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Renewable energy sources require switching regulators as an interface to a load with high efficiency, small size, proper output regulation, and fast transient response. Moreover, due to the nonlinear behavior and switching nature of DC-DC power electronic converters, there is a need for high-performance control strategies. This work summarized the dynamic behavior for the three basic switch-mode DC-DC power converters operating in continuous conduction mode, i.e. buck, boost, and buck-boost. A controller was designed using loop-shaping based on current-mode control that consists of two feedback loops. A high-gain compensator with wide bandwidth was used in the inner current loop for fast transient response. A proportional-integral controller was used in the outer voltage loop for regulation purposes. A proce dure was proposed for the parameters of the controller that ensures closed-loop stability and output voltage regulation. The design-oriented analysis was applied to the three basic switch-mode DC-DC power converters. Experimental results were obtained for a switching regulator with a boost converter of 150 W, which exhibits non-minimum phase behavior. The performance of the controller was tested for voltage regulation by applying large load changes.

Switch-Mode Power Supplies (SMPS) were first developed for aerospace applications in the 60’s. The interest in DC-DC switched power converters has increased due to the many DC electrical energy applications from renewable energy sources like fuel-cell stacks, photovoltaic arrays, or wind power [

The dynamical study of these converters is crucial for control purposes [

In this paper, a discussion of the steady state and dynamic behavior is given for the three basic converters, i.e. buck, boost and buck-boost. Additionally, a current-mode controller design is given with the corresponding parameter selection criteria to ensure stability and performance for these converters. A physical implementation is shown based on low-cost operational amplifiers. To show the performance of the controller, an experimental 150 W boost prototype was built to validate the results given within, where step changes are applied to the output load.

This work is organized as follows. In Section 2, analysis, description and dynamic behavior of the three basic switch-mode PWM converters are discussed. Section 3, details the design of a current-mode controller with the corresponding parameter selection process. Experimental results validating the controller performance are shown in Section 4, and finally some concluding remarks are addressed in Section 5.

The demand for high-performance PWM power converters has increased due to the use of DC electric renewable sources. The primary role of power conversion equipment is to facilitate the transfer of power from the source to a specified load by adjusting the voltages and currents from one form to another. This equipment must be energy-efficient and reliable with a high power density; thus, reducing their size, weight, and cost. To fulfill the above requirements, it is essential to understand the converter topology’s steady state and transient behavior. In this sense, the operation modes widely used are continuous conduction mode (CCM) and discontinuous conduction mode (DCM), associated with high and low power density applications. In CCM, the inductor current should never cross zero during one switching cycle. In DCM, the inductor current ripple is large enough to cause the inductor current falls to zero. Now, the description and dynamic operation of the three basic power DC-DC converters in CCM are highlighted.

The growth of renewable energy sources integration has brought new applications for buck converter topology, i.e. battery chargers in photovoltaic and wind energy systems [

The basic topology of this converter is show in _{O}.

In this case, the resulting average output voltage V_{O} in steady state is smaller than the input voltage E. Moreover, the average output current I_{O} is greater than the average source current I_{S}. In the conventional buck converter, shown in _{2}.

In steady state CCM operation, the voltage and current ripples due to the switching are

Δ V C = D ( 1 − D ) E 8 f S L C , Δ I L = D ( 1 − D ) E f S L (1)

where f_{S} is the switching frequency. Additionally, the inductor value must be selected such as L > ( 1 − D ) R / 2 f S to ensure CCM.

In this work, state-space averaging is used, which is a modeling technique widely used to “approximate” the behavior of switching converters [

i ˙ L = 1 L ( − v O + E d ) (2a)

v ˙ O = 1 C ( i L − v O R ) (2b)

where the state variables are i L and v O , and the control input is the duty cycle d. Assuming that each state variable and the control signal are the sum of DC and AC components, they can be decomposed as

i L = I L + i ˜ L v O = V O + v ˜ O d = D + d ˜ (3)

where (˜) stands for AC variables. Note that the AC terms are equal to zero in steady state. The steady state values for the average output voltage and average inductor current are

V O = E D , I L = E D R (4)

Consequently, taking into account the AC terms, the linear average small-signal model for the buck converter is given by

[ i ˜ ˙ L v ˜ ˙ O ] = [ 0 − 1 L 1 C − 1 R C ] [ i ˜ L v ˜ O ] + [ E L 0 ] d ˜ (5)

The model shown in (5) describes the behavior for frequencies up to f S / 2 ; thus, the corresponding transfer functions results as

i ˜ L ( s ) d ˜ ( s ) = V O D R × R C s + 1 L C s 2 + L R s + 1 (6a)

v ˜ O ( s ) d ˜ ( s ) = V O D × 1 L C s 2 + L R s + 1 (6b)

where both transfer functions have a minimum phase behavior, i.e. there is no right-half plane (RHP) zeros; therefore, control design is easy to carry out.

