- Email: [email protected]
An analysis of dimensional and locational variations in surface-mounted component assembly
ELSEVIER
An analysis of dimensional and locational variations in surfacemounted component assembly T Radhakrishnan Mechanical Engineering Department, Villanova Univeniq,
Villanova, PA 19085, USA
This study focuses on the errors involved in the component fetching and placement phase of pc board assembly with surface-mounted components. Various critical dimensional and locational parameters involved in this phase, which can lead to mis-assembly, are identified. The complex relationships between these parameters, which govern the proper component (pin) placement on the pc board (pad), are mathematically modeled, using both additive and statistical error compounding methods. The use of the model is shown with a quantitative example. 0 1998 Elsevier Science Ltd. All rights reserved
addressed’-‘. The accuracy of placement of such components on the pc board is very critical to achieve a high yield rate of properly assembled boards and to reduce the amount of scrap or rework needed. Earlier published workXm7provides a study of some of the parameters that affect the placement of throughhole components on pc boards. SMCs pose greater assembly problems, partly due to the lack of positive placement registration found in the case of throughhole components. Further, the problem is more pronounced owing to the decreasing component sizes and pin-to-pin distances for SMCs, with very little room for errors. An in-depth analysis of this problem is therefore required. In order to improve the accuracy of the process and maintain an efficient assembly system, the various critical parameters involved in the process should be first identified. This study focuses on the phase of the assembly process involving picking up a component from the component storage area and placing it on the pc board. The critical parameters involved here include various dimensions, motions and related tolerances. These parameters, once identified, should then be mathematically related to provide a scientific tool for analyzing their effects on the proper placement (and hence assembly) of components on the pc board. Such a tool would then be very useful in the understanding, design/development and control of such assembly systems, to reduce errors in the placement of SMCs on pc boards.
pin dimensions pad dimensions angular errors for pin angular errors for board/pad x, y linear errors for pin location and part-fetching process X, y linear errors for board and pad locations angular quantities used in the derivations x and y distances between board and ideal pad centers X- and y-shifts of pin or pad corners (indicated by the subscripts) corners of a rectangular pin foot (in xy plane) corners of a rectangular plane)
pad (in q
Introduction The automation of printed circuit (PC) board assembly with surface-mounted components (SMCs) is an important area of electronic assembly today. Two desirable features in such a process are high speed and flexibility to handle a variety of components. SMCs are widely used and their assembly involves various critical issues that need to be 191
Analysis of dimensional and locational variations: T Radhakrishnan
192
Assembly process and critical parameters Figure
I illustrates a general pc board assembly system with SMCs and shows a typical SMC placement on the pc board. Proper placement of the component means that the feet of the pins (or leads) are located properly on their corresponding contact pads on the pc board. For a good assembly, it is desirable to have at least 75% of the pin’s foot area on the padx. The process involves a robot or other component fetching and placement mechanism (Figure I), to move to the component storage area and pick up a typically using vacuum suction. A component, number of different SMCs may be stored in the storage area, in trays or reels. A process control program would determine the sequence of components to be picked and the location/orientation of placement on the pc board. Once picked, a component is moved to the pc board area and placed so that the pins are positioned over their respective contact pads. The assembler system can be
considered to be set up and involve motions along a set of Cartesian coordinate axes as shown. The sources of errors involved in this process would therefore consist of: dimensional variations for the component pins and pads locational variations for the components at storage and for the pads on the board angular errors in the pin configurations due to bending or twisting inaccuracies in the motions involved in fetching and placing a component. It may be noted that the variations or inaccuracies indicated here have a direct bearing on the specification of various tolerances and required motion control involved in the system. The next section details these critical parameters. By considering the placement of a typical pin on its pad, mathematical relationships involving these parameters can be developed.
Development of mathematical relationships Figure 2 shows a typical pin and the corresponding rr7__-____--____-7
II II
I I
component storage area
I I# I I& I I 0’ LI_A____--__--_-__J
I I
I I I I
component fetching/ ’ placing unit
pad on the board, and some of the associated error sources. Including the effects of tolerances (errors), let 1, be the minimum length of the pin (foot), wN be the minimum width of the pin (foot), h, be the maximum height of the pin (body), &O, be the angular error (in the YZ plane) due to pin bend, &8, be the angular error (in the XZ plane) due to pin bend, +8, be the angular error (in the XY plane) due to pin twist, 1, be the minimum length of the pad, wD be the minimum width of the pad. These extreme pin and pad sizes are used in the analysis, along with appropriate angular error directions, to consider the maximum effect of the errors in the placement of the pin over the pad. Let E,,F,,G,,H,, and P,,Q,,R,,S,, denote the rectangular pin and pad configurations, respectively, on the XY plane, under ideal placement conditions as shown in Figure 2. (The pin foot is assumed to be slightly less in size than the corresponding pad. Also, the thicknesses of the pin and pad are assumed to be negligible, compared with their other dimensions.) Figure 3 shows the effect of the angular error U,, on the configuration of the pin. Corners E,, and F,, of the pin foot are now displaced to locations El and F, as shown (the other two corners G,, and H,, have similar displacements). Hence, the displacements (&) for the pin corners, along the X axis, can be expressed as
Figure 1 General
SMC assembly system.
