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Eleg. 2111 Laboratory 8 Average and RMS Values of Periodic Waveforms Pre-Lab I. Average (or „DC‟) Value of a Waveform Periodic signals are signals that repeat after a fixed interval of time „ T ‟, known as the period of the signal. Thus, a periodic voltage signal, v t , may be described mathematically by the equation, vt vt T . This equation simply states that all waveform voltages separated in time by T seconds have the same value, which is the same as saying that the signal repeats itself with a period T . The average value of any time varying quantity F t , taken over a specified interval A t B , is defined by the equation: 1 B FAVG F t dt B AA 1 As an example, suppose F t represents a velocity (meters per second) in the „ x ‟ direction. This velocity varies with time, increasing, decreasing, and perhaps becoming negative at times, over the time interval A t B . The integral of the velocity over this interval is the „ x ‟ total displacement in meters that occurs during this time. The average velocity is defined to be a constant velocity, which, when multiplied by the time, will give that same total displacement. It can be seen that the definition above gives the required number, and can be used to find the average value of any1 desired function of time. In particular, this definition may be used to find the average value of a periodic voltage v t over one period, and this value will be the same for any period we choose. Though the average may be taken over any period, for definiteness we will choose the period 0 t T , and the average value of a periodic function is given by: 1T v AVG vt dt T0 2 This average value of a waveform is identically the same as the DC value associated with the waveform. It is the voltage that would be measured by a DC voltmeter connected to measure the voltage represented by this waveform. For example, a “DC reading” voltmeter will read zero if a sinusoidal wave applied to it, because a sinusoidal wave has zero average value. A function having a zero average value is called a “zero-mean function.” A zero-mean function can be formed from any periodic function by subtracting the periodic function‟s mean from the periodic function itself. 1 reasonable Eleg. 2111 Laboratory X Of course, the average of any periodic quantity, current, power, or any other, may be found using the definition of equation 2 . The student should note that the average value of the product of a constant and a zero-mean function is zero. An interesting result of this fact is that a sinusoidal current passing through a DC voltage source dissipates zero average power. Example #1: Find the average value of the periodic function Vp Cos t . The period of this function is T 2 , so, using equation 2 , 1T V p 2 v AVG V p Cos t dt T0 2 0 Cos t dt A change of variable to u t requires dt du , an integral lower limit of u , and an integral upper limit of u 2 , and 2 Vp Vp v AVG Cosu du Sin2 Sin 0 2 2 So Vp Cos t is a zero-mean function. II. Effective (or RMS) Value of a Waveform A non-zero periodic voltage will cause positive average power to be dissipated in a passive resistance „ R ,‟ even if its average value is zero. This is because the instantaneous power p t is given (for v t ) by the relation v 2 t pt 3 R which must always be greater than or equal to zero, so the average power must be positive. Eleg. 2111 Laboratory X Since the definition of equation 2 will work for any periodic quantity, we may apply it to find the average power as: 1T 1 T v 2 t 1 1 T 2 PAVG pt dt dt v t dt . 4 T0 T0 R R T 0 Next, let us define the “effective value” of v t as the value of DC voltage that, when applied to a resistor „ R ,‟ will deliver the same average power to that resistor as the voltage waveform v t , when v t is applied to that resistor. The “effective value” of v t can be found by finding an expression for the average power delivered to a resistor „ R ‟ by a DC voltage „ V E ‟ and setting that power equal to the average power delivered by v t , as given by equation 4 . Carrying this out, VE2 1 1 T 2 PAVG v t dt . 5 R R T 0 1 T 2 Equation 5 shows that V v t dt , or that the required effective 2 T 0 E value is: 1T 2 VE v t dt . T0 6 This effective value is most often called the RMS value because it is found by taking the Root of the Mean of the Square of the waveform v t . This idea is very general, and we may find the RMS value associated with many other periodic quantities: current, electric field, magnetic field, sound pressure, and many others. Finally let us consider finding the effective (or RMS) value of the sum of a zero- mean voltage and a DC voltage: vt v z t VDC 7 Eleg. 2111 Laboratory X Applying equation 6 , 1T T VE VRMS v z t VDC 2 dt 1 v z2 t 2v z t VDC VDC dt . 2 8 T0 T0 1T 2 1T 1T 2 VE v z t dt 2v z t VDC VDC dt . 9 T0 T0 T0 Noting that the first term under the radical is just the square of the RMS value of the zero-mean function, that the middle term is equal to zero (since it is the average of the product of a constant and a zero-mean function), and that the third term is just equal to 2 VDC , we find: VRMS VzRMS 2 2 VDC . 10 where, V zRMS has been written to represent the effective (or RMS) value of the zero-mean function. III. Some Examples: Example #2: Find the RMS value of a sinusoid given by Vp Cos t . Apply equation 6 and the trigonometric identity Cos 2 1 Cos2 yields 2 1T 2 V p2 1 T Vp Cos t dt 2 T 1 Cos2 t 2 dt 2 VRMS T0 0 Or V p2 1 T V p2 1 T VRMS 2 T0 dt 2 T Cos2 t 2 dt 0 . 