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In digital current mode controlled DC-DC converters, conventional current sensors might not provide isolation at a minimized price, power loss and size. Therefore, a current observer which can be realized based on the digital circuit itself, is a possible substitute. However, the observed current may diverge due to the parasitic resistors and the forward conduction voltage of the diode. Moreover, the divergence of the observed current will cause steady state errors in the output voltage. In this paper, an optimal current observer is proposed. It achieves the highest observation accuracy by compensating for all the known parasitic parameters. By employing the optimal current observer-based predictive current controller, a buck converter is implemented. The converter has a convergently and accurately observed inductor current, and shows preferable transient response than the conventional voltage mode controlled converter. Besides, costs, power loss and size are minimized since the strategy requires no additional hardware for current sensing. The effectiveness of the proposed optimal current observer is demonstrated experimentally.

In recent years, predictive current control (PCC) was found to be a robust, fast and easy control strategy for digitally controlled DC-DC converters in continuous current mode (CCM). Therefore, it has been extensively studied by many researchers [

Different current sensing techniques meet different applications in terms of cost, size, accuracy, isolation,

The current sensors mentioned above all have different advantages and drawbacks, and thus could meet different requirements. However, existing techniques might not suit applications which require isolation with minimal price, power loss and size. Therefore, a current observer (CO) turns out to be a suitable substitute for conventional current sensors in digitally controlled converters. The cost, size and power consumption can be reduced since it does not need any auxiliary hardware, even though the accuracy might be affected by the voltage ripple or the mismatches between the observer and the converter.

Current observers are used for motor controls, fault detection, and were firstly introduced for DC-DC applications by Midya [

For PCC controllers with the basic current observer, the observed current will diverge due to the forward voltage of the diode and the parasitic resistors in the converter. The divergence of the observed current degrades the reliability of the converter, and will cause calculation result overflow in the digital circuit. More importantly, the divergence further induces steady state errors in the output voltage. In our previously published paper, these issues were discussed in detail, and a compensation strategy based on a boost converter was proposed [

The paper is organized as follows: Section 2 introduces the basic CO-based PCC algorithm, which includes the observer algorithm and the PCC strategy. In Section 3, the relationship between the convergence of the observed current and the parasitic parameters is derived, and the steady state error of the output voltage is analyzed. In Section 4, the OCO strategy is proposed, which can converge the observed current and eliminate the steady state error by compensating the observed current and the sampled voltage. Experimental results are shown in Section 5 to support the proposed theory, and to prove the effectiveness of the OCO based PCC strategy. Finally, a brief conclusion is given in Section 6.

The construction of buck DC-DC converter with the CO based PCC controller, is shown in _{REF}

Supposed that the converter works in CCM. Ignoring the parasitic parameters, a current differential equation can be derived based on the average voltage on the inductor, shown as _{L}, V_{O}, D, R_{IN}

Based on _{O}, V_{IN}, L_{IN}_{O}_{O}_{1}(_{2}(_{1}(_{2}(_{OB}

Employing valley current control and trailing edge (TE) modulation, the inductor current waveform is shown in

In _{REF}_{L}_{REF}_{L}_{REF}

As the switching cycle _{OB}_{REF}

For the PCC controller with the basic CO, the observed current will diverge due to the diode forward voltage and the parasitic resistors in the converter. The divergence of the observed current degrades the reliability of the converter, and will cause the steady state errors in the output voltage. In the following section, the relationship among the parasitic parameters, the convergence of the observed current and the steady state error of the output voltage will be studied in detail.

The basic CO is based on

However, _{F}

_{F}_{F}

Other parasitic parameters like the inductor winding resistor, the diode forward resistor and the MOS on-resistor will also diverge the observed current. The divergence speed is proportional to those parasitic parameters. The divergence of the observed current further causes the steady state error of the output voltage, which will be analyzed in the following.

To demonstrate the steady state error of the output voltage, the mechanism of PI regulator must be taken into consideration. Adopting Laplacian method, the transfer function of PI controller can be written as (_{P}sT_{I}_{P}_{I}_{P}_{I}_{O}_{REF}

In steady state, the output voltage must be constant, which guarantees _{O} = 0, so the

Rationally, the PI regulator can eliminate the voltage error, and the output _{REF}_{REF}_{OB}

Finally, the steady state error of the output voltage occurs due to the divergence of _{OB}

From _{O}_{F}

_{F}_{F}

The slope of the voltage error _{F}_{P}_{I}_{P}_{I}

Other parasitic parameters such as the equivalent serial resistor of the inductor, the diode forward resistor,

For the PCC controller with the basic CO, the observed current is found to be diverging, which further induces the steady state error of the output voltage. In this section, the OCO strategy is proposed, which can converge the observed current to the valley value of the inductor current, and eliminate the steady state error of the output voltage.

