Unit 4 - Students Absolutely Must Learn… Weekly Activity 7: RC Circuits - DC Source    

The properties of exponential functions. How to describe the time dependence of charging and discharging capacitors with exponential functions. What differential equations are and how to check their function-solutions. Advanced features of oscilloscopes.

Weekly Activity 8: RC Circuits - AC Source   

The behavior of a sinusoidally driven system including phase shifts. How to use the solutions to the RC circuit driven by a sinusoidal voltage source and what they mean. Advanced features of oscilloscopes.

1

Unit 4 - Grading Guidelines Staple the lab report, then graphs, and finally worksheets together. Please put your worksheets in order. Turn in your work to your TA at the beginning of the next lab meeting following the completion of the unit. Unit Lab Report [50%, graded out of 25 points] Write a separate section using the section titles below (be sure to label these sections in your report). In order to save time, you may add diagrams and equations by hand to your final printout. However, images, text or equations plagiarized from the internet are not allowed! Remember to write your report alone as collaborating with a lab partner may make you both guilty of plagiarism. Pay close attention to your teacher for any changes to these guidelines. 

Title [0 points] – A catchy title worth zero points so make it fun.



Concepts & Equations [9 points] – {One small paragraph for each important concept, as many paragraphs as it takes, 2+ pages.} Go over the lab activities and make a list of all the different concepts and equations that were covered. Then simply one at a time write a short paragraph explaining them. You must write using sentences & paragraphs; bulleted lists are unacceptable. Some example concepts for this unit report include (but are not limited to):  Describe the operation and features of the digital oscilloscope.  Discuss what differential equations are and how to check their solutions. You should give an example (keep it simple).  Discuss the construction process of the two differential equations that model RC circuits with DC source. Discuss the solutions to these differential equations.  Discuss all you know about capacitors (should be a lot from lecture).  Discuss how to find the capacitance of a capacitor using an RC circuit with a DC source.  Discuss how to determine if data has an exponential relationship.  Discuss what the half-life is of an exponential relationship is and how it works.  Compare and contrast how to find the capacitance of a capacitor using a DC source (square wave) versus a sinusoidal source.  Discuss at length the three time dependent voltage equations that describe the AC-driven RC circuit. Be sure to explain:  impedance  reactive capacitance  phase shifts  Discuss how to find the amplitude of the current through the resistor and what combination of parameters gives this value.  Any other equations that were used in the activities will need explained.  Any other specific TA requests: 2





____________________________________________________________



____________________________________________________________



____________________________________________________________



____________________________________________________________

Selected In-Class Section [6 points]: {3-5 paragraphs, ~1 page}

This week's selection is: Weekly Activity 7, In-Lab Section 3 Write a "mini-report" for this section of the lab manual. Describe what you did succinctly, and then what you found accurately. Then explain what the result means and how it relates to some of the concepts in the previous section. You must write using sentences & paragraphs; bulleted lists are unacceptable. o Procedure: Do not provide a lot of specific details, but rather you should summarize the procedure so that a student who took the course a few years ago would understand what you did. o Results: Do not bother to rewrite tables of data, but rather refer to the page number on which it is found. State any measured values, slopes of ilnes-of-bestfit, etc. Do not interpret your results, save any interpretation for the discussion. o Discussion: Analyze and interpret the results you observed/measured in terms of some of the concepts and equations of this unit. It is all right to sound repetitive with other parts of the report. 

Open-Ended / Creative Design [6 points] – {3-5 paragraph, ~1 page} Choose one of the open-ended experiments from the two weekly activities to write about. Describe your experimental goal and the question you were trying to answer. Explain the ideas you came up with and what you tried. If your attempts were successful, explain your results. If your attempts resulted in failure, explain what went wrong and what you would do differently in the future. You must write using sentences & paragraphs; bulleted lists are unacceptable.



Graphs [4 points] - {attach to typed report} Graphs must be neatly hand-drawn during lab and placed directly after your typed discussion (before your quizzes and selected worksheets). Your graphs must fill the entire page (requires planning ahead) and must include: a descriptive title, labeled axes, numeric tic marks on the axes, unit labels on the axes, and if the graph is linear, the line of best fit written directly onto the graph.

3

Thoroughly Completed Activity Worksheets [30%, graded out of 15 points] 

Week 7 In-Class [7 points]: Pages assigned to turn in: _TA signature page, Post-lab pages, ____________________________________ ___________________________________________________________________



Week 8 In-Class [8 points]: Pages assigned to turn in: _TA signature page, Post-lab pages, ____________________________________ ___________________________________________________________________

The above lab report and worksheets account for 80% of your unit grade. The other 20% comes from your weekly quizzes, each worth 10%. These will be entered into D2L separately.

4

Weekly Activity 7: RC Circuits - DC Source Pre-Lab

!

You must complete this pre-lab section before you attend your lab to prepare for a short quiz. Be sure to complete all pages of the pre-lab. Continue until you see the stop pre-lab picture: Subsection 0-A (If you have not yet covered capacitors in circuits in the lecture, you will need to read the first two sections on capacitance in circuits in your text at this time.) You will investigate two similar RC circuits: charging the capacitor and then discharging the capacitor. First examine charging up the capacitor in an RC circuit. In this circuit (shown below) the capacitor begins without any charge on it and is wired in series with a resistor and a constant voltage source. The voltage source begins charging the capacitor until the capacitor is fully charged. The charging up equation that describes the time dependence of the charge on   t  the capacitor is Q Cap (t)  Qmax 1 e RC . The final charge on the capacitor, Qmax is determined   by the internal structure of the capacitor (i.e. its capacitance): Qmax  C  Vsource.





