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The objective of this lab is to continue developing proficiency in the use of the digital multimeter in the context of verifying Kirchhoff’s Voltage and Current Laws (KVL and KCL). In the process you will investigate both the voltage-divider and current-divider circuit, you will become familiar with the use of the breadboard, and you will learn how to build light-sensor circuits.

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ECE2205: Circuits and Systems I Lab 2–1 Department of Electrical and Computer Engineering University of Colorado at Colorado Springs "Engineering for the Future" Lab 2: Kirchoff’s Laws 2.1 Objective The objective of this lab is to continue developing proficiency in the use of the digital multimeter in the context of verifying Kirchhoff’s Voltage and Current Laws (KVL and KCL). In the process you will investigate both the voltage-divider and current-divider circuit, you will become familiar with the use of the breadboard, and you will learn how to build light-sensor circuits. 2.2 Pre-Lab Preparation Read the lab overview in section 2.3 and answer the questions below. The instructor is to review your answers before you begin the lab tasks. 1. Let the voltmeter in Fig. 2.3 be represented by a resistance Rm . Derive Eq. (2.2) for this circuit. 2. Recall that an ideal voltmeter has infinite resistance. Letting the value of Rm in Eq. (2.2) be infinite should result in the familiar voltage-divider equation (2.1). Derive Eq. (2.1) from Eq. (2.2) by taking the limit as Rm → ∞. L’Hôpital’s Rule may be helpful. 3. Suppose that you measure the full-light and full-dark resistances of two CdS cells and find: R1,low = 90Ä, R2,low = 100Ä, R1,high = 32kÄ, and R2,high = 37kÄ. Find resistances R3 and R4 to match the characteristics of these CdS cells. Which scenario is this? 4. Suppose that you measure the full-light and full-dark resistances of two CdS cells and find: R1,low = 90Ä, R2,low = 100Ä, R1,high = 37kÄ, and R2,high = 32kÄ. Find resistances R3 and R4 to match the characteristics of these CdS cells. Which scenario is this? Be sure to bring your Matlab code minfn1.m and minfn2.m to the lab. If you have a bright flashlight, bring that too. 2.3 Background Prototyping a circuit using a breadboard. The solderless breadboard (sometimes called a protoboard) is the most common type of prototyping circuit board. Prototyping a circuit is valuable for complete evaluation of its design and performance. This requires the circuit to be designed, built and tested in the laboratory. Theoretical calculations and computer simulation are generally part of the design process. Once the circuit configuration is determined, the circuit is built on a prototyping board. There are two main types of prototyping circuit boards: 1. Solderless breadboards, 2. Perfboard. c 2006, Dr. Gregory L. Plett, ECE Dept., CU Colorado Springs Lab reader prepared by & Copyright ° Lab 2–2 ECE2205, Lab 2: Kirchoff’s Laws Perfboard is a thin slab of either epoxy glass or phenolic with small holes punched through it at a 0.1" spacing. A circuit built on perfboard requires either soldering or wire-wrapping the connections. A circuit built on a breadboard requires neither soldering nor wire wrapping the connections. Your laboratory instructor will assign to you a breadboard on which you will build your circuits throughout the semester. This breadboard has a single (two-sided) terminal strip, two bus strips, and three binding posts as shown in Fig. 2.1. The terminal strips and bus strips have many holes (contact receptacles) with a 0.1" spacing where wires or circuit-element terminals may be inserted. The real value of a breadboard is not as a pincushion, however, but as a wiring aid. The secret is in the hidden wiring inside the breadboard that helps you connect components together. Binding posts BREADBOARD MB−102−PLT R.S.R. ELECTRONICS Vb Va Bus strips 1 Terminal strip 5 10 15 20 25 30 35 40 45 50 55 60 A A B B C C D D E E F F G G H H I I J J 1 5 10 15 Figure 2.1 20 25 30 35 40 45 50 55 60 R.S.R. Electronics MB-102-PLT solderless breadboard. Each bus strip has two rows of contacts. Each of the two rows of contacts on the bus strips comprise a single node. That is, every contact along a row on a bus strip is connected together with wiring hidden inside the breadboard. Bus strips are used primarily for power supply connections but are also used for any node requiring a large number of connections. The terminal strip has 5 rows and 63 columns of contacts on each side of the center gap. Each column of 5 contacts is a node. The internal connections of the breadboard are illustrated in the zoomed cutout view in Fig. 2.2 as orange (grey) lines. 