## 1 Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 3.1 Laboratory Objective The objective of this laboratory is to understand the basic properties of sinusoids and sinusoid measurements. 3.2 Educational Objectives After performing this experiment, students should be able to: 1. Understand the properties of sinusoids. 2. Understand sinusoidal manipulation 3. Use a function generator 4. Obtain measurements using an oscilloscope 3.3 Background Sinusoids are sine or cosine waveforms that can describe many engineering phenomena. Any oscillatory motion can be described using sinusoids. Many types of electrical signals such as square, triangle, and sawtooth waves are modeled using sinusoids. Their manipulation incurs the understanding of certain quantities that describe sinusoidal behavior. These quantities are described below. 3.3.1 Sinusoid Characteristics Amplitude The amplitude A of a sine wave describes the height of the hills and valleys of a sinusoid. It carries the physical units of what the sinusoid is describing (volts, amps, meters, etc.). Frequency There are two types of frequencies that can describe a sinusoid. The normal frequency f is how many times the sinusoid repeats per unit time. It has units of cycles per second (s-1) or Hertz (Hz). The angular frequency ω is how many radians pass per second. Consequently, ω has units of radians per second. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 2 Period The period T is how long a sinusoid takes to repeat one complete cycle. The period is measured in seconds. Phase The phase φ of a sinusoid causes a horizontal shift along the t-axis. The phase has units of radians. TimeShift The time shift ts of a sinusoid is a horizontal shift along the t-axis and is a time measurement of the phase. The time shift has units of seconds. NOTE: A sine wave and a cosine wave only differ by a phase shift of 90° or ?2 radians. In reality, they are the same waveform but with a different φ value. 3.3.2 Sinusoidal Relationships Figure 3.1: Sinusoid The general equation of a sinusoid is given below and refers to Figure 3.1. ?(?) = ????(?? +?) (3.1) The angular frequency is related to the normal frequency by Equation 3.2. ?= 2?? (3.2) The angular frequency is also related to the period by Equation 3.3. ?=2?? (3.3) By inspection, the normal frequency is related to the period by Equation 3.4. ? =1? (3.4) ?? Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 3 The time shift is related to the phase (radians) and the frequency by Equation 3.5. ??= ∅2?? (3.5) 3.3.3 Equipment 3.3.3.1 Inductors Inductors are electrical components that resist a change in the flow of current passing through them. They are essentially coils of wire. Inductors are electromagnets too. They are represented in schematics using the following symbol and physically using the following equipment (with or without exposed wire): Figure 3.2: Symbol and Physical Example for Inductors 3.3.3.2 Capacitors Capacitors are electrical components that store energy. This enables engineers to store electrical energy from an input source such as a battery. Some capacitors are polarized and therefore have a negative and positive plate. One plate is straight, representing the positive terminal on the device, and the other is curved, representing the negative one. Polarized capacitors are represented in schematics using the following symbol and physically using the following equipment: Figure 3.3: Symbol and Physical Example for Capacitors 3.3.3.3 Function Generator A function generator is used to create different types of electrical waveforms over a wide range of frequencies. It generates standard sine, square, and triangle waveforms and uses the analog output channel. 3.3.3.5 Oscilloscope An oscilloscope is a type of electronic test instrument that allows observation of constantly varying voltages, usually as a two-dimensional plot of one or more signals as a function of time. It displays voltage data over time for the analysis of one or two voltage measurements taken from the analog input channels of the Oscilloscope. The observed waveform can be analyzed for amplitude, frequency, time interval and more. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 4 3.4 Procedure Follow the steps outlined below after the instructor has explained how to use the laboratory equipment 3.4.1 Sinusoidal Measurements 1. Connect the output channel of the Function Generator to the channel one of the Oscilloscope. 2. Complete Table 3.1 using the given values for voltage and frequency. Table 3.1: Sinusoid Measurements Function Generator Oscilloscope (Measured) Calculated Voltage Amplitude, A (V ) Frequency (Hz) 2*A (Vp−p ) f (Hz) T (sec) ω (rad/sec) T (sec) 2.5 1000 3 5000 3.4.2 Circuit Measurements 1. Connect the circuit in figure 3.4 below with the given resistor and capacitor NOTE: Vs from the circuit comes from the Function Generator using a BNC connector. Figure 3.4: RC Circuit Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 5 2. Using the alligator to BNC cables, connect channel one of the Oscilloscope across the capacitor and complete Table 3.2 Table 3.2: Capacitor Sinusoid Function Generator Oscilloscope (Measured) Calculated Vs (Volts) Frequency (Hz) Vc (volts) f (Hz) T (sec) ω (rad/sec) 2.5 100 3. Disconnect channel one and connect channel two of the oscilloscope across the resistor and complete table 3.3. Table 3.3: Resistor Sinusoid Function Generator Oscilloscope (Measured) Calculated Vs (Volts) Frequency (Hz) VR (volts) f (Hz) T (sec) ω (rad/sec) 2.5 100 4. Leaving channel two connected across the resistor, clip the positive lead to the positive side of the capacitor and complete table 3.4 Table 3.4: Phase Difference Function Generator Oscilloscope (Measured) Calculated Vs (volts) Frequency (Hz) Divisions Time/Div (sec) ts (sec) ɸ (rad) ɸ (degrees) 2.5 100 5. Using the data from Tables 3.2, 3.3, and 3.4, plot the capacitor sinusoidal equation and the resistor sinusoidal equation on the same graph using MATLAB. HINT: Plot over one period. 6. Kirchoff’s Voltage Law states that ??(?)=??(?)+??(?). Calculate Vs by hand using the following equation and Tables 3.2 and 3.3 ??(?)=√??2+??2???(??−???−1(????)) Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 6 3.5 New MATLAB Commands hold on This command allows multiple graphs to be placed on the same XY axis and is placed after the first plot statement. legend (’string 1’, ’string2’, ‘string3’) This command adds a legend to the plot. Strings must be placed in the order as the plots were generated. plot (x, y, ‘line specifiers’) This command plots the data and uses line specifiers to differentiate between different plots on the same XY axis. In this lab, only use different line styles from the table below. Table 3.5: Line specifiers for the plot() command sqrt(X) This command produces the square root of the elements of X. NOTE: The “help” command in MATLAB can be used to find a description and example for functions such as input. For example, type “help input” in the command window to learn more about the input function. NOTE: Refer to section the “MATLAB Commands” sections from prior labs for previously discussed material that you may also need in order to complete this assignment. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 7 3.6 Lab Report Requirements 1. Complete Tables 3.1, 3.2, 3.3, 3.4 (5 points each) 2. Show hand calculations for all four tables. Insert after this page (5 points each) 3. Draw the two sinusoids by hand from table 3.1. Label amplitude, period, and phase. Insert after this page. (5 points) 4. Insert MATLAB plot of Vc and VR as obtained from data in Tables 3.2 and 3.3 after this page. (5 points each) 5. Show hand calculations for Vs(t). Insert after this page. (5 points) 6. Using the data from the Tables, write: (10 points) a) Vc(t) = b) VR(t) = 7. Also, ???(?)=2.5???(628?). Write your Vs below and give reasons why they are different. (10 points) a) Vs(t) = b) Reasons: 8. Write an executive summary for this lab describing what you have done, and learned. (20 points)

info@checkyourstudy.com