Laboratory 3: Thevenin Equivalent Circuits and Maximum Power Transfer

Transcription

1 ES 3: Introduction to Electrical Systems Laboratory 3: Thevenin Equivalent Circuits and Maximum Power Transfer I. GOALS: In this laboratory experiment, you will learn how to determine the Thevenin equivalent circuit of a complex instrument, a battery, and an acoustic sensor. In addition, you will experimentally investigate the Maximum Power Transfer theorem. Understanding these two concepts is very important if you are combining sensors, electronics, and actuators to form an engineering system. You will also expand your repertoire of commonly used lab devices by learning how to use a breadboard to build circuits. II. PRELAB QUESTIONS: In this lab a breadboard is used to quicky prototype and test several circuits. Please read how the breadboard is used to build circuits without soldering in Section V (Supplementary Information) before coming to the lab. Question 1: After reading Section V, look at Figure 1 and demonstrate that you understand the breadboard by drawing the circuit schematic (wiring diagram) for the circuit shown. The circuit consists of (1) a 9 volt battery, (2) an LED, (3) a photoresistor, (4) a 10kΩ resistor, and (5) an ammeter. The ammeter is not shown in the photo, but the black and red alligator clips are connected to the meter. 9 volt battery LED Photoresistor ma Figure 1. A simple circuit built on a breadboard. The wires have been enhanced with black lines to improve visibility. Pre-lab assignment: determine and sketch the circuit schematic. ES 3 Laboratory #3 Page 1 of 13

2 Pre-lab Question 2 Throughout this lab experiment, you will be asked to model complex devices as Thevenin equivalent circuits. This is a very powerful concept because it allows us to simplify circuits containing millions of components by representing that circuit using only two components: a voltage source and a resistor. Consider an automotive sound system. When the speakers are disconnected, the open circuit voltage produced by the sound system is measured by an oscilloscope and found to be v oc (t) = 16sinωt (volts). Automotive Sound System Next, an 8 Ω speaker is connected to the sound system and the voltage across the speaker is V spkr (t) = 8sinωt (volts). Automotive Sound System What is the Thevenin equivalent circuit of the sound system (excluding the speaker)? V TH = sinωt (volts) R TH = Ω If we connected two 8 Ω speakers in parallel to the output of this sound system, what voltage would appear across these two speakers? (Hint: use the Thevenin equivalent circuit!) Important point: Notice how a Thevenin equivalent circuit allows us to predict the behavior of the sound system, even though we don t have any idea what the actual internal circuit is! ES 3 Laboratory #3 Page 2 of 13

3 III. EXPERIMENTS: A. Thevenin Equivalent Circuit of the Function Generator The function generator is a complex electronic instrument, but it is possible to model the function generator with a simple Thevenin equivalent circuit (TEC). In class, you often determine the TEC by finding the open circuit voltage and the short circuit current. In the lab, it is easy to measure the open circuit voltage, but short-circuiting can be a dangerous and destructive procedure. Instead of using a short circuit, we will apply known resistors to the output of the function generator. Then we can measure how the voltage changes as a function of these load resistances. From these data, it is possible to determine the Thevenin resistance, R TH. (Refer to the Pre-Lab questions to review this method.) Use the function generator and the oscilloscope to create a voltage signal of v(t) = sin(2π. 1000t) volts Please refer to Laboratory #2 if you need to review the use of these two instruments. Since the resistance of the oscilloscope is very large, it accurately measures 1 the open circuit voltage of the function generator. This means that we now know the Thevenin voltage: V TH = sin(2π. 1000t) volts. Complete the model of the function generator by finding the Thevenin equivalent resistance, R TH. Short-circuiting the function generator and trying to measure the current might damage the instrument. A safer alternative, is to attach the load resistances listed in Table 1 to the function generator and measure the resulting output voltage (V out ). You will only be given 4 resistors, so you will need to create the tabulated resistances using multiple resistors connected in series. Use the breadboard to securely hold these resistors in place. Two resistors are connected in series in Figure 2 below. 1 Strictly speaking, this is only true if you are measuring low frequency signals. The oscilloscope and the probes have parasitic (unwanted) capacitance and the impedance of this capacitance will be much less than the probe s resistance at high frequency ( Z C = 1/ωC << 1MΩ). ES 3 Laboratory #3 Page 3 of 13

