# measuring rms values of voltage and current

```AN101
Dataforth Corporation
Page 1 of 6
DID YOU KNOW ?
Gustav Robert Kirchhoff (1824-1887) the German Physicist who gave us "Kirchhoff's voltage (current) law"
invented the Bunsen Burner working together with Robert Wilhelm Bunsen, a German Chemist.
Measuring RMS Values of Voltage and Current
Voltage (Current) Measurements
Effective Value
Standard classic measurements of voltage (current)
values are based on two fundamental techniques
either "average" or "effective".
The "effective" value of symmetrical periodic
voltage (current) functions of time is based on the
concept of "heating capability". Consider the test
fixture shown in Figure 1.
The "average" value of a function of time is the net
area of the function calculated over a specific
interval of time divided by that time interval.
Specifically,
F 1 IJ *
Vavg = G
H T2 − T1K
z
T2
T1
V( t )dt
Eqn 1
If a voltage (current) is either constant or periodic,
then measuring its average is independent of the
interval over which a measurement is made. If, on
the other hand, the voltage (current) function grows
without bound over time, the average value is
dependent on the measurement interval and will not
necessarily be constant, i.e. no average value exists.
Fortunately in the practical electrical world values
of voltage (current) do not grow in a boundless
manner and, therefore, have well behaved averages.
This is a result of the fact that real voltage (current)
sources are generally either; (1) batteries with
constant or slowly (exponentially) decaying values,
(2) bounded sinusoidal functions of time, or (3)
combinations of the above. Constant amplitude
sinusoidal functions have a net zero average over
time intervals, which are equal to integer multiples
of the sinusoidal period. Moreover, averages can be
calculated over an infinite number of intervals,
which are not equal to the sinusoidal period. These
averages are also zero. Although the average of a
bounded sinusoidal function is zero, the "effective"
value is not zero. For example, electric hot water
heaters work very well on sinusoidal voltages, with
zero average values.
Vessel
Voltage
Source
Vx
Equilibrium
Temperature
Tx
Figure 1
Test Fixture
This vessel is insulated and filled with some stable
liquid (transformer oil for example) capable of
reaching thermodynamic equilibrium. If a DC
voltage Vx is applied to the vessel's internal heater,
the liquid temperature will rise. Eventually, the
electrical energy applied to this vessel will establish
an equilibrium condition where energy input equals
energy (heat) lost and the vessel liquid will arrive at
an equilibrium temperature, Tx degrees.
Next in this experimental scenario, replace the DC
voltage source Vx with a time varying voltage which
does not increase without bound. Eventually, in
some time Tfinal , thermal equilibrium will again be
established. If this equilibrium condition establishes
the same temperature Tx as reached before with the
applied DC voltage Vx, then one can say that the
"effective" value of this time varying function is Vx.
Hence the definition of "effective value".
AN101
Dataforth Corporation
Equation 2 illustrates this thermal equilibrium.
z
Tfinal
(( VEffective ) / R ) * Tfinal =
2
( V ( t ) 2 / R ) dt
Eqn 2
0
If V(t) is a periodic function of time with a cycle
period of Tp, and Tfinal is an integer "n" times the
period (n*Tp) then the integral over Tfinal is simply
n times the integral over Tp. The results of these
substitutions are shown in Equation 3.
z
Tp
VEffective = (1/ Tp)* V(t)2 dt ,
RMS
Eqn 3
0
Equation 3 illustrates that the effective equivalent
heating capacity of a bounded periodic voltage
(current) function can be determined over just one
cycle. This equation is recognized as the old familiar
form of "square Root of the Mean (average)
Squared"; hence, the name, "RMS".
Examples of Using the "RMS" Equation
The following results can be shown by direct
application of Eqn 3.
1. Sinusoidal function, peak of Vp
VRMS = Vp ÷ 2 ; Vp*0.707
2. Symmetrical Periodic Pulse Wave, peak of Vp
VRMS = Vp (Symmetric Square Wave)
3. Non-symmetrical Periodic Pulse Wave, all
positive peaks of Vp, with duty cycle D
Page 2 of 6
Note: These examples illustrate that the shape of a
periodic function can determine its RMS value. The
peak (crest) of a voltage (current) function of time
divided by 2 is often mistakenly used to calculate
the RMS value. This technique can result in errors
and clearly should be avoided.
Effective (RMS) Values of Complex Functions
An extremely useful fact in determining RMS values
is that any well behaved bounded periodic function
of time can be expressed as an average value plus a
sum of sinusoids (Fourier's Theorem), for example;
V(t) = Ao + ∑ [ An*Cos(nωot) +Bn*Sin(nωot) ]
Summed over all "n" values
Eqn 4
Where ωo is the radian frequency of V(t) and An,
Bn, Ao are Fourier Amplitude Coefficients.
