Oscilloscope

Using X-Y Mode An instruction guide for the HP 54602B Oscilloscope Written by Elliott Wood

Table of Contents Introduction ...................................................................................................................................... 1 The Oscilloscope ......................................................................................................................... 1 X-Y Mode Measurements................................................................................................................ 2 Device Setup................................................................................................................................ 2 Voltage Gain ................................................................................................................................ 2 Phase Shift................................................................................................................................... 3 Frequency Calibration.................................................................................................................. 4 Troubleshooting ............................................................................................................................... 7 X-Y Plot Is Not Stable .................................................................................................................. 7 Lissajous Plot Is Spinning Too Quickly........................................................................................ 7 Lissajous Plots Are Not Symmetric.............................................................................................. 7 Appendix A – Phase Shift Examples ............................................................................................A-1 Appendix B – Calibration Examples .............................................................................................B-1

The Oscilloscope

Using X-Y Mode

Introduction The Oscilloscope The oscilloscope is one of the most common instruments found in a laboratory today. Its prevalence is due to its ability to measure periodic waveforms of all kinds, giving it applications in fields ranging from electronics to mechanics to acoustics. Most people recognize an oscilloscope by the characteristic sinusoidal waveform plotted against time on its screen. This standard display is useful for determining characteristics of a single waveform such as frequency and amplitude, or for measuring timing in electronic circuits. However, most are unaware that an extremely useful mode exists for comparing two sinusoidal signals. This mode, known to experienced oscilloscope users as the X-Y mode, helps to calibrate signals in reference to another and can also be used to measure the gain and phase shift of linear two-port networks. This guide will instruct the reader in using X-Y mode to measure and compare two sinusoidal signals using the Hewlett Packard 54602B Oscilloscope, shown below in Figure I-1.

Figure I-1. HP54602B Oscilloscope

1

Device Setup

Using X-Y Mode

X-Y Mode Measurements Device Setup Before the HP 54602B Oscilloscope can be used to compare two signals, its inputs must be properly connected. To begin, turn on the oscilloscope and press the button labeled “Main/Display” (located under the “Horizontal” controls section on the right side). This will bring up a menu of soft-keys on the bottom of the screen. Press the soft key under “XY” to switch the oscilloscope to X-Y mode.

Main i Delayed

XY

Next, two sinusoidal signals should be connected to the oscilloscope. For voltage gain and phase shift measurements, the input voltage (vi) of the linear two-port network you are measuring should be connected to Channel 1, and the network’s output signal (vo) should be connected to Channel 2. For frequency calibration, the reference frequency should be connected to Channel 1, and the signal to be calibrated should be connected to Channel 2. A summary of these connections is shown below in Table 1-1. TIP: Make sure your signals are both properly connected to a common ground. Table 1-1. Signal Connections for Different Measurements Channel One Voltage Gain Phase Shift Frequency Calibration

Channel Two

Vin

Vout

Reference

Calibration

Voltage Gain The voltage gain of a linear two-port network is the ratio of the output voltage vo to the input voltage vi. This is easily measured in X-Y Mode. Since the x-axis represents the voltage of the input signal, and the y-axis represents the voltage of the output signal, the voltage gain is simply the ratio of the maximum points in each direction. This is pictured in Figure 2-1. In the figure, vi,max is about 1.05 Volts and vo,max is about 0.7 Volts. Therefore the voltage gain for this circuit is 0.7 / 1.05, or 0.667.

Steps to measure voltage gain 1. Measure the maximum value of the plot on the x-axis as vi,max. 2. Measure the maximum value of the plot on the y-axis as vo,max. 3. Calculate the voltage gain in decibels with Equation 2-1.

 vo, max  Gain = 20 ⋅ log   vi, max 

(Eq. 2-1)

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Using X-Y Mode

Figure 2-1. Measurement of Voltage Gain in X-Y Mode

Phase Shift The phase shift of a linear two-port network is defined as the phase difference between vi and vo. The X-Y mode can be used to measure this shift. The input signal (vi) and output signal (vo) of the circuit can be connected to the oscilloscope to precisely measure phase shift. To measure phase shift, first configure the HP 54602B Oscilloscope as described in the Device Setup section, using the connections listed for phase shift measurement. The display on the oscilloscope will show a circle or oval pattern. Assume that v1(t) and v2(t) are the two signals input into the oscilloscope. Then: v1(t) = sin(ωt)

