EVALUATION

Evaluation

Evaluation

Let us consider the transmission of energy from a source to a load, through a given surface as in Fig. 1. In the network of Fig. 1, the voltage waveform v(t) (not necessarily sinusoidal) is given by the source, and the current waveform is determined by the response of the load. In the more general case in which the source output impedance is significant, then v(t) and i(t) both depend on the characteristics of the source and load. If v(t) and i(t) are periodic, then they may be expressed as Fourier series:

where the period of the ac line voltage waveform is defined as T = 2p/w. In general, the instantaneous power p(t) = v(t) i(t) can assume both positive and negative values at various points during the ac line cycle. Energy then flows in both directions between the source and load. It is of interest to determine the net energy transmitted to the load over one cycle, or

This is directly related to the average power as follows:

Let us investigate the relationship between the harmonic content of the voltage and current waveforms, and the average power. Substitution of the Fourier series, Eq. (1), into Eq. (3) yields

To evaluate this integral, we must multiply out the infinite series. It can be shown that the integrals of cross-product terms are zero, and the only contributions to the integral comes from the products of voltage and current harmonics of the same frequency:

The average power is therefore net energy is transmitted at the third harmonic frequency, with average power equal to

Here, V3I3/2 is equal to the rms volt-amperes of the third harmonic current and voltage. The cos(f3-q3) term is a displacement term which accounts for the phase difference between the third harmonic voltage and current. Some examples of power flow in systems containing harmonics are illustrated in Figs. 2 to 4. In example 1, Fig. 2, the voltage contains fundamental only, while the current contains third harmonic only. It can be seen that the instantaneous power waveform p(t) has a zero average value, and hence Pav is zero. Energy circulates between the source and load, but over one cycle the net energy transferred to the load is zero. In example 2, Fig. 3, the voltage and current each contain only third harmonic. The average power is given by Eq. (7) in this case. In example 3, Fig. 4, the voltage waveform contains fundamental, third harmonic, and fifth harmonic, while the current contains fundamental, fifth harmonic, and seventh harmonic, as follows:

Average power is transmitted at the fundamental and fifth harmonic frequencies, since only these frequencies are present in both waveforms. The average power is found by evaluation of Eq. (6); all terms are zero except for the fundamental and fifth harmonic terms, as follows:

The instantaneous power and its average are illustrated in Fig. 4(b).

Root-mean-square (RMS) value of a waveform

The RMS value of a periodic waveform v(t) with period T is defined as

The rms value can also be expressed in terms of ...