# Exercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION

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1 Exercise 1-1 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You will lso be fmilir with the different prmeters of sine wve, such s its period, frequency, mplitude, nd instntneous vlues. You will see how to express current nd voltge in c power circuits s sine wves. You will be introduced to the concept of rms (effective) vlues nd know how to clculte the rms vlue from the mplitude of voltge or current sine wve. The hnds-on exercises in this mnul require you to be fmilir with the computer-bsed instruments. Refer to the user guides titled Dt Acquisition nd Control System (86716-E0) nd Computer-Bsed Instruments for EMS (86718-E0) to become fmilir with the opertion nd use of the computerbsed instruments. DISCUSSION OUTLINE The Discussion of this exercise covers the following points: Reltionship between rotting phsor nd sine wve Period nd frequency of sinusoidl voltge or current Amplitude nd instntneous vlue of sinusoidl voltge or current Effective or root-men-squre (rms) vlue nd heting cpcity Effective (rms) vlue of sinusoidl voltge or current DISCUSSION Reltionship between rotting phsor nd sine wve A direct reltionship exists between circulr motion nd sine wve. Thus, sinusoidl voltge or current cn be grphed using the verticl-xis deflection s function of the ngulr position of rotting phsor. A phsor is vector hving its origin t the center (0, 0) of crtesin plne. Ech cycle of sinusoidl signl corresponds to one complete revolution (one turn) of the phsor round the circle, which is equl to 360 or 2 rdins. Figure 1-4 shows sine wve grphed using the verticl-xis deflection s function of the ngulr position of rotting phsor. Festo Didctic

2 Ex. 1-1 The Sine Wve Discussion (90 ) + Voltge V or current (A) 0 rd (180 ) 225 (270 ) rd (360 ) Angulr position ( ) One complete cycle (2 rdins or 360 ) Figure 1-4. Sine wve grphed using the verticl-xis deflection s function of the ngulr position of rotting phsor. Figure 1-5 shows n exmple in which phsor is rotting counterclockwise round the center of circle t constnt velocity. As the phsor is rotting, the verticl distnce between the tip of the phsor nd the horizontl xis of the circle continuously vries. A grphicl projection of the distnce vlue, on the XY-plne s function of the ngulr position gives sine wve. The following cn be observed in Figure 1-5: Initilly, the phsor is t 0 (horizontl position), so the verticl distnce between the phsor nd the horizontl xis is null. As the phsor is rotting from 0 to 90, the verticl distnce between the tip of the phsor nd the horizontl xis grdully increses from zero to mximum positive (pek) vlue. As the phsor is rotting from 90 to 180, the verticl distnce between the tip of the phsor nd the horizontl xis grdully decreses from the mximum positive vlue to zero. As the phsor is rotting from 180 to 270, the verticl distnce between the tip of the phsor nd the horizontl xis increses from zero to mximum negtive (pek) vlue. As the phsor is rotting from 270 to 360, the verticl distnce between the tip of the phsor nd the horizontl xis decreses from the mximum negtive vlue to zero. One complete revolution of the phsor (i.e., 360 ) corresponds to one complete cycle of the sine wve. 6 Festo Didctic

