2 What is phase shift and phase error?

Phase shift refers to the horizontal shift of a sinewave along the x-axis. Equation 1 specifies the standard sinewave equation centered around the origin with no phase shift:

Where:

• A_max is the max amplitude of the wave
• ωt is the angular frequency of the waveform

Figure 2-1 shows this equation graphically as a single sinewave.

Figure 2-1 Graph of standard Sinewave equation

A phase shift can exist between two measured sinewave signals. This phase shift is denoted by $\mathrm{\phi }$ and the equation for a sinewave with a phase shift is given by Equation 2:

Where:

• A_max is the max amplitude of the waveform
• ωt is the angular frequency of the waveform
• φ is the load phase angle between two waveforms in degrees or radians

Figure 2-2 shows two sinewaves plotted with a phase shift of φ between them.

Figure 2-2 Two Sine wave graphs with shift of φ

As an example, a load phase angle can exist between the voltage and current waveforms of a power measurement system. This phase shift breaks into two distinct components:

• Load phase angle (φ) – an expected phase shift caused by the reactive nature of the load. Use the load phase angle to determine the power factor and real power calculation.
• Phase error (∆φ $)$ – an unwanted phase shift introduced by measurement hardware such as CTs, signal conditioning circuits, or ADC timing mismatches. Unlike the load phase angle, this component disrupts the power measurement and needs to be compensated.component disrupts the power measurement and needs to be compensated.

Figure 2‑3 shows typical power measurement systems waveforms, with a green voltage signal V(t) and a red current signal I(t) :

Figure 2-3 Voltage and Current graphs with shift of $\mathbf{\phi }$

The signals in Figure 2-3 can be expressed as:

Where:

• V_max is the max voltage amplitude
• I_max is the max current amplitude

Equation 4 shows that the load phase angle is subtracted from the angular frequency ωt – φ. This subtraction shifts I(t) to the right along the x-axis relative to V(t) as shown in Figure 2-3.This relationship is referred to as I(t) lagging V(t) and is common for an inductive load. The phase shift expressed as $(\mathrm{\omega }\mathrm{t}+\mathrm{\phi })$ shifts $\mathrm{I}\left(\mathrm{t}\right)$ leftward along the x-axis relative $\mathrm{V}\left(\mathrm{t}\right)$. This relationship is referred to as $\mathrm{I}\left(\mathrm{t}\right)$ leading the $\mathrm{V}\left(\mathrm{t}\right)$ and is common for a capacitive load. In a purely resistive load $\mathrm{\phi }=0$ and the two waveforms are aligned. As the reactive component of the load increases, $\mathrm{\phi }$ increases and the misalignment between the waveforms grows.

Equation 5 uses Equation 3 and Equation 4 to calculate instantaneous power - the power at any single point in time:

Equation 5 calculates the ideal instantaneous power with a phase shift resulting from only the load phase angle. As described earlier, however, phase error also contributes to the total phase shift. Equation 6 calculates the instantaneous power with a phase shift resulting from both the load phase angle and phase error:

Where:

• Equation 7. $∆\mathrm{\phi }$ is the phase error from the system measurement hardware

Equation 6 provides a power value at a single point in time that will change as the voltage and current waveforms oscillate. Therefore, an actual power measurement system needs to average the instantaneous power over a complete cycle to determine the real power consumed by the load during operation. Averaging Equation 6 over a complete cycle yields the average real power formula shown in Equation 7:

A phase error, $∆\mathrm{\phi }$, remaining in the system without calibration introduces systematic error into the power measurement. Equation 8 calculates the resulting power measurement error due to the phase error:

For example, apply Equation 8 to a 50 Hz system with a 0.5 power factor (60° phase angle) to observe the impact of a 0.1° phase error on a power measurement.

A phase error of 0.1° produces a 0.3% measured power error compared to the real power. A phase error of 1° - common among class 1.0 CTs at a power factor of 0.5 -produces a 3% measured power error compared to the real power. Calibrating out the phase error is critical to ensuring accurate power measurements.