A Random Trimming Approach for Obtaining High-Precision Embedded Resistors

Abstract

Embedded resistors will potentially allow electronic applications to cost less and perform better. However, it is difficult to fabricate embedded resistors to the correct resistance value, so embedded resistors are often fabricated with a lower value and then trimmed to raise the resistance to the desired value. A computer simulation for the trimming process of an embedded resistor has been developed that has been verified and calibrated against experimental results. A study of embedded resistors containing random voids of varying size has been performed. A new trimming strategy in which the trims are made randomly (rather than conventional L-shaped trims) is proposed and the results of the analysis demonstrate that single-dive trimming combined with random trimming allows higher precision embedded resistors to be obtained than conventional trimming patterns.

Full text

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1
A RANDOM TRIMMING APPROACH FOR OBTAINING HIGH-PRECISION
EMBEDDED RESISTORS

Phillip Sandborn*
Wilde Lake High School, Columbia, MD

Peter Sandborn
CALCE, Department of Mechanical Engineering, University of Maryland, College Park, MD


ABSTRACT
Embedded resistors are resistors fabricated inside of
printed circuit boards used as an alternative to discrete
resistor components that are mounted on the surface of
the boards. However, it is difficult to fabricate embedded
resistors to the required resistance value, so embedded
resistors are often fabricated with a lower value and then
trimmed to raise their resistance to the desired value. A
study of embedded resistors containing random voids of
varying size and density has been performed. A new
trimming strategy in which the trims are made randomly
(rather than conventional L-shaped trims) is proposed in
this paper. Analysis results demonstrate that single-dive
trimming combined with random trimming enables the
manufacturing of embedded resistors with higher
precision and producibility (Cpk) than can be obtained
with conventional trimming patterns.
The resistor trimming pattern impacts the distribution
of heat in the resistor. Study results show that the highest
temperature reached in randomly trimmed NiCr resistors
is 7.5% lower than the highest temperature in single-dive
trimmed resistors and 1.2% lower than the highest
temperature in L-cut trimmed resistors.

1. INTRODUCTION
The world demands a continuous stream of smaller,
better performing, and less expensive electronic systems,
such as cellular phones. One technology that enables
these systems to become smaller and better performing is
embedding passives (Ulrich, 2003). Embedded passives
are electrical components (most commonly resistors and
capacitors) that are fabricated inside or on the surface of
printed circuit boards instead of being mounted on them
(i.e., “discrete” components).
Embedded resistors are fabricated in one of two
ways: 1) subtractive processes – a layer with resistive
material plated on it is etched to form specific resistors
and then included within the printed circuit board layers,
(e.g., Ohmega-Ply; Wang and Clouser, 2001); or 2)
additive processes – resistive material is printed or plated
onto a layer with other printed circuit board features to
form specific resistors, (e.g., D’Ambrisi et al., 2001).
Embedded resistors, however, have several problems
that are preventing their widespread adoption. The most
serious drawback is that the processes for making resistor
geometries (whether subtractive or additive) are inexact,
requiring the resistors to be “trimmed” if high precision is
required. Resistors are normally fabricated with lower
resistance values than required and are trimmed by cutting
holes in them with lasers to increase their resistance value
(Fjeldsted and Chase, 2002). There are several desirable
trimming characteristics: getting as close to the target
resistance as possible (precision); minimizing the risk of
trimming too much (if a resistor is “over-trimmed,” fixing
it is impractical, and the entire board that it is in may have
to be thrown away); and minimizing the length of each
trim (a longer trim takes more time to make and impacts
the manufacturing throughput thereby costing more
money). Without trimming, embedded resistors can
achieve ~10% tolerance (±10% of the target resistance),
(Wang et al., 2002; Cheng et al., 2007a; Cheng et al.,
2007b), which is insufficient precision for many
applications.
Various trimming patterns are used to meet desired
trimming characteristics. Common practice is to use “L”
shaped trims (“L-cut”) (see Figure 1), where a cut
perpendicular to the current flow in the resistor quickly
increases the resistance to near the target value, then a
change in direction of the trim to parallel to the current
flow slows down the change in resistance until the trim
reaches the target.
Voids (bubbles) in the resistive material complicate
the trimming problem, especially for high-precision
resistors; if a void is encountered during trimming, the
resistance may “jump” uncontrollably to a higher value
beyond the target resistance, Figure 2. Although the
density and size of voids is a function of the process and
materials used to create the embedded resistors, all types
of embedded resistors have reported some amount of
voiding, e.g., polymer thick-film resistors (Narayana et
al., 1992; Chinoy and Langlois, 2004), and thin-film
resistors (Snogren, 2004).
Double Dive
L-Cut
Serpentine
Scan
Single Dive

Fig. 1. Common embedded resistor trimming patterns.

