The electrical power for direct electrical current is P = UI. So, the power that consumes a load is equal to the product of the voltage between it's ends and the current which flows through it. In case of alternating electrical current, the power is equal to product of RMS values of the voltage and current, and since they can be phase shifted one from another, there is an extra member in the product, and that's the cosinus of the phase angle between voltage and current (Fi). In the case when voltage and current are in phase (cos(0) = 1), the formula for the power will be same as for the DC current. The cos(Fi) in this formula is called power factor.
Picture 1: Formulas for DC and AC Electrical Power
In case of AC electrical load, we have: Apparent power S = UI; Barren Power Q = UI sin(fi); Real Power P = UI cos(fi). The formulas for these powers are also shown on Picture 1. The real power is measured in watts [W], the barren power in voltamperes-reactive [VAr] and the apparent power in voltamperes [VA]. Actually, the unit of measure for all three powers is watt, but for better expression and for emphasizing their different "nature" we use different units (W = VA).
The methods for measurement of the AC Electrical power depends on the parameters of the electrical circuit, the frequency, the voltage, the power, and so on. Since the barren and the real power depends on the phase angle between the voltage and the current, the instruments used for their measurement must be phase-sensitive.
Measurement of the Real power of three-phase electrical system
Measurement method with one Wattmeter
Wattmeter is an instrument used for direct measurement of real power P. Using Watt-meter we can measure the real power of the AC load with power factor cos(fi) < 1. With one Wattmeter we can measure the real power of the three-phase AC load only if the load is symmetrical, which means that amplitudes of the current of each phase are equal. The advantage of this method for measurement is that we use only one Wattmeter, but the disadvantage is that we can only measure power of symmetrical systems.
Measurement method with two Wattmeters (Aron circuit)
With this method we can measure the real power of three-wire three-phase system, and the Wattmeters W1 and W2 are connected as in circuit shown on Picture 2. Here we measure the total average real power of the system, which is defined with formula shown on Picture 3. If the phase angles Fi1 and Fi2 are less than 90 degrees, then the total average power is equal to sum of the results from both Wattmeters, because the the both articles in the power formula are positive. If only one of the phase angles is bigger than 90 degrees, and the other phase angle is less than 90 degrees, then the average power is equal to the difference between the results from the Wattmeters, because one of the articles is negative (in this case we have a negative slant of the Wattmeter). So, the total measured power of the system can be calculated with addition or subtraction of the results from Wattmeters, but the single result from one Wattmeter is not the power in single phase.
The advantage of this method is that we are saving one Wattmeter and we can measure the power of unsymmetrical systems, but the disadvantage is that we do not measure the powers by each phase but we measure the total power.
Picture 2: Measurement method with two Wattmeters (Aron circuit)
Picture 3: Formula for calculating the real power using two Wattmeters
Measurement method with three Wattmeters
With this method we can measure the real power of three-wire three-phase system, and the three Wattmeters W1, W2 and W3 are connected as in circuit shown on Picture 4. Here we measure the total average real power of the system, which is defined with formula shown on Picture 5. The total power is equal to the arithmetic sum of the all three results from each Wattmeter, and each result from each Wattmeter individually is equal to the power that is wasted in that phase.
The advantage of this method is that we can measure the power of unsymmetrical systems and the individual power of each phase, but the disadvantage is that we need three Wattmeters.
Picture 4: Measurement method with three Wattmeters
Picture 5: Formula for calculating the real power using three Wattmeters
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