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# Reliability of a simple R-L-C electronic circuit

Authors:
Name Affiliation Phone Number Email Address
The NAFEMS Stochastics Group NAFEMS mary.e.fortier@gm.com
ian.symington@nafems.org
Industrial Sectors:

Electronics

1. DESCRIPTION OF USE CASE

The Use Case presented here was developed by the NAFEMS Stochastics Working Group (SWG) in order to enhancing collaboration with Academia, Government and Industry. By reviewing the methods used to tackle the challenge problem the intention was to provide a platform for future SWG activities.   The Problem A typical electronic component or device in a subsystem may be represented by an equivalent resistive, inductive and capacitive (R-L-C) series circuit. The electric transients that occur within the subsystem are of interest. This particular challenge problem was chosen because the fundamental equations based on Kirchhoff's current and voltage laws are well known and R-L-C parameters are usually readily available to electrically characterize components or systems. This problem is also common among automotive and industrial application where simple devices are part of a much larger electrical architecture. The underlying physics and mathematical model in the form of ordinary differential equation (ODE) can be solved in a variety of software tools. Other analogies to such network include mass attached to a spring and damper or hydraulic pipe system with a dynamic pump and paddle wheel. A schematic of the R-L-C network for the device is shown in Figure 1. The input signal to the device is a step voltage source (V) shown in the circle in Figure 1 while the output signal is the voltage across the capacitor. This capacitive voltage is sensitive to the R-L-C parameters. For this system of interest, the network parameters are assumed to be not known precisely. Uncertain estimates of R-L-C parameters were made available to the participants of this challenge. The goal is to evaluate the reliability of the device using two different criteria and quantify the value of the information provided regarding R-L-C parameters.

Requirements The first functional requirement specifies a minimum voltage drop of 0.9 Volts across the capacitor element at a particular time, 10 milliseconds in this case. The second requirement states the capacitive voltage rise should occur within a specified duration, 8 milliseconds in this application. The voltage rise time is defined as the time from 0% to 90% of the input voltage. These two requirements can be mathematically represented using Equations 1 and 2 where Vc is the capacitive voltage and t is time. $$V_c(t = 0.01) \geq 0.9~~~(1)$$ $$t(V_c = 0.9) \leq 0.008~~~(2)$$ Mathematical Model Equations representing this model are stated in the form of ordinary differential equations with zero initial conditions. For the sake of simplicity, the solution to the network was directly provided here. The system transfer function is defined as: $$\frac{V_c}{V}=\frac{\frac{1}{(LC)}}{s^2+\left(\frac{R}{L}\right)s+\frac{1}{LC}}~~~(3)$$ Depending on the values of R, L and C, the system may be classified as underdamped, critically damped or over damped. The solution for each case may be obtained as:   Under damped: $$V_c(t) = V + \left(A_1\cos\omega t + A_2\sin\omega t\right) e^{-\alpha t}~~~(4)$$ $$A_1 = -1, A_2 = -\frac{\alpha}{\omega}$$     Critically damped: $$V_c(t) = V + \left(A_1 + A_2 t\right) e^{-\alpha t}~~~(5)$$ $$A_1 = -1, A_2 = -\alpha$$   Over damped:  $$V_c(t) = V + \left(A_1 e^{s_1 t} + A_2 e^{s_2 t}\right)~~~(6)$$ $$A_1 = \frac{s_2}{s_1 - s_2}, A_2 = -\frac{s_1}{s_1 - s_2}$$ $$S_{1,2} = \frac{-R}{2L}\pm\sqrt{\left(\frac{R}{2L}\right)^2 - \frac{1}{LC}}, \alpha = \frac{R}{2L}, \omega = \frac{1}{\sqrt{LC}}, \zeta = \frac{\alpha}{\omega}$$       Step voltage source V = 1.0 when t > 0. The coefficient A1, A2 in each case are solved from the initial conditions: $$V_c(0) = 0, \frac{dV_c(0)}{dt}=0~~~(7)$$

2. KEY UQ&M CONSIDERATIONS
2.1 Process Inputs

Four cases of uncertainty estimates on the R-L-C parameters are presented in this challenge problem   Case A: Intervals

 R(Ω) L(mH) C(μF) [40, 1000] [1, 10] [1, 10]
Case B: Multiple Intervals
 Source R(Ω) L(mH) C(μF) 1 [40, 1000] [1, 10] [1, 10] 2 [600, 1200] [10, 100] [1, 10] 3 [10, 1500] [4, 8] [0.5, 4]
Case C: Sampled Points
 R(Ω) L(mH) C(μF) {861,87,430,798,219, 152 64, 361, 224, 614} {4.1, 8.8, 4.0, 7.6, 0.7, 3.9, 7.1, 5.9, 8.2, 5.1} {9.0, 5.2, 3.8, 4.9, 2.9, 8.3, 7.7, 5.8, 10.0, 0.7}
Case D: Incomplete Data
 Source R(Ω) L(mH) C(μF) Interval [40, Ru] [1, Lu] [CL, 10] Other Information Ru > 650 Lu > 6 CL < 7 Nominal 650 6 7

2.2 Propagation

N/A

2.3 Interpretation and Communication of Results

The following tasks should be addressed:   Task 1: Assess the reliability of the device meeting the functional requirements described in the requirements section (section 1.2).   Task 2: Quantify the value of the information provided in cases A to D.   The problem is expected to raise a number of issues for further consideration including:

• How can we quantify or even qualify the value of information obtained through uncertainty quantification approaches, especially when details are not clear or complete?
• How do we handle inconsistent information when describing sources of variation?
• How do we ‘fit’ distributions to limited data?

3. CURRENT STATE OF MATURITY

Not applicable – A summary of the results received to data addressing this challenge problem has been created by the SWG. This summary has not been included but can be made available for future discussion.

References:

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML Guide to the expression of uncertainty in measurement (GUM:1995 with minor corrections) Bureau International des Poids et Mesures, JCGM 100, 2008.

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML Evaluation of Measurement Data - Supplement 1 to the 'Guide to the Expression of Uncertainty in Measurement' - Propagation of distributions using a Monte Carlo method Bureau International des Poids et Mesures, JCGM 101, 2008.

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML Evaluation of Measurement Data - Supplement 2 to the 'Guide to the Expression of Uncertainty in Measurement' - Extension to any number of output quantities Bureau International des Poids et Mesures, JCGM 101, 2011.

Patelli, E., Broggi, M., de Angelis, M., Beer, M. "OpenCossan: An efficient open tool for dealing with epistemic and aleatory Uncertainties”. Liverpool: Institute for Risk and Uncertainty, University of Liverpool.