LAB 7 - RC Circuits in series and parallel PDF

Preview

Summary

Practice...

Description

LAB 7: RC Circuits in Series and in Parallel Manual

PHYSICS

1 1310/1410 Introductory Physics, Lab 7. October, 2020

LAB 7 OBJECTIVES  Access shared iOLab data using your repository account.  Demonstrate the ability to appropriately scale data and zoom in on only the relevant sections of data.  Utilize the iOLab sensors to understand how resistors and capacitors affect the voltage in a circuit.  Experimentally determine the time constant of an RC circuit with two capacitors in series.  Experimentally determine the time constant of an RC circuit with two capacitors in parallel.  Calculate the capacitances of the two capacitors using the two experimentally determined equivalent capacitances.

2 1310/1410 Introductory Physics, Lab 7. October, 2020

LAB 7: RC Circuits in Series and in Parallel RESISTORS Two common electrical circuit elements are resistors and capacitors. In today’s lab you will use both. A resistor is a device that opposes the flow of electrical current. The bigger the value of a resistor the more it opposes the flow of electricity. The value of a resistor is given in ohms (Ω) and is often referred to as its ‘resistance’. Resistors have metal leads coming off of both ends and it does not matter which way current flows through them, so any lead can be connected to any part of a circuit.

coloured bands

resistor

metal leads

Resistors of the sort used in this lab.

Resistors are typically marked with four different colour bands. 3 of the bands identify the rated value of the resistor. The fourth band of color, at one end of the resistor, tells you how accurate that rated value of resistance likely is (this is called the “tolerance.”) The coloured bands can be seen in the image above on a variety of resistors. Some resistors have a total of five bands of colours, where the first four identify the resistance and the fifth band identifies the tolerance.

3 1310/1410 Introductory Physics, Lab 7. October, 2020

Two examples of standard colour codes and how they are used to calculate the value of the resistor are shown in the tables below.

EXAMPLE: Do you see how brown-black-red-gold gives a resistance of 1.0 k +/- 5% ?

EXAMPLE: Do you see how red-green-yellow-blackbrown gives a resistance of 254  +/- 1% ?

EXAMPLE: In this case, what resistance is indicated by black-red-green-silver? Black=0 Red=2 Green=105 Silver=+/- 10% 02x105 = 2x105  = 200 k +/- 10%

Note that the tolerance bands are the only ones that are metallic coloured (gold, silver) making them easier to identify. Tolerance is also sometimes indicated with red or brown bands. 4 1310/1410 Introductory Physics, Lab 7. October, 2020

black gold

brown

yellow

For example, in the picture shown above, the resistor has the colours (reading from the right because the metallic band is always read last) of “brown – black – yellow – gold”. Brown=1. Black=0. Yellow=104. Gold=5%. So the resistance of this resistor is 10x104  = 100x103  = 100 k. The tolerance on this particular resistor would be 5% (gold) of 100 k, or 5 k. So this resistor is 100 k ± 5 k. When measured with a device to directly measure resistance, it was measured to be 100.6 k which is well within tolerance.

No metallic band, just red. The lone red band is well-separated from the other three, so it is the red 2% tolerance band.

The resistor in the picture above has no metallic band, but the tolerance band (red) is on the end and is well-separated from the other bands so it is clear what it is. CAPACITORS Capacitors are circuit elements that store an amount of charge (Q) proportional to the voltage applied across the capacitor. This is given by Q  CV or V 

Q C

The units of capacitance are farads (F), and capacitors with values in the microfarad, µF, range are very common in typical circuits. Most common capacitors are polarized, which means that they are directional and that it does matter in which direction electrical current flows through them. To denote this, capacitors are usually marked with a minus sign (-) to denote the negative lead and an arrow indicating which way current should flow (the arrow 5 1310/1410 Introductory Physics, Lab 7. October, 2020

points toward the negative lead.) Many types of capacitors have a longer lead and a shorter lead. The longer lead is always connected to the higher voltage (positive) side of the circuit with the shorter lead connected to the lower voltage (negative) side of the circuit. Current runs into the capacitor through the positive lead and out through the negative lead. Unlike resistors, capacitors usually have their values printed directly on them (it will read something like 100 µF). The rated capacitance is usually not known as accurately as a rated resistance, so this value has a tolerance somewhere around 20%. This means that a capacitor that has a rated value of 100 µF could be anywhere in the range of 80 – 120 µF.

axial capacitor: one lead sticks out from either end positive lead

negative lead

Arrow on capacitor points in direction of negative lead

Arrow on capacitor or white band points in direction of negative lead (also marked with a minus sign)

positive lead (longer)

negative lead (shorter)

negative lead (shorter)

positive lead (longer)

radial capacitor: both leads stick out from bottom Capacitors of the sort used in this lab.

