Using the definition of a time average of a periodic function over the period T
the average power dissipated is found to be
The time average of sine squared may be worked out directly by evaluating the integral implied by the <sin2> in the preceding formula. Alternatively, you may argue that since sin2 + cos2 = 1 and each term has the same shape, shifted by 90 degrees, that it must be true that <sin2> = <cos2> = 1/2. This is a result worth remembering!
The rms value of any periodic time-varying function may be found by squaring the function, evaluating the mean or time-average of the squared function, and using the square root as the result. Note that for sinusoidal functions, the time average of the signal is simply zero, clearly not a useful representation of the function's magnuitude. The rms value of a sinusoidal function is 1/sqrt(2) = 0.707 times the magnitude of the function as shown below.
Multimeters used in ac mode report rms values. For example, the electrical power we commonly encounter at the wall socket is typically 110 to 115 V rms at 60 Hz. An rms emf of 110 V corresponds to a magnitude of 156 V or peak-to-peak value of 312 V.
The other day I mentioned that the cosine of the phase angle, equal to R/Z from the impedance phasor diagram, was called the phase factor. Its appearance in the formulas involving the transfer of energy from the emf to the resistance in a series RLC circuit is shown below.
The power factor is the effectiveness of the emf in supplying power to the load; the maximum value of the power factor is unity. Keep in mind that the phase angle may vary only from -90 to +90 degrees -- the power factor is never negative!
The power factor is an important consideration in effectively delivering power to large inductive loads such as powerful motors. If XL is large compared to R in a circuit, the phase angle will approach 90 degrees, with a low power factor resulting. Power available at the motor is reduced; the current required to operate the machinery properly will be much greater than if the power factor were closer to unity. The power lost in the transmission lines will be greater; the billable power delivered to the factory might involve a surcharge for power wasted in transmission. More typically, power factors are "corrected" for large inductive loads by connecting a low-loss capacitive load to balance the inductive reactance. You may have noticed large capacitor banks outside of large industrial installations...
Correction of the power factor brings the circuit to resonance. So-called impedance matching is often needed when two devices of differing impedances are connected for power transfer, such as an audio amplifier to a speaker. We will find in our next class that transformers are useful in this regard.
Last updated Dec. 17, 1997.
Copyright George Watson, Univ. of Delaware, 1997.