Connecting Time and Frequency in the RC Circuit

Charging and discharging processes of a capacitor through a resistor, as well as the concept of impedance in alternating current circuits, are topics covered in introductory physics courses. The experimental study of the charge and discharge of a capacitor through a resistor is a well-established lab exercise that is used to introduce concepts such as exponential increase or decrease and time constant. Determining the time constant of the RC circuit has important practical applications because, for example, it can be used to measure unknown values of resistance or capacitance. The transient experiment can be done by using a voltmeter and stopwatch, signal generator and oscilloscope, or even low-cost data acquisition systems such as Arduino. An equivalent topic when studying alternating current circuits arises from the characterization of the impedance of the series or parallel combination of the capacitor and the resistor as a function of frequency. Determining the time constant of the RC circuit by means of impedance measurements for different frequencies is a known experimental technique that can be done using not only LCR meters but also basic instrumentation in the physics lab such as a signal generator, frequency counter, and multimeter. However, lab exercises dealing with RC circuits in alternating current usually focus on their use as filters, and the potential applications in the field of the electrical characterization of material systems are ignored. In this work, we describe a simple exercise showing how the time constant of the RC circuit can easily be determined in the introductory physics lab by means of impedance measurements as a function of frequency. This exercise allows students to learn experimental techniques that find application to characterize the time constants of the charge transport processes in material systems. Moreover, comparison of the time constants obtained from transient and frequency analysis allows us to relate the time and frequency domains, which plays a central role in the advanced analysis of electric circuits, once the concept of Laplace transform has been introduced in order to simplify the problem of dealing with differential equations in the time domain by converting them into algebraic equations within the frequency domain.