Ax,, = AxH, = hN sin I),
(I)
Ax,, = Ax,, = I, sin ($ +(),)/sin Ic/- I,
(2)
Analysis
of dimensional and locational variations: T Radhakrishnan
193
6 El
A%2-
A%2 Z
I-
Y
Figure 4 Effect of angular error 0, on pin configuration.
where 1,9= tan ‘(IN/hN) It is assumed that displacements along the Z direction will not affect the final assembly, since soldering can fill any gaps between the pin and pad in this direction. Figure 4 shows the effect of angular error OXon the further shift of the pin (compounded with the effects of previously mentioned linear and angular errors). The corners of the pin, as seen on the YZ plane, shift to locations E2F2G2Hz. Hence, the Y displacements (Ay) can be expressed as Figure 2 Some typical error sources associated with the pin and pad.
Ay,. = 1/2w, -h,
cos 0, sin (/I~- (I,)
(3)
AyFz= 1/2w, - I, cos($ + O,.) sin (p, - (I,y)/sin$
(4)
A.vti2= I, cos($ + Oy) sin (p, + (),)/sin $ - 1/2w,
(5)
Ay,,, = h, cos 0, sin (pz + (IV)- 1/2w,
(6)
where p, = sin ~‘[1/2w,l(/,
cos($+(I,)lsinII/)]
pz= sin~‘[l/2w,/{h,cos0,]1 Figure
5 shows the effect of angular error (I,, compounded on the effects of previous errors. The pin corners now shift to locations E3,F,,G3,H,, as seen on the XY plane. The X and Y shifts for the corners can therefore be expressed as
z
L
AxE7= Axm = 1/2w, cos 0, sin 0,
(7)
A.v,g= Ay,, = 1/2w, cos Ox(1 - cos 0,)
(8)
AxFT= I, cos 0, cos (7 - O,)/cos 7 - I, cos (I,
(9)
AyFq= 1/2w, cosO,X
Figure 3 Effect of angular error OY on pin configuration.
I, COS()~sin(y-O,)/cosy
Axo3 = I, cos 0, - I, cos 0,. cos (;, + (I,)lcos ;I
(10) (II)
A?‘(;1= I, cos 0, sin (;q+ 0,)lcos ;’- 1/2w, cos Ox (12)
Analysis of dimensional and locational variations: T Radhakrishnan
194
I-
the ideal position, the pad undergoes linear shifts in the X and Y directions, due to errors in board location on the assembler table (&?, fiy,) and due to errors in the location of the pad on the board (bx4, by,). P,Q,R,S, (F @u-e 6) indicates the shift of the pad due to the linear errors alone. In addition, both the board and the pad on the board can have angular errors of their own (about their central axis). Let I: and w indicate the angular errors of the board a+ pad, respectively. Then, the combined effects of these various linear and angular errors for the pad (and board) yield the final pad configuration PQRS, displaced from the initial configuration P,,Q,,R,,S,,, with the pertinent dimensional errors of the pad. (In Figure 6, P’Q’R’S’ indicates the pad configuration with the effect of all errors except that of u.) Hence, the total shift in the corners of the pad can be algebraically expressed as:
Y
,,
Ax, = - Ax,,;,~, - I,, cos (R - w) + I/21,
(21)
A_v,,= - Aypad+ 1/2w, - I,, sin (Sz- w)
(22)
Ax, = - AXIS,<, - I /21, + I,, cos (0 + o)
(23)
A_vu= - A?+ - I,, sin (Cl + o) + I /2w,,
(24)
A
Figure 5 Effect of angular error 0, on pin configuration.
where y= tan-‘[1/2w,
cosU,/(/,
COSU,}]
Furthermore, let &,, iiy, indicate the errors in the location of a pin on the body of component along X and Y directions, and &, by2 indicate the errors in each of the X and Y motions (to-and-fro motions) involved in the fetching and placing of a component (which will therefore affect its position during placement). Then, the total displacement of the corners of the pin projected on the XY plane, from the original configuration E,,F,,G,,H,, to a final configuration EFGH, can be algebraically expressed as: Ax, = Ax,, + Ax,, + 6x
(13)
AyE= Ayw + AY,, + dy
(14)
Ax, = Ax,, + Ax,, + Sx
(15)
Ay, = Ay,, + AY,, + dy
(16)
Ax, = AxG, - AxG3+ 6x
(17)
Ay, = Aycjr+ Ay,, + 6.~
(18)
AxH = AxHI- AxH3+6x
(19)
AyH= AyH>- AY,.,~ + &v
(20)
where
board center
pc b,,,,
I
+
7-
-
L______, ay,+ SY”-l %
i---
AYpad
I+ R”
4
n
k
Note: PO0, R,S,
is the
original
pad configuration.
s Q, R, S,
is the
pad with the
6x=6x, +26x* effect
of linear
errors
6y = 6_Y,+ 2hy, Figure 6 shows the effect of errors relating to the pad,
on its configuration (in the Xy plane). Here m and n are the X and Y distances from the ideal board center to the ideal pad center (for the farthest pad). From
P’Q’R’S’ is the pad with the effects and the board’s angular error E.
of linear
errors
and
PQRS is the final pad configuration with the effects of all linear and angular errors (pertaining to the pad).
