Worst case analysis methods for electronic
systems to reduce technical risk and
improve system reliability
Ray Kendall, P.E.
Intuitive Research and Technology Corporation
6767 Old Madison Pike
Building 4, Suite 440
Huntsville, AL 35806
www.irtc-hq.com
256-971-1992 x121
Fax 256-971-1992
ray.kendall@irtc-hq.com
April 17-19, 2007
Design margin and worst case analysis of electronic circuits and systems
Design margin is the margin between the worst case performance of an electrical
system’s design implementation and the performance required by specification
•Design margin is analogous to the margin of safety determined analytically for
mechanical structures
Performing worst case analysis to determine the design margin is the only way to
verify that electronic circuits and systems will function properly under all conditions
•Worst case analysis determines the maximum variation of electrical design
performance by analytically selecting the worst combination of conditions that the
design can experience during its operational lifetime
Worst case analysis is critical for high-reliability Defense electronic systems
•Most current Defense contracts are written so the contractor only has to verify
compliance with electronic system specifications by test
•Testing cannot determine the design margin of the hardware design
•Reliability analyses generally assume that electronic designs work properly and meet
their specifications under all conditions
•Not performing worst case analysis places electronic systems at risk for an
unidentified quantity of reduced system reliability
•Worst case analysis can be used to reduce technical risk and improve system
reliability of critical Defense electronic systems
1
Worst case analysis methods for electronic circuits and systems
Analysis tools
•Hand analysis (non-simulation) methods can be used for worst case
analysis of electrical circuits and systems of limited complexity
•Computer simulation programs provide the most powerful methods for
performing worst case analysis of electronic circuits and systems
•Spice (Simulation program with integrated circuit emphasis) is an
industry-standard electronic circuit simulation tool that can produce
accurate high-fidelity simulations of complex electronic circuits
•Spice and other computer simulation programs in combination with
sound analytical methods are recommended for performing worst
case analysis of electronic circuits and systems
2
Worst case analysis methods for electronic circuits and systems
Factors affecting worst case performance of electronic circuits and systems
•Worst case analysis must consider variations in the following factors and
conditions affecting design performance:
•Part tolerance and parameter variation
•Aging effects
•Environmental conditions, including temperature extremes
•Input and output interface conditions
•Some of these factors are difficult or impossible to take into account
during test, especially part tolerance
•Worst case performance cannot be determined by test
•All these factors must be considered for worst case analysis regardless
of the analysis tools and methods used
3
Worst case analysis methods for electronic circuits and systems
Extreme value analysis
•A worst case analysis uses all of the factors affecting performance to
determine the worst case parameters for each part and condition in an
electrical circuit
•An extreme value analysis adjusts all of the circuit parameters to their
worst case values that produce the worst case circuit output
characteristic
•An extreme value analysis is the most conservative worst case analysis,
showing what the performance would be at the outer edge of probability
•If a positive design margin results from performing an extreme value
analysis, the design is good
4
Worst case analysis methods for electronic circuits and systems
Monte Carlo analysis
•If a positive design margin does not result from performing an extreme
value analysis, it may be appropriate to perform a Monte Carlo analysis
•A Monte Carlo analysis is a statistical analysis performed by a
computer simulation program by randomly adjusting the defined circuit
parameters within their worst case limits
•A Monte Carlo analysis can give the probability of a circuit output
