Abstract
In this lab, an RC circuit was set up and powered using a square wave from a function generator. The output was manipulated and examined and the results viewed on an oscilloscope. On the oscilloscope, the energy was seen in the way it is stored, delivered and manipulated in circuit elements. In the lab session also, the usefulness of an oscilloscope as a high-speed voltmeter was learned. How it monitors signals occurring in time intervals of microseconds and milliseconds.
Introduction
Electric circuits perform by separating the positive (+) and the negative (-) electricity. It works against the Coulomb forces of attraction between the two separated electricity. It then allows the Coulombs forces to control and combine the separate charges. Electric charges can be separated temporary and recombined. The capacitance of a piece of metal in a space – or in any conductor – is defined as the capacity of the conductor (plate) to acquire electric charge in a given voltage difference ( ). In an isolated piece of metal, is between the plate and infinity (Serway et al. 67). In an arrangement of conductors nearby each other, is the variation in potential from one conductor to the other.
A capacitor has a simple geometry in that metals of any shape and number will generate capacitance. The figure below shows the simplest geometry of a typical capacitor where two flat metal plates are used.
Figure 1. Capacitor symbol in a circuit figure 2. Simplest model of a capacitor
When the switch is closed in figure 1 a, an electric field is created nearly instantly in the circuit, and free electrons start to move towards plate #1 and away from plate #2 toward the positive terminal as shown in the figure below. The current is transient and dies out quickly when the electric charge on the plates generates an electric field between the two plates over a distance d, such that . This is precisely equal to the potential difference provided by the battery in the opposite sense. Plate #1 is negatively charged (-Q) whereas plate #2 is positively charged (+Q). At this moment, the electric field in the circuit drops to zero thus no current flows through the cables. Despite the fact that the capacitor resembles an open circuit, the charge is held indefinitely in storage by the two plates with a potential difference between them. Current flows from the battery until the whole charge deposits on the plates.
Figure 3. Electric circuit after the switch is closed; no current flows at this time when KVL is followed the net potential difference is zero.
Q available in the conductor arrangement is proportional to the energy in the circuit, that is, where the constant of proportionality is different for the arrangement of conductors. The capacitance of the geometric arrangement makes the constant: When more voltage is added to the power supply, the charge will go up proportionately.
Or [units: Coulomb/volt]
The function generator in the experiment generates a square wave where its width is controlled by manipulating the frequency of the cycles (Nilsson, 75). When the circuit battery is replaced with the function generator, square waves are shown on the oscilloscope.
Figure 4. Battery replaced with a function generator
Objectives
Use the 0.01F capacitor and a 10k resistor and wire. Carefully adjust the frequency of the square wave and determine the highest frequency at which the capacitor will fully charge and discharge.
Apparatus
Capacitors (I microfarad)
10k resistor.
Digital Multi-meter
Breadboard to connect circuit components
Jumper wires for use with the breadboard
Pliers to insert resistors and jumper wires into the breadboard.
Function generator
Oscilloscopes
Procedure
Figure 5 below was used as the guide: the input signals were connected into “input A” on the oscilloscope and the “output signal” into “input B” on the oscilloscope.
Figure 5 a). Schematic diagram of the RC circuit.
Figure 5 b) Port connection of the devices
Using an oscilloscope, the time constant for C=0.01microFarads and R = 10 Kiloohms was measured. Both R and C were set as shown in figure 5 a. a square wave was created using the function generator at a 200 Hz frequency and a peak-to-peak voltage of 6.0 0.1 volts. Data was recorded in data sheet #1.
Figure 6. How to magnify
The “Timebase Controls” and “X” were adjusted, so that passed through the upper-left part of the screen. Figure 7 was used to create a precise adjustment of the controls.
Figure 7. Adjusting the Time base Controls and X
Results
The voltage in the circuit was monitored by the oscilloscope throughout the experiment. On the oscilloscope, the capacitor responded to the square wave voltage input in charging and discharging. During the charging cycle, the voltage across the across the capacitor was Vc(t) = Vo(1-e-t/RC). When the switch was made to open, a zero voltage output was generated by the square wave generator. At this moment, the capacitor is discharging, the voltage across the capacitor at this time was found to be Vc=Vo e-t/RC where RC is the time constant. Figure 8 below shows the results seen on the oscilloscope.
