Probability model of the exponentially rising transient response of a failed RC circuit

Abstract

An electric circuit may be analyzed by means of a number of available theorems and techniques. However, the exact value of a circuit element parameter may not be known sometimes, and only partial information, such as range of values in which the parameter lies, may be available. This situation may arise when a circuit component develops a fault while it is deployed in a harsh environment. In this work, we exploit probability methods to explore such an electric circuit where only partial information about the range of values in which a circuit element parameter lies is available. We treat the circuit element parameter lying in a range of values as a random variable. Thereafter, any circuit parameter, such as a voltage or current, is analyzed using probability theory. Probability model of a desired parameter is derived in terms of the cumulative distribution function and probability density function. We demonstrate this technique by using the example of a resistor capacitor electric circuit whose transient response is an exponential rise. The capacitance value of the capacitor is known to lie in a range of values and is treated as a random variable. Probability distributions of the voltage across the capacitor and the voltage at the output terminals of the electric circuit are derived. Use of the probability distributions for the calculation of probabilities is also demonstrated.