When this series is substituted in the integral
expression Equation 2 for RMS, one obtains the
following;
Vrms =
{ ∑ [(A 0 ) 2 + (A n ) 2 / 2 + ( B n ) 2 / 2]}
Summed over all "n" values
Eqn 5
Note: (A n ) 2 / 2 and ( Bn ) 2 / 2 are the squares of
RMS values for each nth Sin and Cosine component.
The important conclusion is;
A bounded periodic function of time has a RMS
value equal to the square root of the sum of the
square of each individual component's RMS value.
VRMS = Vp * D
D ≡ Td/Tp, Pulse duration Td ÷ Period Tp
4. Symmetrical Periodic Triangle Wave, peak Vp
VRMS = Vp ÷ 3 ; Vp * 0.5774 (Saw-Tooth)
5. Full wave Rectified Sinusoid, peak Vp
VRMS = Vp ÷ 2 ; Vp * 0.707
6. Half Wave Rectified Sinusoid, peak Vp
VRMS = Vp ÷2; Vp * 0.5
Practical Considerations
Figure 2 illustrates composite curves formed by
adding two sinusoids, one at 60 Hz and one at
180Hz. Curve 1 is for zero phase difference and
Curve 2 is for a 90-degree phase difference.
Specifically;
Curve 1
Curve 2
V(t) = 170*Sin(377*t) +50*Sin(1131*t)
V(t) = 170*Sin(377*t) +50*Cos(1131*t)
Note: Composite curve shape is determined by
phase and frequency harmonics.
AN101
Dataforth Corporation
200
100.0
0
-100.0
(2) 170*Sin 60Hz +50*Cos 180Hz Volts
(1) 170*Sin 60Hz +50*Sin 180Hz Volts
200
-200
1
100.0
2
0
-100.0
-200
17.0M
21.0M
25.0M
29.0M
33.0M
TIME in Secs
Figure 2
Curve 2 170*Sin(377*t) +50*Cos(1131*t)
Curve 1 170*Sin(377*t) +50*Sin(1131*t)
Industrial sinusoidal functions of voltage (current)
often contain harmonics that impact wave shape and
peak (crest) values. For example, Curve 2 is typical
of the magnetizing currents in 60 Hz transformers
and motors. Inexpensive RMS reading devices often
use a rectifier circuits that capture the peak value,
which is then scaled by 0.707 and displayed as
RMS. Clearly this technique can give incorrect RMS
readings. In this example, using Vpeak ÷ 2
clearly gives incorrect values.
Curve 1: 203*0.707 = 144 volts, not true RMS
Curve 2: 155*0.707 = 110 volts, not true RMS
The correct RMS value for both of these composite
sinusoidal functions is;
2
2
1/2
[ (170) /2 + (50) /2 ]
= 125.3 volts RMS
Table 1 illustrates two examples of RMS
calculations by using individual Fourier coefficients
and Eqn 5. Example one is a full wave rectified 1volt peak sinusoid. Note that for a full wave rectified
function the measurement device needed to achieve
a RMS reading within 0.01% error requires a
bandwidth, which includes the fifth (5) harmonic
and the resolution to read 10 mV levels.
The other example illustrated in Table 1 is a sawtooth 1-volt peak function. For this example, the
Page 3 of 6
measurement device for a saw-tooth function needed
to achieve an RMS reading within 0.3% error
requires a bandwidth, which includes the twentyfifth (25) harmonic and the resolution to read 10 mV
levels.
Assume, for illustration purposes, that an AC ripple
on the DC output of a rectifier can be approximated
by a saw-tooth function. Table 1 illustrates that to
measure within a 0.3% error the AC RMS ripple on
the DC output of a 20 kHz rectifier the measurement
device must have a bandwidth in excess of 500 kHz
and a resolution to read voltage levels down by 40
dB (100 microvolts for a peak 10 mV ripple). This
example clearly illustrates that signal shape, together
with the measurement bandwidth and resolution are
extremely important in determining the accuracy of
measuring true RMS.
Any "true RMS" measurement device must be
capable of accurately implement Eqn 3. The subtlety
in this statement is that electronically implementing
Eqn 3, requires a device to have a very large
bandwidth and be able to resolve small magnitudes.
