An Investigation of the Transient Response of an RC Circuit with an Unknown Capacitance Value Using Probability Theory

In this research, we investigate a resistor capacitor electric circuit that exhibits an exponentially decaying transient response. Due to uncertainty in the precise capacitance value, we treat the capacitance as a continuous uniformly distributed random variable. Using this approach, we derive the desired transient current response of the circuit as a function of the capacitance. Subsequently, we develop a probability model for the response current, expressed in terms of probability density function and cumulative distribution function. The model's validity and correctness are verified, and it is further utilized for probabilistic analysis of the transient current. We demonstrate the application of the model for determining the probability of the transient current response reaching a specific value. By following the same procedure used to derive the probability model of the transient current, probability distributions for other circuit parameters, such as voltages and currents, can also be obtained. Furthermore, for parameters that are functions of the transient current, the probability model can also be obtained from the already derived probability model. To illustrate this, we derive the probability models of three other parameters in the circuit from the already obtained models. We also present examples to demonstrate the usage of the developed probability models.