¿

0-A-1 Since the voltage source is constant in time, an example may be written as Vsource (t )  9 [V] (so it is 9 V for all times). For a capacitor with capacitance of C  1.0 x10 3 [F] , what will the initial charge Q Cap (0) and final charge Q Cap () on the capacitor be (in SI units)? {Hint: use Q  C V .}

5

¿



0-A-2 Use a graphing calculator (or mad graphing skills) and make a quick sketch of Q Cap (t) vs. t on the axes. Assume that the source voltage is 9 [V], the resistance is R  1.0 x10 3 [] and the capacitance is C  1.0 x10 3 [F] . The amount of time that equals the resistance times the capacitance is called the time constant:   R C ( is the Greek letter 'tau'). Create your sketch so that Q(t=) is sketched above the delineated tic mark. Be sure to include charge values along the y-axis. 

6

The second circuit is the discharging of the capacitor in an RC circuit without an applied voltage. In this circuit (shown below) the capacitor begins with some initial charge and is wired in series with a resistor. The capacitor begins discharging through the resistor until no charge remains on the capacitor plates. The discharging equation that describes the time dependence of the charge on the capacitor is Q Cap (t)  Qoe



t RC

.



It helps to think of an RC circuit with a charged capacitor that has a switch that is about to be closed so that the capacitor can discharge:

¿

0-A-3 For a capacitor with capacitance of C  1.0 x10 3 [F] and an initial voltage across it of Vcapacitor  9 [V] what will the initial charge Q Cap (0) on the capacitor be (in SI units)? {Hint: use Q  C V .}

7

¿



0-A-4 Use a graphing calculator (or mad graphing skills) and sketch a graph of Q Cap (t) vs. t on the axes below. Assume that the resistance is R  1.0 x10 3 [] and the capacitance is C  1.0 x10 3 [F] . Find the initial charge on the capacitor by assuming the capacitor had been charged to 9 volts Vcapacitor  9 [V] by a battery before being discharged through the resistor. Create your sketch so that Q() is sketched above the delineated tic mark. Be sure to include charge values along the y-axis.

¿

0-A-5 What is the decimal value of e-1 to 3 decimal places? _________ Engineers usually approximate this number as 1/3 (.333) in order to think quickly about exponential decay. For example, if you plug in t= (one decay time constant), the amount of charge left on the capacitor has decayed to approximately 1/3 of its initial value. Approximately how much of the initial charge is left on the capacitor in the circuit after it has discharged for three time constants t=3 seconds? 8



Now examine the time dependence of the voltage across the capacitor for the same discharging capacitor in previous questions. As the charge on the capacitor changes, the voltage difference across the capacitor plates also changes. In fact, the definition of capacitance easily relates Q (t) VCap (t) and QCap (t) by a constant: VCap (t)  Cap . Therefore, the equation describing the time C t  Q dependent decay of the voltage across the capacitor is simply VCap (t)  Voe RC , where Vo  o . C You will experimentally test  this equation later in this lab.

  0-A-6 Sketch a graph of VCap (t) vs. t on the axes below using your answer to the previous question (graph of QCAP). Be sure to include voltage values along the yaxis.

¿



9

As the capacitor discharges, it causes a current to flow through the resistor. Because energy must be conserved, the magnitude of the voltage across the resistor is the same as the voltage across the capacitor (they are the only circuit components!). Because the resistor is ohmic, the current through the resistor can be related to its voltage and resistance, I=V/R. This gives a time dependent equation for the current through the resistor of I Res(t)  Ioe



t RC

.

You should notice that the time dependence of the charge on the capacitor, the voltage across the capacitor, and the current through the resistor all exhibit the same exponential decay  function, and the constant values in front of the exponentials are simply related to each other using properties you already know, V=Q/C and I =V/R.

¿

0-A-7 Relate the time dependent equation for the current going through the resistor I Res(t)  Ioe



t RC

Q Cap (t)  Qoe

to the time dependent equation for the charge on the capacitor

t  RC

. In other words, relate I o to Q o .

 

10

Subsection 0-B A differential equation is an equation that involves derivatives. Most equations designed to model reality in the physical sciences make use of differential equations, so a good working knowledge of this type of mathematics is essential to any working scientist or engineer, especially if they need to model something that changes in time (electronics, animal populations, chemical reactions, biological processes,…). The following table compares algebraic equations to differential equations using two specific examples:

¿

0-B-1

Examine the differential equation

d 2 y(t)  9y(t) . dt 2

One solution to this

differential equation is y(t)  4 sin3t . Check the solution by plugging 4 sin3t  into the differential equation for y (t ) , perform the mathematical operations on each side of the equation, andsee if the left hand side equals the right hand have  36 sin(3t )  36 sin(3t ) .} side. {You should

11

When analyzing circuits, you often must write a differential equation describing the behavior of the circuit. This is most easily done by using conservation of energy to write a voltage equation. Then one would use V=Q/C and V=IR to relate voltage to charge on the capacitor or current through the resistor to create a new differential equation for Q(t). Examine the discharging circuit for today’s lab and the construction of the differential equation that describes it. The Discharging RC Circuit (no voltage source)

1) First use conservation of energy to write a voltage equation:

0  Vres (t )  Vcap (t )

2) Next V=Q/C and V=IR to relate voltage to Q and I:

0  R  I res (t ) 

3) Notice that I resistor (t ) 

dQ cap (t ) dt

Qcap (t ) C

because charge leaving the capacitor must travel through the 0R

resistor:

dQcap (t )

dQcap (t )

4) Rearrange to have "attractive looking" equation:

dt

dt





Qcap (t )

C Qcap (t )

R C

Since many students have not been trained on how to solve differential equations, the solution is provided: Q Cap (t)  Qoe

¿



t RC

, which should look familiar.

