Page 76 - Modern Control of DC-Based Power Systems

Small-Signal Analysis of Cascaded Systems 41

2.5.1.2 Modeling and Design Procedure of Voltage Mode Control
With PID Controller
This procedure allows designing the voltage controller G c ðsÞ of a con-
verter in VMC (see Fig. 2.4). The procedure is very similar to that one
for the outer voltage controller G c V sðÞ of the converter in PICM_FB.
Of course, the designer is required to use the proper plant transfer func-
tion, i.e., the control-to-output transfer function G vd ðsÞ defined in (2.2).
However, the phase of G vd ðsÞ for the basic converter topologies at high
frequencies either approaches 180 degrees for the buck converter or
270 degrees for the buck boost and boost converters [22]. Therefore,
for the frequency range where the designer wants to close the voltage
loop, there is no way with a PI control to “bump the phase up” to obtain
a positive phase margin. The only available option is PID control. In fact,
the derivative component of a PID control has the “phase lead effect” on
the converter closed-loop characteristics [22]. This allows obtaining a
positive phase margin. However, the derivative component slightly com-
plicates the procedure to design the voltage controller since a third coeffi-
cient needs to be found.
The voltage control loop gain is the following expression:
(2.69)
T VM ðsÞ 5 G c ðsÞ G vd ðsÞ
As usual, it is required that the loop gain of T VM ðsÞ has a certain phase
margin PM VM at the control bandwidth ω c VM .From (2.69), the follow-
ing equations are derived.

(2.70)
T VM ðjω c VM Þ 5 1 5 G c ðjω c VM Þ G vd ðjω c VM Þ

arg T VM jω c VM 52 180 1 PM VM
(2.71)

5 arg G c ðjω c VM Þ 1 arg G vd ðjω c VM Þ
The latter equation can be rewritten as follows.

(2.72)
arg G c ðjω c VM Þ 52 180 1 PM VM 2 arg G vd ðjω c VM Þ

Notice, again, that arg G c ðjω c VM Þ is function of known quantities.
Classical PID controllers have the following transfer function
K i
G c sðÞ 5 K p 1 1 K d s (2.73)
s