An Investigation of Discharge Rates of RC Circuits in Different Configurations

Video Abstract

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Introduction

A capacitor refers to an electrical component that stores electrical energy in an electric field; the effect of the capacitor is known as capacitance. A capacitor is constructed with two plates of conducting materials, with an insulator separating in between. As an electrical potential difference is applied across the two plates, electrical charge begins to build up on each plate, and the charge creates an electric field between the plates. While no current flows through the conductor plates, there is a flow of charge in the circuit. If the voltage is applied across long enough, the charges on the plates approaches a peak value, and current stops flowing through the circuit. The voltage across the plates approaches the voltage of the power source.

Depending on the design of the circuit, the capacitance varies. In this study, I set up different configurations of capacitors and measured the combined capacitance in order to investigate how different configurations alter the combined capacitance.

Methods and Materials

In order to calculate the combined capacitance of the system, a resistor of relatively high resistance was selected, since a higher resistance in the circuit corresponds to a slower discharge and a more precise exponential curve fit. The resistance of the ohmic resistor was calculated by applying different voltages across it, and the voltages were graphed against current. By Ohm's law, the resistance of the resistor is the constant rate of change V/I.

The capacitors were then set up according to the run, connected with alligator clamps. They were first charged up to 12 volts, an arbitrary value irrelevant to the calculation of the combined capacitance. The power source was disconnected, and the circuit was reconnected without the power source. The voltage versus time was recorded with the voltmeter through Logger Pro, and the exponential function was fitted (Fig. 1). The C-Value was then extracted and recorded in excel.

The C-Value was converted to capacitance by using the derived exponential decay formula, and the capacitances of each configuration was graphed.

Figure 1. An example of the exponential voltage decay curve, fitted with the function V = A * e^(-C * (t + B))

Figure 2a. Three capacitors set in series configuration, with an ohmic resistor of 151.6 Ω. A voltmeter is connected across the resistor, and the voltage decay curve is recorded and fitted with an exponential function.

Figure 2b. Three capacitors set in parallel configuration, with an ohmic resistor of 151.6 Ω. A voltmeter is connected across the resistor, and the voltage decay curve is recorded and fitted with an exponential function.

Derivation of the Voltage Decay

Results

Figure 3a. Combined capacitances of capacitors in series arrangement

Figure 3b. Combined capacitances of capacitors in parallel arrangement

Figure 3c. Combined capacitance versus number of capacitors in series

Figure 3d. Combined capacitance versus number of capacitors in parallel

Discussion

Based on figures 3c and 3d, the conclusion can be made, that for capacitors in series, the combined capacitance is inversely proportional to the number of capacitors, and for capacitors in parallel, the combined capacitance is positively proportional to the number of capacitors.

However, the figures 3c and 3d both show lower observed capacitance than expected for higher numbers of capacitors. One explanation to this deviation could be that since the setup is not an ideal setup, there are energy losses with each capacitor. Energy lost means that the voltage decay happens faster than in an ideal scenario, which reflects a lower capacitance than if there were no energy losses.

This experiment could be extended to examine RL (resistor-inductor) and LC (inductor-capacitor) circuits, and the properties of those setups.

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