Phase measurement on the oscilloscope

Joan Mendoza
February 2015

Abstract
This practical lab aims to measure the phase shift of two sinusoidal
waveshape with two dierent methods using an oscilloscope. Both meth-
ods are: Dual-Trace and Lissajous-Pattern. In the case of the Lissajous
method, the gures generated in the oscilloscope are analized in order
to establish a relation between the gures and the ratio of the analized
frequencies. Also these two methods are compared on their accuracy de-
termining the phase angle of voltage signals for a resistor and voltage
source on a RC-series circuit.

1 Introduction.

A phase shift is a change in the phase of one quantity or in the phase dierence
between two or more signals. In general a phase shift is referred from the zero
phase, and when is related with time is called time delay.
Two oscillations that have same frequency and dieres on their phases have a
phase shift, this shift is expressed in degrees from 0o to 360o or in radians from 0
to 2 . If the phase dierence is 180 degrees or radians, then, both oscillations
are said to be in antiphase.
The phase of an sinusoidal function refers such as the following:
S(t) = Asin (2f t + ) (1)
where t is de independent variable: time, A is the amplitd of the function, f
is the frequency of oscillations and is the phase of S(t).
One method of measure a phase shift if through the lissajous gures with
an oscilloscope. The rst time when this method turn up was in 1815 by the
American mathematician Nathaniel Bowditch, but the curves were formally
determined by Jules-Antoine Lissajous. Lissajous used a narrow stream of sand
pouring from the base of a compound pendulum.
Nowdays, Lissajous-patter has a particular value in electronics, the curves
can be made to appear on an oscilloscope, the shape of the curve serving to
identify the characteristics of an unknown electric signal. For this purpose, one
of the two curves is a signal of known characteristics. In general, the curves can
be used to analyze the properties of any pair of simple harmonic motions that
are at right angles to each other.

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 A good way to test and compare both methods for measure phase shift
is on electrical devices. In the case of a RC-series circuit or high pass lter
too, the signal of voltage and current is shifted from the source. From circuit
theory is found that the phase angle () is dependent of the reactance (Xc )
and the resistance of whole circuit, its relation is shown in the next equation:
= tan1 (Xc /R). It Is important to remember that reactance is the opposition
to a change of electric current or voltage on a electric element; this fact and the
relation between the shift angle and the reactance generates a leading of the
signal with the source reference. In order to make the calculus easier, last
equation can also be in the form:
 
1
= tan 1
(2)
wCR

2 Methods for phase measurement.

2.1 Dual-trace method of phase measurement.

This method compared two signals with same frequency, it did not matter if they
were dierent in shape neither in amplitud. It was necesary to choose one as
reference, that is, with no phase angle y1 = A1 sin (2f t). The compared signal
had a phase shift with the reference: y2 = A2 sin (2f t + ). The measured
phase shift can be visualized in sketch on gure 1.

Figure 1: Phase shift of /2 between two sinusoidal waves.

Frequency, shape and amplitud of both signals out of the generator are shown
in table N.1

Table 1: Caracteristics of signals used in the phase measurement

Wave shape Amplitud (mV) Frequency (Hz)
Signal 1 sinusoidal 100,0 1000 1500 2000
Signal 2 sinusoidal 100,0 1000 1500 2000

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 Table 2: Phase shift 4t measured and generated by the source
Phase shiftdegrees (o )
error %
Oscilloscope Source
1 27 75 64,00
2 54 104 48,08
3 81 127 36,22
4 108 155 30,32
5 135 182 25,82
7 162 207 21,74
8 189 231 18,18
9 216 257 15,95
10 297 311 4,50

Table 3: Phase shift in degrees fot two signals and it correspondence Lissajous
patter
Phase shift  degrees (o ) Lissajous pattern
0
Line with positive slopo
355
180 Line with negative slope
270
Circle
89

The measurements obtained from the phase shift are shown in the table 2.
It can be seen by the percentage error that dual trace method is not accurate
since the dierence between of measured value and the real, provided by the
source, is considerable

2.2 Lissajous-pattern phase measurement.

Lissajous method, produced by the intersection of two sinusoidal curves, the axes
of which are at right angles to each other. This method allows to determine the
phase dierence for two signals y1 = A1 sin (2f1 t) and y2 = A2 sin (2f2 t). In
this case, amplitud, phase shift and frequency can be arbitrary. In this practical
lab was analized the two cases: when frequency wass the same and when it was
dierent for both signals

1. Signal of same frequencies:

The Lissajous patterns took the shape of a circle and a line with specic
values of phase shift. The values of these phase are shown in the table N.3
When the Lissajous pattern was not a line and a cirlce they formed an
ellipse. In order to obtain the phase shift it was necesary to measure these
values from the oscilloscope: Y0 and Ym . They are sketched on gure 2.