50 55 60 A B C D E F G H Figure 2.2 Zoomed cutout view of breadboard, showing internal connections. You will build your circuits on the terminal strips by inserting the leads of circuit components into the contact receptacles and making connections with 22 AWG (American Wire Gauge) wire. There are wire cutter/strippers and spools of wire in the lab. You will be using the red and black binding posts for power supply connections. Hence, it is a good idea to wire them to a bus strip. c 2006, Dr. Gregory L. Plett, ECE Dept., CU Colorado Springs Lab reader prepared by & Copyright ° Lab 2–3 ECE2205, Lab 2: Kirchoff’s Laws Using the multimeter as a voltmeter. A voltmeter is a device for measuring voltage. It measures and displays the voltage (potential difference) of the positive (e.g., red) probe with respect to the negative (e.g., black) probes. The voltmeter is placed in parallel with the circuit element whose voltage is to be measured. Recall that two elements are in parallel when they share the same pair of nodes and hence share the same voltage. Consider the voltage divider circuit shown in Fig. 2.3 in which the voltage across R2 is to be measured. If the presence of the voltmeter does not affect the voltage it is intending to measure, the meter must draw no current. That is, it must act as an open circuit. An open circuit may be thought of as an infinite resistance. Hence, an ideal voltmeter has an infinite resistance. You measured the internal resistance of the voltmeter in Lab 1 and found the value to be on the order of 10MÄ, which is large, but certainly not infinite. R1 Red probe vs Voltmeter R2 Black probe Figure 2.3 Voltage divider circuit, with voltage measured by real voltmeter. First consider the circuit with the voltmeter not present. In this case the voltage v2 across the resistor R2 can be expressed in terms of the source voltage vs and the resistors R1 and R2 by R2 v2 = vs . (2.1) R1 + R2 With the voltmeter present, its resistance alters the voltage division equation, which becomes R2 Rm , v2 = vs R2 Rm + R1 (R2 + Rm ) (2.2) where Rm is the resistance of the voltmeter. You will not be able to see how this equation was obtained at first examination. Let the voltmeter in Fig. 2.3 be represented by a resistance Rm . Use resistance reduction and voltage division to obtain an expression for v2 in terms of vs . Then, clear the fractions in the numerator and denominator. Be sure to show your derivation in your lab report. You will now build the voltage divider circuit using the dc power supply as the voltage source vs in Fig. 2.3. Using the Multimeter as an Ammeter An ammeter is a device for measuring current. It measures the current flowing into the positive (e.g., red) probe and out of the negative (e.g., black) probes of the meter. The ammeter is placed in series with the circuit element whose current is to be measured. Recall that two elements are in series when they share in the same branch and hence share the same current. The ideal ammeter will have zero rresistance, thus not alter the resistance or current of the branch whose current is being measured. Consider the current divider circuit shown in Fig. 2.4. The current i 1 through R1 may be expressed as a fraction of the current i s flowing out of the source in terms of R1 and R2 using current division i1 = is 1 R1 1 R1 + 1 R2 = is R2 . R1 + R2 (2.3) In this lab you will build the current divider circuit and make several measurements. Recall that your ammeter is not ideal—in fact you measured its resistance in the first lab. The resistance of the ammeter will be an important consideration when measuring currents in the circuit shown. Record all measured values and present percent error calculations and tables as appropriate. Photoresistors (CdS Cells) Photoresistors are a special kind of resistor that change their value when exposed to light. Some types increase resistance while others decrease resistance. One common photoresistive substance is cadmium-sulfide (CdS), out of which “CdS cell” photoresistors are made. Figure 2.5 shows both a picture of a CdS c 2006, Dr. Gregory L. Plett, ECE Dept., CU Colorado Springs Lab reader prepared by & Copyright ° Lab 2–4 ECE2205, Lab 2: Kirchoff’s Laws 10kÄ vs Figure 2.4 R1 R2 Current divider circuit. cell and its schematic symbol (sometimes the schematic symbol is drawn with arrows pointing at the cell to indicate impinging light). Figure 2.5 CdS photoresistor pictorial representation (left) and schematic representation (right). Photoresistors may be used in a voltage-divider circuit for the purpose of sensing the intensity of light. Figure 2.6 gives an example of how this sensor circuit may be designed. In the circuit, R1 is a known resistor and vs is a known voltage. By measuring vCdS , we can compute RCdS and from there infer the level of light. Here is how: 1. First, a table of CdS resistance versus light level is created. 