The step-up power converter, commonly known as a boost converter, is shown in

inductor L, an active switch M, a diode D1, an output capacitor C and the load R. The main characteristic of this converter is that, in steady state, the average output voltage V_{O} is greater than the input E; henceforth, the name boost.

Due to its nature, this type of converter is used in applications where the source voltage needs to step up to higher levels, i.e. front-end stage for photovoltaic. If a higher power is required, an interleaved converter with two paths can be used.

In steady state CCM operation, the voltage and current ripples for the boost converter due to the switching action are computed by

Δ V C = D I O f S C and Δ I L = D E f S L (7)

additionally, to ensure CCM, the inductor value must be selected as L > D ( 1 − D ) 2 R / 2 f S . Notice that the inductor current is equal to the source current. In contrast to the buck converter, this topology needs a larger filter capacitor C to limit the output voltage ripple.

In the boost converter, it is possible to average the dynamical behavior of this converter by neglecting the ripple phenomena. Thus, applying Kirchhoff laws when M is ON/OFF, the average continuous nonlinear model is obtained as

i ˙ L = 1 L ( − ( 1 − d ) v O + E ) (8a)

v ˙ O = 1 C ( ( 1 − d ) i L − v O R ) (8b)

The nonlinear differential equations in (8) are said to be bilinear, since the input signal d is multiplying the states variables v_{O} and i_{L} directly. The steady state values for the output voltage and inductor current can be easily obtained resulting in

V O = E 1 − D , I L = E ( 1 − D ) 2 R . (9)

Consequently, taking into account the AC terms defined previously in (3), yields to the linear average small-signal model for the boost converter as

[ i ˜ ˙ L v ˜ ˙ O ] = [ 0 − 1 − D L 1 − D C − 1 R C ] [ i ˜ L v ˜ O ] + [ E ( 1 − D ) L − I L C ] d ˜ (10)

where the bilinearity has been eliminated and the resulting matrix has only constant values. The resulting transfer functions of (10) can be expressed as follows

i ˜ L ( s ) d ˜ ( s ) = V O ( 1 − D ) 2 R × R C s + 2 L C ( 1 − D ) 2 s 2 + L ( 1 − D ) 2 R s + 1 (11a)

v ˜ O ( s ) d ˜ ( s ) = V O 1 − D × 1 − L ( 1 − D ) 2 R s L C ( 1 − D ) 2 s 2 + L ( 1 − D ) 2 R s + 1 (11b)

where the transfer function output voltage-to-control signal exhibits a non- minimum phase behavior since it has a RHP zero.

This converter is depicted in

The main feature here is that the average output voltage V_{O} is negative. Its magnitude can be either greater, equal (when D = 0.5), or smaller than the input voltage; hence the name buck-boost has been coined. Thus, the output voltage and inductor ripple are the same as stated in (2).

In steady state CCM operation, the voltage and current ripples for the buck- boost converter due to the switching action can be computed by

Δ V C = D I O f S C , Δ I L = D E f S L (12)

additionally, to ensure CCM, the inductor value must be selected as

L > ( 1 − D ) 2 R 2 f S (13)

It is evident the existence of similarities between the boost and buck-boost converters, where the only remarkable difference is in the current through the capacitor. Considering the average approach, a nonlinear dynamical model for the buck-boost converter is obtained as

i ˙ L = 1 L ( ( 1 − d ) v O + E d ) (14a)

v ˙ O = 1 C ( − ( 1 − d ) i L − v O R ) (14b)

In fact, model (14) is bilinear (11), since the control input d multiplies both state variables. Again, using the decomposition of DC and AC as in (3), the DC relationships for the average output voltage and inductor current are

V O = − E D 1 − D , I L = E D ( 1 − D ) 2 R (15)

On the other hand, the small-signal linear model for the AC component is given by