4 Figure 2. Two resistors in series, connected to the function generator. The function generator s test leads are visible at the top of the photograph and the oscilloscope probe is visible at the bottom. Table 1: Function Generator Output Voltage (V out ) and Average Power Dissipated in R L for various load resistances, R L. R L (Load Resistance) Scope only (1MΩ) 20 Ω 30 Ω 50 Ω 100 Ω 150 Ω V out (zero-pk) R TH Average power (Calculated) P ave = ½*(V 2 out /R L ) 1 volt μw The equivalent circuit, oscilloscope, and the load resistor are shown in Figure 3. Notice that R TH and R L form a simple voltage divider. Therefore, the output voltage is given by: V out = V TH R L RL + R TH Find the Thevenin equivalent resistance of the function generator (R TH ) for each of the load resistances listed in Table 1. In your lab report, describe the measurement method. Also include the complete data set in tabular form (i.e., Table 1). Comment on the invariance of R TH as R L is changed. Function generator S Main Out v out (Scope) V TH R TH 50 Ω R L Figure 3: Equivalent Circuit of the Function Generator with a Load Resistor B. Maximum Power Transfer The delivery of the maximum power to a device is a common engineering problem. Examples of maximum power transfer include delivering the maximum power to your cell ES 3 Laboratory #3 Page 4 of 13

5 phone antenna (to improve the coverage area), and maximum power delivery to an audio speaker (to get the highest sound level from the lowest power amplifier). You have already learned that the DC power dissipated in a resistor (R) is 2 V P diss = watts R If the voltage is sinusoidal, however, the dissipated power is not constant but varies with the sinusoidal voltage. It is common to use the time-averaged power (P ave ) when describing non-dc signals. For a sinusoidal voltage (sine or cosine), the average power dissipated in a resistor (R L ) is P ave T V0 pk sin( ωt) 1 V0 pk = T 0 R L dt = 2 R L watts The coefficient of ½ is due to the integration of sin 2 (ωt) over one full period of the sinusoid (T= 1/f = 2π/ω). Calculate the average power dissipated in R L for the measurements recorded in Table 1. The maximum power transfer theorem states that the maximum power is dissipated in the load (i.e., R L, an antenna, or an audio speaker) when the load resistance is equal to the Thevenin resistance of the source (i.e., the function generator, the cell phone, or the audio amplifier). That is, R L = R TH is the condition that delivers maximum power to R L. Make a graph of P ave vs. R L using Excel. Please refer to Laboratory #1 to review the instructions for using Excel to make a graph. Be certain to properly label the axes of the graph with appropriate units. Include the graph of power dissipation in your lab report. Describe your computation of power delivered to R L and describe how the maximum power transfer theorem is confirmed by these measurements. C. Modeling a 9 volt Battery A battery can also be modeled as a Thevenin equivalent circuit. In this section, you should develop a TEC model for a 9 volt battery by using the following components: 1. 9 volt battery and battery clip Ω resistor 3. DC voltmeter 4. a breadboard CAUTION: the DC power dissipated in the 100Ω resistor will be P DC = V 2 /R 9 2 /100 = 0.81 watts. The resistor is only capable of handling 0.5 watts, so the resistor will become very hot! DO NOT connect the resistor to the battery for more than 5 seconds ES 3 Laboratory #3 Page 5 of 13

