Lab 2: Kirchoff s Laws

Transcription

1 ECE2205: Circuits and Systems I Lab 2 Department of Electrical and Computer Engineering University of Colorado at Colorado Springs "Engineering for the Future" Lab 2: Kirchoff s Laws 2. 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.. Let the voltmeter in Fig. 2.3 be represented by a resistance R m. Derive Eq. (2.2) for this circuit. 2. Recall that an ideal voltmeter has infinite resistance. Letting the value of R m in Eq. (2.2) be infinite should result in the familiar voltage-divider equation (2.). Derive Eq. (2.) from Eq. (2.2) by taking the limit as R m. 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: R,low = 90, R 2,low = 00, R,high = 32k, and R 2,high = 37k. Find resistances R 3 and R 4 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: R,low = 90, R 2,low = 00, R,high = 37k, and R 2,high = 32k. Find resistances R 3 and R 4 to match the characteristics of these CdS cells. Which scenario is this? Be sure to bring your Matlab code minfn.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:. Solderless breadboards, 2. Perfboard.

2 ECE2205, Lab 2: Kirchoff s Laws Lab 2 2 Perfboard is a thin slab of either epoxy glass or phenolic with small holes punched through it at a 0." 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.. The terminal strips and bus strips have many holes (contact receptacles) with a 0." 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 02 PLT R.S.R. ELECTRONICS V a V b Bus strips Terminal strip A B C D E A B C D E F G H I J F G H I J Figure 2. R.S.R. Electronics MB-02-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 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.

3 ECE2205, Lab 2: Kirchoff s Laws Lab 2 3 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 R 2 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 and found the value to be on the order of 0M, which is large, but certainly not infinite. v s R R 2 Voltmeter Red probe 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 v 2 across the resistor R 2 can be expressed in terms of the source voltage v s and the resistors R and R 2 by v 2 = v s R 2 R + R 2. (2.) With the voltmeter present, its resistance alters the voltage division equation, which becomes R 2 R m v 2 = v s R 2 R m + R (R 2 + R m ), (2.2) where R m 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 R m. Use resistance reduction and voltage division to obtain an expression for v 2 in terms of v s. 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 v s in Fig 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 The current i through R may be expressed as a fraction of the current i s flowing out of the source in terms of R and R 2 using current division i = i s R R + R 2 = i s R 2 R + R 2. (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

4 ECE2205, Lab 2: Kirchoff s Laws Lab 2 4 0k v s R R 2 Figure 2.4 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, R is a known resistor and v s is a known voltage. By measuring v CdS, we can compute R CdS and from there infer the level of light. Here is how:. First, a table of CdS resistance versus light level is created. 2. By re-arranging the voltage-divider equation, we find that R CdS = v CdS R. v s v CdS 3. We measure v CdS, compute R CdS, and use the table to look up the light level. v s R v CdS 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 ECE00: 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.

5 ECE2205, Lab 2: Kirchoff s Laws Lab 2 5 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 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: () CdS cell 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 R 3 is a resistor in series with CdS cell to increase its lower resistance value (a side effect that we must consider is that its higher resistance value is also increased). R 4 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: = + R,low + R 3 R 2,low R 4 R,high + R 3 = R 2,high + R 4. Notice that these are two simultaneous equations in two unknowns (R 3 and R 4 ). 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 e = R,low + R 3 R 2,low R 4 e 2 =. R,high + R 3 R 2,high R 4 The new variables e and e 2 are equation errors. If these errors are zero, then both equations are unmodified. We will ask Matlab to pick values for R 3 and R 4 such that J = e 2 + e2 2 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. R 4 R 4 R 3 R 3 CdS Cell Cds Cell 2 CdS Cell Cds Cell 2 Figure 2.7 Two scenarios for matching CdS-cell characteristics. Scenario 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 R 3 is a resistor in series with CdS cell to increase its lower resistance (a side effect that we must consider is that its higher resistance value is also increased). R 4 is a resistor in parallel with CdS cell 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: R 2,low = R 3 + R,low R 4 R,low + R 4 R 2,high = R 3 + R,highR 4 R,high + R 4. Again, we re-arrange these equations into an equation-error format: e = R 3 + R,lowR 4 R,low + R 4 R 2,low e 2 = R 3 + R,highR 4 R,high + R 4 R 2,high.

