Resistors, capacitors, and inductors are the basic passive building blocks of many circuits. These elements can be used to control timing and frequency response. In this experiment we will investigate how circuits comprised of resistors and capacitors behave when their voltages and currents are switched. We will watch the voltage increase on a capacitor as it charges over a relatively long period of time, then observe shorter-term effects with a resistor-capacitor (RC) circuit.
To study how RC circuits behave with switched input voltages. Following completion of this lab you should be able to explain how a capacitor charges when a step voltage is applied to it through a resistor, describe the shape of the voltage across a capacitor when a square voltage wave is applied to it through a resistor, calculate the time constant of a RC circuit and know to what level the capacitor charges in one time constant, determine one of the more commonly-accepted time intervals that it takes to fully charge a capacitor when a step voltage is applied to it through a resistor, observe polarity requirements of electrolytic capacitors, and give a basic explanation of dielectric absorption.
A capacitor is comprised of two conductors, typically called “plates,” separated by an insulating material called a “dielectric.” In electric circuits the capacitors are used in a way in which equal and opposite charges are created on each plate. The charges are created by accumulation or depletion of free electrons in the conductors. The charges produce an electric field in the dielectric, thereby developing a voltage across the plates. For a fixed charge, Q, on the plates, the voltage, V, across the plates is determined by the capacitance value, C, according to the following relationship.
If we want to determine how the voltage and current are related, it is useful to rearrange the equation as follows.
Since current is the time derivative of Q, taking the first time derivative of each side - noting that C is constant - produces an expression that relates current to the rate of change of the voltage in a capacitor.
This can be written explicitly for the rate of change in voltage.
We can use this equation to study the charging of the capacitor. If we had a constant current source, I, the capacitor would charge at a constant rate of I/C volte per second. In the RC circuit, we have a voltage source in series with a resistor instead of a current source. When the 5 V is initially applied to the circuit, it all appears across the resistor because the initially zero voltage across the capacitor cannot change instantaneously (unless, of course, we had an infinite current available). In the first circuit the 5V across the resistor produces a current of 5V/100K amps by Ohm’s law, which begins to charge the capacitor. As the capacitor charges, however, the voltage across the resistor drops by Kirchhoff’s Voltage Law, thereby decreasing the current and the rate of change in capacitor voltage. This process produces an exponential slowing in the rate of increase of the capacitor voltage.
The charging rate in a RC circuit depends on the RC product, which is typically called the “time constant,” represented by the Greek letter tau. In one time constant, a charging capacitor will move 63% of the way from its current voltage to the voltage applied through the resistor. The capacitor voltage, VC, in a RC circuit with an applied step voltage of VT is governed by the following equation.
In one time constant the capacitor reaches approximately 63% of the applied voltage, and in five time constants it reaches approximately 99.3% of the applied voltage. The time it takes to fully charge a capacitor, from a practical perspective, is often taken to be five time constants. In the first circuit, the time constant was (100 KΩ)*(47 μF) = 4.7 seconds, making the time to fully charge the capacitor equal to about 23.5 seconds. Time constants are the same for charging and discharging, so it would take about 23.5 seconds ti fully discharge the capacitor when the voltage out of Channel A was reduced to zero volts.