2. By re-arranging the voltage-divider equation, we find that vCdS R1 . RCdS = vs − vCdS 3. We measure vCdS , compute RCdS , and use the table to look up the light level. R1 vs vCdS Figure 2.6 CdS-cell light sensor using voltage divider. Balancing Two CdS Cells The above method for constructing a light sensor works well if you have a specially calibrated table of resistance versus light level for the CdS cell that you are using. A practical problem, however, is that all CdS cells require slightly different calibration tables. If your circuit is being used in some embedded system, the software can be written to automatically calibrate the sensors (you have had some experience with this in ECE1001: Introduction to Robotics). If you want to match the performance of two CdS cells electronically, however, you must follow a different approach. In this lab, we will consider some circuit modifications so that the two CdS cells have identical resistances at two different reference light settings. For example, if you were constructing a line-following robot, you might want identical readings over the dark part of the line and over the white part of the background. Here, we will calibrate identical readings in maximum darkness and maximum brightness. c 2006, Dr. Gregory L. Plett, ECE Dept., CU Colorado Springs Lab reader prepared by & Copyright ° Lab 2–5 ECE2205, Lab 2: Kirchoff’s Laws First, we measure the minimum resistances of both CdS cells (in maximum brightness). We denote the CdS cell with the lower of these values “CdS cell 1” and the other as “CdS cell 2”. Then, we measure the maximum resistances of both CdS cells (in maximum darkness). There are two possible scenarios: (1) CdS cell 1 has the lower maximum resistance as well; or (2) CdS cell 2 has the lower maximum resistance. In the first scenario, we replace the CdS cells in the voltage-divider circuits with the circuit shown in the left frame of Fig. 2.7. R3 is a resistor in series with CdS cell 1 to increase its lower resistance value (a side effect that we must consider is that its higher resistance value is also increased). R4 is a resistor in parallel with CdS cell 2 to decrease its higher resistance value (a side effect is that its lower resistance value is also decreased). If we match the conductances of these two cells at both low and high ends, we must satisfy the following equations: 1 1 1 = + R1,low + R3 R2,low R4 1 1 1 = + . R1,high + R3 R2,high R4 Notice that these are two simultaneous equations in two unknowns (R3 and R4 ). Further, they are nonlinear! We can do some algebra to solve them, or we can ask Matlab to iteratively solve the equations using an optimization technique. First, re-arrange the equations as 1 1 1 − − e1 = R1,low + R3 R2,low R4 1 1 1 e2 = − − . R1,high + R3 R2,high R4 The new “variables” e1 and e2 are “equation errors”. If these errors are zero, then both equations are unmodified. We will ask Matlab to pick values for R3 and R4 such that J = e12 + e22 is minimized, and if J is “small enough” then we have solved the equations. Before we see how, let’s first consider the second scenario. R4 R3 CdS Cell 1 Cds Cell 2 Figure 2.7 R4 R3 CdS Cell 1 Cds Cell 2 Two scenarios for matching CdS-cell characteristics. “Scenario 1” is on the left; “Scenario 2” is on the right. In the second scenario, we replace the CdS cells in the voltage-divider circuits with the circuit shown in the right frame of Fig. 2.7. R3 is a resistor in series with CdS cell 1 to increase its lower resistance (a side effect that we must consider is that its higher resistance value is also increased). R4 is a resistor in parallel with CdS cell 1 to decrease its higher resistance (a side effect that we must consider is that its lower resistance value is also decreased). If we match the resistances of these two cells at both low and high ends, we must satisfy the following equations: R1,low R4 R2,low = R3 + R1,low + R4 R1,high R4 R2,high = R3 + . R1,high + R4 Again, we re-arrange these equations into an equation-error format: R1,low R4 − R2,low e1 = R3 + R1,low + R4 R1,high R4 e2 = R3 + − R2,high. R1,high + R4 c 2006, Dr. Gregory L. Plett, ECE Dept., CU Colorado Springs Lab reader prepared by & Copyright ° Lab 2–6 ECE2205, Lab 2: Kirchoff’s Laws When J = e12 + e22 , minimizing J is the same as solving for R3 and R4 , provided J is “small enough”. The easy way to solve for R3 and R4 uses Matlab’s optimization toolbox (available in the lab, but not included as part of the student version). Specifically, you will invoke a Matlab procedure that will minimize J = e12 + e22 where e1 and e2 are defined for whichever scenario you encounter. A Matlab function for computing the cost J if you encounter the first scenario is: % % % % % Minimization function to determine cost for case where CdS cell 1 has lower minimum resistance and lower maximum resistance than CdS cell 2. Therefore, CdS cell 1 needs a resistor R3 in series and CdS cell 2 needs a resistor R4 in parallel. % The input "theta" comprises [R3, R4]. % matching error-squared (goal = 0). The output is the function cost = minfn1(theta) global R1low R1high R2low R2high lowerror = 1/(R1low + theta(1)) - 1/R2low - 1/theta(2); higherror = 1/(R1high + theta(1)) - 1/R2high - 1/theta(2); cost = lowerror^2 + higherror^2 end Similarly, a Matlab function for computing the cost J if you encounter the second scenario is: % % % % Minimization function to determine cost for case where CdS cell 