ME 140L Lab 2 Instruction

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Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li Lab 2: Source/Instrument Impedance, Divider Circuits, and Basic AC Measurements (RMS) Required Components  2x 1 kΩ resistors  2x 1 MΩ resistors  1x 2 kΩ resistor Laboratory Equipment Featured in this Lab  benchtop digital multimeter – used here to measure voltage, current, and resistance  benchtop function generator – used here to generate an AC voltage Pre-Laboratory Readings (sections 2.1 through 2.5) 2.1 Learning Objectives In this lab, you will learn about two important basic circuits  voltage divider circuit  current divider circuit These circuits can be used to create customized values for voltage and current sources that are needed to drive various types of sensors and other components in mechatronic systems, such as motors, linear actuators, etc. In addition, you will learn why and when it is important to consider the impedance of both power supplies and other measurement devices, such as the multimeter, that are commonly used in the lab. Specifically, you will investigate:  the output impedance of a real source, such as DC voltage supply or an AC voltage supply (function generator)  the input impedance of a real instrument, such as a multimeter 2.2 Series and Parallel Equivalent Resistance When N resistors are connected in series, as shown in Figure 2.1, the equivalent resistance is the sum of the individual resistances. R eq = R 1 + R 2 + R 3 + ⋯ + R N (2.1) Figure 2.1: Resistors in series For N resistors connected in parallel, as shown in Figure 2.2, the inverse of the equivalent resistance is the sum of the inverses of the individual resistances 1 R eq = 1 R 1 + 1 R 2 + 1 R 3 + ⋯ + 1 R N (2.2)

Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li Figure 2.2: Resistors in parallel 2.3 RMS values for AC voltage and current In the previous lab, you worked with DC circuits and measured voltages and currents using a digital multimeter. In this lab, you will begin working with AC circuits, where voltages and currents vary with time, as shown in figure 2.3. Figure 2.3: A complete cycle of a sinusoidal AC signal (can represent either voltage or current) Note the difference between the amplitude of a signal and its peak-to-peak value. Based on your experience so far with the multimeter, you may think that it is only capable of measuring voltage or current only if it is constant, thus it would not be useful for measuring AC signals, since they are continuously changing. However, when working with AC circuits, we can specify voltage and current values by their effective or root-mean- square (RMS) values, which indeed can be measured by the multimeter. An RMS value is defined as the square root of the average of the square of a signal integrated over one cycle or period, T . For current and voltage signals, the respective RMS relations are V(t) = V m sin(  t) V average = 0.637 V m V rms =0.707 V m Amplitude Peak-to-Peak

Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li i RMS = √ 1 T ∫ 0 T i ¿¿¿ (2.3) V RMS = √ 1 T ∫ 0 T V ¿¿¿ (2.4) where i P and V P are the amplitudes (peak value, as shown in figure 2.3) of the sinusoidal current and voltage waveforms. RMS values are also useful for power calculations. Recall that electrical power is equal to the product of the voltage and current. P = V ⋅ I (2.5) For example, the average AC power dissipated by a resistor can be calculated by using the same equations that are used with DC signals, except that we use the RMS values of voltage and current, as shown in the equation below. P avg = V RMS I RMS = R I 2 RMS = V 2 RMS R (2.6) 2.4 Resistance and Impedance In the previous lab, you worked with DC resistive circuits. In this lab, we will be using the term “impedance” (in place of resistance) which is used in the context of analyzing AC circuits and signals. We can view the impedance of a circuit element essentially as its “resistance” to flow of current. Here, the word “resistance” is used informally and it does not imply that other circuit elements, such as inductors and capacitors, have a resistance in the DC-sense of the word. Without getting into too much detail, you can view impedance as “resistance” when working with AC signals. Figure 2.4 below summarizes this concept nicely. Figure 2.4: The relationship between impedance and resistance

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Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li 2.5 Ideal vs. Real Sources and Meters When analyzing electrical circuits on paper, the concepts of ideal sources and meters are often used. An ideal voltage source has zero output impedance and can supply infinite current. An ideal voltmeter has infinite input impedance and draws no current. An ideal ammeter has zero input impedance and no voltage drop across it. Laboratory sources and meters have terminal characteristics that are somewhat different from the ideal cases. The terminal characteristics of the real sources and meters you will be using in the laboratory may be modeled using ideal sources and meters, plus additional resistances, as illustrated in Figures 2.5 through 2.7. Cases of very large and very small impedances in the circuit compared to the input impedance In some instances, the input impedance of a meter or the output impedance of a source can be neglected and very little error will result. However, in many applications where the impedances of the instruments are of a similar magnitude to those of the circuit, serious errors will occur. As an example of the effect of input impedance, if you use a multimeter to measure the voltage across R 2 , as shown in Figure 2.8, the equivalent circuit is: The equivalent resistance of the parallel combination of R 2 and R i is R eq = R 2 R i R 2 + R i (2.7) Therefore, the actual measured voltage would be Figure 2.5: Real Voltage Source with Output Impedance Figure 2.6: Real Ammeter with Input Impedance Figure 2.7: Real Voltmeter with Input Impedance Figure 2.8: Effect of Input Impedance – +

Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li V O = R eq R 1 + R eq V i (2.8) If R i is large compared to R 2 (which is usually the case), R eq is approximately equal to R 2 and the measured voltage, V 0 would be close to the expected ideal voltage division result of V O = R 2 R 1 + R 2 V i (2.9) However, if R 2 is not small compared to R i , the measured voltage will differ from the ideal result based on Equations 2.7 and 2.8. Calculating Input Impedance If you know values for V i , R 1 , and R 2 in Figure 2.8, and if you measure V O , you can then determine the input impedance, R i , of the measuring device using the following analysis. Equation 2.8 can be solved for R eq , which yields R eq = ( V O V i − V O ) R 1 (2.10) Knowing R eq , we can then determine the input impedance by solving for R i R i = R eq R 2 ( R 2 − R eq ) (2.11)

Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li 2.6 Laboratory Procedures and Experimental Summary Sheet Exercise 1 Select five separate resistors whose nominal values are listed below - connect each resistor to the multimeter and record the measured value for each resistor. Resistor and Nominal Value Measured Value (Ω) absolute error % error R 1 : 1 kΩ 0.986 0.014 R 2 : 1 kΩ 0.984 0.016 R 3 : 1 MΩ 0.995 0.005 R 4 : 1 MΩ 1.005 .005 R 5 : 2 kΩ 1.98 0.02 Exercise 2 Now, construct the voltage divider circuit shown in Figure 2.10 using resistors R 1 and R 2 listed above. Set the DC power supply voltage ( V i ) to 10 V. A voltage divider takes the input voltage, V i , and divides it proportionally, such that V R 1 V R 2 = R 1 R 2 Complete the table below by measuring the appropriate values. In your simulations, use the actual (measured) values for R 1 and R 2 . Note: Hold off on doing your Multisim work until after you’ve finished all of the lab activities – this will save time since you won’t be switching back and forth between your experimental work and the computer. input voltage, V i (volts) output voltage, V o , across R 2 (volts) current, i (mA) actual value from multimeter 9.99 1.6 .999 simulated value from Multisim Figure 2.10: Voltage Divider Circuit – + i V 0 = V R2

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Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li Exercise 3 Using the same circuit from Exercise 2 , use the function generator to supply an AC input voltage to the circuit – check with your TA to ensure that the function generator is set in “High Z mode” (high impedance mode.) The AC voltage should be a sine wave with a frequency of 1 KHz and an amplitude (peak voltage) of 3 V. Complete the table below by measuring and simulating the appropriate values. In your calculations, use the actual (measured) values for R 1 and R 2 . Use RMS values for all table entries. input voltage, V i (RMS value, V RMS ) in volts output voltage, V o (RMS value, V RMS ) in volts current, i (RMS value, i RMS ) in mA actual value from multimeter – with function generator set to 50 Ω impedance 2.12 .0027 39.8 actual value from multimeter – with function generator set to high impedance (high Z) 1.06 .0026 20 simulated value from Multisim Exercise 4 Repeat exercise 2, with V i = 10 V DC, using R 5 and R 4 in place of R 1 and R 2 , as shown in Fig. 2.11. In this case, the impedances of the sources and instruments are close in value to the load resistances and will therefore will affect the measured values. Sketch the equivalent circuit for the source and instrument (voltage supply and voltmeter) along with the circuit that is attached to them, in the space below and at the top of the next page. Use this schematic to explain differences between actual (measured) and theoretical values. input voltage, V i (volts) output voltage, V o (volts) current (mA) actual value from multimeter 9.99 1.165 .999 simulated value from Multisim Figure 2.11: Voltage Divider Circuit – + i R 5 R 4

Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li

Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li Exercise 5 Construct the current divider circuit shown below using resistors R 1 , R 2 , and R 3 listed in exercise 1. Set the source voltage to 6 V DC. Figure 2.12: Current Divider Circuit Complete the table below by calculating and measuring the appropriate values. In your calculations, use the actual (measured) values for R 1 , R 2 , and R 3 . i 1 (mA) i 2 (mA) i 3 (mA) actual value from multimeter 3.02 3.02 .0029 simulated value from Multisim – +

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Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li Exercise 6 Using the same circuit from Exercise 5 , use the function generator to supply an AC input voltage to the circuit. The AC voltage should be a sine wave with a frequency of 500 Hz and an amplitude (peak voltage) of 3 V. Recall that the conversion from frequency in Hertz (cycles per second), f , to frequency in radians per second, ω , is given by ω = 2 πf Therefore, the source voltage is V = 3sin ( ¿ ωt )= 3sin ( ¿ 2 πft )= 3sin ( ¿ 1000 πt ) ¿¿¿ Notice that there is no constant added, so the DC offset is equal to zero – make sure that you use this information in setting up the function generator (DC offset setting = 0). Complete the table below by measuring or calculating the appropriate values. In your calculations, use the actual (measured) values for R 1 , R 2 , and R 3 . Use RMS values for all table entries. You will do this activity twice for two different input impedance mode settings on the function generator, as indicated in the table below. Normally, the input impedance of a meter or the output impedance of a source can be neglected – and very little error will result. However, in some applications where the impedances of the sources and/or instruments are close to those of the circuit, significant errors will occur. i 1 (mA) i 2 (mA) i 3 (mA) actual value using 50Ω output impedance mode on the function generator 1.044 1.04 .0045 actual value using “high Z” (infinite) output impedance mode on the function generator .52 .52 .004 simulated value from Multisim

Portions of this lab were adapted from Laboratory Manual for Introduction to Mechatronics and Measurement Systems , by D. Alciatore – used with permission ME 140 L: Mechatronics Lab Dr. Wei Li 2.7 Post-Lab Questions 1) From the data collected in the lab, calculate the input impedance of the voltmeter. Z in (voltmeter) = _________________________ 2) Suppose that, as part of your Senior Design project, you need power a small motor with a 5V DC voltage, but you can only use a battery that outputs 9 V. Given that you have an assortment of resistors that range from 500 Ω to 1 kΩ, design and sketch the schematic of the circuit that you would use to provide the needed voltage to the motor. Use the space below – and show your work.

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