Crest Factor
Another figure of merit often used to characterize a
periodic time function of voltage (current) is the
Crest Factor (CF). The Crest Factor for a specific
waveform is defined as the peak value divided by
the RMS value. Specifically,
CF = Vpeak / VRMS
Eqn 6
Examples: (from page 2)
1. Pure Sinusoid, CF =
2
2. Symmetrical Periodic Pulses, CF = 1
3. Non-symmetrical Periodic Pulses with duty
cycle D, CF = 1 ÷ D
Example; If D = 5%, CF = 4.47
4. Symmetrical Periodic Triangle, CF = 3
5. Full wave Rectified Sinusoid, CF = 2
6. Half Wave Rectified Sinusoid, CF = 2
From Figure 2;
Curve 1, CF = 1.62
Curve 2,
CF = 1.24
AN101
Dataforth Corporation
Page 4 of 6
DATAFORTH RMS MEASUREMENT DEVICES
True RMS measurements require instrumentation devices that accurately implement Eqn 3, "the" RMS
equation. These devices must have both wide bandwidths and good low level resolution to support high Crest
Factors. Dataforth has developed two products that satisfy these requirements; the SCM5B33 and DSCA33
True RMS Input modules. Both these products provide a 1500Vrms isolation barrier between input and output.
The SCM5B33 is a plug-in-panel module, and the DSCA33 is a DIN rail mount device. Each provide a single
channel of AC input that is converted to its True RMS DC value, filtered, isolated, amplified, and converted to
standard process voltage or current output.
SCM5B33 ISOLATED TRUE RMS
INPUT MODULE, PLUG-IN-PANEL
MOUNT
FEATURES
•
INTERFACES RMS VOLTAGE (0 - 300V) OR
RMS CURRENT (0 - 5A)
•
DESIGNED FOR STANDARD OPERATION
WITH FREQUENCIES OF 45HZ TO 1000HZ
(EXTENDED RANGE TO 20Khz)
•
COMPATIBLE WITH STANDARD CURRENT
AND POTENTIAL TRANSFORMERS
•
INDUSTRY STANDARD OUTPUTS OF EITHER
0-1MA, 0-20ma, 4-20 MA, 0-5V OR 0-10VDC
•
±0.25% FACTORY CALIBRATED ACCURACY
(ACCURACY CLASS 0.2)
•
1500 VRMS CONTINUOUS TRANSFORMER
BASED ISOLATION
•
INPUT OVERLOAD PROTECTED TO 480V MAX
(PEAK AC & DC) OR 10A RMS CONTINUOUS
•
ANSI/IEEE C37.90.1-1989 TRANSIENT
PROTECTION
CSA AND FM APPROVALS PENDING
DESCRIPTION
Each SCM5B33 True RMS input module provides a
single channel of AC input which is converted to its True
RMS dc value, filtered, isolated, amplified, and converted
to a standard process voltage or current output (see
diagram below).
The SCM5B modules are designed with a completely
isolated computer side circuit, which can be floated to
±50V from Power Common, pin 16. This complete
isolation means that no connection is required between
I/O Common and Power Common for proper operation of
the output switch. If desired, the output switch can
be turned on continuously by simply connecting pin 22,
the Read-Enable pin to I/O Common, pin 19.
The field voltage or current input signal is processed
through a pre-amplifier and RMS converter on the field
side of the isolation barrier. The converted dc signal is
then chopped by a proprietary chopper circuit and
transferred across the transformer isolation barrier,
suppressing transmission of common mode spikes and
surges. The computer side circuitry reconstructs filters
and converts the signal to industry standard outputs.
Modules are powered from +5VDC, ±5%.
For current output models an external loop supply is
required having a compliance voltage of 14 to 48VDC.
Connection, with series load, is between Pin 20 (+) and
Pin 19 (-).
AN101
Dataforth Corporation
DSCA33 ISOLATED TRUE RMS
INPUT MODULE, DIN RAIL
MOUNT
FEATURES
•
INTERFACES RMS VOLTAGE (0 - 300V) OR
RMS CURRENT (0 - 5A)
•
DESIGNED FOR STANDARD OPERATION
WITH FREQUENCIES OF 45HZ TO 1000HZ
(EXTENDED RANGE OPERATION TO 20kHZ)
•
COMPATABLE WITH STANDARD CURRENT
AND POTENTIAL TRANSFORMERS
•
INDUSTRY STANDARD OUTPUTS OF EITHER
0-1MA, 0-20MA, 4-20MA, 0-5V, OR 0-10VDC
•
±0.25% FACTORY CALIBRATED ACCURACY
(ACCURACY CLASS 0.2)
•
1500 VRMS CONTINUOUS TRANSFORMER
BASED ISOLATION
(PEAK AC & DC) OR 10A RMS CONTINUOUS
•
•
ANSI/IEEE C37.90.1-1989 TRANSIENT
PROTECTION
•
MOUNTS ON STANDARD DIN RAIL
•
CSA AND FM APPROVALS PENDING
Page 5 of 6
DESCRIPTION
Each DSCA33 True RMS input module provides a single
channel of AC input which is converted to its True RMS
DC value, filtered, isolated, amplified, and converted to
standard process voltage or current output (see diagram
below).