[ i ˜ ˙ L v ˜ ˙ O ] = [ 0 1 − D L − 1 − D C − 1 R C ] [ i ˜ L v ˜ O ] + [ E ( 1 − D ) L I L C ] d ˜ (16)

Note that this dynamical model is no longer bilinear, because the matrix has only constant parameters. For feedback design purposes, the aim is to get the frequency domain representation of (16); thus, the transfer functions for the control signal to each state variable are

i ˜ L ( s ) d ˜ ( s ) = − V O D ( 1 − D ) 2 R × R C s + 1 + D L C ( 1 − D ) 2 s 2 + L ( 1 − D ) 2 R s + 1 (17a)

v ˜ O ( s ) d ˜ ( s ) = V O D ( 1 − D ) × 1 − D L ( 1 − D ) 2 R s L C ( 1 − D ) 2 s 2 + L ( 1 − D ) 2 R s + 1 (17b)

Similarly to the transfer function output voltage-to-control signal of the boost converter, there is a RHP zero; thus, the control task becomes difficult.

Two of the most widely used control techniques are voltage-mode control and current-mode control. The voltage-mode control (VMC) scheme uses just one feedback loop and usually includes two elements: a voltage error amplifier and a voltage comparator [

Current-mode control (CMC) contains two nested feedback loops [

There are two basic schemes reported in the open literature for current-mode control; one is referred to as the peak current-mode control (PCMC) [

Current-mode control has been widely adopted as a useful technique for easing the design and improving regulators’ dynamic performance with switch-mode converters. Early references have discussed the basic principles and advantages of this technique [_{P} the peak magnitude of the ramp used to generate the control pulses. For the voltage loop, H stands for the voltage sensor gain, V_{R} the desired output voltage, and K(s) the transfer function of the PI controller.

The overall controller design procedure for this scheme is a twofold problem: 1) shaping gain for the current loop L_{1}(s), i.e. the product of transfer functions of

inner loop, and 2) shaping gain for the voltage loop L_{V}(s), i.e. the product of transfer functions in the outer loop. For robust stability of each loop the following requirements have to be satisfied: 1) for relative stability, the slope at or near cross-over frequency must be not more than −20 dB/dec; 2) to improve steady-state accuracy, the gain at low frequencies should be high; and 3) for robust stability, appropriate gain and phase margins are required [

As can be seen, the transfer function inductor current-to-control signal has two complex poles and a left-hand zero. When damping is added through the gain N, the behavior looks like a gain and a single pole. Then, the high-gain compensator and low-pass filter are added, which have the following transfer functions

G ( s ) = G P s + ω Z s and F ( s ) = 1 s / ω p + 1 (18)

respectively, where G_{P} is the compensator gain, ω_{Z} stands for the location of the compensator zero and ω_{P} for the location of the filter pole. Notice that both transfer functions can be implemented using a single operational amplifier as shown in

d ˜ = 1 V p ( 1 s / ω p + 1 ) ( G p ( s + ω Z ) s ) ( i ˜ R − N i ˜ L ) . (19)

The design procedure is given now. The high-gain compensator zero ω Z should be placed at least a decade below of half of the PWM switching frequency f S . Practically, the zero is determine by the relationship ω Z = 1 / R a C a where R a and C a are the resistance and capacitance corresponding to the current loop control circuit. The low-pass filter pole f P , on the other hand, should be placed either at half of f S or above. Using the circuitry in

ω p = C a + C b R a C a C b (20)

Here C b is the capacitor of the current loop circuit.

The compensator gain is computed by

G P = R a R b (21)

where the resistance values must be carefully selected such that

G P < 5 V P ( 1 − D ) 2 R 2 N V O (22)

Once the current loop has been tuned, the voltage loop gain has to be designed. The outer loop is designed to provide a suitable steady state correction of the output voltage using a PI controller. The transfer function for the PI controller is given by

K ( s ) = K P ( 1 + 1 T i s ) (23)

where K P = R C / R 1 is the proportional gain and T_{i} the integral time. The resulting reference current for this loop is

i ˜ R ( s ) = K P ( 1 + 1 T i s ) ( v ˜ R − H v ˜ O ) (24)

The design procedure is as follows. The proportional gain is selected such that

K P < 10 N ( 1 − D ) H R (25)

where H = ( R 1 ∥ R 3 ) / ( R 1 ∥ R 3 + R 2 ) .