Figure 6 Effect of errors on pad configuration.
Analysis of dimensional
and locational
variations:
Ax,<= - Ax,;,,,+ I,, cos (Q - W)- l/21,,
(25)
A?,, = - AJ,,:,~, - 1/2~,, + I,, sin (R-Q)
(26)
Ax, = - Ax,;,,,+ l/21, - I,, cos (R + w)
(27)
Ays = - A.vPiIC, + I,, sin (Q + m) - I /2w,
(28)
where Q = tan ‘( w,,//,,) I,,= I /2[ I,)?+ W,)?]“2 Ax,;,,,= rn - I mn~~~C/j+ I:) A!,;,,, = I,,sin (/j + I: - n with I,,, = (m’ +
195
T Radhakrishnan
fz’)“’
Figure 7 shows the final configurations of the pin and pad, with the effects of all the errors, compounded additively (worst-case tolerancing). Under these conditions, the contact area between the pad and pin configurations (as seen on the XY plane) is the deciding criterion for proper placement of the pin (component) on the pad (board). By giving values to the various parameters, this area can be determined.
their maximum level. Though this occurrence may be statistically rare, it will provide for the development of a very reliable system, with stringent standards. However, for a more practical approach which provides for less stringent specifications (at the cost of decreased system reliability), statistical error compounding (tolerancing) methods can be used. Assuming a normal distribution for the various individual errors, the total error may be computed in a modified manner”.“‘. Accordingly, each error (or tolerance) is assumed to vary within its _+3a limits about its mean value. The various individual contributions of these errors to the corner shifts of the pin and pad (indicated by the various terms in eqns (l)-(28) for additive error compounding), will now be appropriately squared and added for statistical error compounding. The square roots of these added quantities will then give the corresponding values for the pin and pad corner shifts for the statistical case. This yields the following relationships (with the subscript ‘s’ denoting terms obtained in statistical tolerancing). For the pin, the corner shifts are now given by Ax,, = [(h, sin 0,)’ + 1/4(w, cos 0, sin 0,)’ + (6x,)‘] 1’2
Statistical error compounding
(29)
A.vw= [ ( 1/2~, -(h,
The previous analysis was based on the worst-case tolerancing or additive error compounding method. It considers the extreme case where all errors occur at
cos Oy sin
(L>~ - 0,)))’
+ l/4( k!+ cos (I,( I - cos 0,)’ + (hy,)‘] “?
(30)
Ax,, = [(I, (sin (II/+ O,)lsin $ - I)}? + ( I, cos &(cos (1’- O,)/cos ;’ - 1) )?
r
+ (6x$] “? Y
(31)
A?,+ = [ { l/214+- I, cos($ + O,)sin(p, -O,)/sin rl/)’ + { 1/2w, cos 0, - I, cos Ov sin ()I- O,)/cos :1}? + (dy,)‘]“? (32)
X
Ax,, = [ ( I, (sin ($ + &)/sin $ - 1) 1’ -(I,cosu,(1-cos(~+(~,)/cos~))’ +(6x,)*]“’
(33)
A_JJ~~ = [(I, cos(tj + 0,) sin (p, + 0,)lsin II/]- l/2& + {I, (cos Ousin (7 + U,)l cos1’) - 1/2w, cos 0,)’ + (iiy,)2]“” (34) Ax,, = [(hN sin O,)‘- I /4(w, cos OXsin (I,)’ + (&K,)“]“z
(35)
A_v”,= [(h, cos 0,. sin (p2 + OX) - 1/2w,)‘-
q
contact oreo between pin and pad (actual)
-__-
ideal
_
actual
Oideal &in,
:
configurations
actual
center
of
pin,
pad
cosO,(l
-cos(~,))’
I’?
(36)
where
configurations
ideal
: Opod
+ (&$I
1/4{w,
6x, = [( csx,)’ + 2(6x2)‘] I’? pin or pod centers
Figure 7 Final configuration of pin-pad placement.
ijv,=[(6v,)‘+2(iiv,)‘]“” and $, p,, oa, )’ are as defined earlier.For corner shifts are given by
the pad, the
196
Analysis of dimensional and locational variations: T Radhakrishnan
AxPI= -[{Z,,cos(Q-(~)-11/21,#+(A~~~~)*]”*
(37)
Conclusions
AyPs= [ { 1/2w, - 1,)sin (0 -o)]*
(38)
This study provides a useful mathematical model, relating various critical parameters involved in the fetching and placement of surface-mounted components, for printed circuit board assembly. With a suitable computer program, an analysis can be performed to determine the net effect of various combinations of parameter values, on the critical contact area between a component pin and its pad (on the pc board). This will be helpful in the design and control of such pc board assembly systems. With suitable modifications, these models may be adapted to a specific type of pc board assembly process involving surface-mounted parts.