3 Ex. 1-1 The Sine Wve Discussion Period Figure 1-5. Reltionship between rotting phsor nd sine wve. Festo Didctic

4 Ex. 1-1 The Sine Wve Discussion Period nd frequency of sinusoidl voltge or current The time required for one complete cycle of sine wve to occur is clled the period. The number of cycles tht occur in one second is clled the frequency of the sine wve. Frequency is mesured in hertz (Hz), 1 Hz being equl to 1 cycle per second. The eqution used for clculting the frequency is: (1-1) where is the frequency of the sine wve, expressed in hertz (Hz) or cycles per second (c/s). is the period of the sine wve, expressed in seconds (s). The period is the reciprocl of the frequency: (1-2) In 60 Hz c power system, the cycle of the sinusoidl voltge nd current repets itself 60 times per second. Therefore, the period of one cycle is 1/60 s 16.7 ms. In 50 Hz c power system, the cycle of the sinusoidl voltge nd current repets itself 50 times per second. Therefore, the period of one cycle is 1/50 s 20 ms. Amplitude nd instntneous vlue of sinusoidl voltge or current The eqution of sine wve is: (1-3) where is the instntneous vlue of the sine wve t given instnt. is the mplitude of the sine wve. is the ngulr position of the sine wve t given instnt, expressed in degrees ( ). is the ngulr velocity, expressed in rdins per second (rd/s) of the rotting phsor.. is the time, expressed in seconds (s). The mplitude is the mximum vlue reched by the sine wve during its cycle. The mximum vlue is the sme for both the positive nd negtive hlfwves of the cycle, except for the polrity which chnges from positive (+) to negtive ( ). 8 Festo Didctic

5 Ex. 1-1 The Sine Wve Discussion The generl eqution of sine wve cn be used to describe sinusoidl voltge mthemticlly: (1-4) where is the instntneous vlue of the voltge t given instnt of the cycle, expressed in volts (V). is the mplitude of the sinusoidl voltge, expressed in volts (V). For exmple, if the mplitude (mximum voltge) of the voltge is 100 V, the instntneous vlue of the voltge t ngulr position is: Figure 1-6 shows the voltge sine wve grph resulting from Eqution (1-4). + Voltge (V) Angulr position (û) Figure 1-6. Voltge sine wve. Similrly, the generl eqution of sine wve cn be used to describe sinusoidl current mthemticlly: (1-5) where is the instntneous vlue of the current t given instnt of the cycle, expressed in mperes (A). is the mplitude of the sinusoidl current, expressed in mperes (A). Festo Didctic

6 Ex. 1-1 The Sine Wve Discussion Effective or root-men-squre (rms) vlue nd heting cpcity The effective (rms) vlue of n c voltge is relted to the het produced by resistive element when supplied with this voltge. For exmple, consider n c voltge hving n mplitude pplied cross lod resistor. The pplied voltge cuses current to flow through the resistor, which hs the effect of incresing the resistor s temperture until it stbilizes to certin level (equilibrium temperture). The rms or effective vlue of this c voltge cn be found by pplying dc voltge cross the lod resistor, nd djusting the dc voltge until the resistor stbilizes to the equilibrium temperture previously reched with the c voltge. The vlue of this dc voltge corresponds to the rms or effective vlue of the c voltge. Effective (rms) vlue of sinusoidl voltge or current The effective (rms) vlue of sinusoidl voltge is clculted s follows: (1-6) Similrly, the effective (rms) vlue of sinusoidl current is clculted s follows: (1-7) For exmple, the effective (rms) vlue of sinusoidl voltge hving n mplitude of 100 V is 70.7 V ( V). This mens tht sine-wve voltge hving n mplitude of 100 V hs the sme heting cpcity s dc voltge of 70.7 V. Figure 1-7 shows the rms vlue of voltge sine wve in comprison to its mplitude vlue. 10 Festo Didctic

7 Ex. 1-1 The Sine Wve Procedure Outline + Voltge (V) Time (s) Figure 1-7. Effective, or root-men-squre (rms), vlue of sinusoidl voltge. Stndrd instruments used to mesure c voltge nd current normlly indicte the effective (rms) vlue directly on their disply. The rms vlues of voltge nd current re used to clculte electricl power in c power circuits. The resoning behind rms vlues will be explined in more detil in Exercise 1-3. PROCEDURE OUTLINE The Procedure is divided into the following sections: Setup nd connections Mesuring voltge, current, nd frequency in n c circuit Reltionship between frequency nd period Mesuring voltge, current, nd frequency in series c circuit PROCEDURE High voltges re present in this lbortory exercise. Do not mke or modify ny bnn jck connections with the power on unless otherwise specified. Setup nd connections In this section, you will connect n c circuit contining resistor nd set up the equipment to mesure the circuit current nd voltge using the LVDAC-EMS softwre. You will lso determine the reltionship between the frequency of sine wve nd its period. 1. Refer to the Equipment Utiliztion Chrt in Appendix A to obtin the list of equipment required to perform this exercise. Instll the required equipment in the Worksttion. Festo Didctic