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This paper describes an experimental method for
emulating embedded resistors, which was used to verify
and calibrate a two-dimensional numerical simulation of
the embedded resistor trimming process. The model was
then used to compare the precision and efficiency of L-
Cut trimming patterns with a new random trimming
approach applicable to high-precision embedded resistors.
Section 4 addresses the thermal aspects of randomly
trimmed embedded resistors. A three-dimensional finite-
element model was developed, experimentally verified
and used to simulate NiCr embedded resistors. The
model was used to determine the location and values of
the maximum temperature reached in each resistor.

2. ELECTRICAL EXPERIMENTAL AND
SIMULATION METHODOLOGY
An experimental approach that emulated the
electrical properties of trimmed embedded resistors was
developed and a two-dimensional numerical simulation
model was constructed, verified, and calibrated using the
experimental results. This section describes both the
experimental approach and the simulation model.

2.1 Experimental Approach
To emulate embedded resistors experimentally,
trimming of conductive paper was performed (Sandborn
and Sandborn, 2007). For a constant sheet resistance
(ohms per square), the resistance of a planar resistor does
not change with size (i.e., length or width) as long as the
ratio of length to width (the aspect ratio) is constant. This
relationship is given by,
ρ
TWA
where ρ is the bulk resistivity, A is the cross-sectional
area of the resistor, R□ is the “sheet resistance” (in ohms
per square), and L, W, and T are the length, width, and
thickness of the resistor. From (1) it can be seen that the
resistance (R) depends on the ratio of the resistor length
and width (rather than the magnitudes of the length and
width), therefore, a small planar resistor (dimensions in
fractions of millimeters) has the same resistance

⎟⎠
⎞
⎜⎝
⎛
=
⎟⎠
⎞
⎜⎝
⎛
===
?
W
L
R
W
L
T
ρL
ρL
R
(1)
characteristics as a large area planar resistor (dimensions
of centimeters), and trimming of small resistors can be
emulated using larger area resistors.
To emulate embedded resistors, sheets of conductive
paper (PASCO Scientific, Model Number PK-9025) were
cut into 28 x10 centimeter sheets. Silver contacts were
added to the two ends of each paper resistor using silver
conductive paint. Metal clamps were used to connect to
the resistor contacts and ohmmeter probes were connected
to the metal clamps. The silver contacts make the
resistance measurements less sensitive to the clamp
placement or pressure.
The resistance of the conductive paper was found to
vary from sheet to sheet. In order to make measurements
from multiple sheets comparable, the measurements were
normalized. The “normalization” process scaled all
untrimmed measurements from each sheet to 65 kΩ. To
normalize the various sheets to 65 kΩ, 10-15 initial
measurements from each sheet were made prior to
trimming, the measurements were averaged, and the
average was divided by 65 kΩ to obtain a scaling factor
that was specific to the sheet. Every measurement taken
from the specific sheet during trimming was multiplied by
this scaling factor before comparison with data from other
sheets. The normalization process was performed for
every resistor used in the experimental study. This
normalization procedure accounts for variations in the
sheet resistance from sheet-to-sheet only (variations in
series resistance in the measurement path are not affected
by the normalization, e.g., clamp pressure or placement).

