PHYS 2426 – Engineering Physics II

RC Circuits

 

Leader: _________________________          Recorder: __________________________

Skeptic: _________________________         Encourager: ________________________

 

Materials

Genecon

1 F Capacitor

2 x light bulbs in Sockets

4 x Alligator Clip Cables

Stop Watch

Pasco Circuit Board

Function Generator

BNC cable

BNC – Alligator cable

Oscilloscope Probe

DMM

Digital Storage Oscilloscope

Laptop

 

Part 1 Qualitative Observations with a Genecon

Introduction

      In this activity we will explore the circuit made by resistor and capacitor connected in series known as an RC circuit.  RC circuits have a lot of uses such as controlling the rate at which capacitors charge and discharge and setting rates at which circuits oscillate.  Here we will make some simple qualitative observations with the Genecon and then we will shift to the oscilloscope to make quantitative measurements.

 

Procedure

1.  Qualitative Observations

      Connect the circuit shown in figure 1 below.  Use the Genecon for the potential source, a light bulb for the resistor and the 1 F capacitor.

 

Turn the Genecon handle at about 1 turn per second for at least 20 s.

 

Q1)  Does the light bulb light initially?

 

 

Q2)  Does current flow in the circuit initially?

 

Q3)  Does the light bulb continue to glow as brightly as time goes on?

 

 

Q4)  Does current continue to flow as strongly as time goes on?

 

 

Initially the capacitor is uncharged, so to charge it current must flow in the circuit.

 

Q5)  Why does the light bulb go out as the capacitor charges?

 

 

Once the light bulb goes out, hold the handle of the Genecon.

 

Q6)  Does the light bulb now light?

 

Q7)  Does the light bulb continue to light as brightly over time?

 

 

Q8)  Is current flowing in the circuit?

 

 

Q9)  Does the current continue to flow at the same rate as time goes on?

 

 

When you hold the handle, the Genecon is essentially a long piece of wire through which the capacitor can discharge.  Current must flow in the circuit for the capacitor to discharge.

 

Q10)  Why does the bulb eventually go out?

 

 

Q11)  How does the time it takes the capacitor to charge up compare to the time it takes for the capacitor to discharge.  You may want to use the stop watch to make a comparison of the two.

 

 

Charge the capacitor again, then stop and release the handle.

 

Q12)  Does the handle turn? 

 

Q13)  Does the light bulb visibly glow?

 

 

When you release the handle, the Genecon acts as a motor.  When motors run, they develop a potential difference across them called a back EMF.  This significantly reduces the current flowing in the circuit and thus the brightness of the light bulb.  We will look at this more in a later chapter.

 

Now connect two light bulbs in series with the Genecon and capacitor.

 

Turn the handle again at the same rate as before to keep the potential supplied by the Genecon constant.

 

Q14)  Does the capacitor charge as rapidly?

 

 

Q15)  Does as much current flow in the circuit as before?

 

 

Q16)  Does this surprise you?  Explain.

 

 

Once the capacitor is charged, hold the handle.

 

Q17)  How does the time the capacitor takes to discharge compare to the time it took to charge in this case.

 

Q18)  When you add a second light bulb in series, how do you change the resistance of the circuit?

 

 

Q19)  When you increased the resistance, how did the time it took to charge the capacitor change?

 

 

We will investigate this dependence more below.

 

Part 2 Quantitative Observations with an Oscilloscope

Introduction

      In this lab we will investigate the behavior of RC circuits subjected to a square wave input as shown in figure 1.  For our purposes this will be equivalent to periodically flipping the switch shown in figure 2 

 

Figure 1  A square wave with Amplitude A, period T, and phase j

 

 

Figure 2 An RC circuit which is switched between charging and discharging

 

Application of Kirchhoff's Voltage Law and Solution of the Circuit

1.  Use Kirchhoff's voltage law to find an equation for the circuit shown in figure 2 when the switch is in the B position.  We will assume that the capacitor is initially full.

 

 

 

 

2.  Use the definition of current to turn your answer for question 1 into a differential equation for charge.

 

 

 

 

3.  Show that Q = Ae-t/τ is a solution to the differential equation only if τ = RC and A is some constant.

 

 

 

 

 

4.  Assume that the capacitor initially had a charge of Q0 at t = 0.  Use this to evaluate the constant A.

 

 

 

5.   Describe in words the type of behavior shown by your solution and sketch a graph of the charge on the capacitor vs. time.