Figure 8 a) the square wave driving the RC circuit. With the switch closed, the input voltage is the peak voltage, Vo. When the capacitor discharges, the input voltage drops to zero.
Figure 8 b) the voltage drop across the capacitor as read by the oscilloscope. The capacitor charges towards Vo and discharges towards zero. The time constant RC changes as time passes thus the curving nature of the function.
When the time constant was changed, manipulated five times the previous one, the relationship in figure 9 below was a result.
Figure 9. A long period of the square wave drives the circuit and voltage response in the capacitor more than in a short period. Here, the capacitor remains charged until the voltage in the input drops again to zero.
When the width of the square wave was adjusted to exceed five times the time constant, the period as well increased, the capacitor was fully charged for a relatively longer time before it started to discharge. When the width of the square wave was made five times lesser than the time constant of the cycle, the capacitor could not charge fully during its positive-volt time interval. It could not as well discharge fully during the zero applied voltage. It was seen to rise only a fraction of the full available voltage and discharge the same fraction as shown in figure 10 below.
Figure 10. Fractional charging and discharging of the capacitor when the time constant is reduced five times.
Calculations
Capacitor of 0.01 microfarads and resistor of 10 kiloohms:
Time constant = capacitance*resistance = (1.0*10-8 )*(10*103)=10-4
After a period of 10 seconds, the voltage rise; Vo*(1-e-t/RC) = Vo*(1-e-t/10-4) will be
Vo*(1-e-10/10-4).
Also, after a period of 10 seconds of voltage drop, the voltage Vo*e-t/RC) = Vo*e-t/10-4) will be Vo*e-10/10-4
Discussion
In this experiment, the relationship that exists between a capacitor and circuit charging was examined. Charging and discharging of the capacitor was compared to a full square wave whereby in the capacitor, the two states occurred with a growing and decaying response respectively. The time constant formed the basis of how fully the capacitor charged and discharged. When the time constant was made large, the period was prolonged thus the capacitor was charged fully. When the time constant was made smaller, the capacitor neither charged nor discharged fully. Only a fraction of the input voltage was charged. The discharged capacitor did not return to zero as well.
In the circuit, the resistor purposely was to limit the flow of current through the capacitor. The voltage across the capacitor, therefore, would not respond immediately to the alteration of voltage in the source. The resistor controls the charging, and discharging rate of the energy of the capacitor - the current flow across the circuit increases as the resistor is added. The oscilloscope generated the fine transient behavior of the RC circuit when the voltage was suddenly changed in the circuit. To understand the behavior of the circuit, the function generator was supposed to be ground with the uncharged capacitor. The generator modified to high state, and the entire voltage appeared across the resistor. As the charge continued to flow, it accumulated on the capacitor, and there was a build in voltage drop in the capacitor. Voltage drop consequently induced a drop in the current flowing in the circuit.
Voltage and charge in the capacitor do not grow linearly with time; they follow the exponential law instead. Initially, the capacitor is uncharged. After a time, t=RC, it is charged with 1/e (0.3679) of the final value. Similarly, when the fully charged capacitor is discharged through the resistor. The voltage across the capacitor falls to zero with time in an exponential rule.
Figure 11. The voltage of a charging capacitor.
Figure 12. Voltage of a discharging capacitor.
Conclusion
The objective of determining the charging and discharging of a capacitor was achieved successfully. The oscilloscope provided a vivid representation of the real growth and decay of the charging and discharging respectively. The function generator replacing the battery enhanced the process as an electric signal was the characteristic of examination. No challenges were observed in the experiment. Arguably, the exercise was a success.
Work cited
Nilsson, James W. Introduction to Circuits Instruments and Electronics. New York: Harcourt, Brace and World, 1968. Print.
Serway, Raymond A, Jerry S. Faughn, and Chris Vuille. College Physics. Belmont, CA: Brooks/Cole, Cengage Learning, 2009. Print.