0-B-2 dQcap (t )

Examine the differential equation  

dt



Qcap (t ) R C

.

The solution to this

t RC



t RC

differential equation is Q Cap (t)  Qoe . Check the solution by plugging Q o e into the differential equation for Q Cap (t ) , perform the mathematical operations on each side of the equation, and see if the left hand side equals the right hand 

side. {You should have 

t  Q o  RCt Q e   o e RC in only two steps.} R C R C

12

Notice that no numerical value is needed for the initial charge Qo in this checking process. This seems to indicate that the functional behavior of the circuit is the same regardless of the specific numerical details, the real life quantities. The capacitor is going to discharge with exponential decay no matter what the starting value of charge is on the capacitor. Technically speaking then, a differential equation describes the general behavior of a system, but it cannot provide all the information needed. One also needs to specify the actual starting values of the system. These are called initial conditions or boundary conditions. This is only mentioned for completeness, you won't have to worry about this for now.

¿

0-B-3 What trivially happens to the initial charge Qo in this checking process? Your answer to this should allow you to see that the initial amount of charge on the capacitor Qo is not determined by the differential equation. Basically you choose the initial amount of charge to put on the capacitor plates and the differential equations determines how quickly that charge discharges through the resistor.

13

The other circuit for today’s lab, charging the capacitor, also has a differential equation to describe its behavior in time: CHARGING WITH CONSTANT SOURCE

1) First use conservation of energy to write a voltage equation:

Vsource  Vres (t )  Vcap (t )

2) Next V=Q/C and V=IR to relate voltage to Q and I:

Vsource  R  I res (t ) 

3) Notice that I resistor (t ) 

dQ cap (t ) dt

Qcap (t ) C

because charge leaving the capacitor must travel through the

resistor: 4) Rearrange to have "attractive looking" equation:

Vsource  R

dQ cap (t )

dQcap (t )

Vsource  R R C

dt



dt



Q cap (t )

C Qcap (t )

Since many students have not been trained on how to solve differential equations, the solution   t  is provided: Q Cap (t)  C  Vsource1 e RC , which should look familiar.  



14

In-Lab Section 1: examining slow RC circuits with a stopwatch A large capacitance C and large resistance R translate into a slow time constant =RC so that you may easily measure the rate of decay with a stopwatch. You are supplied with a 1000 [F] electrolytic capacitor. Electrolytic capacitors are “one-way” capacitors.

!

Be careful to only apply voltage correctly to the electrolytic capacitor or you will damage it (the negative terminal is clearly marked on the capacitor). You will discharge your capacitor in an RC circuit with approximately 10 [k Remember the time dependent equation for the voltage across a discharging capacitor VCap (t)  Voe



t RC

.

¿

1-1 What time constant  should you expect with R = 10 [k and C = 1000 [F]? 

¿

1-2 Since approximately four time constants 4 allows the circuit to discharge to about 2% of its initial value (because

1 1 1 1 1      0.012 or more accurately 3 3 3 3 81

e 4  0.018 ), how long should you measure the decay of the capacitor’s charge in

order to make an accurate graph that doesn't take all day to collect data?

¿

1-3 Charge an electrolytic capacitor without resistance in the correct direction using the 9-Volt battery (this happens quickly since there is very little resistance). Then switch to discharge the capacitor through a ~10 [k resistor (if the resistance is too small, the capacitor will discharge too rapidly to measure). Collect (voltage, time) data by having the DMM measure voltage across the capacitor while it discharges through the resistor using a stopwatch. You should collect more data at the beginning when there is rapid voltage change. Record your data here:

15

¿

1-4 Make a “raw graph” of your data by plotting Vcap(t) vs. t. {On separate graph paper.} Next you will linearize your data by taking the natural logarithm of your voltages. Since VCap (t)  Voe





t RC

, taking the natural logarithm of the function cancels the exponential:   t    t  1 RC lnVoe  lnVo   lne RC   t  lnVo .     RC 1 t  lnVo  is the equation of a line with a slope of -1/(RC) and yThe function y(t)   RC intercept of ln(Vo).  Thus, if you make the graph of lnVCap (t) vs. t on regular (Cartesian) graph paper, you will obtain a line with a slope equal to -1/RC if your data is exponentially related. 

¿

 1-5 Linearize your data by taking the natural logarithm of your measured voltages ln(V). Record your data here:

¿

1-6 Graph your linearized data by taking the natural logarithm of your measured voltages ln(V) and plot these vs. t on regular graph paper. This should give you a line with slope equal to -1/RC. {On separate graph paper, then calculate slope and record here.}

¿

1-7 Find your experimentally measured value of capacitance C from the slope of your linearized data graph and the value of the resistor's resistance R.

16

In-Lab Section 2: more oscilloscope practice The following picture shows the digital oscilloscope and labels its most common features.

You now need to practice using the digital oscilloscope so that you are prepared to make measurements with it. Keep in mind that the oscilloscope is simply a tool that allows you to analyze the details of a rapidly changing voltage. With that in mind, you will now practice the more common measurements that are made as well as their uses.

