### Introduction

From the classic Lissajous pattern to state transition diagrams for modern quadrature communications systems X-Y plots provide a view of functional relationships between to waveforms. X-Y plots can be displayed alone or in conjunction with the X-T and Y-T components. Combined with X-Y cursors it is possible to make measurements using both Cartesian and polar coordinates. In this application brief we will look at some commonly used X-Y measurements.

### Lissajous Patterns

For most engineers introduction to X-Y displays is via the classic Lissajous pattern where two sine waves are plotted, one against the other. This display can show the relative phase of the sine waves. In the case where the sine waves are at different frequencies, we can determine the ratio of the two frequencies as shown in Figure 1 where a 10 and 25 MHz sine waves are used as inputs to the Lissajous pattern. Note that X-Y displays are set up using the Display dialog box as shown in Figure 1.

The user can select which waveform is connected to the vertical axis and which is associated with the horizontal axis. The number of vertical peaks (5 in this case) and horizontal peaks (2) indicate the frequency ratio of the vertical (Y) axis to horizontal (X) axis. In this case 5/2 (25/10 MHz).

### Amplitude Modulation Measurement

A slightly more sophisticated version of the Lissajous pattern is the trapezoidal diagram of Figure 2.

In this X-Y plot the Y axis is the amplitude modulated carrier (C2) . The X axis is the modulation signal (C1). The modulation index of the amplitude modulated signal can be determined by measuring the lengths of the vertical sides of the trapezoid. If the length of the left hand side is called Q and that of the right hand segment called P then the modulation index, m, can be computed as;

$$m = {(P-Q)/(P+Q)}$$

In our example P=4 and Q=1 division respectively. The value of m is 0.6 which represents 60 % modulation.

Moving into the domain of power measurements we can plot the current through a power field effect transistor (FEt) as a function of the drain to source voltage in a switched mode power supply. The resultant plot is known as a safe operating area (SOA) plot. An example is shown in Figure 3. In this screen image channel 1 is the current waveform and channel 2 is the voltage waveform. When the FET is on the drain-source voltage is close

to zero and the current ramps up from zero to a maximum value. This forms the vertical segment on the left of the SOA plot. When the current is zero the voltage waveform plots as the horizontal segment on the bottom of the SOA plot. When the FET is changing state we get simultaneous non-zero current and voltage values which show as the curved sections joining the linear elements previously discussed. These are regions of finite power dissipation. Higher power levels are indicated by moving upward and to the right on the SOA plot.

Another common power measurement is the hysteresis or B/H curve for an electromagnetic component like an inductor of transformer. Magnetic materials are characterized by plotting magnetic flux density (B) as a function of magnetic field intensity (H).

Figure 4 shows how an inductor is connected to generate a B/H curve.

In Figure 5 we have a screen image of such a B/H curve.

Note that the voltage waveform has to be integrated in order to determine the magnetic flux density. In Figure 5 we have a screen image of such a B/H curve.

Trace M3 contains the current waveform which is applied to the X input. The voltage waveform is integrated using the Math function and applied to the Y input.

The area within the hysteresis loop is proportional to the energy loss per cycle. This area can also be measured as described in LeCroy Application brief LAB_707A.

Our final example of X-Y plots comes from the world of communications. Modern communications systems employ a variety of quadrature modulation schemes. These are phase modulation systems that use baseband in phase (I) and quadrature (Q) components to modulate a carrier. Cross plots of the I and Q components, known as state transition diagrams, provide information on the amplitude and phase of the resultant vectors. In addition to viewing the vector components one can measure the magnitude and phase suing X-Y cursors.

Figure 6 is an example of a state transition diagram for a PHP cellular phone system. The I waveform is applied to the horizontal or X axis and the quadrature (Q) component is applied to the vertical or Y input. This type of X-Y plot also employs display persistence to maintain a history of multiple data values on the screen.

### Summary

There are many applications which benefit from X-Y displays. These are a few of the more commonly used applications. LeCroy oscilloscopes incorporate X-Y displays along with persistence and specialized cursors to get the most value from this useful tool.