X-Y Displays, also known as scatter or cross plots, provide a means of plotting one trace against another. This type of display shows comparisons and relationships between the two waveforms. A third parameter, such as frequency of occurrence, may be introduced by adding it as a variation of color or intensity to the two-dimensional plot.

The X-Y display mode finds many classical and current applications. From the classic Lissajous pattern to measure phase or frequency ratios, to state transition diagrams for modern quadrature communications systems. X-Y plots can show if there is a relationship between two variables. If such a relationship is found it can show if the relationship is linear or nonlinear and the direction of the relationship. This article will look at some of these applications.

**Lissajous Patterns**

The classic introduction to X-Y plots is the Lissajous plot. Two sine waves are plotted against each other. For sine wave of the same frequency the plot is used to measure the phase difference between the sine waves. With modern oscilloscopes it is easier to just read the relative phase with a measurement parameter. If the sine waves are at different frequencies the ratio of their frequencies can be determined as shown in Figure 1.

**Figure 1** An X-Y plot of two harmonically related sine waves the ratio of the number of horizontal peaks to the number of vertical peaks shows the frequency ratio of the inputs in this case 2 to 5.

This plot shows the relationship between the frequencies of the two inputs is 2 to 5. The measured frequencies are 1 and 2.5 MHz.

**Curve tracer**

Using an oscilloscope along with an arbitrary waveform generator (AWG) you can characterize semiconductor device using the X-Y display to plot their V-I characteristics. Figure 2 shows the result of measuring a silicon diode.

**Figure 2** The V-I curve for a silicon diode based on twelve applied voltage values generated by an AWG and the resultant current through the device.

The AWG generated twelve pulses with voltage amplitudes incremented from -5 to +5 Volts which was applied to the diode. The voltage across the diode and the current through the diode were acquired using the oscilloscope’s sequence mode with one pulse per sequence segment. Measurement voltage and current parameters were summarized in a trend plot with one parameter value per point. The voltage trend was applied to the X-Y display horizontal input with a scale factor of 1 Volt per division while the current trend was applied to the vertical axis at 12.4 mA per division. The resulting X-Y display shows the diode V-I characteristic. A quick test like this is useful for things like matching components.

**Quadratic signal measurements**

Quadratic signal generation uses two signals with a ninety-degree phase difference to generate a signal with a variable phase. Two signals added in quadrature define a vector with a magnitude and a phase. The magnitude of the input signals determines both vector attributes. X-Y plots, supported by X-Y cursors allow you to view and measure the vector phase and magnitude as it is generated by the quadrature inputs as shown in Figure 3.

**Figure 3** The X-Y display of two quadrature signals shows the phase trajectory of the vector generated as the quadratic sum of the signals. X-Y cursors read the vector magnitude (Radius) and the phase (Angle) as well as the Cartesian coordinates.

The X-Y display allows you to visualize the path traced by a rotating vector defined by the quadratic sum of the two exponentially weighted RF pulses. The X-Y cursors read the vector magnitude (Radius) and Phase relative to the positive X axis in addition to the voltages in the source sample points in channel 1 and 2. These cursor readings track across the X-Y and X-T and Y-T waveform displays. This means that the source components for any cursor measured vector are measured at the same time. The 401mV vector magnitude is the result of the 349.6mV X component and the 196.9 mV Y component. This information is useful in applications that use quadratic signal generation such as radar and digital communications where an error in the vector parameters can be traced easily to the source components.

The X-Y display can also be rendered as persistence display which retains multiple traces overlaid on the display. Persistence displays show the frequency of occurrence of the displayed pixels either using displayed intensity or color. Figure 4 shows the in-phase(I) and quadrature (Q) components of a 16QAM signal along with the X-Y display showing I plotted against Q, the state transition diagram. Monochrome persistence is used on the X-Y display.

**Figure 4** Rendering the X-Y display in monochrome persistence shows the data states of the state transition diagram as brighter dots while the multiple transitioned phase paths also are shown as brighter.

The state transition diagram shows the data states at the end of each transition path as well as marking the path between data the states. The waveform spends more time at each data state than on the transition path and therefore the data states are shown as brighter points on the X-Y display. The four phase states at 45, 135, 225, and 315 degrees are written over twice with two different vector amplitudes and also appear brighter. Persistence thus provides additional information about this measurement including more clearly showing the overlapping vectors.

**Power related X-Y measurements**

X-Y displays are also used for measurements on switched mode power devices. One example plots the voltage across a power FET versus the current through it in order to assure that the device operates within its safe operating area (SOA) as shown in Figure 5.

**Figure 5** An X-Y display used to measure the SOA of a power FET by plotting drain current versus drain to source voltage. Pass/Fail testing compares the X-Y display against a mask, red circles indicate failures.

The drain to source voltage (V_{DS}) across the FET is applied to the horizontal axis and the drain current is applied to the vertical axis. The vertical section of the X-Y display represents the FET in conduction, the V_{DS }is nearly constant while the current increases. The horizontal loops show the time when the FET is off, current is zero and the voltage is oscillatory. The traces in between represent the switching transitions where power is dissipated by the device. Testing can verify that the devices voltage, current, and power limits are not exceeded.

The blue area is a Pass/Fail mask which is used to monitor the measurement. The X-Y trace should not intersect the mask. If it does, it indicates a failure, it is marked as a failure by a red circle. Additionally, Pass/Fail testing has several response options, both hardware and software to indicate the state of the test for response actions.

Another power related X-Y measurement is measuring the magnetic properties of inductive devices. Figure 6 shows a magnetic hysteresis plot of an inductor.

**Figure 6** The magnetic hysteresis plot plots magnetic flux density as a function of magnetic field strength.

The hysteresis plot inputs are magnetic field strength and flux density. Magnetic field strength is calculated from the current through the inductor while flux density is derived from the integral of the applied voltage. The oscilloscope used in this article has a software option for power measurements which performs these calculations based on the know coil geometry (cross sectional area and magnetic path length), voltage and current. The area inside the hysteresis loop is the energy lost per cycle called, appropriately, the hysteresis loss.

**Mechanical measurements**

X-Y displays can be used to analyze mechanical devices. This requires appropriate transducers to convert mechanical parameters into proportional electrical signals. An example of an X-Y display of a mechanical device is a pressure-volume plot for an engine as shown in Figure 7.

**Figure 7** An X-Y display plotting pressure as a function of cylinder volume in an internal combustion engine.

The pressure is read using a pressure transducer ported through the spark plug for the cylinder. The volume is calculated based on the measured crank angle using a rotary encoder. Both transducers use the oscilloscope’s rescale function to allow measurements in standard units of pressure and volume in this case Pascals and liters. There are two loops in the PV plot. The upper, larger, one represents the power stroke and the lower one is the exhaust stroke. The mechanical work generated for each cycle of the engine is proportional to the areas inside the PV diagram loops. The power stroke being positive work and the exhaust stroke is negative work.

**Conclusion**

It should be obvious that the X-Y display is an extremely useful tool for interpreting measurements. It allows you to compare signals, it shows the relationships between the plotted variables, and it graphically displays the vector magnitude and phase for quadratic inputs. It also can provide information of energy contributions or losses in cyclic processes. It is a tool that you should keep in mind for studying dual trace applications.

*Arthur Pini** is a technical support specialist and electrical engineer with over 50 years experience in electronics test and measurement.*

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