17

¿

2-1 Hook the output of the function generator directly to one of your oscilloscope channels (and be sure the other channel is shut off). Create a sinusoidal wave with your function generator with a very small voltage (i.e. use a special feature of the function generator and a frequency in the 1-100 [kHz] range. Use the autoset button to quickly get your signal on the screen so you can adjust your function generator DC offset correctly. Be sure that your channel is on “1x probe” and that your trigger is set to the correct source. Do this now and check you work by interacting with students in other groups.

¿

2-2 Use the oscilloscopes measure feature to determine the average voltage of your sine wave. Be sure to have about 7-10 full oscillations appear on the oscilloscope screen as the oscilloscope measure feature actually uses the screen for its data (and too few oscillations will create error in the averaging). Record your measurement here.

¿

2-3 Get the digital oscilloscope to tell you on its screen the wave’s period and frequency using the oscilloscope's features. Record your measurements here.

¿

2-4 Get the digital oscilloscope to tell you on its screen the wave’s amplitude. Remember that the 'peak-to-peak' voltage value is twice the amplitude. Record your measurement here.

18

¿

2-5 Adjust the amplitude of the wave on the function generator until you see that the wave spends more of its time being negative than positive using the 'DCoffset' feature on the function generator. This will change your average value for the voltage. Use a two cursor measurement of time and get the oscilloscope to tell you on its screen how much time the sine wave spends being positive. Then do the same thing to find out how much time the wave spends being negative. Record your measurement here.

¿

2-6 Now use a two-cursor measurement in voltage and get the oscilloscope to tell you on its screen the voltage drop of the wave from its maximum positive value to zero. Record your measurement here.

¿

2-7 Change your sine wave to a triangle wave of 500,000 [Hz] and use the DC offset so that the minimum of the triangle wave is zero volts. Examine a part of the triangle wave that is decreasing. Use a two cursor measurement to find how long it takes for the triangle wave to decrease from its highest value to one half of that value. Record your measurement here. Initially, many students become confused about the cause and effect relationship between function generator and the oscilloscope. The function generator is creating the oscillating voltage while the oscilloscope is merely observing it.

¿

2-8 A student is asked to change the amplitude of a voltage source and begins to push buttons on the oscilloscope. Why is the student's TA disappointed?

¿

2-9 A student is analyzing a working circuit and is asked after to measure some other feature of the circuit. The student starts turning knobs on the function generator. Explain why the student's TA is yelling in a panicky voice. {Hint: the phrase 'ruined the previous measurements' should appear in your answer.}

19

In-Lab Section 3: examining fast RC circuits with an oscilloscope Most digital electronics make extensive use of capacitors. However, the decay rates are typically much too rapid to measure with a DMM. In this part of the lab you will create an RC circuit using a 0.1 F capacitor and a 1 k resistor and you will rapidly charge and discharge the capacitor with an oscillating square wave.

¿

3-1 Calculate the time constant  that an RC circuit using a 0.1 [F] capacitor and a 1 [k resistor produce.

¿

3-2 You should choose a period of 20 (or rather a frequency of 1/(20) [Hz]) so that there is plenty of time for the capacitor to discharge fully. Calculate this frequency.

¿

3-3 Set up the RC circuit shown below powered by a 0 [V] to 3 [V] square wave at the frequency you calculated in the previous question with R=1 [k and C=0.1 [F]. Use your function generator to create a square wave with a voltage alternating between VMIN = 0 [V] and VMAX = 3 [V] by 1st setting the wave to oscillate between +1.5 [V] and -1.5 [V] and then using the DC-offset to shift your signal to have VMIN = 0 volts. The voltage across the capacitor should look like 'shark fins' on your oscilloscope as the capacitor exponentially charges and then exponentially discharges. Do this now and check/discuss with students in other groups to make sure you are getting it correct and understanding fully. Use the same bottom ground setup as shown below:

20

¿

3-4 Based on the bottom ground setup shown previously, which oscilloscope channel gives the voltage of the function generator source and which the voltage across the capacitor?

¿

3-5 In this setup, is there any special feature of the oscilloscope that will allow you to view the voltage on resistor? {The answer is 'yes', but explain.}

¿

3-6 During the time interval that the square wave source voltage is at +3 [V], is the capacitor being charged or discharged? Which circuit below describes the situation?

¿

3-7 During the time interval that the square wave source voltage is at +0 [V], is the capacitor being charged or discharged?

21

¿

3-8 The following is an important reminder that you won’t need in today’s lab, but is important to remember. Many times you may need to find the current in a circuit. The component you must measure is the resistor because it is the only ohmic device. if you want to determine the current of the circuit and use the oscilloscope to measure the resistor's voltage amplitude, how could you then turn this value into the current amplitude? {Hint: 'ohmic'.}

Observe the voltage across the capacitor and the total circuit voltage simultaneously using a bottom ground configuration. You should see the “shark fin” pattern that is modulated by the alternating square wave source voltage (turning on, then off).

¿

3-9 Use a double cursor measurement to find the time it takes for your charged capacitor to decrease by half (in SI units so use scientific notation!).

¿

3-10 When a physical quantity decays exponentially, the time it takes for it to decay to ½ its original value is called the half-life t½. Solve the half-life equation for t½ 1

to find what t½ should be in this circuit in terms of R and C:

 t half 1 Vo  Voe RC . 2

Show your work. 

22

¿

3-11 Combine the results of the previous questions and calculate the experimentally determined capacitance C of your capacitor using your half-life measurement in 3-9.