2.2 Numerical Simulation
A numerical simulation of embedded resistors that
allows the study of variations in trimming patterns was
implemented, calibrated,
experimental results from trimming conductive paper.
Previously reported numerical simulations of trimmed
embedded resistors include: models for predicting the
performance of laser-trimmed resistors taking into
account the heat-affected zone around the trim (Ramirez-
Angulo et al., 1987; Ramirez-Angulo and Geiger, 1988);
numerical trim simulations that were used in the resistor
design process to predict trim results (Schimmanz and
Jacobson, 2002; Poslethwaite, 1984; Schimmanz and
Kost, 2004); and three-dimensional simulators that were
utilized in modeling electronic interconnect structures,
(Martins et al., 1998). The model developed here is a
finite difference model formulated similarly to Ramirez-
Angulo et al. (1987) and is described in this section.
A grid is superimposed on the resistor. The resistor
is assumed to be uniform in the third dimension (into the
board), in order to isolate the two-dimensional problem.
To determine the resistance of the embedded resistor, the
voltage at each of the grid points must first be determined.
The voltage is found by solving Laplace’s Equation in
two dimensions at each grid point,
and verified using the
3
2
11111
4
3
2222
Voids VoidsVoids Voids
3
2
1111
4
3
222
TrimTrim TrimTrim
Single-Dive TrimSingle-Dive Trim Single-Dive Trim
Resistor with 3 voidsResistor with 3 voidsResistor with 3 voids
Overtrimmed resistor when void encounteredOvertrimmed resistor when void encountered Overtrimmed resistor when void encounteredOvertrimmed resistor when void encountered Overtrimmed resistor when void encountered Overtrimmed resistor when void encounteredOvertrimmed resistor when void encounteredOvertrimmed resistor when void encountered
Length of TrimLength of TrimLength of Trim Length of Trim
1111111111111111222222222222222233333333333333334444444444444444555555555555555566666666666666667777777777777777
Resistance
Target Resistance Target ResistanceTarget ResistanceTarget Resistance
Single-Dive TrimSingle-Dive Trim Single-Dive TrimSingle-Dive Trim
Resistance
4
333
4
33
44
44
ResistanceResistance


Fig. 2. An embedded resistor containing 3 voids (“bubbles”
in the resistive material). If a void is encountered during
trimming, the resistance value jumps uncontrollably.

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3

0
y
V
2
x
V
2
V
22
2
=
∂
∂
+
∂
∂
=∇
(2)
where V is the voltage. Assuming that the distances
between the grid points in the plane of the resistor are the
same, a 5-point finite difference approximation of
Laplace’s Equation reduces to,

4VVV
j i,j 1,i1j i,
−+
−−
An equation like the one above is written for every
grid point. The system of equations is solved by writing
the set of equations in the form of a matrix equation. A
Gaussian Elimination technique is then used solve the
matrix equation for the voltages at every point.
Once the voltages at all the points have been solved
for, the electric field (
E
−∇=
found by differencing, and the current density at every
point is given by,
0VV
1j i,j1,i
=++
++
(3)
V
) at every point can be

R
E
E
σ
J
==
(4)
where σ is the conductivity and R□ is the resistivity (Ω/ )
of the material. Using Ohm’s Law, the resistance is,
V
R
∑

ΔyJ
γ
x
applied
=
(5)
where γ is any line connecting the top and bottom (non-
contact edges) of the resistor and Δy is the grid spacing in
the non-contact direction.
On the contact ends of the resistor, the voltage is
known (Dirichlet boundary condition). On the top and
bottom of the resistor, the electric field perpendicular to
the boundary is zero (Neumann boundary condition),

y ˆ
y
V
x ˆ
x
V
VE
∂
∂
−
∂
∂
−=−∇=
(6)
so, at the top and bottom boundary (y is perpendicular the
top and bottom),
∂
−=

0
y
V
Ey
=
∂
(7)
Trims into the resistor and voids in the resistive material
require applying a Neumann boundary condition on the
edges of the trim or the void.
The simulator was verified against experimental
results using conductive paper by both probing the
voltage in the resistor and measuring the resistance as a
function of trimming the resistor, Figure 3. The trim spot
size in the simulator also had to be calibrated. This was
performed by experimentally trimming one spot
successively larger in an experimental resistor sheet until
the resulting resistance matched the resistance on a 1 x 1
grid cell trimmed in the simulator.

