2.2. Power of Distorted Waves: Active Power Is the Sum of the Products of Voltage, Current, and Phase of Each Frequency Component
Ideal active power is expressed as the product of the instantaneous voltages and currents averaged over one period of voltage or current. However when voltage, current, and power from distorted waves are included, the voltage, current, and active power are expressed as follows.
*n is the number of harmonic orders, U and I are the nth order component voltage and current RMS values, and φn is the nth order component voltage and current phase difference.
We know that the active power from the voltage and current of the distorted wave is the sum of the active powers obtained from the products of the voltages, currents, and power factors of the same harmonic component (frequency). The value of the products of voltages and currents from differing frequency components is 0, which tells us that this can not be the active power. When measuring the active power, it is enough to use a measuring instrument having lower-frequency bandwidth characteristics even if either the voltage or current has a high frequency component.
2.3. Three-Phase Power
· Three-Phase Power Is the Sum of Each Phase
Three-phase power can be determined by summing the three phases of power as measured by a wattmeter (see figure 3). However, as shown in figure 4, a neutral wire is not always present in actual power systems. In such cases, Blondel’s Theorem allows one to find the power by summing the results from two wattmeters, as in figure 4.
Figure 3: Method of measuring three-phase power with three wattmeters
Figure 4: Three-phase, three-wire measurement using the two wattmeter method
· Blondel’s Theorem
If energy be supplied to any system of conductors through N wires, the total power in the system is given by the algebraic sum of the readings of N wattmeters, so arranged that each of the N wires contains one current coil, the corresponding voltage coil being connected between that wire and some common point. If this common point is on one of the N wires, the measurement may be made by use of N-1 wattmeters.
Rather than using three wattmeters to measure power from each phase of voltage and current with reference to a neutral wire, the two wattmeter method involves measurement of power from two line voltage between the terminal and two phase current. In theory, the total three-phase power value will be the same with either method. This is explained in figure 5 below using vectors.
With the two wattmeter method, the measured value on each power meter is different because the phase difference of each line voltage and phase current is different. Depending on the relationship between phase voltage and phase current, negative power can also be measured because the phase difference between line voltage and phase current can exceed 90 degrees. Thus, ultimately the power value represents only the total value of three-phase power. Also, if the two wattmeter method is used—even with three-phase unbalanced—the active power can generally be measured accurately based on the three-phase voltage and current vector diagram in figure 5. But in cases where the sum of the vectors of currents of each phase do not equal 0 (such as when current is flowing in the neutral wire), the UT x (IR + IS + IT) portion of the equation above does not equal zero, with the result that this portion is a measurement error relative to the value displayed on the wattmeter.
Figure 5: Vector diagram of three-phase voltage and current
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