DIGITAL CONTROL IN POWER ELECTRONICS
where tr = nok TS is the desired response time (evaluated between 10% and 90% of a step response) for the generic harmonic k and nok is the number of supply periods TS . There is, however, a bandwidth limitation that applies to each harmonic component, given by angular frequency ωL . Indeed, even for angular frequencies below ωL , the transient response of the harmonic component may be lightly damped. As an example, using the parameters of Aside 7, we have set the harmonic component at 75% of ωL (i.e., k = 17). The result is reported in Fig. A8.1, which clearly shows a lightly damped behavior.
[A]
2
IOREFF
0 -2 0
10
20 [ms]
30
40
[A]
2
IO
0 -2 0
10
20 [ms]
30
40
2 [A]
104
εI
0 -2 0
10
20 [ms]
30
40
FIGURE A8.1: From top to bottom: current reference IOREF , current IO , and current error εI when the reference current is at kω0 and a resonant filter tuned at harmonic k is used.
e-j(θ+φk)
ejθ x
K Ik s
x
K Ik s
x
+
I
x e-jθ
mI
+
ej(θ+φk)
FIGURE A8.2: Rotating reference frame controller with phase lead φ k .
This problem can be easily attenuated compensating the delay of the feedback loop by introducing a phase lead effect in the controller. As shown in Fig. A8.2, the phase lead φ k is added when the outputs of the synchronous frame regulators RkDC (s ) are transformed back to the stationary reference frame coordinates. Using theorem (4.20), the relation between synchronous reference frame regulators RkDC (s ) and stationary reference frame regulators