6 1310/1410 Introductory Physics, Lab 7. October, 2020

 , THE TIME CONSTANT When a capacitor has a voltage (V) applied across it, it takes some time for the charge (Q) to accumulate given by the capacitance (C). Eventually all of the charge will accumulate. This process can be made even slower by adding a resistor with resistance (R) to the circuit. The product RC equals the “time constant”, , of the circuit. RC = . The product RC tells you how quickly and how slowly the capacitor will charge up when voltage is connected, or discharge when voltage is disconnected. Although RC does have units of time (seconds), the time constant is not exactly the time it takes the capacitor to charge or discharge. Rather, it is the exponent in an exponential increase or decrease in the voltage across the capacitor. The equations are given below, where t = time and RC = .

Equations describing the exponential rise in voltage while a capacitor is charging up from zero volts (left) and the exponential fall in voltage while a capacitor is discharging to zero volts (right). At a time equal to 3RC = 3, the voltage has risen to approximately 95% of its maximum (while charging) or dropped to 5% of its maximum (while discharging). So a capacitor takes roughly 3 to charge or discharge.

7 1310/1410 Introductory Physics, Lab 7. October, 2020

voltage turned on voltage measured on A7 slowly increases to maximum

voltage measured on A7 slowly decreases to zero

voltage turned off

Unlike past labs, when an RC circuit is used the voltage measured at A7 will not “jump up” to 3.3 V. It takes some time to rise slowly up to its maximum. The length of time this takes is determined by the time constant,  = RC.

8 1310/1410 Introductory Physics, Lab 7. October, 2020

CALCULATION OF  = RC FROM A DISCHARGE CURVE In this lab, discharge curves like the Discharging a Capacitor one on the right, “Discharging a Capacitor,” will be created by voltage measured on A7 slowly decreases to zero constructing various RC circuits. To calculate  = RC you will be asked to obtain the voltage and time for two points on such a curve (you will measure V1 , t1 and V2 , t2 .)

voltage turned off

By plugging those measured values into the equation given for discharging a capacitor you will get   t1   t2  V1  V0  e   and V 2  V 0  e      

By dividing V1 by V2, the constant V0 goes away. If you then take the natural logarithm (ln) of each side, you can derive an equation for the experimental value of the time constant, . You will be asked to do this derivation during the experiment. You will wind up with an equation that depends ONLY on t2, t1, V2, and V1. You may need to remember how exponentials divide and multiply to do this. e a b  e a  e b  e a b  e    . Also, ln ex   x . b e a

Remember,

9 1310/1410 Introductory Physics, Lab 7. October, 2020

Objectives of Exercises In this lab you will perform the following two tasks, the second depending on values obtained in the first. An RC circuit with one resistor and two capacitors in series will be constructed and tested for you. Exercise 1 Summary: 1. Analyze a voltage discharge curve obtained from the repository. 2. Measure three pairs of points to obtain three calculations of the time constant, series. 3. Average those three to obtain an average value of series with the standard deviation of the three being its uncertainty. series 4. Calculate the equivalent capacitance of the circuit (call it Cequivalent ) with its uncertainty, using the known R and the measured  (with their uncertainties). The circuit will then be rewired using the same two capacitors but they will be wired in parallel. You will then: Exercise 2 Summary: 1. Analyze a new voltage discharge curve obtained from the new circuit. 2. Measure three pairs of points to obtain three calculations of the time constant, parallel. 3. Average those three to obtain an average value of parallel with the standard deviation of the three being its uncertainty. parallel 4. Calculate the equivalent capacitance of the circuit (call it Cequivalent ) with its uncertainty, using the known R and the measured parallel (with their uncertainties). 5. Using an equation you will derive, calculate the capacitances of the two series parallel and Cequivalent . capacitors using your measured values of Cequivalent

10 1310/1410 Introductory Physics, Lab 7. October, 2020

Exercise 1, Determine the time constant of an RC circuit with two capacitors in series and calculate an equivalent capacitance Using the iOLab device and a breadboard, an RC circuit was created using one resistor and two capacitors in series. The iOLab was wired up as shown in the video at right.