1 has lower minimum resistance and higher maximum resistance than CdS cell 2. Therefore, CdS cell 1 needs a resistor R3 in series and a resistor R4 in parallel. % The input "theta" comprises [R3, R4]. % matching error-squared (goal = 0). The output is the function cost = minfn2(theta) global R1low R1high R2low R2high lowerror = theta(1) + R1low*theta(2)/(R1low + theta(2)) - R2low; higherror = theta(1) + R1high*theta(2)/(R1high + theta(2)) - R2high; cost = lowerror^2 + higherror^2 end The actual work gets done in Matlab using the following code segment: global R1low R1high R2low R2high R1low = ; R1high = ; R2low = ; R2high = ; g3 = ; % initial guess for value of R3 g4 = ; % initial guess for value of R4 fminsearch(’minfn1’,[g3 g4]) Use minfn2 instead of minfn1 in the last line if you encounter the second scenario. Values of R3 and R4 will be returned from the fminsearch procedure. c 2006, Dr. Gregory L. Plett, ECE Dept., CU Colorado Springs Lab reader prepared by & Copyright ° Lab 2–7 ECE2205, Lab 2: Kirchoff’s Laws Variable Resistors In order to match your two CdS cells, you will need to be able to construct the resistance values returned by Matlab’s optimization routine. Since fixed-value resistors are only manufactured with certain nominal values, it is necessary to use a variable resistor (perhaps in combination with a fixed-value resistor) to achieve the desired resistance. A variable resistor is a three-terminal device, depicted schematically in the left frame of Fig. 2.8. Figure 2.8 The variable resistor, or potentiometer (“pot”). Two terminals are connected across the full resistance. The third terminal is connected to a sliding contact that can sweep across the resistive surface to achieve any value between zero resistance and the full resistance of the device. A common configuration for wiring a variable resistor is shown in the right frame of Fig. 2.8. The slider terminal is hard-wired to one of the end terminals. Now, the end-to-end resistance is variable. There are different kinds of mechanisms for adjusting variable resistors. Some have linear sliders; others have rotational sliders. The kind we will use in this lab use a screwdriver to rotate a small rotational slider. If you are careful, you should be able to adjust the resistance in increments of roughly 1% of the full-scale resistance of the device. 2.4 Lab Assignment Task 1: Prelab Certification. Have the Lab Assistant/Instructor review your answers to the prelab assignment questions and sign the certifications page. Task 2: Check out a Breadboard Check out an “R.S.R. Electronics MB-102-PLT” breadboard from your lab instructor. You will be using this board throughout the semester. Obtain a piece of masking tape and affix it to the top of your board. Write your name on the tape. After you have checked out your breadboard, examine it closely and compare with Figs. 2.1 and 2.2. Use the ohmmeter to verify the hidden wiring of Fig. 2.2. Task 3: Voltage Divider with Moderate-Valued Resistors. 1. Obtain two 1kÄ resistors from the parts bin. Designate one of the resistors as R1 and the other as R2 2. Measure and record the resistor values using the multimeter as an ohmmeter. Be sure to keep track of which resistor corresponds to which value measured! 3. Build the circuit in Fig. 2.3 on your breadboard using the 1kÄ resistors for R1 and R2 . 4. Set the power supply to 5V. Use the voltmeter, not the front panel display of the power supply to ensure the proper setting. Important note: You built the circuit before you set the power supply voltage to 5V. If the current limiter is set to a value lower than than the current demanded by the circuit, the constant current (cc) indicator will light up and the voltage control knob will no longer adjust the output voltage. If this happens, simply increase the current limiter until you are able to achieve 5V in the constant voltage (cv) mode. 5. Using the voltmeter, measure the voltage across resistor R1 , and then across resistor R2 . Record these values, as always, and verify Kirchhoff’s Voltage Law (KVL). 6. Comment on the accuracy of measurements made considering the internal resistance of the voltmeter. 7. Create a table presenting theoretical and measured voltages along with percent error. Consider whether your theoretical values for the voltages across R1 and R2 should include the effect of Rm . Important Note: When you are calculating percent error, you should avoid cases in which the theoretical value is zero since the percent error is meaningless. To calculate percent error between theoretical and experimental verification of KVL, use the source voltage as the reference. For example, in the measurements made in this section, the theoretical value (and measured value!) for the voltage across the supply is 5V. The measured value is the same as the c 2006, Dr. Gregory L. Plett, ECE Dept., CU Colorado Springs Lab reader prepared by & Copyright ° ECE2205, Lab 2: Kirchoff’s Laws Lab 2–8 theore ...
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