¿

3-12 Now use the double cursor method to find the time it takes for your capacitor to discharge from ½ of its initial value to ¼ of its initial value. The decaying exponential function has the unique property that each consecutive halving of its value occurs in the same amount of time. Thus the half life is an important feature in exponentially decaying systems because no matter when you begin measuring, you know that each half life of time that passes, the value will have decreased by half.

¿

3-13 Using this knowledge, predict how long it should take for your capacitor to discharge to 1/128 of its initial value (which is approximately 1% of its initial value).

¿

3-14 Now use the cursors to collect voltage vs. time data for your decaying capacitor. Then linearize your data, graph it on regular graph paper, and compute to the capacitance C from the slope. Be sure your value is close to the labeled value. Record your data below, make your graph on separate graph paper, linear your data below, make your linearized graph on separate graph paper, find the slope, and calculate C below.

23

(This page intentionally left blank.)

24

In-Lab Section 4: authentic assessment Quickly set up a working circuit that simultaneously uses a capacitor and resistor in series powered by a function generator. Use the concept of a half-life and a single measurement to determine the capacitance of the capacitor. {Note using thalf=RC is much quicker than finding the slope of a linearized graph with several data points, though much less accurate.} Sketch your circuit and label the resistance of the resistor.

¿ 4-1 Show a student in a different group that you can successfully measure the capacitance of a capacitor with only one measurement. Once you are successful, have them sign below. Note: if someone is stuck, please give them advice!

"Yes, I have seen this student successfully find a capacitor's capacitance using the measurement of the half-life. They are able to use this property of exponential decay!"

Student Signature:___________________________________________________

25

In-Lab Section 5: open-ended / creative design When capacitors are added in series, they have a combined capacitance determined by one of the two following equations:

Ceffective  C1  C2 or Ceffective 

1

1 1  C1 C2

.

Design an experiment to determine which mathematical relationship is correct and which is incorrect.



You are allowed to "cheat" by talking to other groups for ideas, but are not allowed to "cheat" by just stating an answer you may already know, looking it up online or asking your TA. Below you are given three prompts: hypothesizing/planning, observations/data, calculations/conclusion. Your job is to figure out the answer using these prompts as your problem-solving model. In the event that you should run out of time, you may not discover the correct answer, but you should make an attempt at each prompt. Grades are based on honest effort. Your open-ended solution should probably include some of the following items: sketches of circuit diagrams, tables of data, calculations, recorded observations, random ideas, etc. Write at the prompts on the next page.

26

¿

5-1 hypothesizing/planning:

¿

5-2 observations/data:

¿

5-3 calculations/conclusion

I, the physics 241 laboratory TA, have examined this student's Weekly Activity pages and found them to be thoroughly completed.

!

TA signature: _______________________________________________________________ 27

Post-Lab: RC Circuits - DC Source

!

You must complete this post-lab section after you attend your lab. You may work on this post-lab during lab if you have time and have finished all the other lab sections.

¿

X-1 A capacitor is charged by connecting it to a battery as shown below:

a) What is the direction of the current in the circuit as the capacitor charges, clockwise or counterclockwise? b) What is the current in the space between the capacitor plates as the capacitor charges? c) Which plate of the capacitor becomes positively charged, the upper or lower?

d) What happens to the magnitude of the current in the circuit as the capacitor charges, increase, decrease or stay constant?

e) After the capacitor charges, which plate of the capacitor has a higher voltage, the upper or lower? f) After the capacitor charges, what is the direction of the electric field in between the plates, upward or downward?

28

¿

X-2 A capacitor is charged with a battery and then placed in series with a resistor to discharge.

a) What is the direction of the current in the discharging RC circuit as the capacitor discharges, clockwise or counterclockwise? b) Compare the currents at points a, b, c, d, e, and f in the discharging RC circuit. How are they related?

c) For the charging capacitor circuit, compare the magnitude between point pairs a-b and c-d.

d) For the discharging capacitor circuit, compare the magnitude between point pairs a-b and c-d.

¿

X-3 Why do capacitors in series have the same magnitude of charge on each plate. Draw a diagram to help explain. {Hint: focus on the charge separation of the conducting 'island'.}

29

¿

X-4 What affect does resistance have on the time is takes for a capacitor in an RC circuit to discharge?

¿

X-5 If it takes 10 seconds for a cully charges capacitor to discharge though a 10 [k resistor, how long would it take to discharge through two 10 [k resistors connected in series?

¿

X-6 If it takes 10 seconds for a cully charges capacitor to discharge though a 10 [k resistor, how long would it take to discharge through two 10 [k resistors connected in parallel?

¿

X-7 If it takes 10 seconds for a cully charges capacitor to discharge though a 10 [k resistor, how long would it take to discharge through two 10 [k resistors connected in series?

30

Weekly Activity 8: RC Circuit - AC Source Pre-Lab

!

You must complete this pre-lab section before you attend your lab to prepare for a short quiz. Be sure to complete all pages of the pre-lab. Continue until you see the stop pre-lab picture: Subsection 0-A Last week you studied RC circuits, examining the exponential time dependence of the capacitor voltage as you charged and discharged the capacitor with a constant source voltage. To do this you used a square wave with a DC offset. Now you will examine the behavior of a capacitor when a sinusoidal source voltage is applied: Vsource(T)  Vsource sin( D t) , where D is called the amplitude

angular driving frequency of the circuit. 