- (Ay,,J*]“*
Axes= -[{ 1/21,-I,,cos(R+u)]2+(Ax,,,)2]“2
(39)
Ay,, = - [I,, sin (Q + o) - 1/2w,}*+(Aypad)*]“* AxRs= [{ Z,,cos(CJ-CU)-
(40)
l/21,)*
- ( Axpad)*]“* AyRb= - [ ( 1/2w,-I,,
(41) sin@ -o))*
+ (Ay,,J*]“*
Ax,, = [ ( l/21, - 1,,cos (CI+ w) } * -(Ax,;,,)*]“*
(42) (43)
By,,= -[{/,,sin(R+w)-1/2w,)* - (Ay,,d)*l
“*
(44)
where A.q,,&AY,,~~l,b Q are as defined earlier, with
Additive Toleroncing
fi = tan~‘[(n+((Sy,)2+(~y~)2)“2}/((~~j)2+(S~~)2)”2] 66.00
Using these expressions, the contact area between the pin and the pad, under the effects of these errors, can be determined as before.
67.50 b?
Quantitative analysis To illustrate the use of this analysis, a computer program was developed to incorporate both additive error-compounding and statistical error-compounding methods. Using some practical values for the various parameters, the contact or overlapping area between a pin and its pad (as seen on the XY plane) was calculated and expressed as a percentage of the ideal contact area ( = nominal pin foot length x nominal pin foot width). The following values were used:
z < -c, 66.50
deg.
5 g 66.00 0 65.50 4
0,
1, = 0.71 mm (0.028”), w,=O.46 mm (0.018”), h, = 1.57 mm (0.062”), l,, = 1.22 mm (0.048”), w, = 0.7 1 mm (0.028”), m =n = 101.6 mm (4”), d;~~=~x~=~x~=dx~=~y~=6y~=t3y,=~y~= 0.025 mm
76.50
(O.OOl”), c = 0.05”, 0 = 0.04”
76.45
Figure 8 shows plots of how the contact area percentage varies with 0, and 0, for this example (with 0, = 0 for simplicity), by the two errorcompounding methods. Similar plots can be obtained with various combinations of chosen parameter values. Such plots show the relative effects of the different parameters on the contact area. They can indicate which parameters are more critical and need to be better controlled. It can also be seen that the statistical method of error compounding yields better results for the contact area (over 75% for this particular set of parameter values), though this will be at the cost of some reliability. Either error compounding method may be used in practice, depending on the economics of relaxed tolerances vs increased reliability.
deg.
067.00
degrees
Statistical
Toleroncing
R76.40 ii $ 76.35 % o 76.50 -w 5 0 76.25
76.20
76.15 ,,,vv,...~,,,,,,, 0.00 0.05 0.10 0,
0.15
"0?2'0"
degrees
Figure 8 Variation of contact area with 0, and 0,.
O.&i
Analysis of’dimensional and locational variations: T Radhakrishnan
197
Acknowledgements
5 Cusik,
This project was funded by the Faculty Summer Research Grant Program of Villanova University.
326-335 6 Radhakrishnan,
References Schmits, J, Heiser, G and Kukowski, J ‘Electronic assembly in the next millennium’, CircuitsAssembly Vol 8 (1997) 32-36 Belmonte, J and Zarrow, P ‘Advanced surface-mount manufacturing methods’, CircuitsAssembly Vol 7 (1996) 36-41 Rodgers, B ‘Coplanarity problems’, Circuits Asscn~h/y Vol 7 ( 1996) 46-49 Deley, M, Ramsey, C and Smith, D ‘Putting fine-pitch errors on a budget’, Circuits MururfacturingVol 30 ( 1990) 40-45
W T ‘Design parameters for automatic printed circuit board assembly’, NEPCON- WestCoufercrlce Proceedirgs ( 1987)
7
8
9 IO
T and Hegde, S M ‘An analysis of the errors involved in component pick up and registration for automated printed circuit board assembly’, Trar~~.ASME-J. Eqg. ,for Ind. Vol 113 (1991) 101-107 Radhakrishnan, T ‘Combined effects of linear and angular errors in PC board assemblv’. Iw. J. Prod. Rex Vol 30 (19921 . I 1037- 1044 Duffey, B M and Roberts, L ‘Applying machine vision to tinepitch pick-and-place’, Circuits MurlufizcturiugVol 29 ( 1989) 22-23 Chase, K W and Greenwood, W H ‘Design issues in mechanical tolerance analysis’, Mnrlufac/urirq Review Vol I (1988) 50-59 Stout, K Quality control in automation. Prentice Hall (1985) 103-l 13
An analysis of dimensional and locational variations in surfacemounted component assembly T Radhakrishnan Mechanical Engineering Department, Villanova Univeniq,
Villanova, PA 19085, USA
This study focuses on the errors involved in the component fetching and placement phase of pc board assembly with surface-mounted components. Various critical dimensional and locational parameters involved in this phase, which can lead to mis-assembly, are identified. The complex relationships between these parameters, which govern the proper component (pin) placement on the pc board (pad), are mathematically modeled, using both additive and statistical error compounding methods. The use of the model is shown with a quantitative example. 0 1998 Elsevier Science Ltd. All rights reserved
