*4.5. Feedback Compensator Design*

Stability, steady state error, and response time are three important criteria for feedback compensator design. The feedback compensator is designed based on the following steps:

**Figure 9.** Bode-plot of open loop and compensated loop transfer functions: (**a**) magnitude, (**b**) phase.

4.5.1. Crossover Frequency Selection

The first step of controller design is to select crossover frequency, *fc*. To make a stable controller *fc* must be slightly higher than the resonance frequency of *LC* filter, *fLC*. The resonance frequency for the *LC* filter is defined by (41).

$$f\_{\rm LC} = \frac{1}{2\pi\sqrt{\rm L\dot{C}}}\tag{41}$$

The rule of thumb to design an efficient and stable controller is to choose crossover frequency less than 1/10th of switching frequency *fSW*. The crossover frequency selection criterion can be expressed by (42).

$$f\_{\rm LC} < f\_{\rm c} \le \frac{1}{10} f\_{\rm SW} \tag{42}$$

Considering the value of *fLC* and *fSW* the value of *fc* = 2.5 kHz is selected.

## 4.5.2. Gain Adjustment

The compensated loop gain at cross over frequency, |*GidGc*| *f c*, should be 0 dB which can be expressed by (43).

$$|G\_{id} G\_c|\_{fc} = 1\tag{43}$$

where, *Gc* is controllers transfer function. (43) can be re-written as (44)

$$k = |\mathbf{G}\_{\mathbf{c}}|\_{fc} = \frac{1}{|\mathbf{G}\_{id}|\_{fc}}\tag{44}$$

where, *k* is the factor for gain adjustment.

The value of |*Gid*| *f c* is 18.96 dB i.e., 8.87, therefore the value of *k* is 0.11. This system is reducing gain instead of boosting. Gain reduction is necessary to improve stability.