In order to model the system precisely, several parasitic parameters are considered. Then the current slopes are compensated as:
_{AV}_{AV}_{V}_{pp}_{T}_{L}_{DS}_{F}

Employing first order differential approximation _{OB}_{OB}_{OB}

Therefore, the observed current can be solved as:

To be strict, the parasitic parameters are always changing with the temperature, the time,

It is a natural character of DC-DC converters that the output voltage contains ripples, so the sampled voltage may deviate from the average value of the output voltage, depending on the sampling point. The maximum deviation range is the value of the output voltage ripple. Nevertheless, all the industrial applications take the average value as the reference voltage. What is more, the current observer also prefers _{AV}_{pp}

As _{O}_{C}_{ESR}_{AV}_{C,pp}_{ESR,pp}

In _{C.pp}_{pp}T_{pp}_{ESR,pp}_{pp}R_{C}_{O,pp}

The sampled voltage is determined by _{O}_{S}_{C}_{AV}_{S}_{pp}_{COMP}

To calculate the accurate error between _{COMP}_{AV}_{C}_{O}

Meanwhile, the transfer function from the inductor current to _{O}

And the transfer function from the inductor current to _{C}

Employing Laplacian method, the expressions for _{O}_{C}

By means of computer assistance analysis, _{C}_{O}_{AV}_{C}_{O}_{COMP}_{AV}

As _{S}_{COMP}_{S}_{AV}_{AV}_{S}_{AV}_{COMP}_{COMP}_{AV}_{AV}_{COMP}_{AV}_{COMP}_{AV}_{COMP}_{AV}_{S}

In order to prove the theories about the convergence of the observed current and the steady state error of the output voltage, both the basic CO based PCC controller and the OCO based PCC controller are experimentally demonstrated. Meanwhile, comparison between the OCO based PCC controller and a conventional voltage mode controller is also carried out. In this section, the experimental settings which include the design specifications and the measured parasitic parameters are introduced first. Then the test results are given and analyzed.

The design specifications and the measured parasitic parameters are shown in _{OB}

As shown in Section 3, for the basic CO based PCC controlled converter, the observed current will diverge from the actual inductor current due to the parasitic parameters. An experiment is carried out to demonstrate this conclusion. With the CO based PCC controller, the observed current and the actual inductor current are shown in

As

The OCO strategy is proposed in Section 4. In the following, an experiment is carried out to demonstrate the effectiveness of the OCO based PCC controller. The observed current is shown in

As

To demonstrate the advantage of the OCO-based PCC controller in dynamic response speed, a comparison is made between the proposed controller and a conventional voltage mode controller. The test results are as follows: when the load steps from 3 to 5 Ω, the output voltage of the conventional voltage mode controlled converter increases to 6.55 V for a short time, and re-stabilizes in 400 μs, which is shown in

The output voltage transient of the conventional voltage controlled converter when the input voltage steps from 10 to 12 V is shown in

As the experimental results show, the proposed strategy can improve the transient response of the buck DC-DC converter.

A current observer is a possible substitute for conventional current sensors in digitally controlled DC-DC converters. This paper studied the divergence problem of the observed current based on the PCC controlled buck DC-DC converter. Meanwhile, the divergence-induced voltage steady state error is analyzed. In order to solve these issues, the OCO strategy is proposed. It achieves the highest observation accuracy based on compensations for all the known parasitic parameters. Experimental results demonstrate that by employing the OCO-based PCC controller, the observed current converges to the valley value of the inductor current, and the steady state error of the output voltage is also eliminated. Moreover, compared to the conventional voltage mode controller, the proposed algorithm can improve the transient response, which is proven to have a good theoretical and practical application potential for CCM buck converters.

This work was supported by the National Natural Science Foundation of China under Grant 61202469, 61376031.

The authors declare no conflicts of interest.

Buck DC-DC converter with the CO based PCC controller.

Current waveform under valley current control and TE modulation.

The observed current with different _{F}

Simulated steady state error of the output voltage under different PI parameters and _{F}

Output voltage and voltage between the ideal capacitor.

Simulated _{C}_{O}

Error between _{COMP}_{AV}

(

(

The output voltage transient when the load steps from 3 to 5 Ω—a conventional voltage mode controlled converter.

(

The output voltage transient when the input voltage steps from 10 to 12 V—a conventional voltage mode controlled converter.

(

Design specifications of the buck converter.

Input voltage | 10 V |

Output voltage | 6 V |

Rated output current (ROC) | 1.2 A |

Voltage ripple under ROC | 0.12% |

Switching frequency | 100 kHz |

Inductance of the power inductor | 100 μH |

Capacitance of the output capacitor | 50 μF |

Measured parasitic parameters.

Inductor winding resistance _{L} |
200 mΩ |

MOSFET _{DS} |
100 mΩ |

Diode forward resistance _{F} |
100 mΩ |

Diode forward Voltage _{F} |
0.7 V |

ESR value of the output capacitor _{C} |
70 mΩ |