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Figure 2: Lissajous-pattern for two signals of same frequency with phase shift

The relation between the values Y0 and Ym and the phase shift of two
signals in degrees is obtained as follows:
From gure 2, let x = Asin(wt) and y = Ym sin(wt + ), when x = 0,
|y| = Bsin() = Y0 , it means that when x = 0, wt must be multiple of .
 
Thus, the phase dierence is given by: sin () = YYm0 = = sin1 YYm0
 
Y0
arcsin = (3)
Ym

The values obtained are in the table N.4

Table 4: Phase shift, Y0 and Ym

Phase shift  degrees (o )
values Y0 Ym
4o = 0,01
1 44,43 7 10
2 36,87 6 10
3 23,58 4 10
4 11,54 2 10
5 90,00 10 10
6 53,13 8 10
7 36,87 6 10
8 23,58 4 10
9 11,54 2 10

2. Signal of dierent frequencies:

In this case, one of the signals was changed its frequency. These started
with 2000 [Hz] up to 10000[Hz] with a variation of 1[KHz], the other signal
keep a value of 1000 [Hz].
Ten dierent characteristic Lissajous gures are plotted in gure 3 and
the ratios between their frequencies are shown in table 3.

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 1a 1b

2a 2b

3a 3b

4a 4b

5a 5b

6a 6b

7a 7b

8a 8b

9a 9b

Figure 3: Lissajous patterns when the frequency of one of the signal was dier-
ent. The frequencies were xed in order to obtain the same ratio as the values
on the table 5.

From table and gure 5 it can be noted the relation between the ratio and

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 Table 5: Ratio of frequencies between two signals
Pair of signals Ratio (R)
1a 1/2
1b 2/1
2a 3/1
2b 1/3
3a 4/1
3b 1/4
4a 5/1
4b 1/5
5a 6/1
5b 1/6
6a 7/1
6b 1/7
7a 8/1
7b 1/8
8a 9/1
8b 1/9
9a 10/1
9b 1/10

the Lissajous pattern. All gures that the ratio is bigger than 1 (R>1)
are oriented with the horizontal axis. On the other hand, when the ratio
is smaller than 1 (R<1) gures were oriented with the vertical axis. In
the same way, the gures showed a pattern with their peaks. The bigger
frequency determinate the number of peaks that are adressed depending
of the value of R

2.3 Phase shift of a RC circuit.

A RC circuit in series was built as gure 4. The phase shift for two signals
was measured using both methods: Dual trace and Lissajous-Pattern. The rst
signal came out from the generator Vg and the laged signal out of the resistor.
The phase shift, in degrees, theorically calculed is given by
The circuit was built with a capacitance C =0,01F , and three dierent
values of resistances were analized.

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Figure 4: RC circuit for the measurement of the phase shift between the voltage
signals of the source and the resistance.

The results of measuring a phase shift with two methods are summarized in
table 6.

Table 6: Measurement of phase shift of two signals with their correspondence

error in relation with the theorical values
Theorical phasedegrees Dual Tracedegrees Lissajousdegrees
ResistanceOhm () D.T. error% L error%
4o = 0, 01 4o = 0, 01 4o = 0, 01
10000 17,44 14,40 25,37 17,43 15,55
20000 32,14 10,80 44,43 66,39 38,22
30500 43,77 64,80 34,51 48,02 21,14

3 Conclusion

Both methods, Dual-trace and Lissajous-Patterns can measure phase shift of

two signals using an oscilloscope. Despite two methods give a considerable
percentage error, the Lissajous-pattern is more accurate than the Dual-trace.
Thes error is primarlly originated by the noise generated on the circuit which
yielded about 4 to 10 degrees of uncertainty. This error reduces in the case of
the Lissajous method because the noise seems to have less inuence when the
ellipses where plotted, even more, the distances Ym and Yo were much more
easy to measure because the vertical axis was bigger than the X axis.
In the Lissajous pattern is important to highlight this fact: when the dif-
ference of frequencies were minimum, in the order of 1Hz the gures keep their
shape but they moved as turning around on a vertical axis. On the other hand,
when the dierence of frequencies was in the order of 1KHz the shape of the
pattern changed depending of the ratio of their frequencies leading to more
complex shapes.

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4 References

Impedance 1: Phasor relationships. (n.d.). Retrieved February 6, 2015, from

http://www.dejazzer.com/ece311/labs/lab3.pdf
Lissajous gure. (n.d.). Retrieved February 6, 2015, from http://www.britannica.com/EBchecked/topic/343
gure