Finally, the integral time is computed by T i = R C C C where R C and C C are the resistance and capacitance values of the PI controller, which must be selected such that 1 / T i should be placed at least one decade below f S .

In this sequel, for the sake of clear and straightforward exposition, the attention has been focused on a boost converter since its transfer function (14) has a non-minimum behavior; thus, a controller is more difficult to design. However, without loss of generality, the proposed control scheme can be extended to both buck and buck-boost converters. The resulting expressions are shown in

A boost converter with the corresponding current-mode controller is shown in _{O} = 24 V and an average inductor current of I_{L} = 12.6 A to a nominal resistive load of R = 3.8 Ω, which results in an output power of about 150 W.

Converter | f Z _{ } | f P | G P _{ } | T i _{ } | K P _{ } |
---|---|---|---|---|---|

Buck | _____ | _____ | _____ | 10 / f S < T i | K P < 5 V P D H V O |

Boost | f Z < f S / 2 10 | f S / 2 < f P | G P < 5 V P ( 1 − D ) 2 R 2 N V O | 10 / f S < T i | K P < 10 N ( 1 − D ) H R |

Buck-boost | f Z < f S / 2 10 | f S / 2 < f P | G P < 5 V P D ( 1 − D ) R N V O | 10 / f S < T i | K P < 5 N H R |

Capacitor C | 135 µF |
---|---|

Inductor L | 22 µH |

Nominal load R | 3.8 Ω |

Diode D | Schotky SBL3045PT |

MOSFET M | IRFP4468 |

MOSFET M_{1} | IRF150 |

Modulator | LM311 |

Finally, the switching frequency is selected to be f_{S} = 75 kHz, the peak magnitude of the ramp is V_{P} = 5 V, and the voltage sensor gain is H = 0.033. According to (7), the capacitor voltage and inductor current ripples are Δ V C = 0.308 V and Δ I L = 3.6 A , respectively. To ensure CCM, the inductor L has been selected such that (8) is fulfilled. The controller parameters are f Z = ω Z / 2 π = 267.93 Hz , G_{P} = 1, f P = ω P / 2 π = 40.4 kHz , K_{P} = 7.7 and T_{i} = 0.0136 ms.

Open and closed-loop experimental tests were performed considering nominal and step changes in the load resistance through the switch M_{1}. These variations range from 3.8 Ω to 38.5 Ω; that is from full to 10% of load at a frequency of 10 Hz, which evidently modifies the load current profile.

Frequency responses of the theoretical transfer functions and the corresponding experimental from the prototype are shown in _{P} is included in (12) for a coherent comparison.

The experimental time response of the boost converter behavior in open loop is shown in _{1} is turned OFF. It is clear that the output voltage is unregulated since, for every change in the load, the output voltage changes as well, as can be seen in

The experimental frequency responses in closed-loop for the current and voltage loop gains are shown in

The output voltage regulation is shown in _{CON} to be compared with the ramp signal is shown in

This paper deals with a practical methodology for output voltage regulation for the three basic switch-mode converters. The scheme feds back the inductor current to implement an inner control loop using a high-gain compensator and a low-pass filter. The sensed current can also be used for over-current load protection. Afterward, the voltage loop is designed by implementing a PI controller for steady-state error regulation. Furthermore, a well-defined and straightforward procedure for the selection of the controller parameters is given, which ensures system stability and output voltage regulation. The resulting voltage loop gain is shaped into L V ( s ) ≈ ω C / s , which provides good gain and phase margins. The

simplicity of the approach is of significant value in which the analytic results can be used to make design choices and tradeoffs between the inner and outer loops. This methodology was explicitly implemented in a current-mode controller for a boost converter, but it could be easily extended to the buck and buck-boost

converters. This procedure can also be extended to other kinds of converters like the quadratic or cascaded buck or boost converters widely used to step-up or step-down voltages from renewable energy sources.

The authors declare no conflicts of interest regarding the publication of this paper.

Ortiz-Lopez, M.G., Diaz-Saldierna, L.H., Langarica-Cordoba, D. and Leyva-Ramos, J. (2020) Loop-Shaping Based Control for Switch Mode DC-DC Power Converters. Journal of Power and Energy Engineering, 8, 66-83. https://doi.org/10.4236/jpee.2020.810006