addressed’-‘. The accuracy of placement of such components on the pc board is very critical to achieve a high yield rate of properly assembled boards and to reduce the amount of scrap or rework needed. Earlier published workXm7provides a study of some of the parameters that affect the placement of throughhole components on pc boards. SMCs pose greater assembly problems, partly due to the lack of positive placement registration found in the case of throughhole components. Further, the problem is more pronounced owing to the decreasing component sizes and pin-to-pin distances for SMCs, with very little room for errors. An in-depth analysis of this problem is therefore required. In order to improve the accuracy of the process and maintain an efficient assembly system, the various critical parameters involved in the process should be first identified. This study focuses on the phase of the assembly process involving picking up a component from the component storage area and placing it on the pc board. The critical parameters involved here include various dimensions, motions and related tolerances. These parameters, once identified, should then be mathematically related to provide a scientific tool for analyzing their effects on the proper placement (and hence assembly) of components on the pc board. Such a tool would then be very useful in the understanding, design/development and control of such assembly systems, to reduce errors in the placement of SMCs on pc boards.
pin dimensions pad dimensions angular errors for pin angular errors for board/pad x, y linear errors for pin location and part-fetching process X, y linear errors for board and pad locations angular quantities used in the derivations x and y distances between board and ideal pad centers X- and y-shifts of pin or pad corners (indicated by the subscripts) corners of a rectangular pin foot (in xy plane) corners of a rectangular plane)
pad (in q
Introduction The automation of printed circuit (PC) board assembly with surface-mounted components (SMCs) is an important area of electronic assembly today. Two desirable features in such a process are high speed and flexibility to handle a variety of components. SMCs are widely used and their assembly involves various critical issues that need to be 191
Analysis of dimensional and locational variations: T Radhakrishnan
192
Assembly process and critical parameters Figure
I illustrates a general pc board assembly system with SMCs and shows a typical SMC placement on the pc board. Proper placement of the component means that the feet of the pins (or leads) are located properly on their corresponding contact pads on the pc board. For a good assembly, it is desirable to have at least 75% of the pin’s foot area on the padx. The process involves a robot or other component fetching and placement mechanism (Figure I), to move to the component storage area and pick up a typically using vacuum suction. A component, number of different SMCs may be stored in the storage area, in trays or reels. A process control program would determine the sequence of components to be picked and the location/orientation of placement on the pc board. Once picked, a component is moved to the pc board area and placed so that the pins are positioned over their respective contact pads. The assembler system can be
considered to be set up and involve motions along a set of Cartesian coordinate axes as shown. The sources of errors involved in this process would therefore consist of: dimensional variations for the component pins and pads locational variations for the components at storage and for the pads on the board angular errors in the pin configurations due to bending or twisting inaccuracies in the motions involved in fetching and placing a component. It may be noted that the variations or inaccuracies indicated here have a direct bearing on the specification of various tolerances and required motion control involved in the system. The next section details these critical parameters. By considering the placement of a typical pin on its pad, mathematical relationships involving these parameters can be developed.
Development of mathematical relationships Figure 2 shows a typical pin and the corresponding rr7__-____--____-7
II II
I I
component storage area
I I# I I& I I 0’ LI_A____--__--_-__J
I I
I I I I
component fetching/ ’ placing unit
pad on the board, and some of the associated error sources. Including the effects of tolerances (errors), let 1, be the minimum length of the pin (foot), wN be the minimum width of the pin (foot), h, be the maximum height of the pin (body), &O, be the angular error (in the YZ plane) due to pin bend, &8, be the angular error (in the XZ plane) due to pin bend, +8, be the angular error (in the XY plane) due to pin twist, 1, be the minimum length of the pad, wD be the minimum width of the pad. These extreme pin and pad sizes are used in the analysis, along with appropriate angular error directions, to consider the maximum effect of the errors in the placement of the pin over the pad. Let E,,F,,G,,H,, and P,,Q,,R,,S,, denote the rectangular pin and pad configurations, respectively, on the XY plane, under ideal placement conditions as shown in Figure 2. (The pin foot is assumed to be slightly less in size than the corresponding pad. Also, the thicknesses of the pin and pad are assumed to be negligible, compared with their other dimensions.) Figure 3 shows the effect of the angular error U,, on the configuration of the pin. Corners E,, and F,, of the pin foot are now displaced to locations El and F, as shown (the other two corners G,, and H,, have similar displacements). Hence, the displacements (&) for the pin corners, along the X axis, can be expressed as
Figure 1 General
SMC assembly system.