The capacitor voltage will no longer exhibit exponential time behavior. Instead the capacitor voltage will oscillate sinusoidally with the same frequency as the source driving frequency. (This can be proven by writing the differential equation for the circuit, finding its solution, and checking the solution. However, this requires knowledge of solving inhomogenous differential equations. This will not be done in this course.) Instead, the most useful results of that calculation are provided: the time dependent voltages across each component. Thus, you are not required to be able to derive the solutions to the AC-driven RC circuit, but you must memorize, understand and be able to use these results. 31

Each component of the sinusoidally driven RC circuit has a sinusoidally varying voltage across it, but each peaks at a different time determined by a phase shift. Did you catch that? Different components reach their maximum voltages at different times than other components. The solutions for the time dependent voltages of each component are given by the equations:

Vsource (t )  Vsource sin(D t  shift   ) amplitude

R VResistor(t )   Vsource sin(D t )  Z  amplitude  VCapacitor (t )   C  Z

   V sin  t   source  D  2  amplitude 

There are several new parameters to discuss. First notice that the source voltage is now written with a source phase shift shift, the capacitor voltage has a phase shift of –/2, and the resistor voltage has no phase shift. This can be seen in the graph below.

32

What this means in practice is that we will use the resistor voltage as a reference for all other components in the circuit: i.e. we will measure the phases of each component in relation to what is happening inside the resistor. This is because the resistor is ohmic and can always provide the time dependent current via Ohm’s law simply by dividing the resistor voltage by V (t ) resistance, I circuit (t )  R . R

!

This choice of measuring voltage phase shifts from the perspective of the resistor's voltage is not a uniform convention, and other teachers and texts may make a different choice (so be sure you understand the principles). In the previous graph, the source voltage can be seen as the negative sum of the components' voltages VS (t )  VR (t )  VC (t ) . This is due to the conservation of energy. If you add the electrical potential energy (per unit charge) in a circle, the sum should be zero. This is just like making a loop on a staircase; if you end up at the same point, then you will have gained as much gravitational potential energy as you have lost. Thus

VS (t )  VR (t )  VC (t )  0 , or reminding you that component voltage is the change in voltage across the component,

VS (t )  VR (t )  VC (t )  0 .

The Source Voltage Equation:

Vsource (t )  Vsource sin(D t     ) amplitude

The source voltage equation is straightforward. It oscillates sinusoidally, i.e. it is a sine function of time. The maximum voltage applied across the whole circuit is Vsource . The source oscillates amplitude

 

with an angular driving frequency D  2f D (which you will set later with your function generator). The source voltage is phase shifted from the resistor voltage by an amount   of the capacitor given by the equation   arctan C  where XC is the reactive capacitance   R   1 C  (more on this later). Note that this x-like variables are really the capital Greek letter  DC Chi (pronounced kai). The  is also included as an additional phase shift, but it is equivalent to multiplying by -1, sin(t   )   sin(t ) , which emphasizes that electric potential in the circuit is conserved VS (t )  VR (t )  VC (t ) .

33

If you look at the equation for resistor voltage, you will see no phase shift. Again, what this means is that we measure all phases in relation to the resistor not the source. The resistor will have its maximum voltage at a different time than when the source voltage is maximum.

¿

0-A-1 Imagine a sinusoidally driven RC circuit. If the source voltage has an amplitude of Vsource =1.8 [volts], a linear driving frequency fD=555 [Hz], a resistance R=150 amplitude

[, and a capacitance C=1.5x10-5 [F], find the phase shift of the source voltage compared to the resistor. 

The Resistor Voltage Equation:

R  VR (t)   Vsource sin( Dt) Z  amplitude

The resistor voltage oscillates sinusoidally without a phase shift while R is simply the resistance. Z is the impedance of the whole circuit. Z acts like the “total resistance” of the circuit. Z is measured in SI units of Ohms and is given by the equation Z  R 2   L  C  . 2



This definition has new stuff, too. XL and XC are like the “resistances” of the inductor and capacitor, respectively. We won’t study inductors until later in the semester, but it is easier to  memorize the complete equation. Since we don’t have an inductor (coil) in the circuit, you can set this to zero. So we have Z  R 2  C2 . C is called the reactive capacitance and is measured in ohms [].

¿

0-A-2  Now examine the resistor equation as VR (t)  Vresistor sin( Dt). The maximum and amplitude

R Z

minimum voltage would oscillate across the resistor is Vresistor  Vsource . amplitude

amplitude

Imagine a sinusoidally driven RC circuit. If the capacitance is increased, explain what happens to the amplitude of the resistor voltage? 

34

¿

0-A-3 If the frequency is increased what happens to the amplitude of the resistor voltage?

¿

0-A-4 Explain what happens to the current through the circuit if the resistor voltage amplitude decreases?

¿

0-A-5 Explain what happens to the power lost through heating the resistor if the resistor voltage amplitude decreases? (Remember that PR=IRVR.)

¿

0-A-6 Imagine a sinusoidally driven RC circuit with source voltage amplitude VS, resistance R, and capacitance C. Explain whether the resistor will become hotter if you increase the driving frequency? Use the concept that Z = total impedance of the circuit.

35

The Capacitor Voltage Equation:

C     VC (t)   Vsource sin Dt    Z  amplitude  2 

The capacitor voltage oscillates sinusoidally and lags behind the resistor voltage by 90o. The reactive capacitance C is like the resistance of the capacitor and is measured in SI units of ohms [].



