BLONDEL'S THEOREM - MATHEMATICS OF ELECTRICITY METERING

The theory of polyphase watthour metering was first set forth on a scientific basis in 1893 by Andre E. Blondel, engineer and mathematician. His theorem applies to the measurement of real power in a polyphase system of any number of wires.

The Blondel's theorem is as follows:
If energy is 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 the use of N-1 wattmeters.

The receiving and generating circuits may be arranged in any desired manner and there are no restrictions as to balance among the voltages, currents, or power factor values.

From this theorem it follows that basically a meter containing two elements or stators is necessary for a three-wire, two- or three-phase circuit and a meter with three stators for a four-wire, three-phase circuit. Some deviations from this rule are commercially possible, but resultant metering accuracy, which may be decreased, is dependent upon circuit conditions that are not under the control of the meter technician.

An example of such a deviation is the three-wire, single stator meter.

The circuit shown in below may be used to prove Blondel’s Theorem.

Three watthour meters, or wattmeters, have their voltage sensors connected to a common point D, which may differ in voltage from the neutral point N of the load, by an amount equal to EN. The true instantaneous load power is:

WattsLoad = EAIA + EBIB + ECIC

Inspection of the circuit shows:
EA = E'A + EN
EB = E'B + EN
EC = E'C + EN

Substituting in the equation for total load power:
WattsLoad = (E'A + EN)IA + (E'B + EN)IB + (E'C + EN)IC
WattsLoad = E'AIA+  E'BIB + E'CIC + EN(IA  IB  IC)


Since from Kirchhoff’s Law, IA + IB + IC = 0, the last term in the preceding equation becomes zero, leaving

WattsLoad  = E'AIA + E'BIB + E'CIC = W1+ W2+ W3

Thus, the three watthour meters correctly measure the true load power. If, instead of connecting the three voltage coils at a common point removed from the supply system, the common point is placed on any one line, the voltage becomes zero on the meter connected in that line.

If, for example, the common point is on line C, E'C becomes zero and the preceding formula simplifies to:

WattsLoad = E'AIA + E'BIB = W1+ W2

proving that one less metering unit than the number of lines will provide correct metering regardless of load conditions.