(Eq. 3-1)

v2(t) = sin(ωt + φ)

(Eq. 3-2)

In Eq. 2, φ is the angular phase shift. This value can be measured using X-Y mode on the oscilloscope. Given the circle or oval display, let A be the peak in the Y direction, and let B be the point at which the plot intersects the Y-axis. Then: B = A sin(φ)

(Eq. 3-3)

φ = sin-1(B/A)

(Eq. 3-4)

This is demonstrated in Figure 3-1 below.

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Using X-Y Mode

Frequency Calibration

Figure 3-1. Measurement of Phase Shift in X-Y Mode In this plot, A is measured to be 750mV and B is measured to be 530mV. Using Eq. 3-4, φ is then calculated as 45°. This indicates a 45° phase shift between v1(t) and v2(t).

Steps to calculate phase shift 1. Measure A, the maximum value in the y-direction of the plot. 2. Measure B, the intersection of the plot on the positive y-axis. 3. Take the inverse sine of B divided by A. This is the phase shift. If B intersects the Y-axis at the origin (in the case of an apparent line), then the phase shift is either zero or 180°. A positively sloping line intersecting at the origin indicates a phase shift of zero. A negatively sloping line intersecting at the origin indicates a phase shift of 180°, or an inverted signal. Finally as the plot begins to look more like a circle, then B approaches A. In this case the phase shift approaches 90° as sin-1(B/A) approaches sin-1(1). Examples of these conditions are included in Appendix A.

Frequency Calibration Suppose a signal is required that is exactly twice the frequency of another. While a pure sinusoid signal near 60Hz may be easily accessible from a given source, what if a signal at twice that frequency is required? A signal generator could be set to output a signal at 120Hz, but this may not be precise enough if the source is actually 59.995Hz. How can a signal that is an exact multiple of another be obtained? The X-Y mode of an oscilloscope can calibrate a signal to an exact rational multiple of a reference signal. First, configure the HP 54602B to the state described in the Device Setup section, using the connections listed for frequency calibration.

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Using X-Y Mode

Frequency Calibration

Lissajous figures will be used to calibrate the frequency of a signal on channel two to a multiple of the frequency of the signal on channel one. A Lissajous figure is a series of connected horizontal and vertical loops. When these loops stabilize (stop rotating), the frequency of the calibrated signal is an exact rational multiple of the frequency of the reference signal. For example, the Lissajous plot shown in Figure 4-1 shows the case in which the frequency of the calibrated (fcal) signal exactly matches that of the reference signal (fref).

Figure 4-1. Lissajous Plot for fcal = fref TIP: If the loops are rotating or moving, the signals are not stabilized and therefore need a small adjustment to the frequency for precise calibration. For more complex multiples, the number of loops can be counted. In the general case, the ratio of fcal to fref can be calculated as the ratio between the number of horizontal loops to the number of vertical loops. More specifically, suppose a box was drawn tangent to the loops of the Lissajous plot. The number of vertical loops is the number of times the plot would touch the top of the box. The number of horizontal loops is the number of times the plot would touch the left side of the box. Figure 4-3 demonstrates this.

5

Frequency Calibration

Using X-Y Mode

Figure 4-2. Lissajous Plot for fcal = 3/2 · fref In Figure 4-3 above, the loops touch the top of the box three times and the side of the box twice. Therefore, the ratio of fcal to fref is 3 to 2. In this example, if fref is 60Hz, then fcal is 90Hz.

Steps to find the frequency given a reference signal 1. Stabilize the Lissajous plot by fine-tuning the frequency fcal. 2. Draw a box tangent to the loops of the plot. 3. Count A, the number of times the loops touch the top of the box. 4. Count B, the number of times the loops touch the left side of the box. 5. Divide A by B, and multiply by the reference frequency. frequency.

This is the calibrated

For more examples, three additional Lissajous plots are included in Appendix B.