The “resistance” of the capacitor is related to the capacitance of the capacitor and the driving 1 frequency. This relationship C  can be derived from the differential equation modeling  DC the circuit, but you must memorize it. The larger the capacitance, the less “resistance” in the capacitor. But just as importantly if the driving frequency is increased, the “resistance” of the capacitor decreases. This is why a capacitor is often used as a high pass filter in electronics: the  capacitor has less resistance to more quickly oscillating currents. BE SURE TO REMEMBER THIS DURING THE LAB!    If we rewrite the capacitor equation as VC (t)  Vcapacitor sin Dt  , the capacitor voltage  2  amplitude C amplitude is given by Vcapacitor  Vsource . That means that the ratio of the capacitive Z amplitude amplitude reactance and the total circuit impedance times the source amplitude gives the amplitude of  the voltage across the capacitor.

 In a previous equation, you found that the resistor voltage amplitude increases when the frequency is increased. Since the voltage across the resistor and capacitor must add to the voltage across the source, if the resistor voltage amplitude increases, then the capacitor voltage must decrease. Therefore, as you increase the driving frequency, the resistor voltage amplitude increases while the capacitor voltage amplitude decreases.

36

Subsection 0-B Work though an example before beginning. Remember the equations below as you work: Vsource (t )  Vsource sin( D t     ) amplitude

R  VR (t)   Vsource sin( Dt) Z  amplitude C     VC (t)   Vsource sin Dt    Z  amplitude  2  

¿

0-B-1 If your

circuit

has

Vsource

 2 [V] ,

R  10,000 []

,

amplitude   D  1,500 [radians/s ec] find the following values (in SI units):

C  1x10 -7 [F] ,

and

XC = Z= = VR,amplitude. = VC,amplitude. =

¿

0-B-2 Now compute VR,amplitude + VC,amplitude = Your answer to this question has a sum that is greater than Vsource amplitude!!! No, you didn’t make a mistake. Since the voltages are out of phase, their maximums do not add together at the same time.

¿

0-B-3 At any instant of time, explain what should the component voltages add to, V R (t )  VC (t )  ?

37

Now let’s try and visualize this circuit’s behavior.

¿

0-B-4 Write the functions for VS (t) , VR (t) and VC (t) using the numerical solutions to the previous questions. 





¿

0-B-5 Quickly sketch VR (t) and VC (t) on the oscilloscope screen below using a graphing calculator. Don’t worry about providing the scale of the time axis. Then sketch VR (t) + VC (t) onto the screen using a dotted line. This should equal the function check it using VS (t) so  your graphing calculator.  



38

In-Lab Section 1: examining the components Set up the sinusoidally driven RC circuit with R  10,000 [] , and C  1x10 -7 [F] . Set your function generator to create a sin wave with a voltage amplitude of a nice round number like 3 [V]. You may want to adjust your frequency later, but start at about 400 [Hz]. Set up a middle ground to view the voltage across both the resistor and the capacitor simultaneously making sure to invert the correct channel (a necessary step when using a middle ground). Check your setup with other students in the lab.

¿

1-1 Make a sketch of the oscillating resistor and capacitor voltages on the oscilloscope screen below. Label the signals VR (t) and VC (t) on your sketch





39

¿

1-2 Explain which signal is phase shifted to lag by 90o using calculus (derivatives).

¿

1-3 Find the amplitudes of each signal by measuring the peak-to-peak voltage of each signal.

¿

1-4 Use the labeled values to determine the impedance of your circuit for this





2 2 driving frequency. Remember Z  R  C . .



¿

1-5 Use your previous answer to determine what the signal amplitudes should be and then compare these predicted (calculated) amplitudes to your measured amplitudes in the other previous question (they should be close).

¿

1-6 Find the frequencies f of each signal using oscilloscope measurements. 40

¿

1-7 Use your answers to the previous questions to write equations for VR (t) , VC (t) and VS (t) entirely with numerical values (no free parameters). (Don’t forget the phase shift.) 





¿

1-8 Set your oscilloscope to plot VR (t) on the x-axis and VC (t) on the y-axis (an XY plot). Sketch the result on the oscilloscope screen below. 



41

¿

1-9

In an XY plot, if the signal on the y-axis oscillates twice as fast as the signal on the x-axis and the signals are 90o out of phase, then sketch what will appear on the oscilloscope screen below.

42

In-Lab Section 2: experimentally finding the capacitance 1 by observing a sinusoidally driven RC circuit using  DC many different driving frequencies. Use the same circuit set up as in the previous part of the lab. Next you will test the relationship C 

¿

 2-1 As you increase the driving frequency, the amplitude of the resistor voltage will

R Z

increase because the total circuit impedance is decreasing, i.e. Vresistor  Vsource . amplitude

amplitude

Work through this logic so that you are sure you understand it. 

Meanwhile, as the driving frequency increases, the capacitor amplitude decreases. This makes sense because the resistor and the capacitor are the only two components in the circuit other than the source. Since the voltages across both must add up to the source voltage at any instant in time, if the voltage amplitude of one increases, then the other must decrease. Therefore, there must be some specific driving frequency when the amplitude of the resistor voltage is the same as the capacitor voltage: Vresistor  Vcapacitor for a specific angular driving amplitude

amplitude

frequency D.  R  Realizing that Vresistor  Vsource and Vcapacitor  C Vsource , setting these two voltages equal Z amplitude Z amplitude amplitude amplitude assuming the circuit is being driven at some specific angular driving frequency D,equal you get C R Vsource  Vsource , which simplifies to  C  R . In other words, the voltage across the Z amplitude Z amplitude   capacitor equals the voltage across the resistor if their "resistances" are equal.