2 The first term under the radical reduces to V p 2 and the second is zero because any sinusoid is a zero-mean function (you should satisfy yourself that this is true). Thus, for a sinusoid, Vp VRMS 11 2 Eleg. 2111 Laboratory X Example #3: Find the RMS value of the sum of a DC voltage of 6 volts and a sinusoid having the value 8 2Cos377 t 8 . The sinusoid is a zero-mean waveform and its RMS value 8 volts RMS. By equation 10 , then, the RMS voltage of the sum is: VRMS 82 6 2 10 Volts RMS . IV. Prelab calculations a) Find the RMS value of the sinusoid: vt 8Cos 6283 t 8 . b) Express this sinusoid as a phasor and find its period and frequency. c) What would the RMS value of this sinusoid be if the frequency were changed to 1000 rad/sed? To 10000 rad/sec? d) What would the RMS value of this sinusoid be if the phase were changed to 2? e) Suppose a new voltage waveform v s t 4 8Cos 6283 t 8 is formed by adding a DC voltage of 4 volts to v t . Find the average and RMS values of this new waveform. f) Suppose that the sinusoid of part „a‟ is half-wave rectified to produce a series of sinusoidal half-waves, each one-half period long, separated by half-wave intervals during which the voltage is zero. Compute the average value of this waveform. Compute the RMS value of this waveform. g) A triangle voltage waveform is described by the equation: vt t 10000 t on the interval 0.0005 s t 0.0005 s , and repeats with period T 0.001 s for all time „ t ‟. This is sometimes called a „sawtooth wave.‟ Find the average value of this waveform. Do a calculation to show that the RMS value of this waveform is VRMS 5 3. h) Suppose a new voltage waveform v st t vt t 4.1 is formed by adding a DC voltage of 4.1 volts to v t . Find the average and RMS values of this new waveform. Laboratory I. Average and RMS Values of Periodic Waveforms Required Parts and Equipment 1. One digital storage oscilloscope and probes 2. One true reading DC voltmeter 3. One true reading RMS voltmeter 2. One signal generator Required Information 1. User manual for the cathode ray oscilloscope. 2. User manual for the signal generator. 3. User manual for the true reading DC voltmeter. 4. User manual for the true reading RMS voltmeter. II. Laboratory procedure a) Set up the signal generator to produce the sinusoid: vt 8Cos 6283 t . b) Use the „measure‟ function of the oscilloscope to measure the RMS value of this sinusoid. c) Change the radian frequency to 628.3 rad/sec. Leave the amplitude the same. (Be careful. What is the corresponding frequency in hertz?) d) Use the oscilloscope to find the RMS value of the sinusoid of part „c‟. e) Change the radian frequency to 12566 rad/sec. Leave the amplitude the same. (Again, be sure to use the correct frequency in hertz?) f) Use the oscilloscope to find the RMS value of the sinusoid of part „e‟. g) Form a new voltage waveform v s t 4 8Cos 6283 t by adding a DC voltage of 4 volts to v t . Measure the RMS value of this new waveform using the „measure‟ function of the oscilloscope. h) Measure the voltage of part (g using a true RMS reading meter. What value is found? Is this reasonable and consistent? i) Measure the voltage of part (g with a true DC reading meter. What value is found? Is this reasonable? j) Use the „precision rectifier‟ provided by the lab instructor to half-wave rectify the sinusoid of part „a.‟ This should produce a series of sinusoidal half-waves, each one-half period long, separated by half-wave intervals during which the voltage is zero. Measure the DC value of this waveform, using a true reading DC voltmeter. Measure the RMS value of this waveform using the „measure‟ function of the oscilloscope. Then measure its RMS value again, using a true RMS reading meter. Record these values. k) Adjust the signal generator to produce a triangle-voltage waveform described (as nearly as possible) by the equation: vt t 10000 t on the interval 0.0005 s t 0.0005 s and repeats with period T 0.001 s . l) Use a true DC voltmeter to measure the DC value of this waveform. Then measure the RMS value of this waveform using the „measure‟ function of the oscilloscope. Finally, measure its RMS value again, using a true RMS reading meter. Record these values. m) Adjust the voltage generator to produce a new voltage waveform vst t vt t 4.1 volts by using the „DC offset‟ function of the generator. n) Use a true DC voltmeter to measure the DC value of this waveform. Then measure the RMS value of this waveform using the „measure‟ function of the oscilloscope. Finally, measure its RMS value again, using a true RMS reading meter. Record these values. III. Laboratory Report 1. Present your prelab calculations in a neat and readable form with comments. Type these calculations using Microsoft Equation 3.0 or an equivalent equation editor. 2. Present a table similar to the one shown below. For each waveform, fill in the measured and calculated values. Briefly discuss the results in light of the stated accuracy the measuring devices. Take special note of, and suggest explanations for, any major discrepancies. Measured Measured Calculated Average Measured Calculated RMS Value Waveform Average Value RMS Value RMS Value (True RMS Value (True DC (Oscilloscope) Meter) meter) Sinusoid Sinusoid with DC added Half-wave Rectified Sinusoid Triangle Wave Trangle Wave with DC