A Novel Hysteresis Control Strategy Based on Ampere-Second Balance of the Modulate Capacitor

Abstract

A novel hysteresis PWM control strategy for synchronous buck converter is proposed. The proposed control strategy is based on ampere-second balance of the modulate capacitor, which not only offers faster transient response to meet the challenges of the power supply requirements of fast dynamic load changes, but also provides better stability and solves the compensation problem of error amplifier in the conversional voltage PWM control. Finally, the steady-state and dynamic operation of the control method is analyzed and verified by simulation and experimental results.

I. INTRODUCTION

Nowadays, DC-DC switching converters have been increasingly used in a wide range of applications, such as LEDs, air-space industry, portable systems, and power factor correction [1]-[6]. The requirements of the transient response performance of the DC-DC power switching converters are stringent. This performance is consistently expected to have a lower dynamic output voltage deviation and shorter settling time during load current transients. As one of the widely used converter topologies, buck converters have been investigated with different control strategies [7]-[18]. Voltage mode converter is a traditional PWM control buck converter that operates in continuous conduction mode (CCM), which requires a complicated compensation network to ensure stable operation challenged by complex poles in the loop gain transfer function [7]. This challenge not only increases the design difficulty of the control circuit, but also causes poor dynamic load performance. Many investigations have been carried out on this subject to avoid these difficulties. The current mode control is applied with voltage feedback to improve stability and dynamic performance [8], [9], which needs complex slope compensation and response speed limited by voltage loop controller. In [10]-[12], time-optimal digital and digital controllers are presented. These techniques significantly improve transient response, but suffer from complex implementation. In [13]-[15], nonlinear sliding-mode are presented to improve the dynamic response. However, the main drawback of sliding mode is variable frequency. Control strategies such as hysteresis control have received extensive attention. These strategies do not need compensation circuit and provide almost instantaneous load transient response [16]-[18].

A novel hysteresis PWM control strategy based on ampere-second balance of the modulate capacitor is proposed. The proposed control strategy imports the inductor current information in the feedback loop of single output voltage as well as adds inductor current and output voltage as feedback variables. These variables improve capacitor charging and discharging rates effectively. Moreover, only a comparator is used with hysteresis, feedback resistors, and current sensor. Hence, the number of components will be obviously reduced in the control circuit. No error amplifier and complex compensating network is found. Hence, faster response was achieved when the load suddenly changes.

The operating principles of the proposed control strategy are introduced in Section II. The steady-state was analyzed. A small-signal model is derived and analyzed to show the advantages of the proposed control strategy in Section III. The prototype board is tested. The simulation and experimental results are verified in Section IV. The conclusion is given in Section V.

 

II. OPERATING PRINCIPLES

The configuration of a buck converter with a synchronous rectifier that uses the proposed control strategy is shown in Fig. 1. The key waveforms of comparator U in one switching cycle are shown in Fig. 2. The control circuit consists of a comparator U with hysteresis and feedback resistors Rf and RL. The output voltage and inductor current are returned to capacitor C for the triangular wave generator through resistor Rf and RL, respectively. In addition, the output of the hysteresis comparator was connected with the noninverting input terminal through resistor R2 and then connected with Vref through resistor R1. Therefore, the upper and lower thresholds of hysteresis controlled buck converter were obtained when the output of comparator U changed.

Fig. 1.Circuit diagram of the proposed converter.

Fig. 2.Key waveforms of the comparator.

The working principle of the control circuit is as follows: When the output voltage becomes large (small), the charging current of capacitor C in the on mode period increases (decreases), and the discharging current of capacitor C in the off mode period decreases (increases). Hence, the on time duration of the pulse decreases (increases) and off time duration increases (decreases). Therefore, the switching duty was changed. Moreover, the output voltage can be regulated.

 

III. OPERATION ANALYSIS

A. Steady-State Analysis

To simplify the analysis, all circuit elements are assumed to be ideal. In Fig. 1, Vf is the voltage across capacitor C. VL and VH are the threshold levels of comparator U. VU is the output voltage of comparator U. The switching cycle starts at instant t=0.