Ax,, = AxH, = hN sin I),
(I)
Ax,, = Ax,, = I, sin ($ +(),)/sin Ic/- I,
(2)
Analysis
of dimensional and locational variations: T Radhakrishnan
193
6 El
A%2-
A%2 Z
I-
Y
Figure 4 Effect of angular error 0, on pin configuration.
where 1,9= tan ‘(IN/hN) It is assumed that displacements along the Z direction will not affect the final assembly, since soldering can fill any gaps between the pin and pad in this direction. Figure 4 shows the effect of angular error OXon the further shift of the pin (compounded with the effects of previously mentioned linear and angular errors). The corners of the pin, as seen on the YZ plane, shift to locations E2F2G2Hz. Hence, the Y displacements (Ay) can be expressed as Figure 2 Some typical error sources associated with the pin and pad.
Ay,. = 1/2w, -h,
cos 0, sin (/I~- (I,)
(3)
AyFz= 1/2w, - I, cos($ + O,.) sin (p, - (I,y)/sin$
(4)
A.vti2= I, cos($ + Oy) sin (p, + (),)/sin $ - 1/2w,
(5)
Ay,,, = h, cos 0, sin (pz + (IV)- 1/2w,
(6)
where p, = sin ~‘[1/2w,l(/,
cos($+(I,)lsinII/)]
pz= sin~‘[l/2w,/{h,cos0,]1 Figure
5 shows the effect of angular error (I,, compounded on the effects of previous errors. The pin corners now shift to locations E3,F,,G3,H,, as seen on the XY plane. The X and Y shifts for the corners can therefore be expressed as
z
L
AxE7= Axm = 1/2w, cos 0, sin 0,
(7)
A.v,g= Ay,, = 1/2w, cos Ox(1 - cos 0,)
(8)
AxFT= I, cos 0, cos (7 - O,)/cos 7 - I, cos (I,
(9)
AyFq= 1/2w, cosO,X
Figure 3 Effect of angular error OY on pin configuration.
I, COS()~sin(y-O,)/cosy
Axo3 = I, cos 0, - I, cos 0,. cos (;, + (I,)lcos ;I
(10) (II)
A?‘(;1= I, cos 0, sin (;q+ 0,)lcos ;’- 1/2w, cos Ox (12)
Analysis of dimensional and locational variations: T Radhakrishnan
194
I-
the ideal position, the pad undergoes linear shifts in the X and Y directions, due to errors in board location on the assembler table (&?, fiy,) and due to errors in the location of the pad on the board (bx4, by,). P,Q,R,S, (F @u-e 6) indicates the shift of the pad due to the linear errors alone. In addition, both the board and the pad on the board can have angular errors of their own (about their central axis). Let I: and w indicate the angular errors of the board a+ pad, respectively. Then, the combined effects of these various linear and angular errors for the pad (and board) yield the final pad configuration PQRS, displaced from the initial configuration P,,Q,,R,,S,,, with the pertinent dimensional errors of the pad. (In Figure 6, P’Q’R’S’ indicates the pad configuration with the effect of all errors except that of u.) Hence, the total shift in the corners of the pad can be algebraically expressed as:
Y
,,
Ax, = - Ax,,;,~, - I,, cos (R - w) + I/21,
(21)
A_v,,= - Aypad+ 1/2w, - I,, sin (Sz- w)
(22)
Ax, = - AXIS,<, - I /21, + I,, cos (0 + o)
(23)
A_vu= - A?+ - I,, sin (Cl + o) + I /2w,,
(24)
A
Figure 5 Effect of angular error 0, on pin configuration.
where y= tan-‘[1/2w,
cosU,/(/,
COSU,}]
Furthermore, let &,, iiy, indicate the errors in the location of a pin on the body of component along X and Y directions, and &, by2 indicate the errors in each of the X and Y motions (to-and-fro motions) involved in the fetching and placing of a component (which will therefore affect its position during placement). Then, the total displacement of the corners of the pin projected on the XY plane, from the original configuration E,,F,,G,,H,, to a final configuration EFGH, can be algebraically expressed as: Ax, = Ax,, + Ax,, + 6x
(13)
AyE= Ayw + AY,, + dy
(14)
Ax, = Ax,, + Ax,, + Sx
(15)
Ay, = Ay,, + AY,, + dy
(16)
Ax, = AxG, - AxG3+ 6x
(17)
Ay, = Aycjr+ Ay,, + 6.~
(18)
AxH = AxHI- AxH3+6x
(19)
AyH= AyH>- AY,.,~ + &v
(20)
where
board center
pc b,,,,
I
+
7-
-
L______, ay,+ SY”-l %
i---
AYpad
I+ R”
4
n
k
Note: PO0, R,S,
is the
original
pad configuration.
s Q, R, S,
is the
pad with the
6x=6x, +26x* effect
of linear
errors
6y = 6_Y,+ 2hy, Figure 6 shows the effect of errors relating to the pad,
on its configuration (in the Xy plane). Here m and n are the X and Y distances from the ideal board center to the ideal pad center (for the farthest pad). From
P’Q’R’S’ is the pad with the effects and the board’s angular error E.
of linear
errors
and
PQRS is the final pad configuration with the effects of all linear and angular errors (pertaining to the pad).
