2. DC ANALYSIS 16
2.11 DESIGN TOLERANCES, SENSITIVITY ANALYSIS
A MIMO circuit of Fig. 2.7.3 is uniquely characterized by its L constants (primary parameters) . Any transfer function (secondary parameter) is a function of these constants, , any output signal is a function of these constants and input signals, . Both transfer functions and output signals can be considered as circuit variables designated by circuit constants and/or circuit inputs
(2.11.1)
Example 2.11.1
A voltage divider presented in Fig. 2.11.1 is characterized by two parameters, resistances . Then, two circuit variables, transfer functions are selected:
input resistance: (2.11.2a)
gain: (2.11.2b)
Fig. 2.11.1 Voltage divider
Then, ideal and practical circuits are distinguished.
Ideal circuit: All circuit parameters have nominal values:
(2.11.3)
Practical circuit: All circuit parameters are characterized by nominal values and design tolerances, i.e. their values lay within tolerance margins:
(2.11.4)
Based on tests, made during the manufacturing process, the probability distribution for each parameter can be found. Presented in Fig. 2.11.2 normal or Gauss distribution is the most commonly used. This distribution is described by the following equation (index i has been omitted for simplicity of description):
where is the so called standard deviation.
Fig. 2.11.2 Normal distribution of circuit (element) parameter
For the given standard deviation, tolerance margins are related with the probability distribution by the following equation:
(2.11.6) where q is the production yield, e.g. for , 95% of the production is classified as
“healthy”.
For the assumed parameter deviation X X Xn Xn X 0, production yield q can be calculated from (2.11.6). If q is assumed, then the acceptable deviation can be calculated, practically deviation of
For P parameters characterizing a circuit, its tolerance region can be defined.
Tolerance region (tolerance box) is a parallelepiped in the parameter space with planes parallel with coordinate axes, and designated by tolerance margins of circuit parameters
.
Example 2.11.1 – cont.
The nominal values of resistances and their tolerances are: , . Graph the tolerance region.
The tolerance region is presented in Fig. 2.11.3, nominal point is denoted.
Fig. 2.11.3 Tolerance region for Example 2.11.1
Presence of design tolerances has to be taken into account at a circuit design stage. Two approaches are possible:
1. Parameter design tolerances are given by the design-engineer. Finding of maximum deviations of circuit variables, caused by these tolerances, is the task.
2. Design specifications, acceptable deviations of circuit variables, are given by the design-engineer. Finding of parameter design tolerances is the task.
Designation of the maximum design deviation of circuit variable
For each circuit variable F (index has been omitted for simplicity of description), its maximum deviations, due to design tolerances of circuit parameters, can be found. Two different techniques are possible:
worst case analysis,
sensitivity analysis.
P
XP
X ,...,1
2 1
1 n
n R
R 05
. 0 ,
1 .
0 2
1 tol
tol
1.05 0.95
0.5 0.9 1.1
Worst case analysis
It is assumed that within the tolerance region, first derivatives of a circuit variable function (2.11.1) do not change sign:
for (2.11.9)
Then, the boundary values of a circuit variable, due to a presence of parameter design tolerances, are calculated by setting in function (2.11.1) the boundary values of parameters:
(2.11.10a) is the 1st derivative calculated at the nominal point , the so called sensitivity of a circuit variable F with respect to small changes of parameter in a close neighborhood of the nominal point, the 1st order sensitivity. For M circuit variables and P circuit parameters, the
sensitivity matrix can be created
(2.11.11a)
Finally, the maximum deviation caused by parameter tolerances is
(2.11.12)
Example 2.11.1 – cont.
The boundary values of circuit variables are calculated from the following equations:
(2.11.13a)
(2.11.13b)
and the maximum deviations, caused by the design deviations of parameters are:
const
(2.11.14)
Sensitivity analysis
Consider the 1st order approximation of the circuit variable function (2.11.1), its Taylor’s series expansion around the nominal point:
(2.11.15) Then, the deviation of a circuit variable can be expressed by the 1st order sensitivities and parameter deviations:
(2.11.16) The relative sensitivity can be introduced:
(2.11.17) and then, the relative deviation of a circuit variable is
(2.11.18) To find the maximum deviation, signs of sensitivities should be disregarded:
(2.11.19)
(2.11.20)
Example 2.11.1 – cont.
Sensitivities of the selected circuit variables are:
(2.11.21a)
(2.11.21b) Then, the maximum deviations are:
, .
Normally, an analytic form of a circuit variable function (2.11.1) is not known. Then, two different methods of sensitivity calculations, other than the explicit one, are used.
Adjoint Circuit method, based on Tellegen’s Theorem.
Direct method.
In the Tellegen’s theorem based adjoint circuit method, an adjoint circuit is created. This circuit is obtained from the original (nominal) one by zeroing all sources (shorting voltage sources, opening current sources). Its excitation is designated by the considered circuit variable. Next, based on analyses of two circuits, the adjoint and the original one, all sensitivities of this variable, one row of the sensitivity matrix, are calculated, Tellegen’s Theorem is applied. Details of this method are not discussed.
As M circuit variables are considered, then, to find all sensitivities, analyses have to be performed: original circuit analysis + M analyses of adjoint circuits.
In the direct method, two analyses are performed, the original (nominal) circuit analysis and analysis of the nominal circuit with an increment added to one parameter:
, (2.11.22)
From the results of analyses, increments of all circuit variables are designated and sensitivities with respect to small increment of , one column of the sensitivity matrix, are calculated:
(2.11.23) As P circuit variables are considered, then, to find all sensitivities, analyses have to be performed: original circuit analysis + P analyses of circuits with one parameter incremented.
Designation of parameter design tolerances
It is assumed that for the selected M circuit variables, design specifications are set by the designated. Various methods of mapping and then design centering and tolerancing are used and they are not discussed here.
Example 2.11.2
Then, the acceptability region in the parameter space is defined by the following inequalities:
(2.11.25a) (2.11.25b) From (2.11.25a) the following two boundary lines are designated:
(2.11.26a)
From (2.11.25b) the other two boundary lines are designated:
(2.11.26b)
This boundary lines and the obtained acceptability region are presented in Fig. 2.11.4.
Fig. 2.11.4 Example 2.11.2 acceptability region with marked tolerance regions
Design centering and tolerancing is the next step. For this simple example, central location of 2
. 2 8
.
1 R1 R2 55 . 0 45
. 0
2 1
2
R R
R
2 . 2
: 2 1
max R R
Rin
8 . 1
: 2 1
min R R Rin
1 2
max :R 1.22R
K
1 2
min :R 0.82R
K
n
n
2.2 2.0 1.8
1.6
1.4 1.2
1.0 0.8 0.6 0.4 0.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
assumed 100% yield, the greatest tolerance box included in the acceptability region can be found: , as marked by the dotted lines. Another tolerance box:
, marked by the dashed lines, overlaps the acceptability region. For this box, the production yield is less than 100%.
Drill problems 2.11
1. Given the voltage divider: , find all four voltage sensitivities
2. In the voltage divider of Problem 2.11.1, and resistors have 10% tolerance.
Find the design deviations of both voltages. Use both the worst case and sensitivity approach.
3. Given the current divider: , find all four current sensitivities.
4. In the current divider of Problem 2.11.3, and resistors have 10% tolerance.
Find the design deviations of currents. Use both the worst case and sensitivity approach.
R1 R2 0.1
R1 R2 0.2
10 , 2 20
1 R
R
V
30 Uin
10 , 2 20
1 R
R
A
30 Iin