Electrical Energy Meters – Principles and Applications
●The following applies regarding power in alternating current systems:
■P = U · I or P = U · I · cos φ
●For energy in alternating current systems the following:
■W = P · t
●For power in three-phase systems the following:
■P = U₁ · I₁ + U₂ · I₂ + U₃ · I₃
●For power in load balanced three-phase systems the following:
■P = 3 · U · I (1) or P = 3 · ( U-NPD · I ) · cos φ(2)
●And for energy in three-phase systems the following:
■W = P · t
●Delta Voltage - Neutral-Point Displacement Voltage:
■As a rule, delta voltage is used for the calculation of values in three-phase systems.
Delta voltage and neutral-point displacement voltage can be derived from one another using a factor of √3.
▲U-NPD = U-Delta / √3 (3)
●Derivation of the Factor √3:
■The following ensues from the cosine law and the angular relationships demonstrated by the voltage triangle shown below:
▲a = b = U-NPD = 1 ; c = U-Delta ; φ = 120° ; c² = a² + b² - 2 · a · b · cos φ⇒U-Delta = √( 1² + 1² - 2 · 1 · 1 · cos 120° ) = √( 1 + 1 - 2 · ( -0,5 ) ) = √3
●If the above equation (3) is applied to the power equation, (1) or (2), the following results:
■P = 3 · U-Delta / √3 · I · cos φ
●Because it is generally assumed that we are concerned with delta voltage, or phase-to-phase voltage, when dealing with three-phase systems, the index is ignored.
●The quotient equation 3 / √3 = √3 results in the following:
■P = √3 · U · I · cos φ
●Respective power „P“ must be multiplied by time „t“ for the measurement of energy.
●The Technical Realization of Energy Measurement:
■The measurement of energy is accomplished by means of a voltage-frequency converter connected downstream from the power meter. The individual pulses are then summated through the use of an electromechanical meter, and are made available at a pulse output as well.
A single-phase meter is used in alternating current systems.
A measuring system with three multipliers is required for 4-wire three-phase systems of unbalanced load with neutral conductor (N). The „three-wattmeter method“ can also be used in the absence of a neutral conductor if an artificial neutral is available. This method results in a highly accurate measurement if a precision wattmeter is used.
However, 3-wire three-phase systems are commonly found in industrial applications, for which ARON circuits are used. This type of measuring circuit offers cost advantages because it allows for the measurement of power and energy with only two current transformers at phase conductors L1 and L3. However, this measurement only provides correct results if the vector sums of all currents to be measured in the system are equal to zero (I1 + I2 + I3 = 0). This condition can only be fulfilled when no currents flow to earth (leakage current of a capacitive, an inductive or a resistive nature).
●The theory of the ARON circuit can be demonstrated with the following equations:
■The terms (UL1 - UL2) and (UL3 - UL2) represent delta voltage.
UL1, UL2 and UL3 are the corresponding conductor voltages to earth or any desired virtual reference point. The above equations are to be interpreted vectorially.
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3/10.09 |
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