Download preview - SOL*R

PHYSICS COURSE NAME
Appendix 5
AVERAGING PERIODIC VOLTAGES
Figure 1. Periodic voltage applied to a resistor
AC AND DC COMPONENTS — V AC and V DC
If a resistor is connected to a periodic emf, the voltage across the resistor will be a periodic function of
time V(t). V(t) can be separated into two pieces called AC and DC components. The DC component is
constant, while the AC component oscillates about zero.
V t   VAC  VDC
Figure 2. AC and DC components to an electrical signal
AVERAGE VOLTAGE — V ave
To be precise, let us define VDC to be equal to the average value of V(t) over one cycle.
VDC  Vave
(Alternate notation for Vave is V .) Vave is calculated by integrating V(t) over one cycle and dividing by the
length of one cycle.
 V t dt

T
Vave
0
T
Creative Commons Attribution 3.0 Unported License
1
PHYSICS COURSE NAME
Appendix 5
ROOT MEAN SQUARE VOLTAGE — V rms
Also used for quantities which oscillate is a second kind of average, called the root-mean-square (Vrms).
Vrms represents the average of both the AC and the DC components together. First, the average value of
V2(t) is calculated for one cycle.
V
2


T
0
V 2 t dt
T
Then, the square root is taken
Vrms  V 
2

T
0
V 2 t dt
T
ROOT MEAN SQUARE OF AC COMPONENT — V ACrms
A third type of average is also important. This is the root-mean-square of VAC. This quantity describes
the size of oscillations of the waveform without the DC component. VAC is defined as whatever is left
after the DC component is removed.
VAC  V t   VDC
VAC has an average value of zero. However, a root-mean-square can be computed for VAC.
 V t   V  dt    V   V 
T
VACrms 
2
DC
0
T
2
rms
2
ave
POWER
An investigation of the power dissipated by a resistor will reveal one reason Vrms is important. Recall
that the power dissipated by a resistor is
V 2 t 
Pt  
R
The average power is then
P

T
0
V 2 t 
2
dt 
Vrms 
R

T
R
But this equation has the same form as the power dissipated when the voltage is constant.
Creative Commons Attribution 3.0 Unported License
2
PHYSICS COURSE NAME
Appendix 5
P
V2
R
Therefore, we must conclude that the rms voltage corresponds to an “effective” DC voltage. For
example, a periodic voltage of Vrms = 115 V will produce Joule heating in a resistor as the same rate as if
it were a constant 115 V. (As a result, alternate notation for Vrms is Veff.)
DC METERS
DC meters measure Vave. As a result, a pure AC waveform will read as zero on a DC meter.
AC METERS
AC meters usually read VACrms. Most digital multimeters and galvanometers give VACrms correctly only for
sinusoids. The determination is made indirectly as follows.
1.
2.
3.
4.
VDC is removed by capacitive coupling.
VAC is rectified (full-wave or half-wave).
A DC meter records the average value of the rectified waveform.
This average recitified value is scaled to Vrms of the unrectified waveform (which is the value
indicated on the meter).
Since the scaling factor used is for sinusoids, the meter will read incorrectly (or at best approximately)
for other waveforms.
Some meters, called “True rms” meters, measure the rms more directly (and therefore
correctly) for all waveforms. These include electrodynamometers (which respond to the square of the
current) and thermocouple meters (which respond to Joule heating). Quite often, however, these
meters still give VACrms as opposed to Vrms. The manual for a particular meter should be consulted to
determine exactly what is being measured.
Lab manual by John Wonghen & David Everitt, KPU.
Imported by T. Sato for the Remote Science Labs for Second Year Physics Project (2012 –
2013) funded by BCcampus.
Creative Commons Attribution 3.0 Unported License
3