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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, vt   vt  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      vt 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             Cosu  du                Sin2     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 
                             pt                                                      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     pt 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:

                       vt   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  Cos2  yields
                                                                             2

                 1T 2                    V p2 1 T
                 Vp Cos  t   dt  2 T 1  Cos2 t  2 dt
                         2
        VRMS
                 T0                             0


   Or

                         V p2 1 T      V p2 1 T
                VRMS   
                          2 T0
                                 dt  2 T  Cos2 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 2Cos377 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: vt   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: vt   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

				
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