Instrumentation & Measurement Magazine 23-2 - 26

physical measurement results, and online visualization versus time of these results.
Before starting the application, the gyroscope offset and
gain errors were compensated. For each axis the offset was
compensated by subtracting from each sample the mean value
of all acquired samples achieved in the stationary position of
the sensor, while the gain was compensated by multiplying the
angular position values by a suitable factor, which depends on
the analyzed data [12].
The measurements were performed in different scenarios.
A scenario was the following: the CC2650 Sensor Tag is held
in the hand with z axis aligned with gravity force and then it
is rotated across the y axis in both senses. A similar rotation is
executed then across the x axis. Then, by using the data from
gyroscope and the accelerometer, the pitch and role angles are
computed. Fig. 5 shows the achieved results. The first graph
shows the angles generated by the gyroscope and the second
graph shows the accelerations, while the third graph shows
the roll and pitch angles computed by (1) and (2).
In Fig. 5 it can be seen that the angle across the y axis of the
gyroscope (blue crosses in the first graph) corresponds to the
inverted of pitch angle (blue crosses in the third graph), while
the angle across the x axis of the gyroscope (red circles in the
first graph) corresponds to the roll angle (red circles in the third
graph). These results are in very good agreement with the position and the sign of the axis corresponding to the gyroscope
and accelerometer packages (Fig. 4a).

Lessons Learned
These activities teach students to develop an algorithm that
implements the data acquisition, the signal processing, and the
data presentation in a suitable format using the MATLAB environment. Also, the achieved graphical results related to the
movement scenarios are analyzed and discussed.
The difficulty in this application was the handling of the
messages from sensor to the PC. Besides measurement data,
they contain information such as the opcode, the status, and
the data length. This problem was solved with the help of the
teaching coordinator.

Fig. 5. Experimental results achieved: angular positions computed by
gyroscope, accelerations, and roll and pitch angles.
26	

Phasor Measurements
Phasor Parameters
A real-life electrical waveform (voltage or current) can be defined as:
	

(

)

     x ( t ) = a1 ( t ) cos 2π ft + φ1 ( t ) + ϑ ( t )
	

{

} + ϑ (t )	

	

= Re a1 ( t ) e

	

= Re p ( t ) e j 2π ft + ϑ ( t ) ,
	(3)

{

jφ1 ( t ) j 2π ft

e

}

where f, a1(t), ϕ1(t) are the waveform frequency, amplitude,
jφ ( t )
and phase, p ( t ) ≡ a1 ( t ) e 1 represents the waveform phasor,
and ϑ(t) is the disturbance signal, which contains the waveform harmonics, interharmonics, and wideband noise. The
frequency f may differ from the nominal frequency fn (50 Hz
or 60 Hz).
In practice, due to the electromechanical transients, the
waveform (3) can be modulated in amplitude and/or phase
(dynamic conditions), while due to the electromagnetic transients, it can be affected by the step changes in amplitude and/
or phase or ramp of frequency (transient conditions) [16].
Therefore, the waveform amplitude and phase (phasor components) are time varying.
The off-nominal frequency (f ≠ fn), harmonics and interharmonics represent the steady-state conditions.
It is worth noting that the phasor measurements are performed at the reference frequency f0, which can have any value.
The difference between the frequencies f and f0 represents the
Frequency Deviation (FD), that is FD = f - f0. The derivative of
FD with respect to time represents the Rate-Of-Change of Frequency (ROCOF), that is ROCOF = dFD .
dt
In modern power systems the so-called Phasor Measurement Units (PMUs) are used for monitoring and control
operations. PMUs are advanced devices which provide in realtime and synchronized with the Coordinated Universal Time
(UTC) the estimates of the waveform phasor parameters. In
this case, when f0 = fn, the phasor is called synchrophasor. A
PMU should provide at a constant rate accurate estimates for
the phasor components (a1 and ϕ1), FD, and ROCOF [17]. To
this aim the PMUs use different time-domain or frequencydomain algorithms. They are based on the static or dynamic
model of the phasor p(t). In the static model the waveform
amplitude and phase are assumed constant, while in the dynamic model they are assumed time varying [17]. By using the
dynamic model, the phasor measurements achieved under dynamic and transient conditions are more accurate than those
achieved by using the static model [18], [19].
According to the IEEE Standard C37.118.1-2011 for synchrophasor measurements for power systems there are two
classes of performances for the PMUs, which are the P-class
for protection purposes and the M-class for measurement applications [20]. Also, in that Standard there are suggested as
accuracy parameters the Total Vector Error (TVE), Frequency
Error (FE), and Rate-of-change of Frequency Error (RFE). The

IEEE Instrumentation & Measurement Magazine	

April 2020
Instrumentation & Measurement Magazine 23-2

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