7 gmf (_{}), for almost 60%. Also note that at _{}, the maximum insertion force switches to the right spring.
In the mean time, The maximum normal force depends on the stiffness of the
spring, and does not change significantly as the coefficient of friction
increases. The important role of the coefficient of friction was noticed in
reducing the insertion force without sacrificing the contact normal force of a
connector. For example, Scholz, et al. [1992] suggested the application of
thiol-based 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 of several samples of the connector were measured. The insertion forces
of the connectors ranged from

In the current design, the normal contact force 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 shape of the contact springs to further reduce the maximum insertion force, while the normal force and the maximum stress are kept within the 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

The design variables used in the shape optimization process are the positions of the control points that describe the geometry of the contact spring. 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.

Figure 8. The geometry of the spring is divided into three portions

The geometry of the contact surface is very crucial to the insertion force of the contact spring. For each spring, 7 control points are used to describe this profile. 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. 6 control points are used to describe the geometry of the bending arm. 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. 9 control points are used to describe this profile. For a pair of contact springs, there are a total of 44 control points.

To avoid
generating unreasonable shape during the optimization process, the control
points are allowed to move only in the normal direction, as shown in Figure 8.
The amount of movements _{} of the total 44
control points are the design variables in the optimization process. This
optimization problem can be written into the following mathematical format:

min. _{}

s.t. _{}

_{}

_{} (4)

where *N* is the specified
minimum normal force, *S* is the
allowable stress of the material of the spring. The functions _{}, _{}, _{}, and _{} are evaluated by
finite element analysis as described in the previous sections. Method of
centers [Vanderplatts, 1884] is used as the optimization algorithm. The
sensitivities of functions _{}, _{}, _{}, and _{} required in the
algorithm are calculated by finite differences.

At this point we have not obtained the optimal shape design for the contact springs.

This paper shows how the insertion of a PCB into the contact springs of a connector is modeled by finite element analysis. The insertion force, normal force, and maximum stress of the contact springs can be calculated. The effect of the coefficient of friction is also discussed.

The process of finding the optimal shape of the contact springs of a connector is proposed. At this point we have not obtained the optimal shape design for the contact springs. It is expected that the insertion force can be further reduced while the normal contact force is maintained within a specified value.

The authors gratefully acknowledge Nextronics Engineering Corporation, especially Ms. Angela Liu, for providing valuable technical data and professional experience for this research.

Bertoncini, D.,
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Engel, P.A.,
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