The behavior of a single-phase ac system coupled to a dc system via a pulse converter with commutation on the dc side shall now be examined more closely [11.14, 11.15, 11.18, 11.30].
Figure 11.16a shows the circuit considered. It is assumed that the commutator is driven at constant pulse frequency fp. The relative duty factor A. = T tiT p is variable. To generate sinusoidal variables on the ac side A. must vary sinusoidally with the system frequency (see Sects. 8.3.3 and 8.3.4).
To study the limiting values of very high pulse frequency (fp--+oo) the inductance L on the ac side and the smoothing capacitor Cd on the dc side can be very small (L--+O and Cd--+O). From the power balance between the ac and dc sides it follows that
( 11.37) The assumed characteristics of the commutator permit three states for the voltage v ( t) on the ac side of the commutator
ud(t)
v(t) = 0 (11.38 )
Accordingly for the current id ( t) on the dc side of the commutator i(t)
id(t) = 0 (11.39 )
-i(t)
Assuming very high pulse frequency and hence any size of store L and Cd where Ud~V2U the current i(t) can be set to any value. The only actuating variable necessary for this is the variable duty factor of the commutator. Generally a sinusoidal waveform of current i ( t) is desired, i ( t) = i sin ( rot - cp ). In this case the power p ( t) on the ac side is
fti [ cos ( 2rot - cp ) ]
p ( t) = -2 [cos cp - cos ( 2rot - cp )] = P d 1 - . ( 11.40 )
. cos cp
0.5
o
-0.5
-T.O
~ ..L
u :
T'Ol
0.5
o
a
r-o--l.L __ -+-(x----. --
'---+-0=::.-- --
~ i u :
T'Ol 1.0
D.; 0.5
b c -T.O
Fig. 11.168 - c. Voltage and current waveforms for a pulse converter with commutation on the dc side. 8 Circuit with equivalent commutator and filter; b with active power; c with inductive reactive power
The power p ( t) therefore oscillates about the mean value P d at twice the system frequency and with an amplitude of P wlcos cp (see Eq. (11.23)).
From the power balance between the ac and dc sides it follows that at constant dc voltage Ud the current id (t) at the dc side output of the commutator
(11.41)
Under the assumed conditions this current therefore pulses about the mean value Id as does the power on the ac side at twice the system frequency. In order that only the desired constant current flows in the dc voltage source Ud a filter circuit tuned to twice the system frequency must be added in parallel with the dc voltage source.
This absorbs the ac portion of id ( t ). This filter circuit must absorb the power pulsations of the single-phase ac voltage source independent of the pulse frequency.
Pulse Frequency 225 Figures 11.16b and c illustrate the voltage and current waveforms when active power and inductive reactive power are transmitted via a pulse converter in single- phase bridge connection. The current and voltage waveforms were determined by simulation. A pulse frequency fp = 42fN, sinusoidal variation of the duty factor A.
between the values 0 and 1, and a reactance factor Uk = roLIdU on the ac side of 30% are assumed. Making these quantitative assumptions the current i on the ac side is approximately sinusoidal. The remaining harmonics are determined by the areas under the curve of the difference between the voltage v and its fundamental component v t and by the magnitude of the inductance L on the ac side.
On the dc side the ac current i2 superimposed on Id and pulsating at twice the system frequency is absorbed by the filter circuit. The magnitude of this current is independent of whether active current or reactive current is being transmitted. The ripple of the dc voltage Ud is determined by the area under the waveforms of the current in the smoothing capacitor Cd'
The coupling of a multi-phase ac system with a dc system via a pulse converter with commutation on the dc side shall also be considered. Such a pulse converter can for example be realized in three-phase bridge connection with semiconductor switches for both directions of current (capable of being turned off in one direction and uncontrollable in the other).
Figure 11.17a shows the principle connection with equivalent commutator.
The commutator works at pulse frequency fp and connects each ac terminal alternately to the positive and negative dc terminal, the duty factor A. determining the actual positions of the three commutator switches. First ideal conditions shall be assumed, namely that the commutator operates with a very high pulse frequency (fp -+00 ) . In this case the inductances on the ac side and the smoothing capacitor on the dc side can shrink to any size as they can with the single-phase bridge connection [11.13, 11.19, 11.20, 11.21, 11.22].
With sinusoidal and symmetrical currents on the three-phase ac side the sum of the power flow from there is constant.
p(t) =Ut (t)it (t) +u2(t)i2(t) +u3(t)i3(t) =3UI cos cp=const=Pd (11.42 ) The power balance is therefore satisfied at any instant in this case without supplementary energy stores. The filter circuit on the dc side is omitted. The amplitude and phase displacement of the current i ( t) in the ac voltage sources are, in principle, continuously adjustable under the above assumptions so long as the condition Ud~2V2u is satisfied.
In Figs. 11.17b, c, and d, the current and voltage waveforms are reproduced again for operation with active power and with inductive and capacitive reactive power. The waveforms were determined quantitively by simulation. The pulse frequency fp is again 42 fN' the duty factor A. varies sinusoidally with the system frequency between the v~lues 0 and 1, the reactance factor nk on the ac side is 30%.
The current it on the ac side is approximately sinusoidal, the remaining current harmonics are determined by the area under the curve of the difference between the converter phase voltage v t and its fundamental component and by the size of the inductance L on the ac side.
a
1.0 0.5
o 0 -0.5 -0.5 -1.0
-1.0
T Ud -.-
I U
2 1.0 0.5 0 -0.5 c
1.0 0.5 0.5
o 0 -0.5 -0.5
-1.0 -1.0 i Ud
wt ---
;:t:-
b wl---
1.0 0.5
o
-1.0
1.0 0.5
o
-0.5 d
Fig. 11.17a - d. Voltage and current waveforms for a pulse converter with commutation on the dc side (three-phase bridge connection). a Circuit with equivalent commutator and filter; b with active power; c with inductive reactive power; d with capacitive reactive power
On the dc side there is still only a high-frequency power pulsation at the pulse frequency which is absorbed by the smoothing capacitor Cd' On reactive power operation the mean value ofthe dc current Id disappears. The mean value of the dc voltage Ud drops with inductive reactive power in the three-phase system and rises with capacitive reactive power. Approximately
(11.43 )
Pulse Frequency 227
Since the harmonics superimposed on the ac current vary with the magnitude of the dc voltage U d, they attain their maximum value with capacitive reactive power in the three-phase ac system.