Appendix F
Sinusoid Measurement Techniques

F.1 Amplitude and Phase Measurements

Figure F.1 shows a pair of sinusoidal signals as displayed on an oscilloscope. The “Reference” sinusoid serves as the phase reference for the “Signal of Interest” and has a phase of zero degrees. Both sinusoids oscillate at the same frequency f₀. Three measurements suffice to determine the phasor representation of each sinusoid as follows:

1. The reference sinusoid is A0^∘,
2. The phase shift θ in degrees is ±θ = C × f₀ × 360^∘ where C is the absolute value of the time shift between the two sinusoids in seconds and f₀ is sinusoidal frequency in Hz; choose positive sign when the signal of interest leads the reference (its zero crossing occurs before the reference as pictured in Figure  F.1) and negative sign otherwise, and
3. The signal of interest is Bθ^∘.

Consider the NI ELVISmx Oscilloscope display of Figure F.2 as an example of this measurement technique. Note that the “Volts per Division” scales have been adjusted to make both sinusoids fill as much of the screen as possible. The green trace on Channel 0 serves as the reference while the blue trace on Channel 1 is the signal of interest. The cursors have been adjusted to measure the time shift between the two signals. Note that the numerical display under the traces provides all necessary measurements:

1. The reference sinusoid amplitude is 1.696 volts divided by 2 (to convert from peak-to-peak) or 848 mV with a phase of zero degrees,
2. The phase shift is -65 μs × 2.502 kHz × 360^∘ = -59^∘; the phase is negative because the signal of interest (blue trace) lags the reference, i.e., the zero crossing occurs after the zero crossing of the reference, and
3. The amplitude of the signal of interest is 427 mV divided by 2 or 214 mV.

From these measurements the phasor form of the reference is 8480^∘ mV and the phasor form of the signal of interest is 214- 59^∘ mV.