3.1a Monitoring and measuring alternating current
Alternating current
An alternating current (a.c.) periodically changes direction. Contrast this to
direct current (d.c.), which only flows in one direction. Although it is possible
that current could vary in any way, we will consider a sinusoidal function, ie the
voltage constantly changes in the form of a sine function. This is the most
common form of a.c. since electricity generators work by spinning in circles,
which means that the current is pushed one way then the other, with the
instantaneous value of voltage constantly varying. Examples of each type of
current when displayed on an oscilloscope are shown below.
a.c. waveform
d.c. waveform
Measuring frequency and peak voltage
An oscilloscope plots a graph of changing voltage on the y-axis versus time on
the x-axis. Just like a graph we need to read the scale, except the numbers aren't
along the axis, they are on a dial underneath the screen. The voltage axis scale is
usually called the volts/div or volts/cm and the time axis scale is called the
timebase.
To calculate the frequency of an a.c. signal, we first of all have to find its period.
This is the time for one complete cycle of to and fro current, so we measure the
horizontal distance on the screen between crests.
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An alternating voltage varies between the same maximum negative and positive
value as the voltage pushes first one way (positive) then the other (negative).
This maximum is defined as the peak voltage. It can be measured from an
oscilloscope by measuring the vertical distance on the screen from the bottom of
the signal to the top. This then has to be halved to get the peak voltage.
Worked example
In the picture below each box on the CRO screen has a side of 1 cm.
Timebase = 5 ms cm–1
Volts/div = 2 V cm–1
The distance between crests is 4 cm.
The distance from bottom to top is 8 cm.
The time base was set at 5 ms cm –1 ,
then the period of the wave is:
T = 4 × 5 ms = 4 × 0.005 = 0.02 s
1
f 
T
1
f 
0  02
f = 50 Hz
The volts/div was set at 2 V cm –1 ,
then the peak voltage is:
V peak = ½ × 8 × 2
2
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V peak = 8 V
If the time base is switched off, the a.c. signal will not be spread along the xaxis. However, the voltage variation will continue, meaning a straight vertical
line will be displayed on the screen. The height of this line from bottom to top
can be processed in exactly the same way as above, ie halved then converted into
voltage using the volts/div.
Alternating current – peak and rms
Electricity is a method of transferring energy, so a.c. is capable of transferring
energy in the same way as d.c. The instantaneous amount of power being
supplied can be calculated using the value of voltage and/or current at a
particular point. However, in a.c. these values change constantly, completin g 50
cycles every second in the UK. Our eye couldn’t possibly follow such quick
changes. We need an average, but the average value of the voltage during any
complete cycle is zero!
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The amount of energy transmitted by a.c. will depend on some ‘average’ value of
the voltage, but as we have seen above the average value is zero! A different
‘average’ must be used. The ‘average’ value of an a.c. voltage is called the root
mean square (rms) voltage (V rms ).
The definition of the rms voltage is that it is that value of direct voltage that
produces the same power (eg heating or lighting) as the alternating voltage. The
rms value is what is quoted on a power supply so that a fair comparison between
a.c. and d.c. can be made, eg a 6 V battery will produce the same brightness of
light bulb as a 6 V rms a.c. supply. The rms current has a similar definition.
Consider the following two circuits, which contain identical lamps.
The variable resistors are altered until the lamps are of equal brightness. As a
result the d.c. has the same value as the effective a.c. (ie the lamps have the same
power output). Both voltages are measured using an oscilloscope, giving the
voltage equation below. Also, since V = IR applies to the rms valves and to the
peak values, a similar equation for currents can be deduced.
V rms = 1 V peak
2
and
Irms = 1 Ipeak
2
Note: A multimeter switched to a.c. mode will display rms values.
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Graphical method to derive the relationship between peak and rms values of
alternating current
The power produced by a current I in a resistor of resistance R is given by I 2 R. A
graph of I 2 against t for an alternating current is shown below. A similar method
can be used for voltage.
2
The average value of I is
is
2
I peak
2
2
I peak
2
and therefore the average power supplied
R.
An identical heating effect (power output) for a d.c. supply is I rms 2 R (since I rms is
defined as the value of d.c. current that will supply equivalent power).
Setting both of these equal to each other gi ves:
2
I peak
2
2
I peak
2
I peak
2
2
R  I rms
R
2
 I rms
 I rms
Important notes:
1.
2.
Readings on meters that measure a.c. are rms values, not peak values.
For power calculations involving a.c. always use rms values:
2
P  I rmsVrms  I rms
R
2
Vrms
R
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3.
The mains supply is usually quoted as 230 V a.c. This is of course 230 V
rms. The peak voltage rises to approximately 325 V. Insulation must be
provided to withstand this peak voltage.
Example
A transformer is labelled with a primary of 230 V rms and secondary of 12 V rms.
What is the peak voltage which would occur in the secondary?
V peak = 2 × V rms
V peak = 1.41 × 12
V peak = 17.0 V
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