8 Ex. 1-1 The Sine Wve Procedure 2. Mke sure tht the min power switch on the Four-Qudrnt Dynmometer/ Power Supply is set to the O (off) position, then connect its Power Input to n c power outlet. Connect the Power Input of the Dt Acquisition nd Control Interfce to 24 V c power supply. Turn the 24 V c power supply on. 3. Connect the USB port of the Dt Acquisition nd Control Interfce to USB port of the host computer. Connect the USB port of the Four-Qudrnt Dynmometer/Power Supply to USB port of the host computer. 4. Turn the Four-Qudrnt Dynmometer/Power Supply on, then set the Operting Mode switch to Power Supply. This setting llows the Four- Qudrnt Dynmometer/Power Supply to operte s power supply. 5. Turn the host computer on, then strt the LVDAC-EMS softwre. In the LVDAC-EMS Strt-Up window, mke sure tht the Dt Acquisition nd Control Interfce nd the Four-Qudrnt Dynmometer/Power Supply re detected. Mke sure tht the Computer-Bsed Instrumenttion function for the Dt Acquisition nd Control Interfce is vilble. Select the network voltge nd frequency tht correspond to the voltge nd frequency of your locl c power network, then click the OK button to close the LVDAC-EMS Strt-Up window. 6. Set up the circuit shown in Figure 1-8. Note the symbol used to indicte vrible-voltge c source in this circuit. In order to obtin the resistnce vlue indicted next to the resistor in Figure 1-8, mke the necessry switch settings on the Resistive Lod. Throughout this mnul, keep in mind tht the resistnce vlues shown in the figures re equivlent resistnces. To obtin given resistnce vlue with the Resistive Lod, you my need to connect two or more resistors of resistor section in prllel s well s to interconnect resistor sections in prllel. Appendix C lists the switch settings to implement on the Resistive Lod in order to obtin vrious resistnce vlues. Use inputs I1 nd E1 of the Dt Acquisition nd Control Interfce to mesure the source current, nd the voltge cross the resistor. 12 Festo Didctic

9 Ex. 1-1 The Sine Wve Procedure 100 V 300 Figure 1-8. AC circuit contining resistor. 7. In LVDAC-EMS, open the Four-Qudrnt Dynmometer/Power Supply window, then mke the following settings: Set the Function prmeter to AC Power Source. This setting mkes the internl power source operte s n c power source (i.e., source producing sinusoidl voltge). Mke sure tht the Voltge Control prmeter is set to Knob. This llows the c power source to be controlled mnully. Set the No Lod Voltge prmeter to 100 V. This sets the no-lod effective (rms) voltge of the c power source to 100 V. Set the Frequency prmeter to the frequency of your locl c power network. This mkes the frequency of the c power source equl to the frequency of your locl c power network. Leve the other prmeters set s they re. Notice tht the Sttus prmeter is set to Stopped. This indictes tht the c power source is disbled (i.e., it produces no output voltge). The c power source will be enbled in the next section of the procedure. Festo Didctic

