Multi-terminal network power measurement

International Journal of Electrical Engineering Education, Apr 2002 by Quintela, Felix Redondo, Melchor, Norberto Redondo

Abstract This paper presents a simple and general theory on power absorbed by multi-terminal networks and the way to measure it, which has direct application to unification of active and reactive power measurement procedures of polyphase loads. This approach to the subject is highly efficient, not only for its simplicity, but also because it eliminates all particular demonstrations or proofs of different measurement methods, with consequent time and effort savings for students. It also offers absolute certainty about the scope of every power measurement procedure and their possible variations.

Keywords active power; multi-terminal network; power measurement; reactive power; wattmeter

Different power measurement procedures applied to polyphase loads have historically been developed separately, normally to give single solutions to specific problems1,2 and extending the arrangement for monophase loads to three-phase systems.3,4 Text books and manuals still present this subject with particular case-by-- case demonstrations.5,6 This form of presentation consequently needs to justify the adequacy of the proposed measurement methods for each type of three-phase load (wye, delta, balanced) and the same for other polyphase loads. In addition to assimilation difficulty for students, this approach generates great uncertainty with respect to possible or impossible wattmeter connections, depending on the actual load configuration. For example, the student knows that, in order to measure the power of a Y-connected three-phase load with neutral, he can use the three wattmeters method as shown in Fig. 1 (a). But he might ignore whether the configuration of connections of Fig. 1(b), where one of the wattmeters is placed on the neutral and some monophase, wye and delta loads are connected, will offer the power absorbed by the whole set. The student may also know that two wattmeters connected as shown in Fig. 10, with their respective current coils on the phases, can measure the two-phase load power, but he might not know if the same procedure is applicable when a Z^sub 3^ extra impedance is added to the former load (Fig. 12), or when a wattmeter is placed on the neutral instead of on one of the phases (Fig. 11(a)). In general, the information the student receives contributes to his uncertainty on matters such as these.

As will be proven, we can arrive at a unified and very easy-to-follow demonstration valid for all the cases of polyphase receivers and, in fact, for every multiterminal network, whatever its excitation may be. The following demonstration would eradicate the possibility of common doubts. From the concept of multiterminal network power, every possible connection of wattmeters for power measurement is shown at once and with minimum effort.

Conclusion

The key issue of this paper is the important and trivial theorem represented by expression (2), duly demonstrated in the accompanying text, which gives the single formula to calculate the instantaneous power absorbed by a multi-terminal network. It serves in turn as essential basis to develop its homologue in complex power of expression (9). Thus, a general theory of multi-terminal network power measurement is formally deduced, and is immediately applicable to poly-phase load power measurement.

Due to its simplicity, it is preferable to present the subject of multi-terminal network power measurement in this unified manner. In particular, if the multiterminal network is a poly-phase load, the need to justify each measurement method separately, as is usually done in textbooks, is eliminated, with consequent time and effort savings for the students. Furthermore, this presentation provides great certainty with respect to the adequacy of all possible connections of wattmeters and varmeters in order to measure polyphase load power. This certainty has never been supplied by the particular proof of each individual case.

We have also illustrated that measurement procedures with wattmeters, based on this method, always give the average value of the power, whatever the waveform of the excitation may be.

References

1 Moeller-Werr, Elektrische Messtechnik (B. G. Teubner, Stuttgart, 1960).

2 J. Brenner, Analisis de Circuitos Electricos (Ediciones del Castillo, Madrid, 1966).

3 S. Franco, Electric Circuits Fundamentals (Saunders College, Fort Worth, 1994).

4 J. David Inving, Basic Engineering Circuit Analysis (Macmillan, New York, 1990).

5 R. C. Dorf, Introduction to Electric Circuits (John Wiley, Chichester, 1993).

6 D. E. Johnson, J. L. Hilburn and R. J. Johnson, Basic Electric Circuit Analysis (Prentice Hall, Englewood Cliffs, 1990).

7 P. Penfield, R. Spence and S. Duinker, A Generalized Form of Tellegen ' Theorem, IEEE Trans. Circuit Theory, 17(3) August (1970).

8 V. Parra, J. Ortega, A. Pastor and A. Perez, Teoria de Circuitos (UNED, Madrid, 1992).

Felix Redondo Quintela and Norberto Redondo Melchor

Escuela Tecnica Superior de Ingenieria Industrial, Universidad de Salamanca, Bejar, Salamanca, Spain