The instantaneous value of power p is the product of the instantaneous value of the voltage u across a circuit component and the instantaneous value of the current i in that component. With single-phase ac voltages and currents this product generally assumes positive and negative values during each cycle. Positive values of p indicate energy flow in one direction and negative values indicate energy flow in the opposite direction [11.23].
For a circuit with sinusoidal ac voltage and sinusoidal current displaced in phase by an angle <p, the waveform of power p against time is illustrated in Fig.
11.2. The result is a power pulsation about the mean value at a frequency of twice the system frequency (see Sect. 7.1.7). The mean power (active power) is
1 T
P= -T S 0 uidt. (11.1)
The amplitude of the alternating component of the power oscillation is the apparent power
S=UI. (11.2 )
u(t) t
i(t)
t--
r - - - . 1
t--
Fig. 11.2. Waveform of power p against time with sinusoidal phase-displaced current
The mean power (active power) is
P= VI cos <P (11.3)
where V and I are the rms values of the voltage u and current i. The momentary value p of power is given by
p=ui=P-S cos(2rot-<p). ( 11.4 )
In converters generally non-sinusoidal electrical quantities occur but these are usually periodic. The currents generated in the single-phase power system by the periodic switching processes are only sinusoidal in the special case of ideal filters.
In systems with an impressed sinusoidal voltage, non-sinusoidal currents also cause non-sinusoidal voltage drops across the internal impedance (see Sect. 9.3).
The waveform of power against time as the product of voltage and current is likewise not sinusoidal. The usual definitions of active power, apparent power, and reactive power for sinusoidal voltage and current curves are not, therefore, adequate for the description of the energy conditions in the case of converters. It is necessary to consider the waveform of power against time
p(t) =u(t)i(t) with the resultant energy
w (t) = S p (t) dt = S u (t) i (t) dt.
(11.5 )
( 11.6) Presupposing sinusoidal voltage and non-sinusoidal current the following defi- nitions apply
active power P = VII cos <PI (11.7)
where V is the rms value of the voltage, 11 the rms value of the fundamental component of the current and <PI the phase displacement between the voltage and the fundamental component of the current.
Further definitions which have already been introduced in Sect. 7.1.7, are apparent power S = VI
as the product of the rms values of voltage and current,
and
reactive power Q=VS2_p2,
fundamental frequency reactive power Ql = Ql = VII sin <PI' distortion power D = VVI~ + n .. ,
fundamental frequency content gj = II I'
( 11.8 )
(11.9 ) (11.10) (11.11)
(11.12) The apparent power S, active power P, fundamental frequency reactive power Ql' and distortion power D are related as follows:
(11.13) This can be illustrated graphically by the power tetrahedron (see Fig. 7.16).
Waweform of Power against Time 207
The relationship between active and apparent power has special significance. It is known as the
power factor A= S P = giCOS <Plã (11.14)
The fundamental power factor cos <Pl which is usually known as the displacement factor therefore produces the power factor A only by multiplication with the fundamental frequency content gl i.e. with non-sinusoidal currents A is smaller than the displacement factor cos <Pl. The definitions listed above apply only when a sinusoidal voltage is presupposed. With a non-sinusoidal voltage they are no longer valid.
Other power definitions have been suggested [11.3,11.4,11.5] that can also be employed with non-sinusoidal periodic quantities. The relationship ( 11.1 ) always applies for the active power P.
The apparent power S can also be defined as the product of the rms value of voltage and current (Eq. (11.2)). In addition to this,Troger [11.2, 11.20]
suggested the terms
u(t) t
i (t)
1 T
regenerative power Pr= ~ S [lu(t)i(t)l-u(t)i(t)]dt, (11.15) 2T 0
1 T
transmitted power P d = - S lu (t) i( t) Idt = P + 2Pr ,
T 0
r
(11.16)
t -
Fig. 11.3. Waveform of power p against time with a line-commutated converter (single-phase bridge connection)
and
(11.17) The power quantities are mean values and, for systems loaded with harmonics, lead to unambiguous and physically understandable terms. They can, moreover, be measured.
Figure 11.3 illustrates the waveform of power p for a line-commutated converter. In this example a sinusoidal voltage waveform u ( t) was assumed. The ac current i ( t) is sinusoidal during commutation but is otherwise constant and lags behind the voltage by the angle <Pl. The waveform of p (t) is obtained by mUltiplying voltage and current, the active power P as the mean value of p ( t ) , and the regenerative power Pr as the mean value of the areas under the negative power/time curve.
Summarizing we reiterate that only the waveform of p (t) and the mean power resulting from it have any physical significance. The other terms such as reactive power, distortion power or apparent power are calculated quantities resulting from mathematical analysis of voltage and current into their fundamental and harmonics and active and reactive components.