The value of the resistor used in this experiment is NOT what was mentioned in the video. In fact it was the same resistor used in Lab 6. For students who are color blind, visually impaired, or if the file/display quality is not clear enough, this resistor’s bands are (from left): brown-orangeorange. The tolerance band at the far right is red.

REPORT 1.1 Report the value of the resistor used in this circuit with its uncertainty, using the colour code guides in the instructions. The values of the capacitors used in this experiment are what you will be determining but are NOT what are stated in the video if any values are stated. The iOLab was then used to monitor the voltage drop across the two capacitors as measured at the point indicated by the yellow A7 arrow below. The circuit diagram for the circuit being analyzed is shown at right.

iOLab Vdc (On-Off)

A7 D6 +

C1

-

C2

ND

11 1310/1410 Introductory Physics, Lab 7. October, 2020

R

C2 C1

A schematic of what your circuit with two capacitors in series will look like. Images adapted from Hayden-McNeil, LLC.

ACTION As described in the introduction on page 9, derive an equation for the time constant   RC that depends ONLY on the values obtained in one pair of measurements: t2, t1, V2, and V1. REPORT 1.2 Report your equation for the time constant, . You do not need to show your derivation, just provide the equation you will be using. This is the same equation you were using in Lab 6 to analyze discharging RC circuits. ACTION Log-in to your iOLab repository and access the data file, “1310/1410 Lab 7 Series Capacitors RC Circuit Discharge.” This is the data acquired from the previous video, but with different component values. You will be analyzing it and turning in your results. https://iolabrepository.azurewebsites.net/ Following the instruction shown in the video at right, measure your first pair of points (V1, t1; V2, t2). Record these in an Excel sheet, use these to calculate a time constant, , using the equation you derived and reported in 1.2. If you calculate this in Excel, not on your calculator, you will save a lot of time. Record this value of  in an Excel sheet. 12 1310/1410 Introductory Physics, Lab 7. October, 2020

Pay particular attention to where you pick your two points to calculate . You do not want it to be up on the high-voltage flat part and you do not want it to be way down where it goes to zero. This is discussed at 2:36 in the video. ACTION Record a screen capture of your data analysis of one point on the voltage discharge curve for inclusion in your report. REPORT 1.3 Include in your report a screen capture showing a well-scaled discharge curve which shows your data cursor measuring the voltage at one time.

ACTION Choose a DIFFERENT pair of points (V1, t1; V2, t2). Record these in your Excel sheet and use these to calculate the time constant, , again. Record this second value of , which will probably not be the same as the first, in your spreadsheet. ACTION Choose a THIRD DIFFERENT pair of points (V1, t1; V2, t2). Record these in your Excel sheet and use these to calculate the time constant, , again. Record this third value of , which will not be the same as the first two, in your spread sheet. ACTION Using Excel, calculate the average and the standard deviation of these three measurements of . As this is the time constant for your capacitors in series circuit, call this series.

13 1310/1410 Introductory Physics, Lab 7. October, 2020

REPORT 1.4 Report your Excel table that should show the raw data, which are your three pairs of V, t measurements, and also your calculations of the time constant  (in seconds) along with its average and uncertainty, which is the standard deviation of your three measurements. Your Excel table could look like the table below. The numbers in this table are not real, so pay no attention to those. Be sure to indicate clearly that this is the SERIES CAPACITORS data. Measurement 1 Measurement 2 Measurement 3

v1 (V) 8.802 7.523 6.132

t1 (s) 10.05 10.13 10.28

v2 (V) 1.338 2.303 1.32

t2 (s) tau (s) 11.21 0.61577564 10.98 0.71805558 11.01 0.47529469 AVG 0.60304197 STDEV 0.12188036

ACTION Using the value of the resistance from the resistor’s color-code (reported in 1.1) and your just-calculated average value of series, calculate the equivalent capacitance of the two capacitors in series and its uncertainty. Call this value series C equivalent . series series  If  series  RCequivalent , then Cequivalent

 series R

. Calculate this with its uncertainty.

