conductor, if we measure the current in amperes and the difference of potential between the ends of the circuit in volts, the product of these two numbers gives us the power taken up in the circuit measured in watts. In this case two simple measurements give the required rate of dissipation of energy in the conductor. If, however, we have to deal with an alternating current circuit, in which the current strength is varying from instant to instant, according to a periodic law, and if likewise, the difference of potential between the ends of the circuit is varying in the same periodic manner, we cannot always obtain the measurement of the mean power taken up in the circuit, by taking the product of the root-mean-square value of the amperes and rootmean-square value of the volts. What we really require in this case is the mean value of the power taken up in the circuit. We can obtain the measurement of the mean power if we can measure at every instant the true value of the current strength and the difference of potential. Suppose these instantaneous values of the current and pressure known at equidistant intervals taken throughout one complete period. If we then multiply the instantaneous value of the current by the corresponding value of the pressure, or difference of potential, we obtain a number representing the instantaneous value of the power, and if we imagine the period divided into a large number of equidistant intervals of time, and those products taken at every such instant, then the mean value of these products taken throughout the period will give us a close approximation to the mean value of the power being absorbed by that circuit..

We have seen in a previous chapter that it is possible to determine and describe curves representing the instantaneous values of the current and electromotive force in the case of an alternating current circuit, but this in general is not a simple matter to do, and we have therefore to resort to other methods of obtaining the required power measurement.

In dealing with the power taken up in alternating current circuits, there are two cases to be considered.

The first case is that in which the circuit is noninductive. In that case, as before explained, the impedance of the circuit is the same as its resistance, numerically speaking. For such circuits the alternating current flowing in the circuit is in step, as regards phase, with the alternating potential difference between its extremities. When this is the case the power taken up in that circuit can very easily be measured. If we measure the root-mean-square value of the alternating current by means of any of the balances or dynamometers already described, and if by means of any of the electrostatic or thermal voltmeters we measure the root-mean-square difference of potential between the ends of the circuit, and multiply these two mean-square values together, we obtain the mean value of the power taken up in the circuit, and we arrive at the same result as if we had been able to measure separately the instantaneous values of the current and potential difference at numerous equidistant intervals throughout the phase, and taken the mean value of their products.

As an instance of this, it may be pointed out that an incandescent lamp may be treated as a practical noninductive circuit. If an incandescent lamp is traversed by an alternating current, and we measure the current flowing through the lamp by means of, say, a Siemens dynamometer, and the potential difference between the terminals of the lamp by means of an electrostatic voltmeter, and if we multiply the two scale readings of these instruments together, we obtain the mean value of the power measured in watts taken up in the lamp.

So far then all is quite simple, and in dealing with any circuit which we know, or can prove to be, practically non-inductive, we have no difficulty, by means of two instruments of the proper kind, in determining the real mean power taken up in the circuit. Our difficulties come in when we have to deal with circuits such as

Y

transformers, which, when not fully loaded, we know to be inductive. If, in this case, we can determine numerous instantaneous equidistant values of the current, and difference of potential between the ends of the circuit, then proceeding as above described we can graphically find the mean value of the power taken up in the circuit. If, however, it is not convenient to do this, we cannot

Fig. 107.-Power Curves for a Non-inductive Circuit.

proceed to measure the root-mean-square values of the current and electromotive force, and then multiply them together. Such a proceeding would lead to a considerable over-estimate of the real power taken up in the circuit. Without going into elaborate proof of this, it may be simply sufficient to present the following figures.

In Fig. 107 are shown two simple harmonic curves

in step with one another. Let the semi-period of each curve be divided into eight equidistant parts, and ordinates be erected at each point. The values of these ordinates for the two curves represent equidistant instantaneous values of these periodic current and electromotive force curves. By squaring each of the values of the ordinates, and taking the square root of the mean of the squares, we obtain for each curve a number which would represent the instrumental value obtained by an alternating current dynamometer or voltmeter. If we multiply together the simultaneous values of current and electromotive force, we obtain a number, given by the dotted curve, which represents the instantaneous value of the power taken up in the circuit, and if we take the mean value of all these separate instantaneous values of the power, we obtain the same number as we do if we take the products of the square roots of the mean of the squares of the instantaneous values of the current and electromotive force. Hence, we see that when the two simple harmonic curves are in step with one another, the product of the square roots of the mean of the squares of the separate ordinates is equal to the mean value of the products of the corresponding ordinates.

In Fig. 108 are shown two periodic curves, which may be taken to represent a current and an electromotive force curve, but one of which is displaced backwards relatively to the other. This is what happens in an inductive circuit, where the periodic current always lags in phase behind the periodic electromotive force.

If we perform the same operations on the ordinates of these curves, we find that the product of the root-meansquare values for the two separate curves is in excess of the mean of the product of the instantaneous equidistant values for the two curves. In other words, if in such a circuit we measure the current by means of a dynamometer, and the potential difference between the ends by ineans of an alternating current voltmeter, the product of these two numbers gives us a number which is in

excess of the true value of the mean power taken up in the circuit.

We can at once measure the power-absorption of an energy transforming apparatus, when using continuous currents, by the employment of an ordinary direct cur

rent ammeter to measure the current going into it, and a voltmeter to measure the terminal volts, the instrument being arranged as shown in Fig. 109.

In this use of the instruments, the voltmeter must either be an electrostatic voltmeter or else of very high resistance if a current-taking instrument. Otherwise, if a rather low resistance voltmeter is used, the current which it takes up is reckoned into the reading of the ammeter; and a correction must be made for it before we can obtain the true value of the current taken by the device being tested.

The potentiometer can, of course, be used to make