SINUSOIDS February 4, ELEC-281 Network Theory II Wentworth Institute of Technology. Bradford Powers Ryan Ferguson Richard Lupa Benjamin Wolf

Transcription

1 SINUSOIDS February 4, 28 ELEC-281 Network Theory II Wentworth Institute of Technology Bradford Powers Ryan Ferguson Richard Lupa Benjamin Wolf

2 Abstract: Sinusoidal waveforms are studied in three circuits: a simple resistive circuit, a resistor-capacitor (RC) circuit and a resistor-inductor (RL) circuit. The mathematical wave equation is defined and applied to each wave. A current waveform is derived from the voltage waveform using Ohm s law. The phase angle between two waveforms is described. 2

3 Objectives: The objectives of this laboratory experiment were to examine sinusoidal voltage waveforms using the oscilloscope. Measuring current waveforms with the oscilloscope were also studied. Last, the phase relationship between voltages and currents in RL and RC circuits was covered. Discussion of Theory: A sinusoid is a wave that has the form of a sine or cosine function. A sinusoid can be represented mathematically by Equation 1. V( t) VM Sin( t ) (1.) In Equation 1, V M is the amplitude, ω is the angular velocity, t is time, and φ is the phase angle. The period T and frequency f can be found by the relationship in Equation 2. T 2 1 f (2.) Equipment: Oscilloscope DMM Function Generator Bread Board One 1KΩ resistor One 1mH inductor One 1 μf Capacitor 3

4 Procedure: Part A: To begin the first of three parts of laboratory experiment 1, we assembled the circuit shown in figure 1 of the laboratory handout. The function generator was set to an amplitude of 1V and 1Hz. Channel 1 of the oscilloscope was used to observe the voltage across the resistor. From this measurement, we were able to calculate the frequency and the angular frequency. Part B: Part B of laboratory exercise 1 was begun by assembling the circuit shown in figure 2 of the laboratory handout. Channel 1 on the oscilloscope was used to measure the voltage across the resistor. Channel 2 was used to measure the voltage across the capacitor. Part C: To begin the third part of laboratory experiment 1, we performed the same activities in part b. However, in figure 2, a capacitor is used in series with the resistor. For part B, the capacitor was eliminated and replaced with an inductor of 1mH as is shown in figure 3 of the laboratory handout. Resistive circuit: A 1k Ω resister was placed in series with the signal generator, figure 1. Figure 1 ~ Resistive Circuit 4

5 Volts The signal generator was set to at an amplitude of 1 volt and a frequency of 1 Hz. The oscilloscope was set to measure the voltage across the resistor, Figure 2. Voltage across resister Seconds Figure 2 ~ Voltage Waveform The amplitude was found to be.5 volts, the period was 1 ms, the frequency was 1Hz, and the angular frequency was 2π. The equation of the sinusoid is: V t.5sin2 t The current through the resistor was calculated in excel from the equation current was then plotted verses time, Figure 3. The equation of this sinusoid is: t.1sin2t i V I. The R 5

6 Current Current through the resister Seconds Figure 3 ~ Current Waveform The second procedure (Figure 4) was setup the same way as Figure 1 by utilizing the function generator with the addition of a capacitor in series with the resistor. Then BNC cables were attached to the capacitor and resistor to measure the voltage across the two components and to have a sketch of the voltage waveform. When constructing the circuit as shown in Figure 4, the voltage waveforms didn t show up as they were supposed to on the oscilloscope due to the capacitor being short circuited. Therefore the resistor and the capacitor had to be measured separately in order to get the correct voltage waveform across the circuit. When the components were measured separately the voltage waveforms were sketched on the oscilloscope. Figure 5 is the representation of the resistor voltage waveform having an amplitude of.5 V, a period of 1 ms, a frequency of 1 Hz, and an angular frequency of 2π. Figure 5 is the representation of the capacitor voltage 6

7 Volts waveform having an amplitude of 1V, a period of 1 ms, a frequency of 1 Hz, and an angular frequency of 2π. Figure 4 ~ RC Circuit RC-Circuit Voltage across the resistor Seconds Figure 5 ~ Resistor voltage waveform in the RC Circuit 7

8 Volts RC- Circuit Voltage across the capactior Seconds Figure 6 ~ Capacitor voltage waveform in the RC - Circuit R-L circuit: In the next circuit the voltage was measured across a 1k resister in series with a 1mH inductor, Figure 7. Figure 7 ~ RL Circuit The voltage across the inductor and the resister could not simultaneously. The oscilloscope would have grounded out the resistor in the circuit. The wave form across the entire circuit is shown in figure 8. 8

9 Volts Voltage across RL circuit Seconds Figure 8 ~ Voltage Waveform Results: In the resistive circuit, the amplitude was measured at.5v and the period was measured at 1 ms. Using the wave equation for a sinusoid, the frequency was calculated to be 1 Hz the angular velocity was calculated to be approximately 628 radians per second. The equation for this waveform is shown below. v( t).5sin(2 t) (V) (3.) The plot of the period of the waveform is shown below. 9

10 voltage (V).5 Resistive Circuit - v(t) =.5*sin(2*pi*t) time (ms) Figure 9 ~ Simulated voltage waveform in resistive circuit Using the Ohm s law, the current waveform was found to be i( t) 1 Sin(2 t) (A) (4.) 1 In the resistor-capacitor circuit proved to be impossible to measure directly. Using a Multisim simulation, the electrical circuit was simulated outside of the laboratory. In this simulation, two waveforms were analyzed, V RC over the entire circuit and V C over the capacitor. The period of V RC was measured at 9.9 ms and the amplitude was.5 V. The period of V C was measured at 1.8 ms and the amplitude was.8 V. The waveform equations and plots of both are shown below. V RC V C ( t).5sin(22 t) (V) (5.A) ( t).8sin(185 t) (V) (5.B) 1

11 voltage (V) RC Circuit Vrc Vc time (ms) Figure 1 ~ Two voltage waveform From Figure 1 it is apparent that the voltage waveform over the capacitor leads the voltage waveform over the circuit. For the resistor-inductor circuit, another Multisim simulation was run. The period and amplitude measured over the whole circuit was found to be the same as in the RC circuit, approximately 1 ms and 5 mv. The period of the waveform over the inductor was measured to be 7.4 ms and the amplitude was 627 µv. This extremely small amplitude made the signal elusive in the laboratory. The angular velocity of the wave was calculated to be 27 radians per second. From these results the wave equations were as follows: V RL.5Sin(2 t) (V) (6.A) V L 627Sin(27 t) (μv) (6.B) 11

12 Conclusions: In conclusion, the oscilloscope was used successfully to study simple sinusoid voltages in resistive, resistor-capacitor, and resistor-inductor circuits. It was also shown that Ohm s law can be used to obtain current waveform from a measured voltage waveform. Some problems emerged when attempting to measure the phase difference between two waveforms. Due to the electrical properties of the oscilloscopes ground terminal, it proved to be impossible to measure both the waveform over the resistor and the waveform over the capacitor or inductor simultaneously. A clear solution to this problem would be to measure the voltage waveform over the entire circuit along with the voltage waveform over one of the elements. Applying Kirchhoff s Voltage Law, the following relationship can be established between the three waveforms in the circuit: V RC V V (7.) R C Using this relationship, the waveforms over each element could be found and the phase between them could be calculated. 12