characteristic being within a defined range, including determining the
probability of a negative
design margin
•Monte Carlo analysis
results are commonly
displayed in the form of a
histogram
P
e
r
c
e
n
t
15
o
f
S
a
m 10
p
l
e
s
5
0
10m
15m
n samples
= 200
n divisions = 15
mean
sigma
20m
= 0.0246584
= 0.00414288
25m
Max(I(U14))
minimum
10th %ile
= 0.0149284
= 0.0196616
30m
median
90th %ile
= 0.0242169
= 0.0301862
35m
maximum
3*sigma
40m
= 0.036474
= 0.0124287
Figure 1 – Example of a Monte Carlo histogram
5
Part models for worst case analysis of electronic circuits and systems
Worst case (extreme value) models
•Linear parts – resistors, capacitors, and inductors
•Linear parts have a single
primary characteristic to
model
•Independent factors
affecting worst case
tolerance should be
combined using the rootsum-square (RSS) method
(see Table 1 example)
Parameter
Symbol
Tol
Initial tolerance
TC
Temperature coefficient
TL
Life
TM
Moisture resistance
TB
Bonding exposure
TH
High temperature exposure
TO
Short time overload
TLT
Low temperature operation
TT
Thermal shock
Calculated RSS worst case tolerance:
Maximum value
1%
100 ppm/°C
0.5 %
0.5 %
0.25 %
0.5 %
0.25 %
0.25 %
0.5 %
T = Tol 2 + (TC ⋅ ΔT ) 2 + TL2 + TM 2 + TB 2 + TH 2 + TO 2 + TLT 2 + TT 2 = 1.6%
Table 1 – RSS tolerance calculation for MIL-R-55342F resistors
•Semiconductor parts
•The complex interaction of Spice semiconductor model parameters usually
makes the use of simple model parameter tolerances impractical for worst case
analysis
•Specialized minimum and maximum worst case Spice models can be used to
produce accurate worst case analyses for most scenarios (see model guidelines
in Table 2 and Table 3)
6
Part models for worst case analysis of electronic circuits and systems
Characteristic
Diode
Minimum output
Maximum forward Minimum current
voltage (Vf)
gain (hFE);
maximum
collector-emitter
saturation voltage
(Vce(sat))
Maximum baseemitter voltage
(Vbe)
Maximum junction Maximum junction
capacitance and
capacitances and
switching times
switching times;
minimum
bandwidth
Maximum input
Minimum speed
Transistor
MOSFET
Macromodel
Minimum
transconductance
(gFS); maximum
On resistance
(Rds(on))
Minimum output
voltage swing and
gain; maximum
output resistance
Maximum
threshold voltage
(Vth)
Maximum turn-on
charge,
capacitances, and
switching times
Maximum input
voltage threshold
or offset
Maximum
switching times;
minimum slew
rate and
bandwidth
Table 2 – Minimum worst case semiconductor part model guidelines
Characteristic
Diode
Transistor
MOSFET
Macromodel
Maximum output
Minimum forward
voltage (Vf)
Maximum current
gain (hFE);
minimum
collector-emitter
saturation voltage
(Vce(sat))
Minimum baseemitter voltage
(Vbe)
Minimum junction
capacitances and
switching times;
maximum
bandwidth
Maximum
transconductance
(gFS); minimum
On resistance
(Rds(on))
Maximum output
voltage swing and
gain; minimum
output resistance
Minimum
threshold voltage
(Vth)
Minimum turn-on
charge,
capacitances, and
switching times
Minimum input
voltage threshold
or offset
Minimum
switching times;
maximum slew
rate and
bandwidth
Minimum input
Maximum speed
Minimum junction
capacitance and
switching times
Table 3 – Maximum worst case semiconductor part model guidelines
7
Part models for worst case analysis of electronic circuits and systems
Monte Carlo modeling methods
•For linear parts a probability distribution must be defined for the primary
characteristic using a probability distribution function (see Table 4) with
the worst case tolerance defining the edges of the distribution
Probability distribution
function
Uniform
•Spice semiconductor
models usually cannot
use direct modeling
methods for Monte Carlo
analysis because of the
complex interaction of
their model parameters
Shape
Application
Simplest and most
conservative – all values
equally likely
Gaussian (normal)
Most realistic for most
parameter variations
Triangular
Simple linear
approximation of
Gaussian, works well for
non-uniform variations
Table 4 – Common probability distribution functions for Monte Carlo analysis
•Macromodeling methods can sometimes be used to linearly
approximate the range of worst case operation for a Monte
Carlo analysis
8