Figure 6 Effect of errors on pad configuration.
Analysis of dimensional
and locational
variations:
Ax,<= - Ax,;,,,+ I,, cos (Q - W)- l/21,,
(25)
A?,, = - AJ,,:,~, - 1/2~,, + I,, sin (R-Q)
(26)
Ax, = - Ax,;,,,+ l/21, - I,, cos (R + w)
(27)
Ays = - A.vPiIC, + I,, sin (Q + m) - I /2w,
(28)
where Q = tan ‘( w,,//,,) I,,= I /2[ I,)?+ W,)?]“2 Ax,;,,,= rn - I mn~~~C/j+ I:) A!,;,,, = I,,sin (/j + I: - n with I,,, = (m’ +
195
T Radhakrishnan
fz’)“’
Figure 7 shows the final configurations of the pin and pad, with the effects of all the errors, compounded additively (worst-case tolerancing). Under these conditions, the contact area between the pad and pin configurations (as seen on the XY plane) is the deciding criterion for proper placement of the pin (component) on the pad (board). By giving values to the various parameters, this area can be determined.
their maximum level. Though this occurrence may be statistically rare, it will provide for the development of a very reliable system, with stringent standards. However, for a more practical approach which provides for less stringent specifications (at the cost of decreased system reliability), statistical error compounding (tolerancing) methods can be used. Assuming a normal distribution for the various individual errors, the total error may be computed in a modified manner”.“‘. Accordingly, each error (or tolerance) is assumed to vary within its _+3a limits about its mean value. The various individual contributions of these errors to the corner shifts of the pin and pad (indicated by the various terms in eqns (l)-(28) for additive error compounding), will now be appropriately squared and added for statistical error compounding. The square roots of these added quantities will then give the corresponding values for the pin and pad corner shifts for the statistical case. This yields the following relationships (with the subscript ‘s’ denoting terms obtained in statistical tolerancing). For the pin, the corner shifts are now given by Ax,, = [(h, sin 0,)’ + 1/4(w, cos 0, sin 0,)’ + (6x,)‘] 1’2
Statistical error compounding
(29)
A.vw= [ ( 1/2~, -(h,
The previous analysis was based on the worst-case tolerancing or additive error compounding method. It considers the extreme case where all errors occur at
cos Oy sin
(L>~ - 0,)))’
+ l/4( k!+ cos (I,( I - cos 0,)’ + (hy,)‘] “?
(30)
Ax,, = [(I, (sin (II/+ O,)lsin $ - I)}? + ( I, cos &(cos (1’- O,)/cos ;’ - 1) )?
r
+ (6x$] “? Y
(31)
A?,+ = [ { l/214+- I, cos($ + O,)sin(p, -O,)/sin rl/)’ + { 1/2w, cos 0, - I, cos Ov sin ()I- O,)/cos :1}? + (dy,)‘]“? (32)
X
Ax,, = [ ( I, (sin ($ + &)/sin $ - 1) 1’ -(I,cosu,(1-cos(~+(~,)/cos~))’ +(6x,)*]“’
(33)
A_JJ~~ = [(I, cos(tj + 0,) sin (p, + 0,)lsin II/]- l/2& + {I, (cos Ousin (7 + U,)l cos1’) - 1/2w, cos 0,)’ + (iiy,)2]“” (34) Ax,, = [(hN sin O,)‘- I /4(w, cos OXsin (I,)’ + (&K,)“]“z
(35)
A_v”,= [(h, cos 0,. sin (p2 + OX) - 1/2w,)‘-
q
contact oreo between pin and pad (actual)
-__-
ideal
_
actual
Oideal &in,
:
configurations
actual
center
of
pin,
pad
cosO,(l
-cos(~,))’
I’?
(36)
where
configurations
ideal
: Opod
+ (&$I
1/4{w,
6x, = [( csx,)’ + 2(6x2)‘] I’? pin or pod centers
Figure 7 Final configuration of pin-pad placement.
ijv,=[(6v,)‘+2(iiv,)‘]“” and $, p,, oa, )’ are as defined earlier.For corner shifts are given by
the pad, the
196
Analysis of dimensional and locational variations: T Radhakrishnan
AxPI= -[{Z,,cos(Q-(~)-11/21,#+(A~~~~)*]”*
(37)
Conclusions
AyPs= [ { 1/2w, - 1,)sin (0 -o)]*
(38)
This study provides a useful mathematical model, relating various critical parameters involved in the fetching and placement of surface-mounted components, for printed circuit board assembly. With a suitable computer program, an analysis can be performed to determine the net effect of various combinations of parameter values, on the critical contact area between a component pin and its pad (on the pc board). This will be helpful in the design and control of such pc board assembly systems. With suitable modifications, these models may be adapted to a specific type of pc board assembly process involving surface-mounted parts.