3. RESULTS: TRIMMING ANALYSIS
Random trimming is performed by choosing a point
on the resistor at random, and then “firing” a hole into
that position, Figure 4. Random trimming by itself,
although well controlled, is generally too long of a
process (it takes too much time). So, random trimming
was combined with the single-dive trim in the following
way: a single-dive was performed to reach a target
resistance range quickly, and then the random trim was
implemented to get as close to the target resistance as
possible. If a random trim happens to coincide with or
touch a void, the effect on the resistance is intuitively
much less than it would be if a void is encountered on the
edge of one of the fixed trimming patterns shown in
Figure 1. It was also necessary to define the current
channel region between the end of the single-dive and the
edge of the resistor as an area where random trims were
not allowed in order to better control the resistance.
Note, the random trimming approach suggested
herein differs from the “swiss cheese” approach suggested
in Ramirez-Angulo and Geiger (1988), where a set of trim
targets are initially created as holes in the resistor and
then cuts from the edge of the resistor to the targets are
performed. In the random trimming approach proposed
here, after an initial single-dive trim into the resistor,
random trims are made. This approach decreases the
heat-affected portion of the resistor and provides a
controlled approach to the target resistance that minimizes
the sensitivity of the trim to voids that may be present in
the resistor material. The following sections quantify the
performance and producibility (process capability) of
random trimming relative to L-cuts.

3.1 Trimming Precision Results
For the example shown in Figure 5, the target
resistance was 100 kΩ. When trimming with a single-
dive, the resistance increase per trim step near the target
Experimental VoltagesExperimental VoltagesExperimental Voltages
TrimTrimTrim
Simulation VoltagesSimulation Voltages
1155 total
grid pointsgrid points
TrimTrim
Centered Straight DiveCentered Straight Dive
6060
7070
8080
9090
100100
110110
120120
130130
140140
150150
00224466881010
Length of Trim (cm)Length of Trim (cm)
Resistance (Kohms)
Simulation
ExperimentalExperimental
135 total
probe pointsprobe points
1155 total
Resistance (Kohms)
Simulation
135 total
L-Cut Turn after 6L-Cut Turn after 6
6060
6565
7070
7575
8080
8585
9090
9595
100100
00224466881010
Length of Trim (cm)Length of Trim (cm)
Resistance(Kohms)
Simulation
ExperimentalExperimental
Simulation VoltagesSimulation VoltagesSimulation Voltages
TrimTrimTrim
Simulation Electric FieldsSimulation Electric Fields
CornerCorner
Resistance (kΩ Ω)
Resistance(Kohms)
Simulation
Resistance (kΩ Ω)

Fig. 3. Simulation verification for a single-dive and an L-cut trim by comparison with experimental results. This figure
includes contour plots that show the voltage variations and electric field variations.

Page 5

4
was 3.8 times greater than that of an L-cut, and 15 times
greater than that of a random trim. This result
demonstrates that the random trim results in a significant
increase in precision compared to the single-dive and L-
cut. If the trimming pattern can get closer to the target, a
higher precision resistor can be fabricated. Higher
precision trimming allows a larger fraction of the resistors
in a system to be embedded (as opposed to discrete).
The result in Figure 5 is an example for a single
resistor without voids. To statistically demonstrate
random trimming’s precision advantages, trimming was
performed on resistors with a range of void densities (1-6
voids per resistor) and void sizes in the 28 x10 centimeter
resistors (4 different void diameters: 1.0 centimeter, 1.5
centimeters, 2.0 centimeters, and 2.5 centimeters).
The following steps were performed to obtain
quantitative results from the numerical simulation: for a
resistor with a specific number of voids of a specific size:
1) randomly place the voids in the resistor, 2) perform a
single-dive to a specific depth (specified percentage of
target resistance), and 3) perform either an L-cut
(conventional approach) or random trim (proposed new
approach) to reach the target resistance. Steps 2 and 3
were then repeated for a series of random void
placements. The whole process was repeated for a range
of void densities (voids per resistor) and void sizes.
Analyses were performed by varying the stopping
criterion for the single-dive. Figure 6 shows an analysis
for initial single-dive depths ranging from 82.5% of the
target to 93.4% of the target. If the single-dive was
allowed to go as far as possible without over-trimming
(which can be done with the simulation, but would be
impractical for real resistors; not shown on Figure 6)
random trimming reduces the mean error after trimming
from 1.8% for L-cuts to 0.46%.
The result in Figure 6 shows that the precision
obtained from L-cuts decreases as the initial single-dive
goes further into the resistor; however, the precision that
can be obtained from the random trim is approximately
independent of the length of the initial single-dive. Figure
6 also shows distributions of the L-cut precision and the
random trim precision. The shapes of the distributions
shown in Figure 6 allow the inference to be made that the
precision of random trimming may be, for practical
purposes, better than the calculated mean.
The result in Figure 7 shows that the number of
trimming steps necessary to reach the target decreases as
the initial dive depth increases and is always larger with
random trimming (as would be expected). However, the
result in Figure 7 suggests that the single-dive/random
trimming combination can be practically used with an
initial dive depth of 93.4% while L-cuts are probably
practically limited to an initial dive depth of 87% or less.