 

 

 

 

 

 

 

6.  Note that in your solution to 4, that the resistance and capacitance occur in your solution in the form RC.  This is known as the RC time constant.  Show that the SI unit of RC is s.

 

 

 

 

7.  Interpret the meaning of RC by answering the following question.  By what factor is the charge reduced when t = RC.

 

 

 

8.  Experimentally it is much easier to determine the time that the charge is reduced by half.  Use your solution to 3 to find the time in terms of RC that the capacitor has discharged to half.

 

 

 

 

 

9.  Find an expression for the current in the circuit while the capacitor is discharging.

Sketch a graph of the current in the capacitor vs. time.  How does the decay constant for the current compare to the decay constant for the charge?

 

 

 

 

Now let's turn our attention to the situation when the switch in figure 1 is in the A position.         

 

10.  Use Kirchhoff's voltage law to find an equation for the circuit shown in figure 2 when the switch is in the A position.  We will assume that the capacitor is initially Empty.

11.  Use the definition of current to turn your answer for question 10 into a differential equation for charge.

 

 

 

 

12.  Show that Q = CV – Ae-t/τ is a solution for 11 if τ = RC.

 

 

 

 

 

 

 

13.  Evaluate the constant A in 12 if initially the capacitor is uncharged.

 

 

 

 

14.   Describe in words the type of behavior shown by your solution and sketch a graph of the charge on the capacitor vs. time.

 

 

 

 

 

15.  For this situation, what percentage is the capacitor full when t = RC?

 

 

 

 

 

 

16.  Find the time in terms of RC when the capacitor is half full.

 

 

 

 

 

 

17.  Find an expression for the current in this situation.

 

 

 

18.  Describe in words the behavior of the current and sketch a graph of the Current vs. time.

Procedure

1.  Measure the Resistors

Obtain 100 Ω, 330 Ω, and 560 Ω resistors from the baggie.  Use the DMM to measure the resistance of each of the three resistors.  Record these values.

 

 

2.  Set-up the Function Generator

      Use the BNC cable to connect the output of the function generator to CH 1 on the oscilloscope.  Turn on the function generator and press the 1k button and.  Select the square wave output by pushing the button labeled with a square wave.  Press the Auto Set button on the oscilloscope.  Adjust the amplitude button so that the output of the function generator is 5 V and set the frequency to 500 Hz.

 

3.  Build the circuit

Begin by connecting the circuit shown in figure 3.  At first use the 100 W resistor and the 1 mF capacitor.  Note the connection to ground will be made when you connect the oscilloscope leads to the circuit.   Disconnect the BNC cable from the function generator and replace it with a BNC – Alligator cable.  Connect the red lead on the cable to the resistor and the black lead to the capacitor.

 

Figure 3 Schematic of circuit used for experiment

 

         

Make sure that the switch on the oscilloscope probe is in the 1x position.  Connect the oscilloscope probe to the scope and the positive lead to the + side of the capacitor and the black alligator to the – side of the capacitor.

      Press the Auto Set button.  Adjust the frequency of the frequency generator so that the capacitor fully charges and discharges in a time a little less than 1 cycle.  Adjust the Sec/Div knob so that about two full cycles of charge and discharge are shown on the display.  Press the cursor button and press the type button to choose voltage. Adjust the cursors to measure the amplitude of the square wave.  Press type again to select time, and measure the time for the potential across the capacitor to both grow and decay by ½ the value of the amplitude.  Record your measured values in a properly labeled table attached to the report.

      Repeat the same procedure for each of the other two resistors then repeat for the entire procedure for all three resistors and the 10 mF capacitor.  Tabulate all of your data in a neatly organized and properly labeled table.

 

 

Data Analysis and Questions

1.  For each resistor-capacitor combination, determine both the time constants for the charging and the discharging of the capacitor from the times that you have measured. Tabulate your results in a neatly organized table below.

 

 

 

 

 

 

 

 

 

 

2.  Within a reasonable uncertainty, do your time constants seem to be the same for both charging and discharging?  Explain.

 

 

 

 

3.  Explain a method to determine the value of the capacitance from your measured time constant.

 

 

 

4.  From your measured time constants and your known value of the resistance, determine the value of the capacitance.  Tabulate your results neatly in a table below.  Do the values of the capacitance that you have determined agree within a reasonable level of accuracy to the labeled value of the capacitor?  Typically you can believe the label on a capacitor to about ±10%.

 

 

 

 

 

 

 

 

 

5.  Summarize what your have learned about RC circuits in the space below.