6 and avoid touching the resistor. When designing circuits, it is always important to make certain that all of the circuit elements are sufficiently robust to dissipate the expected power. You could make this experiment safer by using two 200Ω resistors in parallel, or two 50Ω resistors in series, but that wouldn t be as interesting! In your lab report, describe the experiment that you devised to determine V TH and R TH of the 9 volt battery. Remember to measure and record the actual resistance of the 100Ω resistor. Using the TEC of the battery that you determined above, answer the following questions: If the 9 volt battery is connected to a GPS (global positioning system) receiver that requires 250 ma to operate, what is the output voltage of the battery? If the same battery is used to power a cell phone that requires 2.0 amperes, what is the output voltage of the battery? In your lab report, write a description of the problem and its solution. D. Thevenin Equivalent Circuit of a Microphone In addition to electronic circuits and batteries, it is also useful to model sensors as Thevenin equivalent circuits. Sensors include chemical, optical, or mechanical devices that convert a physical action into an electrical signal. A microphone converts sound into an electrical signal and is a common example of a sensor. Recall that a speaker produces sound by converting current flowing through the voice coil into mechanical motion of the speaker cone. By reversing this process, it is also possible to use a speaker as a microphone: Sound waves will cause the speaker cone and voice coil to move. As the voice coil moves through the permanent magnet, a current is induced in the voice coil and a voltage appears across the voice coil. This is commonly called a dynamic microphone (see Figure 4). Notice that the dynamic microphone looks very similar to the speaker described in Laboratory #2. The main difference is that a very thin diaphragm replaces the bulky cone of the speaker. This thin diaphragm greatly improves the microphones ability to accurately convert high frequency sound into an electrical signal. Figure 4. Operation of a dynamic microphone (source: ES 3 Laboratory #3 Page 6 of 13

7 We will use an audio speaker as a simple microphone in the next experiment. The goal of the exercise is to develop a Thevenin equivalent model of this acoustic sensor. In order to provide a stable, controllable sound that is easy to measure, you will excite the dynamic microphone using another 16 Ω speaker connected to the function generator. Couple two speakers together as shown in the Figure 5. You may want to use some wire to tie the speaker frames together through the mounting holes. Using the function generator, apply a voltage of V spkr = 0.5sin(2π. 440t) volts to one of the speakers; we will call this speaker the driver. Use the oscilloscope to measure the voltage generated by the second speaker. Note that this second speaker is performing the function of a microphone (albeit, rather poorly). Determine the TEC for the microphone by measuring the open circuit voltage produced by the microphone. Then add a 20 Ω resistor across the terminals of the microphone such that current flows through the coil and the 20Ω resistor. Deduce the value of R TH based on the change in voltage observed as the 20 Ω resistor is added to the circuit. Describe the experiment and then draw the TEC of the dynamic microphone in your lab report. Driver Microphone Wires to the oscilloscope Figure 5. Two speakers coupled together. The speaker on the left is connected to a function generator and produces sound at 440 Hz. The speaker on the right acts as a dynamic microphone and converts the sound waves into an electrical signal. ES 3 Laboratory #3 Page 7 of 13

8 IV. LAB REPORT: All required data, questions, and written comments can be found within the framed text blocks. Tabulate or plot all of the requested data and answer the questions in a manner that flows naturally. Complete Lab Report Guidelines are listed on the course website. In addition, complete the following problem: Engineers frequently use sensors to monitor a huge variety of devices. The sensor signals are often quite weak and need to be interfaced with an amplifier. As a concrete example, consider interfacing a microphone (acoustic sensor) with an audio amplifier. To solve this circuit problem, we need to have a Thevenin equivalent circuit model for the microphone as well as a model for the amplifier and the audio speaker. The block diagram for this system is shown in Figure 6. The circuit model for the entire audio system is shown in Figure 7. It is important to realize that Figure 7 is not the actual circuit (which is much more complicated); it is only a model that describes how the real circuit responds to inputs and outputs. It is very important to notice, however, that both the microphone and the output of the amplifier can be modeled as simple Thevenin equivalent circuits (an ideal voltage source and a resistor). Microphone Amplifier Speaker Figure 6. A system consisting of a sensor (microphone), electronics (amplifier) and an actuator (speaker). ES 3 Laboratory #3 Page 8 of 13