10 Ex. 1-1 The Sine Wve Procedure Mesuring voltge, current, nd frequency in n c circuit In this section, you will mesure the rms vlues of the voltge nd current wveforms in the c circuit. You will then mesure the mplitude of these wveforms nd clculte the corresponding rms vlues. You will compre the mesured nd clculted rms vlues. Finlly, you will compre the shpes of the voltge nd current wveforms to confirm tht they rech their mximum nd null vlues t the sme time. 8. In the Four-Qudrnt Dynmometer/Power Supply window, enble the c power source (i.e., set the Sttus prmeter to Strted or click on the Strt/Stop button). The c power source now delivers power to the lod,. In this mnul, for brevity purposes, the rms vlue is implied in the denomintion of most prmeters tht re commonly mesured s rms vlues (e.g.,,,, men,, ). A complementry suffix, such s rms or mx, is dded only when extr clrity nd precision re necessry, such s in equtions, for exmple. 9. In LVDAC-EMS, open the Metering window. Set meters E1 nd I1 to mesure the rms vlues of the resistor voltge nd source current. When doing mesurements using either the Metering window, Oscilloscope, or Phsor Anlyzer, lwys select the continuous refresh mode. This enbles updted dt to be seen on screen t ll times. In the Four-Qudrnt Dynmometer/Power Supply window, redjust the vlue of the No Lod Voltge prmeter so tht the c power source voltge (indicted by meter E1 in the Metering window) is equl to 100 V. Record below the mesured vlues of the resistor voltge nd source current (meters E1 nd I1, respectively). Resistor voltge V Source current A 10. In LVDAC-EMS, open the Oscilloscope nd disply (input E1) nd (input I1) on chnnels 1 nd 2, respectively. If necessry, set the time bse to disply t lest two cycles of the sine wves. In the Oscilloscope settings, you cn enble the filtering option to improve the regulrity of the displyed wveforms. 14 Festo Didctic

11 Ex. 1-1 The Sine Wve Procedure 11. On the Oscilloscope, select the continuous refresh mode. Chnge the verticl scles so s to be ble to mesure the mplitude of the resistor voltge nd source current. Record the mesured vlues below. The rms vlue, verge vlue, nd frequency of the signls pplied to the input chnnels of the Oscilloscope re displyed below the Oscilloscope screen. To obtin n ccurte mesurement, you cn use the horizontl cursors of the Oscilloscope to mesure the mplitude of the voltge nd current. Resistor voltge V Source current A 12. Clculte the rms vlues of nd from the voltge nd current mplitudes mesured in the previous step. Resistor voltge V Source current A 13. Compre the rms vlues obtined in the previous step with the rms vlues indicted by the meters (s recorded in step 9). Are they pproximtely the sme? Yes No 14. Compre the source current wveform with the resistive lod voltge wveform. Are they both sine wves? Yes No 15. Wht is the instntneous vlue of the resistor voltge t the ngulr position 225û? Resistor voltge V 16. Do the wveforms of the resistor voltge nd source current rech the mximum positive vlue, null vlues, nd mximum negtive vlue t the sme time? Yes No When wveforms rech the mximum nd null vlues t the sme time, the wveforms re sid to be in phse, which mens there is no phse shift between them. This is covered in more detil in Exercise 1-2. Festo Didctic

12 Ex. 1-1 The Sine Wve Procedure Reltionship between frequency nd period In this section, you will determine the reltionship between the frequency nd the period of sine wve. You will do so by setting the frequency of the c power source to vrious vlues nd mesuring ech time the period of the voltge wveform. You will then clculte the frequency from ech mesured period nd compre it with the c power source frequency. 17. In the Four-Qudrnt Dynmometer/Power Supply window, set the Frequency prmeter to 40 Hz. This chnges the frequency of the c power source to 40 Hz. 18. Using the Oscilloscope, mesure the period of the wveform of the resistor voltge. Record the period vlue in the corresponding cell of Tble 1-1 below. To obtin n ccurte mesurement, use the verticl cursors of the Oscilloscope to mesure the time intervl between two points on wveform. Tble 1-1. Reltionship between the frequency nd period. Source frequency (Hz) Period (ms) Clculted source frequency (Hz) 19. Repet steps 17 nd 18 for ech c power source frequency indicted in Tble 1-1. Record your results in the corresponding cells. 20. Bsed on the mesured period vlues in Tble 1-1, clculte the resulting frequencies. Record your clculted source frequency vlues in the corresponding row of Tble Compre the c power source frequency vlues to your clculted source frequency vlues. Are they the sme? Yes No 22. Does this confirm the reltionship ( ) between the frequency nd period? Yes No 16 Festo Didctic