( i ) state 1: 0˂t˂TON

When the output signal VU is at a high level, the capacitor was charged through the feedback branch. The following equations are obtained:

Solving the above equations under the initial condition of Vf (0)= VL obtains the next equation as follows:

where

From Equation (2), Vf increases exponentially from VL to . Since Vf must be greater than VH to invert the state of the comparator, the next constraint is obtained as follows:

Setting Vf = VH and t = TON in Equation (2) obtains TON as follows:

If VH - VL ˂˂ - VH, the Equation (4) can be approximated as follows:

( ii ) state 2: TON˂t˂Ts

When output signal VU is at a low level, the capacitor was discharged through the feedback branch. The following equations are obtained:

Solving the above equations under the initial condition of Vf(TON)= VH obtains the next equation as follows:

From Eq. (7), Vf decreases exponentially from VH to . Since Vf must be lower than VL to invert the state of the comparator, the next constraint is obtained as follows:

From Equation (7), TOFF is obtained by setting Vf = VL and t= TS as follows:

If VH - VL ˂˂ VL - () Equation (9) can be approximated as follows:

Combining (4) and (9) obtains the duty ratio D= TON /( TON+TOFF) as follows:

Frequency fs is as follows:

Combining (5), (10), and D= TON /( TON+ TOFF) obtains the output voltage as follows:

where VH = Vref + VOH

B. Dynamic Analysis

The small-signal circuit model is obtained by applying PWM switching modeling techniques. This application helps to further analyze the dynamic characteristics of the buck converter circuit. Fig. 3 shows the small-signal equivalent circuit of a buck converter with the proposed control strategy. In Fig. 3, ki is the gain of the inductor current feedback circuit, and kv is the gain of the output voltage feedback circuit. Rc is the series-equivalent resistance of capacitor C. Based on Figure 3, the small-signal transfer functions can be obtained.

Fig. 3.Small signal equivalent circuit.

Combining (5) and (10) obtains the duty ratio D= TON /( TON+TOFF) approximately as follows:

The small signal disturbance of (20) can be represented as follows:

From (20) and (21), the following formula can be obtained:

Combining (20) and (22) obtains the gain of the output voltage feedback and inductor current feedback circuit as follows:

Fig. 4 shows a control block diagram of the circuit proposed in this paper. In Fig. 4, the loop gain transfer function of the proposed controller is expressed as follows:

The closed-loop transfer function can be obtained according to Mason’s rule.

Fig. 4.Control block diagram of the buck converter employing the proposed control scheme.

“Control-to-Output” is as follows:

where a1 = Co(Ro + RC) + RCCo(kvVi + 1) +

“Input voltage susceptibility” is as follows:

If GvvGdi - GviGvd = 0, Equation (26) can be written as follows:

“Output impedance” is as follows:

Fig. 5 shows the results of the frequency response of the loop gain of output voltage feedback control transfer function. The feedback resistor Rf is taken as a parameter. The frequency response of the gain and phase shows that the control loop is steady and possesses good phase margin at different Rf. The smaller Rf has higher gain in all frequencies and better phase margin. In addition, Fig. 6 shows the results of the frequency response of the loop gain of the inductor current feedback control transfer function, which takes the feedback resistor RL as a parameter. The control loop is also seen as steady. The smaller RL has higher gain in all frequencies, but the phase margin is kept unchanged.

Fig. 5.Frequency response of the loop gain of output voltage feedback control with different Rf.

Fig. 6.Frequency response of the loop gain of inductor current feedback control with different RL.

The above analysis shows that taking Rf ,RL between the gain of open loop transfer function and phase margin is a trade-off. Results show that good performance of proposed controller was achieved with Rf (=0.6kΩ) and RL (=40kΩ).

The loop gain transfer function T(s) of the proposed controller is shown in Fig. 7. The frequency response of loop gain for the buck converter with feedback T(s) and without any feedback Gvd(s) is compared. The figure shows that feedback control increases the loop DC gain from 14 dB to 64.6 dB, which also has a larger cut-off frequency of 72.9 kHZ. The phase margin was increased to 35° and T(s) has 40 dB attenuation in middle-high frequency. This margin is effective in high-frequency interference mitigation. Hence, both good stability and wide bandwidth are easily achieved if the feedback loop is added. Therefore, the dynamic characteristic is verified.

Fig. 7.Frequency response of the loop gain T(s).

Fig. 8 describes the relationship of inductor current feedback resistor RL output voltage feedback resistor Rf and output voltage Vo. This figure shows that feedback control parameters must be selected in a certain range to stabilize the output voltage.

Fig. 8.Effect of RL and Rf to Vo.

Fig. 9(a) shows a Nyquist diagram of the transfer function. Point (-1, 0i) is not surrounded by the Nyquist plot. In addition, no pole points of the open transfer function in the right half plane are found. The stability of the buck converter control system is obtained in terms of the Nyquist stability criterion. Fig. 9(b) shows the part of the amplification region at point (-1,0i) of the Nyquist plot. Point (-1,0i) is not be surrounded or passed through by the Nyquist plot.