6 ECE2205, Lab 2: Kirchoff s Laws Lab 2 6 When J = e 2 + e2 2, minimizing J is the same as solving for R 3 and R 4, provided J is small enough. The easy way to solve for R 3 and R 4 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 = e 2 + e2 2 where e and e 2 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 has lower minimum resistance and lower maximum % resistance than CdS cell 2. Therefore, CdS cell needs % a resistor R3 in series and CdS cell 2 needs a resistor R4 in % parallel. % The input "theta" comprises [R3, R4]. The output is the % matching error-squared (goal = 0). function cost = minfn(theta) global Rlow Rhigh R2low R2high lowerror = /(Rlow + theta()) - /R2low - /theta(2); higherror = /(Rhigh + theta()) - /R2high - /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 has lower minimum resistance and higher maximum % resistance than CdS cell 2. Therefore, CdS cell needs % a resistor R3 in series and a resistor R4 in parallel. % The input "theta" comprises [R3, R4]. The output is the % matching error-squared (goal = 0). function cost = minfn2(theta) global Rlow Rhigh R2low R2high lowerror = theta() + Rlow*theta(2)/(Rlow + theta(2)) - R2low; higherror = theta() + Rhigh*theta(2)/(Rhigh + theta(2)) - R2high; cost = lowerror^2 + higherror^2 end The actual work gets done in Matlab using the following code segment: global Rlow Rhigh R2low R2high Rlow = <insert value>; Rhigh = <insert value>; R2low = <insert value>; R2high = <insert value>; g3 = <insert value>; % initial guess for value of R3 g4 = <insert value>; % initial guess for value of R4 fminsearch( minfn,[g3 g4]) Use minfn2 instead of minfn in the last line if you encounter the second scenario. Values of R 3 and R 4 will be returned from the fminsearch procedure.

7 ECE2205, Lab 2: Kirchoff s Laws Lab 2 7 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 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 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 % of the full-scale resistance of the device. 2.4 Lab Assignment Task : 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-02-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. and 2.2. Use the ohmmeter to verify the hidden wiring of Fig Task 3: Voltage Divider with Moderate-Valued Resistors.. Obtain two k resistors from the parts bin. Designate one of the resistors as R and the other as R 2 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 k resistors for R and R 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 R, and then across resistor R 2. 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 R and R 2 should include the effect of R m. 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

8 ECE2205, Lab 2: Kirchoff s Laws Lab 2 8 theoretical value because you used the voltmeter to set the power supply voltage to 5V. To obtain the KVL measured voltage, add the voltage across R to the voltage across R 2. Compare with 5V. Task 4: Voltage Divider with Large-Valued Resistors. Obtain two 0M resistors from the parts bin. Designate one of the resistors as R and the other as R Measure 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 0M resistors for R and R Set the power supply to 5V. 5. Using the voltmeter, measure the voltage across resistor R, and then across resistor R 2. Record these values, and verify Kirchhoff s Voltage Law (KVL). 6. Comment on the accuracy of the voltage measurements made (consider 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 R and R 2 should include the effect of R m. Task 5: Current Divider with Moderate-Valued Resistors. Obtain two k resistors from the parts bin. Designate one of the resistors as R and the other as R 2. (You may use the same resistors that you used in Task 3 if you wish). Also, obtain one 0k resistor. Designate this resistor as R Measure 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. 2.4 on your breadboard using the k resistors for R and R Set the power supply to 0V. Don t forget to set the voltage using the voltmeter rather than depending on the front panel display of the power supply. Important Note: You built the circuit before you set the power supply voltage to 0V. 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 0V in the constant voltage (cv) mode. 5. Using the voltmeter, measure the voltage across the 0k resistor followed by the parallel combination of resistors R and R 2. Record these values, as always, and verify Kirchhoff s Voltage Law (KVL). 6. Configure the multimeter to measure current. Remember that this requires two things: Remove the terminal of the red probe from the voltage/resistance measuring receptacle and insert it in the current measuring receptacle on the front panel of the multimeter. Then press the shift and DC I buttons to select dc current measurement. 7. Measure the current through the 0V source. Remember that you have to break the circuit and insert the ammeter in series with the 0V source to allow the current to flow through the ammeter. 8. Measure the current through R and then the current through R Verify Kirchhoff s Current Law (KCL). Remember that a theoretical value of zero produces a meaningless percent error. 0. Comment on the accuracy of the voltage measurements made (consider the internal resistance of the voltmeter).. Comment on the accuracy of the current measurements made (consider the internal resistance of the ammeter).

