*3.2. Generalization of Controller for the Implementation*

Since the converter of Figure 1 operates in buck and boost conversion modes depending on the value of duty cycle *D*, a general structure for the controller must be determined. The objective is to investigate and determine if the proposed controller appropriately regulates voltage in either conversion mode.

Electrical implementation of the controller (16) requires a simpler mathematical representation. By using the partial fraction expansion of (16), one obtains mathematical expressions whose electrical equivalence is standard and well known. Before synthesizing the electrical arrangement that describes the controller (16), it is necessary to determine the type of roots that will be obtained in a general way. Therefore, by considering the effect of *T<sup>i</sup>* previously described in Figure 4, the controller will be given by (16), *Gc*(*s*) = *kcN*(*s*)/*D*(*s*) (*T<sup>i</sup>* → ∞) or *Gc*(*s*) = *kcD*(*s*)/*N*(*s*) (*T<sup>i</sup>* → 0).

Please note that in all cases, the controller *Gc*(*s*) depends on *N*(*s*) and *D*(*s*), which are numerator and denominator of approximation *s α* . Since both *N*(*s*) and *D*(*s*) of approximation (9) are quadratic polynomials, as long as *a* 2 <sup>1</sup> > 4*a*2*a*0, the roots of controller *Gc*(*s*) will be real. Knowing that *a*0, *a*<sup>1</sup> and *a*<sup>2</sup> depend on 0 < *α* < 1, Figure 8 proofs that condition *a* 2 <sup>1</sup> > 4*a*2*a*<sup>0</sup> holds for every value of *α*, therefore, the partial fraction expansion of *Gc*(*s*) will be given in terms of real poles only as follows,

$$G\_{\mathbb{C}}(s) = \left(\frac{A\_1}{\gamma\_1 s + 1}\right) + \left(\frac{A\_2}{\gamma\_2 s + 1}\right) + \left(\frac{A\_3}{\gamma\_3 s + 1}\right) + \left(\frac{A\_4}{\gamma\_4 s + 1}\right) + A\_5. \tag{18}$$

**Figure 8.** Approximation parameters *a*0(*α*), *a*1(*α*), *a*2(*α*) and values *a* 2 1 , 4*a*2*a*<sup>0</sup> that ensure *a* 2 <sup>1</sup> > 4*a*2*a*0.

Since the first four terms resemble an RC circuit transfer function, the partial fraction expansion of the controller (18) can be directly generated through RC circuits and OPAMPs as inverting amplifiers and in adder configuration. In Figure 9a, the electrical arrangement to implement the controller (18) is shown. Gamma coefficients are the equivalence of multiplying *R*1*C*<sup>1</sup> to *R*4*C*4, and constants *A*'s are the gains of inverting amplifiers obtained by dividing *R*<sup>5</sup> to *R*<sup>9</sup> over *R*.

Constant values to represent fractional-order PID controller approximation (17), whose coefficients are given in Table 3 for buck and boost conversion modes, in its partial fraction expansion (18) are shown in Table 5 columns 1 to 3. Due to resulting gain values for the controller (18), the electrical circuit of Figure 9a is rearranged to consider the sign of *A*<sup>1</sup> and *A*3, thus resulting in the electrical circuit of Figure 9b, whose parameter values are provided in Table 5 columns 4 to 6.

**Figure 9.** (**a**) Electrical representation of partial fraction expansion of controller (18). (**b**) Electrical representation of controller (18) for parameter values of Table 5.

**Table 5.** Constants *A*'s and *γ*'s of controller (18) and parameter values for electrical arrangement of Figure 9b for buck and boost conversion modes.


Note from Table 5 that the value of constants *A*<sup>1</sup> and *A*<sup>3</sup> for both conversion modes are very small and can be neglected. For this reason, the top part of electrical circuit for the controller in Figure 9b can be also omitted in the implementation with no effect in the final result, since its contribution is in the range of *ηV*.

Using PSIM 9.0 (Powersim Inc., 2001-2010), the proposed arrangement was tested through the electrical simulation of the complete system. In Figure 10, the output voltage *V<sup>o</sup>* and inductor current *i<sup>L</sup>* for converter in Figure 1 are shown. Synthesized controllers effectively regulated output voltage in both buck *V<sup>o</sup>* = 15 V (Figure 10a) and boost *V<sup>o</sup>* = 35 V (Figure 10b) conversion modes, while operating the converter in continuous conduction mode, as can be corroborated through inductor current.

On the other hand, implementation results confirmed the viability and effectiveness of the proposed approach. The components for the experiment are all commercial and were obtained from Mouser Electronics, Mexico. The experiment technical characteristics are the following: a very high current capacity inductor 1140-103K-RC of 10mH with ±20%, a DC resistance (DCR)of 2.76 Ω and 10 A. A polypropylene metalized film capacitor of 30 µF with maxDC voltage of 500 V, tolerance of 5% and equivalent series resistance (ESR)of 3.5 mΩ. A high current capability power MOSFET NTP5864NG with maximum drain-to-source voltage of 60 V, continuous drain current of 63 A and *R*ON = 12.4 mΩ. Lastly, a diode SR504 R0 with forward voltage of 0.55 V. The controller was implemented with the high speed, 4 MHz wide bandwidth quad junction field effect transistor (JFET) inputs operational amplifier LF347N and the pulse width modulation (PWM) signal was created with the

traditional TL494. Capacitors and resistances of described values with tolerances of ±5% and ±1%, respectively.

**Figure 10.** Regulated output voltage *V<sup>o</sup>* and inductor current *i<sup>L</sup>* of converter in Figure 1 for (**a**) buck and (**b**) boost conversion modes.

In Figure 11 evidence of the experiment table is shown. From left to right are the oscilloscope, the fractional-order controller approximation, the PWM generator, DC voltage sources and the buck–boost converter with the corresponding load.

**Figure 11.** (**a**) Experiment table with oscilloscope, DC voltage sources and the electrical system. (**b**) Electrical system composed of fractional-order controller approximation, pulse width modulation (PWM) generator and buck–boost converter with the corresponding load.

In Figure 12 the voltage regulation of buck–boost converter can be corroborated in both conversion modes. In Figure 12a,b the output voltage (top signal) corroborates buck mode (*V<sup>o</sup>* = 15 V) and boost mode (*V<sup>o</sup>* = 35 V), respectively.

In Figure 13 the tracking characteristic of buck–boost converter is shown. As can be seen, the controller successfully regulates output voltage with a fast and stable tracking characteristic. It is important to mention the similarity of implementation results with those predicted through Figure 5 and Table 4, column 2, since boost mode regulation exhibits a faster response compared to the one produced in the buck mode.

**Figure 12.** (**a**) Regulation to *V<sup>o</sup>* = 15 V in buck mode. (**b**) Regulation to *V<sup>o</sup>* = 35 V in boost mode.

**Figure 13.** Tracking characteristic of buck–boost converter. (**a**) Buck mode. (**b**) Boost mode.