If the uncertainty on R is R (which you get from the resistor), and the uncertainty on  is  (which you got from the standard deviation of your measurements), then the uncertainty on C is C, and is given by: 2

     C  C  R      R  

2

REPORT 1.5 Report your calculated value of the equivalent capacitance of the two capacitors series in series, C equivalent , and its uncertainty. 14 1310/1410 Introductory Physics, Lab 7. October, 2020

Exercise 2, Determine the time constant of an RC circuit with two capacitors in parallel and calculate an equivalent capacitance The existing circuit was modified by wiring the SAME TWO capacitors (C1 and C2) in parallel with each other, as opposed to in series with each other as they were in experiment 1. The same resistor was used. A video of how this was done is given at right. The link to the video is also a decent picture of what the circuit looked like when completed.

The iOLab was then used to monitor the voltage drop across the two capacitors as measured at the point indicated by the yellow A7 arrow below. The circuit diagram for the circuit being analyzed is shown at right.

iOLab Vdc (On-Off)

A7

D6 + C2

C1 -

ND

R C2

C1

A schematic of what your circuit with two capacitors in parallel will look like. Pay attention to where the leads to D6, A7, and GND are connected. Images adapted from Hayden-McNeil, LLC. 15 1310/1410 Introductory Physics, Lab 7. October, 2020

ACTION Log-in to your iOLab repository and access the data file, “1310/1410 Lab 7 Parallel Capacitors RC Circuit Discharge.” This is the data acquired from the previous video, but with different component values. You will be analyzing it and turning in your results. https://iolabrepository.azurewebsites.net/

ACTION Repeat the experiment of Exercise 1, analyzing the discharge curve by measuring the time constant  three times by choosing three pairs of points. Call this new time constant, parallel. REPORT 2.1 Provide a clear and scaled screenshot of your new discharging RC circuit voltage; make sure this photo shows the cursor measuring at least one of the points V1, t1 that you measured.

REPORT 2.2 Report your Excel table that should show the raw data, which are your three pairs of V, t measurements, and also your calculations of the time constant parallel (in seconds) along with its average and uncertainty, which is the standard deviation of your three measurements. Be sure to indicate clearly this is the PARALLEL CAPACITORS data.

ACTION The two capacitors in parallel have a new “equivalent capacitance” which we will parallel parallel parallel  call Cequivalent . If  parallel  RCequivalent , then Cequivalent

 parallel R

. Calculate this with its

uncertainty. REPORT 2.3 Report your measured value of the equivalent capacitance for the two capacitors parallel , and its uncertainty. in parallel, Cequivalent 16 1310/1410 Introductory Physics, Lab 7. October, 2020

Exercise 3, Determine C1 and C2 series (you reported this in 1.5) and from exercise 2 From exercise 1 you know Cequivalent parallel

you know Cequivalent (you reported this in 2.3). Knowing these two you can now calculate the values of C1 and C2.

ACTION Using your knowledge of how two capacitors in series combine to give an equivalent capacitance (equation one) and also how two capacitors in parallel combine to give an equivalent capacitance (equation two), derive an equation for either C1 or C2. Because these are two equations in two unknowns, it is possible to do this. You will need to algebraically manipulate the two equations until you can use the b  b2  4 ac quadratic formula, C1 or C2  . The two solutions to the quadratic 2a

equation (no matter whether you solve for C1 or C2) are the two capacitances you are looking for. This means when you are algebraically manipulating your equations you are looking for some kind of equation like: 0  aC12  bC1  c or 0  aC22  bC2  c so that you can use the quadratic equation.

You will not actually be able to tell which capacitor is C1 and which is C2, but you will be able to say with certainty that one of them has capacitance C1 and the other must have capacitance C2.

17 1310/1410 Introductory Physics, Lab 7. October, 2020

REPORT 3.1 Submit your derivation of the equation that is then solved using the quadratic equation. This derivation should be algebraic (no numbers) and should only be series parallel written in terms of C1 or C2 and Cequivalent and Cequivalent . Make sure all your work is shown and that the final equation is clearly expressed.
...