The first method for finding the capacitance of an unknown capacitor makes use of the previous equation. All you need to do is adjust the driving frequency of your circuit until the 1 capacitor voltage amplitude and the resistor voltage amplitude are equal. Then use C   DC and  C  R for the specific D,equal to find the capacitance. 

43

¿

2-2

Substitute C 

1 into  C  R , and then solve for C. Be sure to realize that  DC

this equation is only true when the circuit is being driven at the specific frequency D,equal that makes the resistor and capacitor voltages equal. 

¿

2-3 Set up a 3 [volts], 400 [Hz] sinusoidally driven RC circuit with R  10,000 [] , and C  1x10 -7 [farads] . Set up a middle ground to view the voltage across both the resistor and the capacitor simultaneously making sure to invert the correct channel. Adjust the driving frequency until the resistor and capacitor voltages are equal. Then use your formula from 2-2 to find your experimentally determined value for capacitor's capacitance. Remember that the function generator reads the linear frequency.

44

The second method for finding an unknown capacitance is more involved, but more accurate as it involves multiple measurements. The voltage amplitudes of the sinusoidally driven RC are:

Vresistor  amplitude

and

Vcapacitor 

C Z

amplitude



R Vsource Z amplitude Vsource

.

amplitude

Dividing these two equations gives



 C  Vsource  amplitude    Vcapacitor  Z    C amplitude     R  V Vresistor R source amplitude   amplitude   Z  

Therefore,

C  R



.

Vcapacitor

amplitude

Vresistor

.

amplitude

In order to experimentally determine C for your capacitor, simply combine the last equation 1 with the definition C  and rearrange:  driveC





V 1 resistor amplitude  C drive . R Vcapacitor amplitude

 45

V 1 resistor amplitude  C drive R Vcapacitor amplitude

looks like a weird arrangement for this equation, but if you think of y=mx, then you see that if you graph

V 1 resistor amplitude vs.  drive, you should obtain a linear graph with a slope equal to C. R Vcapacitor



amplitude

¿

2-4 Find C by collecting data for multiple driving frequencies, making a graph on  separate graph paper and finding the slope.

46

In-Lab Section 3: authentic assessment

¿

3-1 Quickly set up a working circuit that simultaneously uses a random capacitor and a 1000  resistor in series powered by a sinusoidal source voltage on your function generator. Then make the necessary measurements to determine the capacitance of the capacitor. Be sure your experimentally determined measurements give the correct capacitance. Show your results to a student in a different group:

"Yes, I have seen this student find an unknown capacitance 'the easy way'." Student Signature:___________________________________________________

47

In-Lab Section 4: open-ended / creative design Make a capacitor from the square cardboard pieces covered in conductive aluminum foil. Sandwich a non-foil square of cardboard between the foiled boards, and be sure your makeshift capacitor is not shorted out by accident. Measure the capacitance of your homemade  A capacitor. The equation for the capacitance of two parallel plates is given by C  o . Use d this equation to report the dielectric constant  of the sandwiched cardboard between the C2 plates with correct units. Note: o  8.85x1012 . Design an experiment to determine the N  m2  capacitance of your cardboard capacitor and the dielectric constant of the cardboard. You are allowed to "cheat" by talking to other groups for ideas, but are not allowed to "cheat"  by just stating an answer you may already know, looking it up online or asking your TA. Below you are given three prompts: hypothesizing/planning, observations/data, calculations/conclusion. Your job is to figure out the answer using these prompts as your problem-solving model. In the event that you should run out of time, you may not discover the correct answer, but you should make an attempt at each prompt. Grades are based on honest effort. Your open-ended solution should probably include some of the following items: sketches of circuit diagrams, tables of data, calculations, recorded observations, random ideas, etc. Write at the prompts on the following page.

48

¿

4-1 hypothesizing/planning:

¿

4-2 observations/data:

¿

4-3 calculations/conclusion

I, the physics 241 laboratory TA, have examined this student's Weekly Activity pages and found them to be thoroughly completed.

!

TA signature: _______________________________________________________________ 49

Post-Lab: RC Circuits - AC Source - REVIEW PRACTICE OF GAUSS'S LAW

!

You must complete this post-lab section after you attend your lab. You may work on this post-lab during lab if you have time and have finished all the other lab sections.

¿

X-1 Gauss’s Law applied to systems with spherical symmetry:

A Cartesian representation of the electric vector field, E  E x xˆ  E y yˆ  E z zˆ , is useless. Try ˆ  E ˆ . Because of the using a spherical coordinate system representation, E  E r rˆ  E   symmetry of the system, E  0 and E  0 . So E  E r rˆ . Thus the problem of solving for a vector field reduces to a problem of solving for  a scalar field quantity that only depends on the radial distance, Er. 





50

a. Is the electric field constant in magnitude for a fixed radius? Explain.

b. In which direction does the electric field point, and how does this depend on the sign of ? Explain.

c. Explain why the sphere is the appropriate Gaussian shape to draw?

d. What must the units of the constant A be?

e. Write the integral that defines the total electric flux through a Gaussian sphere. Solve it for each of the three Gaussian spheres shown.

51

f. Write and solve the integrals to find the total charge enclosed within each Gaussian sphere.

g. Use Gauss’s law to equate your answers to e and f to solve for the electric field for all three places: a radial distance r inside the hollow region, inside the charged region and outside the sphere. Be sure to express your answers as vectors.

h. Explain what the resultant electric field would be like if the charge density  had turned out to be “balanced”, i.e. the total charge found by integrating over the entire sphere is equal to zero.

52