9 Microphone Model Amplifier Model Speaker Model R th (mic) R th (amp) + + (Vmic )sin(ωt) V IN R IN + - A V V IN R spkr - _ Figure 7. The equivalent circuit model for the microphone, amplifier, and speaker that are shown schematically in Figure 6. The microphone and speaker in Figure 7 are correctly modeled with the following parameters: Microphone Model: V mic = 0.1 volts R th (mic) = 16 Ω Speaker Model: R spkr = 16 Ω Two different amplifiers are available. As an engineer, your task is to select the amplifier that delivers the highest average power to the speaker. The parameters for each amplifier are: Amplifier Model #1: R IN = 5 Ω. A V = 100 (volts/volts) R th (amp) = 16 Ω Amplifier Model #2: R IN = 500 Ω. A V = 100 (volts/volts) R th (amp) = 16 Ω Analyze the performance of the two amplifiers, choose the best amplifier, and justify your decision in the lab report. Describe the importance of developing an accurate Thevenin equivalent circuit model of sensors. Also, develop a general principle for selecting an appropriate amplifier for a sensor. ES 3 Laboratory #3 Page 9 of 13

10 V. SUPPLEMENTARY INFORMATION: The Breadboard FRONT SIDE Throughout this lab you will use a breadboard. These boards are useful for quickly assembling circuits without soldering. IC The front side of the board consists of long rows of tie points that are all connected together. These rows are useful for distributing power supply voltages throughout the board. BACK SIDE Surrounded by these long rows are columns of five tie points. Each group of five is electrically connected under the surface by metal clips, as shown in the back-side view below. Integrated circuits should straddle the deep groove running along the board such that each pin of the IC is connected to a unique set of five tie points. Figure A. The breadboard consists fo hundreds of holes with underlying metallic connectors. Wires pushed into groupings of holes are connected to each other. To better understand how a breadboard is used to create a circuit prototype, consider the example in Figure B. We wish to use a 9 volt battery to power a red light emitting diode (LED). A 1000Ω resistor must be placed in series with the LED so that the current flowing through the LED is limited, and the LED does not overheat and burn out volts ma Resistor LED -7v + -2v + Figure B. The LED schematic for the breadboard circuit of Figure 8. ES 3 Laboratory #3 Page 10 of 13

11 Figure C shows a photograph of this circuit, as implemented on a breadboard. The battery or power supply in more complicated circuits connects to a large number of nodes, so we usually connect the battery to the long rows on the breadboard (see Figure 8). The black wire from the 9v battery is connected to the line of tie points next to the blue line marked with (-) in the photograph. This set of tie points connects vertically to the 1000Ω resistor. The resistor jumps horizontally to a new set of five tie points. The LED connects to the fifth tie point, joining the LED and the 1kΩ resistor. The other wire of the LED then connects to the vertical line of tie points marked with the red line and the (+) sign. The LED and a short length of red wire are connected by this vertical set of tie points. The end of the red wire is connected to the ammeter by the black alligator clip. Finally, the circuit is completed by the connection between the ammeter and the battery. This connection is between the red alligator clip and the red wire of the battery. MULTIMETER ma COM ma Figure C: One possible configuration for prototyping the circuit schematic in Figure B. ES 3 Laboratory #3 Page 11 of 13

12 Reading Resistor Color Codes The resistance is usually printed on a resistor using four bands of color. Each color represents a digit from 0-9, or, in the case of silver and gold, a manufacturing tolerance. The code is formatted in scientific notation. The first two bands represent the mantissa (with no decimal point!) and the third band represents the power of ten in the exponent. In the example below, a 26kΩ resistor is coded as 26x10 3 which uses the colors redblue-orange. Resistor Color Codes Black 0 Brown 1 Red 2 Orange 3 Yellow 4 Green 5 Blue 6 Violet 7 Gray 8 White 9 Silver ±10% Gold ±5% Example: 26 kω ± 5% = RED + BLUE + ORANGE + GOLD 2 6 x 10 3 ±5% = 26kΩ ES 3 Laboratory #3 Page 12 of 13

13 VI. PARTS LIST: Resistors (10 Ω, 20 Ω, 30 Ω, 50 Ω, 100 Ω) Two small audio speakers (16Ω) 9 volt battery and battery clip Breadboard for prototyping circuits Function Generator (Instek, GFG-8250A) Oscilloscope (Tektronix TDS 2012, or equivalent) with probes Test Leads: BNC-to-Alligator clips (1) Wires with Alligator clips (2) Banana plug-to-alligator clips (2) ES 3 Laboratory #3 Page 13 of 13