9 ECE2205, Lab 2: Kirchoff s Laws Lab 2 9 Task 6: Current Divider with Small-Valued Resistors. Obtain two 0 resistors from the parts bin. Designate one of the resistors as R and the other as R Measure 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. 2.4 using the 0 resistors for R and R Set the power supply to 0V. Don t forget to set the voltage using the voltmeter rather than depending on the front panel display of the power supply. Important Note: You built the circuit before you set the power supply voltage to 0V. 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 0V in the constant voltage (cv) mode. 5. Using the voltmeter, measure the voltage across the 0k resistor followed by the parallel combination of resistors R and R 2. Record these values, as always, and verify Kirchhoff s Voltage Law (KVL). 6. Configure the multimeter to measure current. Remember that this requires two things: Remove the terminal of the red probe from the voltage/resistance measuring receptacle and insert it in the current measuring receptacle on the front panel of the multimeter. Then press the shift and DC I buttons to select dc current measurement. 7. Measure the current through the 0V source. Remember that you have to break the circuit and insert the ammeter in series with the 0V source to allow the current to flow through the ammeter. 8. Measure the current through R and then the current through R Verify Kirchhoff s Current Law (KCL). Remember that a theoretical value of zero produces a meaningless percent error. 0. Comment on the accuracy of the voltage measurements made (consider the internal resistance of the voltmeter).. Comment on the accuracy of the current measurements made (consider the internal resistance of the ammeter). Task 7: Check out a parts box Check out a parts box. Initially, this will only contain the two CdS cells required for this lab exercise, and two variable resistors. However, you will collect other parts over the duration of the semester that you will keep in this box. At the end of the semester you will return the box, and all the parts. Affix a piece of masking tape on the back of the box and write your name on the masking tape. Task 8: Matching CdS cells. Determine a way to differentiate between your two CdS cells. Perhaps you could use masking tape on the leads of one, or perhaps you can keep track using a position on the breadboard. 2. Measure the full-dark resistance of both CdS cells. Cover the cell s surface completely with your finger to make sure that it sees no light. Record both values. 3. Measure the full-light resistance of both CdS cells. It helps if you use a bright flashlight shining directly into the cells. Record both values. 4. Determine whether your cells fall within scenario or scenario Compute the resistances R 3 and R 4 required to match the CdS cells. 6. Using the variable resistors from your parts box and perhaps some fixed resistors from the parts bins in the lab, construct the circuit(s) required to match resistances of the CdS cells. 7. Repeat the full-dark and full-light resistance measurements to verify that your CdS cells are now matched. Record the values. Are the cells matched under ambient light conditions?

10 ECE2205, Lab 2: Kirchoff s Laws Lab 2 0 Task 9: Lab report. Submit your results in the form of a typed report. Refer to Lab for instructions regarding proper format and content of an acceptable lab report. Please also address the following questions: The current division equation (Eq. (2.3)) does not include the resistance of the ammeter. Let the internal resistance of the ammeter be R m. Write the expression for the current i through R, including the resistance of the meter assuming that the ammeter is being used to measure the current i. Then take the limit of this expression as the ammeter internal resistance goes to zero, showing that the limit is given by Eq. (2.3). The voltage source and 0k resistor in Fig. 2.4 form an approximate current source for small load resistances. If the voltage source and 0k resistor formed an ideal current source, then the current i s would be constant, independent of the resistances of R and R 2, which is certainly not the case. Consider the parallel combination of R and R 2 as a single resistance R L. If R L is small compared to 0k, then the current is will be very nearly ma (Recall that v s = 0V) independent of R L. Calculate the range of values of R L such that the current is will deviate from ma by no more than 5%. Consider the circuit shown in Fig Suppose you want to know the value of all voltages and currents in the circuit. Assume that you know nothing at all about the resistor values. You want the results to be as accurate as possible. You have a multimeter that you may use as either a voltmeter or an ammeter. Explain the sequence of measurements that you make. Comment on your level of confidence that your results are accurate. Don t forget that you have Ohm s Law and Kirchhoff s Laws that may be used. Try to find the minimum set of measurements required to solve for all unknowns. R v s R 2 R 3 Figure 2.9 Resistive network.