3.2 Process Capability (Producibility) Results
The previous section addresses how close the L-cut
and random trimming approaches can get to a target
resistance (“precision”), but it does not address the
capability of the trimming process to produce embedded
resistors that fall into an application’s specification range,
e.g., the ability to obtain a 1% or 5% resistor.
The calculation of Cpk will be used to examine the
spread as well as the centricity of a process between
specification limits.
min[HSL
Cpk=

3σ
LSL] -
μμ,-
(8)
where HSL and LSL are high-specification limit and low-
specification limit respectively, µ is the mean of the
process, and σ is the standard deviation.
An analysis of the Cpk for the random trimming
process was performed, and was compared with the Cpk
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
8082848688909294
Initial Single-Dive Depth (% of target resistance)
Variation from Target Resistance (% of
target resistance)
0
5
10
15
20
25
30
35
40
45
00.4
Variation (% of target resistance)
0.81.21.622.42.83.23.64
Precis ion (% of target resistance)
66 Points66 Points
Frequency
Single-Dive/Random Trimming
L-Cut Trimming
Higher Precision
Lower Precision
67 Points
00
55
1010
1515
2020
2525
30 30
3535
4040
45 45
000.40.4
Variation (% of target resistance)
0.80.81.21.21.61.6222.42.42.82.83.23.23.63.644
Precision (% of target res istance)Precision (% of target res istance)
Frequency
66 Points66 Points
Frequency
Fig. 6. Trimming precision as a function of stopping criteria.
Each data point in the graph on the left represents the mean of
55 or more void density/void size combinations. The
histograms on the right show the data from two of the points.
606060 60
65656565
70 707070
757575 75
80 808080
85 858585
9090 9090
95959595
100100100100
105105105 105
0000555510 1010101515 1515 20 20202025252525303030303535 3535 404040404545 4545 50 505050
Length of Trim (cm)Length of Trim (cm)Length of Trim (cm)Length of Trim (cm)
Resistance (Kilo-ohms)
Single-Dive
Single-Dive
Single-DiveSingle-Dive
Lcut
Lcut
L-cut
LcutLcut
L-cut
Single-Dive/Random
Single-Dive/Random
Single-Dive/RandomSingle-Dive/Random
Resistance (kΩ)
Difference =
6.35 kΩ
6.35 kΩ
6.35 kΩ
6.35 kΩ
Difference =
1.66 kΩ
1.66 kΩ
1.66 kΩ
1.66 kΩ
Difference =
0.42 kΩ
0.42 kΩ
0.42 kΩ
0.42 kΩ
Resistance (Kilo-ohms)
Resistance (kΩ)
Difference =
Difference = Difference =
Difference =
Difference = Difference =
Difference =
Difference = Difference =
Resistance (Kilo-ohms)
Resistance (kΩ)
Resistance (Kilo-ohms)
Resistance (kΩ)

Fig. 5. Random trimming allows a significant increase in
precision compared to the single-dive and L-cut in a resistor
without voids. (Resistor dimensions: 28 cm x 10 cm,
dimensions similar to that of conductive paper)
Random Trim Random TrimRandom TrimRandom Trim
1 11 11 11 1
3 33 33 33 3
6 66 66 6 6 6
2 22 22 22 2
5 55 55 55 5
4 44 44 44 4
Void in resistive materialVoid in resistive material
Trim spot Trim spot

Single-Dive Followed by
Random TrimRandom TrimRandom TrimRandom Trim
Region where random trims are
not allowednot allowed
Single-Dive Followed by Single-Dive Followed by Single-Dive Followed by
Region where random trims are

Fig. 4. Random trimming. Left – pure random trimming;
right – single-dive first then random trim to target. Example
voids are also indicated.