Fig. 9.(a) Nyquist diagram of loop gain transfer function. (b) Amplification region at point (-1,0i).

 

IV. SIMULATION AND EXPERIMENTAL RESULTS

A simulation model based on Fig. 1 was built in PSIM to verify the proposed control strategy, and a prototype was designed. The circuit parameters are shown in Table I. The conventional PWM controller is a TL5001 PWM controller. The regulator uses the PI compensation network, as shown in Fig. 10.

Fig. 10.Traditional voltage-mode controlled buck converter.

TABLE IDESIGN PARAMETERS OF THE PROPOSED CONVERTER

Fig. 11 shows the relation between the output current and the output voltage. The variations of the output voltage are extremely small, as seen in the figure. The simulation values of the output voltage are in good agreement with the experimental values. Fig. 12 shows the relation between the input voltage and output voltage. The variation of the output voltage is also extremely small despite significant variation in the input voltage. No steady-state error is observed in the output voltage when changes of the input voltage and load current occur. The same observation is found even when the controller does not employ a high-gain operational amplifier.

Fig. 11.Relation of the output current and output voltage.

Fig. 12.Relation of the input voltage and output voltage.

Fig. 13 shows the relation between the output current and switching frequency. The figure shows that the switching frequency remained at 120 kHz when the load current changed. Moreover, the drawback of variable frequency was solved compared with some control technologies.

Fig. 13.Relation of the output current and switching frequency.

Fig. 14 shows the output voltage and inductor current waveforms during load transients. The load current initially decreased from 5 A to 2 A. The excess inductor current decreased slowly with a slope proportional to Vo. Operations were in the opposite direction when the load current increased from 2 A to 5 A. In addition, the output voltage drops with magnitude proportion and inductor current increased quickly with a slope proportional to Vi-Vo. Therefore, Vo quickly recovered. Moreover, the regulator was consistently stable during the load transients.

Fig. 14.Waveforms of output voltage and inductor current during load transients.

Below is a comparative analysis of conventional and proposed controllers. Fig. 15 shows the loop gain transfer function of the conventional and the proposed controllers. The proposed controller has higher gain in all frequency ranges, a higher cut-off frequency, and a higher phase margin than the conventional controller. This controller is simple to use in outer circuit design, which results in high closed-loop gain and high closed-loop bandwidth. Therefore, the proposed controller has a higher sensitivity to control signal and faster response than the conventional controller.

Fig. 15.Bode diagram of loop gain function.

Fig. 16 shows the “output impedance” transfer function of the two. The output impedance of the circuit is mainly decided by the parasitic resistance of the output filter capacitor at high frequency ranges. Hence, both controllers have the same output impedance at high frequency ranges. In addition, the proposed controller has lower output impedance at medium and low frequency ranges. Therefore, the output voltage is less affected by changes in load current compared with the conventional controller. Consequently, better dynamic load performance is achieved.

Fig. 16.Bode diagram of output impedance.

Fig. 17 shows the “input voltage susceptibility” transfer function of the two. The proposed controller has lower gain than the conventional controller at low and medium frequency ranges. Hence, the proposed controller has better anti-input voltage disturbance capability.

Fig. 17.Bode diagram of input-to-output function.

Figs. 18(a) and 18(b) show the simulated and experimental responses of the proposed and conventional controllers that undergo load current step changes of 5A to 2A and 2A to 5A, respectively. The proposed controller not only has a short response time, but also a small overshoot and undershoot. Hence, this controller has better dynamic response characteristics than the conventional controller.

Fig. 18.(a) Simulated and experimental response to a 5 A to 2 A load current step change. (b) Simulated and experimental response to a 2 A to 5 A load current step change.

 

V. CONCLUSIONS

This paper proposed and analyzed a novel hysteresis PWM control strategy applied to a buck switching converter. The proposed control strategy based on ampere-second balance characteristics of the modulate capacitor uses the output feedback signal for capacitor charging and discharging to generate modulation voltage Vf. This technique is simple and solves the compensation problem of the error amplifier in conventional voltage PWM control. This strategy also achieves faster transient response and quasi-switching frequency to meet the challenges of power supply requirements for fast dynamic load changes. Finally, the steady-state and dynamic operation of the proposed control method are analyzed and verified by simulation and experimental results.