The field voltage or current input signal is processed
through an AC coupled pre-amplifier and RMS converter
on the field side of the isolation barrier. The converted
DC signal is then filtered and chopped by a proprietary
chopper circuit and transferred across the transformer
isolation barrier, suppressing transmission of common
mode spikes and surges.
Module output is either voltage or current. For current
output models a dedicated loop supply is provided at
terminal 3 (+OUT) with loop return located at terminal 4
(-OUT).
Special input circuits provide protection against
accidental connection of power-line voltages up to
480VAC and against transient events as defined by
ANSI/IEEE C37.90.1-1989. Protection circuits are also
present on the signal output and power input terminals to
guard against transient events and power reversal. Signal
and power lines are secured to the module using
pluggable terminal blocks.
DSCA33 modules have excellent stability over time and
do not require recalibration, however, both zero and span
settings are adjustable to accommodate situations where
potentiometers located under the front panel label and are
non-interactive for ease of use.
AN101
Dataforth Corporation
Page 6 of 6
Table 1
RMS Calculated from Individual Fourier Coefficients
Full-wave Rectified,1 Volt Peak
Saw-Tooth Function,1 Volt Peak
n
An
An (rms^2)
Total rms
% Error
Bn
Bn (rms^2)
Total rms
% Error
0
6.36620E-01
4.24413E-01
8.48826E-02
3.63783E-02
2.02102E-02
1.28610E-02
8.90377E-03
6.52943E-03
4.99310E-03
3.94192E-03
3.19108E-03
2.63611E-03
2.21433E-03
1.88628E-03
1.62610E-03
1.41628E-03
1.24461E-03
*
*
*
*
*
*
*
*
*
4.05285E-01
9.00633E-02
3.60253E-03
6.61689E-04
2.04225E-04
8.27027E-05
3.96386E-05
2.13168E-05
1.24655E-05
7.76936E-06
5.09148E-06
3.47453E-06
2.45163E-06
1.77903E-06
1.32211E-06
1.00293E-06
7.74531E-07
*
*
*
*
*
*
*
*
*
6.366198E-01
7.038096E-01
7.063643E-01
7.068325E-01
7.069770E-01
7.070355E-01
7.070635E-01
7.070786E-01
7.070874E-01
7.070929E-01
7.070965E-01
7.070989E-01
7.071007E-01
7.071019E-01
7.071029E-01
7.071036E-01
7.071041E-01
*
*
*
*
*
*
*
*
*
9.9684
0.4663
0.1050
0.0388
0.0184
0.0101
0.0061
0.0040
0.0027
0.0020
0.0015
0.0011
0.0009
0.0007
0.0006
0.0005
0.0004
*
*
*
*
*
*
*
*
*
5.00000E-01
3.18310E-01
1.59155E-01
1.06103E-01
7.95775E-02
6.36620E-02
5.30516E-02
4.54728E-02
3.97887E-02
3.53678E-02
3.18310E-02
2.89373E-02
2.65258E-02
2.44854E-02
2.27364E-02
2.12207E-02
1.98944E-02
1.87241E-02
1.76839E-02
1.67532E-02
1.59155E-02
1.51576E-02
1.44686E-02
1.38396E-02
1.32629E-02
1.27324E-02
2.500E-01
5.066E-02
1.267E-02
5.629E-03
3.166E-03
2.026E-03
1.407E-03
1.034E-03
7.916E-04
6.254E-04
5.066E-04
4.187E-04
3.518E-04
2.998E-04
2.585E-04
2.252E-04
1.979E-04
1.753E-04
1.564E-04
1.403E-04
1.267E-04
1.149E-04
1.047E-04
9.577E-05
8.795E-05
8.106E-05
5.0000E-01
5.4833E-01
5.5976E-01
5.6476E-01
5.6756E-01
5.6934E-01
5.7057E-01
5.7148E-01
5.7217E-01
5.7272E-01
5.7316E-01
5.7352E-01
5.7383E-01
5.7409E-01
5.7432E-01
5.7451E-01
5.7469E-01
5.7484E-01
5.7497E-01
5.7510E-01
5.7521E-01
5.7531E-01
5.7540E-01
5.7548E-01
5.7556E-01
5.7563E-01
13.397
5.027
3.048
2.181
1.696
1.388
1.174
1.017
0.897
0.802
0.726
0.663
0.609
0.564
0.525
0.491
0.461
0.435
0.412
0.390
0.371
0.354
0.338
0.324
0.311
0.298
Exact RMS
7.071068E-01
0
Exact RMS
5.7735E-01
0
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