13 Ex. 1-1 The Sine Wve Procedure Mesuring voltge, current, nd frequency in series c circuit In this section, you will clculte the voltge nd current rms vlues in circuit contining two resistors in series. You will confirm tht Ohm s lw pplies to c power circuits by mesuring the voltge nd current rms vlues nd compring them with those you clculted. You will lso confirm Kirchhoff s lw by verifying tht. Finlly, you will verify tht both the current nd voltge wveforms re in phse with ech other. 23. In the Four-Qudrnt Dynmometer/Power Supply window, disble the c power source (i.e., set the Sttus prmeter to Stopped or click on the Strt/Stop button). 24. Set up the circuit shown in Figure 1-9. To obtin the resistor rrngement of Figure 1-9, mke the necessry connections nd switch settings on the Resistive Lod (refer to Appendix C if necessry). Use inputs I1, E1, E2, nd E3 of the Dt Acquisition nd Control Interfce to mesure the source current, voltge, voltge, nd source voltge respectively V 300 Figure 1-9. AC circuit contining two resistors connected in series. 25. Clculte the equivlent resistnce of the circuit resistors. Using Ohm s lw, you will then be ble to clculte the rms source current. Equivlent resistnce Source current A 26. Clculte the resistor voltges nd cross ech resistor by using the source current vlue obtined in the previous step. Resistor voltge = V Resistor voltge = V Festo Didctic

14 Ex. 1-1 The Sine Wve Procedure 27. In the Four-Qudrnt Dynmometer/Power Supply window, set the Frequency prmeter bck to the frequency of your locl c power network, then enble the c power source. In the Four-Qudrnt Dynmometer/Power Supply window, redjust the vlue of the No Lod Voltge prmeter so tht the c power source voltge (indicted by meter E3 in the Metering window) is equl to 100 V. 28. In the Metering window, set meter E2 to mesure the rms vlue of the resistor voltge. Record voltges nd nd the source current. Resistor voltge V Resistor voltge V Source current A 29. Compre voltges nd mesured in the previous step with the voltges clculted in step 26. Are they pproximtely the sme? Yes No Is the sum of the voltges cross the resistors ( equl to the source voltge? Yes No 30. Compre the source current mesured in step 28 with the source current clculted in step 25. Are they pproximtely the sme? Yes No 31. Clculte nd compre the rtio of the voltges mesured cross the resistors with the rtio of the resistor vlues. Are they pproximtely the sme? Yes No 32. From the results obtined in the previous steps, cn c circuits be solved by using the sme fundmentl rules s those used with dc circuits, which re Ohm s lw, Kirchhoff s voltge nd current lws, nd the formuls used to clculte the equivlent resistnce? Yes No 18 Festo Didctic

15 Ex. 1-1 The Sine Wve Conclusion 33. On the Oscilloscope, disply the voltge (input E1), voltge (input E2), nd source current (input I1) on chnnels 1, 2, nd 3, respectively. Mke sure the time bse is set so s to disply s lest two cycles of the sine wves. 34. Compre the wveforms displyed on the Oscilloscope. Do the source current wveform nd resistor voltge wveforms rech the mximum positive vlue, null vlues, nd mximum negtive vlue t the sme time, indicting tht they re in phse? Yes No 35. In the Four-Qudrnt Dynmometer/Power Supply window, disble the c power source. 36. Close LVDAC-EMS, then turn off ll the equipment. Disconnect ll leds nd return them to their storge loction. CONCLUSION In this exercise, you were introduced to the sine wve nd the wy to express sine wve using phsor rotting round the center of circle. You lerned how to clculte the different prmeters of sine wve, such s the period, the frequency, the mplitude, nd the instntneous vlue. You were introduced to the concept of rms vlue nd lerned how to clculte the rms vlue from the mplitude of sine wve. REVIEW QUESTIONS 1. A voltge sine wve hs n mplitude of 200 V. Clculte its rms vlue. 2. The period of sine wve is 0.02 s. Clculte its frequency. 3. Wht is the effect on the cycle of sine wve of incresing the frequency? Festo Didctic

16 Ex. 1-1 The Sine Wve Review Questions 4. Which ngulr position (in degrees s well s in rdins) on circle corresponds to the positive mximum vlue of voltge sine wve whose cycle begins t the origin? 5. Wht is the difference between the mplitude nd rms vlue of voltge sine wve? 20 Festo Didctic

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