- (Ay,,J*]“*
Axes= -[{ 1/21,-I,,cos(R+u)]2+(Ax,,,)2]“2
(39)
Ay,, = - [I,, sin (Q + o) - 1/2w,}*+(Aypad)*]“* AxRs= [{ Z,,cos(CJ-CU)-
(40)
l/21,)*
- ( Axpad)*]“* AyRb= - [ ( 1/2w,-I,,
(41) sin@ -o))*
+ (Ay,,J*]“*
Ax,, = [ ( l/21, - 1,,cos (CI+ w) } * -(Ax,;,,)*]“*
(42) (43)
By,,= -[{/,,sin(R+w)-1/2w,)* - (Ay,,d)*l
“*
(44)
where A.q,,&AY,,~~l,b Q are as defined earlier, with
Additive Toleroncing
fi = tan~‘[(n+((Sy,)2+(~y~)2)“2}/((~~j)2+(S~~)2)”2] 66.00
Using these expressions, the contact area between the pin and the pad, under the effects of these errors, can be determined as before.
67.50 b?
Quantitative analysis To illustrate the use of this analysis, a computer program was developed to incorporate both additive error-compounding and statistical error-compounding methods. Using some practical values for the various parameters, the contact or overlapping area between a pin and its pad (as seen on the XY plane) was calculated and expressed as a percentage of the ideal contact area ( = nominal pin foot length x nominal pin foot width). The following values were used:
z < -c, 66.50
deg.
5 g 66.00 0 65.50 4
0,
1, = 0.71 mm (0.028”), w,=O.46 mm (0.018”), h, = 1.57 mm (0.062”), l,, = 1.22 mm (0.048”), w, = 0.7 1 mm (0.028”), m =n = 101.6 mm (4”), d;~~=~x~=~x~=dx~=~y~=6y~=t3y,=~y~= 0.025 mm
76.50
(O.OOl”), c = 0.05”, 0 = 0.04”
76.45
Figure 8 shows plots of how the contact area percentage varies with 0, and 0, for this example (with 0, = 0 for simplicity), by the two errorcompounding methods. Similar plots can be obtained with various combinations of chosen parameter values. Such plots show the relative effects of the different parameters on the contact area. They can indicate which parameters are more critical and need to be better controlled. It can also be seen that the statistical method of error compounding yields better results for the contact area (over 75% for this particular set of parameter values), though this will be at the cost of some reliability. Either error compounding method may be used in practice, depending on the economics of relaxed tolerances vs increased reliability.
deg.
067.00
degrees
Statistical
Toleroncing
R76.40 ii $ 76.35 % o 76.50 -w 5 0 76.25
76.20
76.15 ,,,vv,...~,,,,,,, 0.00 0.05 0.10 0,
0.15
"0?2'0"
degrees
Figure 8 Variation of contact area with 0, and 0,.
O.&i
Analysis of’dimensional and locational variations: T Radhakrishnan
197
Acknowledgements
5 Cusik,
This project was funded by the Faculty Summer Research Grant Program of Villanova University.
326-335 6 Radhakrishnan,
References Schmits, J, Heiser, G and Kukowski, J ‘Electronic assembly in the next millennium’, CircuitsAssembly Vol 8 (1997) 32-36 Belmonte, J and Zarrow, P ‘Advanced surface-mount manufacturing methods’, CircuitsAssembly Vol 7 (1996) 36-41 Rodgers, B ‘Coplanarity problems’, Circuits Asscn~h/y Vol 7 ( 1996) 46-49 Deley, M, Ramsey, C and Smith, D ‘Putting fine-pitch errors on a budget’, Circuits MururfacturingVol 30 ( 1990) 40-45
W T ‘Design parameters for automatic printed circuit board assembly’, NEPCON- WestCoufercrlce Proceedirgs ( 1987)
7
8
9 IO
T and Hegde, S M ‘An analysis of the errors involved in component pick up and registration for automated printed circuit board assembly’, Trar~~.ASME-J. Eqg. ,for Ind. Vol 113 (1991) 101-107 Radhakrishnan, T ‘Combined effects of linear and angular errors in PC board assemblv’. Iw. J. Prod. Rex Vol 30 (19921 . I 1037- 1044 Duffey, B M and Roberts, L ‘Applying machine vision to tinepitch pick-and-place’, Circuits MurlufizcturiugVol 29 ( 1989) 22-23 Chase, K W and Greenwood, W H ‘Design issues in mechanical tolerance analysis’, Mnrlufac/urirq Review Vol I (1988) 50-59 Stout, K Quality control in automation. Prentice Hall (1985) 103-l 13