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Using X-Y Mode

Troubleshooting

Troubleshooting X-Y Plot Is Not Stable If the X-Y plot is moving during either voltage gain or phase shift measurements, then the frequencies of vi and vo are not identical. First, make sure that both signals share a common reference ground. Second, make sure that the frequency of either signal is not being altered by some component in the circuit.

Lissajous Plot Is Spinning Too Quickly If the Lissajous plot is spinning faster than can be read, then the frequency has not yet been properly calibrated. This spinning condition is normal until an exact multiple of the reference signal has been reached. Adjust the frequency of the signal being calibrated until the plot slows to a stop, and then count the number of loops. If the desired frequency is not indicated, repeat the procedure by adjusting to a different frequency at which the plot becomes stationary.

Lissajous Plots Are Not Symmetric Lissajous plots may not always be symmetric as those pictured in this guide. The images given here are symmetric for clarity. However, this is not a requirement for an accurate measurement. Symmetry is a function of phase shift, and the plots pictured in this guide all have a phase shift of 90°. This is not necessary though, and loops may still be counted using the method described. Only a stable, non-rotating set of loops is required for an accurate measurement.

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Using X-Y Mode

Appendix A – Phase Shift Examples

Appendix A – Phase Shift Examples

Figure A-1. Phase Shift of Zero

Figure A-2. Phase Shift of 180°

A-1

Using X-Y Mode

Appendix A – Phase Shift Examples

Figure A-3. Phase Shift Approaching 90°

A-2

Using X-Y Mode

Appendix B – Calibration Examples

Appendix B – Calibration Examples The following Lissajous plots assume a reference frequency fref of 60Hz.

Figure B-1. Lissajous Plot for fcal = 2 · fref, or 120Hz.

Figure B-2. Lissajous Plot for fcal = 3 · fref, or 180Hz.

B-1

Using X-Y Mode

Appendix B – Calibration Examples

Figure B-3. Lissajous Plot for fcal = 5/2 · fref, or 150Hz.

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Figure 3 - Use of oscilloscope for phase measurements The CRO may be used to measure phase shift in an electronic circuit, as shown in Fig. 3. An oscillator is connected to the input of the circuit under test. The output of the circuit is connected to the CRO vertical input, whereas the oscillator signal is connected directly to the horizontal input. The phase-shift angle ϕ may be determined from the relation,

where B and A are measured as shown in the figure. For zero phase shift the ellipse will become a straight line with a slope of 45° to the right; for 90° phase shift it will become a circle; and for the 180° phase shift it will become a straight line with a slope of 45° to the left.

B-2

Using X-Y Mode

Appendix B – Calibration Examples

Figure 4 - Lissajous figures for different phases. The CRO offers a convenient means of comparing signal frequencies through the use of Lissajous diagrams. Two frequencies are impressed on the CRO inputs, one on the horizontal input and one on the vertical input. One of these frequencies may be a known frequency as obtained from a variable frequency oscillator or signal generator. If the two input frequencies are multiples of each other then the patterns that are displayed on the CRT screen are called Lissajous diagrams. The frequency ratio is related to number of vertical and horizontal maxima of the diagrams. Some typical shapes for the Lissajous diagrams are shown in Fig. 4. It may be noted that these shapes can vary somewhat depending on the phase relation between the input signals. Oscilloscope traces may be recorded by various photographic methods, and several cameras are manufactured especially for oscilloscope applications.

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Phase-shift measurements: The phase-shift between two signals can be determined by two different methods. One-way is to use a Lissajous figure as shown in Figure 5. The expression for phase-shift is _ = sin-1(a/b) The second method requires a duel-trace oscilloscope (two Y- input channels). The vertical positions of both traces MUST be EXACTLY the same. The greatest accuracy is obtained when the distance between the horizontal crossover of the two signals is the largest. This is controlled by the horizontal sweep setting. An example is shown in Figure 6. The expression for phase-shift is θ = 360o (t/T)

B-3

Using X-Y Mode

Appendix B – Calibration Examples

B-4

Using X-Y Mode

Appendix B – Calibration Examples

B-5

Using X-Y Mode

Appendix B – Calibration Examples

B-6