Direct Charge Control Method for Inverters in Discontinuous Conduction Mode

Abstract

The discontinuous conduction mode (DCM) of the inductor current is an effective way to achieve high efficiency and high power density of inverters because it has the advantages of zero-current switching and small inductance of the filter inductor. Although the DCM can operate at a fixed switching frequency, there has been no linear model for realizing high-performance digital control. This paper provides a solution for this problem. Firstly, a linear first-order charge quantity model (CQM) for a DCM inverter with a fixed switching frequency is established, which uses charge quantities output by the inverter bridge as a control variable. Secondly, a direct charge control (DCC) method is proposed, and the design process is significantly simplified. Finally, the performance of the proposed control method is verified by the experiments using a 1 kW prototype, and a high-quality output voltage waveform with a peak efficiency of above 98% was achieved. The proposed method outperforms existing works because the CQM is a first-order linear model that realizes the high-performance digital control and significantly simplifies the design of controller.

1. Introduction

Inverters have been widely used in industrial and residential applications [1,2]. There are three current modes of the filter inductor that can be used for an inverter: continuous conduction mode (CCM), triangular current mode (TCM), and discontinuous conduction mode (DCM). Considering a half-bridge circuit that is shown in Figure 1, the waveforms of the three current modes are shown in Figure 2, where T_s denotes the switching period. The TCM operation can realize the zero-voltage switching (ZVS), which is also called the boundary conduction mode (BCM) [2,3,4], and the DCM operation can realize the zero-current switching (ZCS) [5,6,7,8,9,10].

Both the DCM and TCM are effective ways for reducing the switching loss by soft-switching and lifting the power density using a filter inductor with small inductance [11]. Although the DCM and TCM provide a higher peak inductor current than the CCM [12], their switching losses are acceptable for small-power applications where devices with low turn-off losses can be used [13,14].

The TCM normally operates at a variable switching frequency [3,15,16], and it is realized by controlling the inductor current, i.e., its peak value and the valley value. It can also be regarded as a type of boundary conduction mode (BCM) because its control methods determine the boundary of the inductor current [16].

The main drawback of the TCM is that its switching frequency can vary significantly, which can further lead to excessive turn-off switching losses in power devices, as well as excessive core losses and winding losses of the inductor [1]. Furthermore, the boundary value instead of the average value of the inductor current is required. Therefore, the required frequency of a high-precision accuracy sampling unit is a few times larger than the switching frequency, which leads to high cost. The variable switching frequency makes the closed-loop digital control relatively difficult [17]. In order to solve this problem, some studies have been focused on averaging the switching frequency [1,18]. The TCM with a fixed switching frequency was proposed in [18], but this method can cause larger conduction losses because the RMS value of the inductor current can increase significantly.

In contrast, the DCM can operate at a fixed switching frequency. Moreover, the DCM does not require detecting the inductor current. Therefore, the digital control for the DCM can be realized by using a microprocessor. However, there is still one difficulty: the state average model of the DCM is nonlinear when the capacitor voltage and inductor current are used as state variables. Therefore, no closed-loop control algorithm has been proposed yet.

This paper aims to develop a linear model for the DCM inverter operating at a fixed switching frequency, which is easy and meaningful for performing the closed-loop controls. A linear first-order model named the charge quantity model (CQM), which uses the charge quantities output by the inverter bridge as a control variable, is proposed. In addition, a closed-loop control method named the direct charge control (DCC) method is developed. Finally, the parameter design of the control system is completed. This method can achieve high stability and low distortion of the output voltage. The proportional control can be used for the mentioned control type. Using the proposed control method, the ZCS high-efficiency DCM inverter can operate stably at a fixed switching frequency and achieve accurate sine wave output through digital control. It is also insensitive to device parameter deviation.

The rest of this paper is organized as follows. In Section 2, related works are mentioned and the purposes of the study are stated. In Section 3, a half-bridge inverter with an LC filter for DCM operation is described. The CQM, which is established by using the charge quantities output by the inverter bridge as a control variable, is presented in Section 4, as well as the calculation of the duty cycle. In Section 5, the design procedure of a DCM inverter is introduced. The simulation and experimental results are given in Section 6. Finally, in Section 7, the conclusions are drawn.

2. Overview of Related Work

Nowadays, in the literature, the research on the control of output voltage for the DCM inverter mainly focuses on two problems. One is the optimization of the output voltage waveform distortion around the zero-crossing point of the inductor current. The other is to lower the average switching frequency.

In the inverters designed for CCM, the inductor current still works under DCM around the zero-crossing point of the load current, which causes the serious distortion of output voltage waveform. Research in [19] introduces a judgment and compensation method for DCM in the closed-loop control of the voltage waveform. Reference [10] proposed a power energy modulation (PEM) technique for both DCM and CCM that divided the energy demanded by the load into energy packets and modulated the bridge inverter on the basis of packets of energy for each switching period.

The control methods for the output voltage of DCM inverters are normally realized by using the hysteresis control or the off-time control. However, because the switching frequency is not constant, the extremely high switching frequency may appear and lead to higher switching losses. Reference [11] proposes a variable off-time control method for a single-phase microinverter operating in the DCM to lower the average switching frequency. Furthermore, a fixed off-time control method for single-phase micro-inverter operating in DCM is proposed in [15]. However, the control of output voltage is not discussed.

For the DCM inverter, a closed-loop controller is difficult to design. In [17], a feedforward control was proposed for low-power-grid-tied inverters under DCM, which may increase the switching frequency. In [20], the control of the DCM inverter is based on measuring the boundary of inductor current, and although the inductor current equation is given, it did not provide the digital control method for the output voltage waveform.

In summary, all the above methods cannot guarantee the good waveform of output voltage for the DCM inverter. Even if the switching frequency is fixed, there is still no linear model that has been established. Compared with the existing literature, the charge quantity model is established in this paper, which can solve the problem of the DCM inverter being unable to achieve high-performance closed-loop control due to the absence of a linear model. Furthermore, the closed-loop control method DCC based on CQM was constructed, which is simple to design and achieves a high-quality control effect.

The unique feature of the study in this paper is that a simple first-order linear model is established for the DCM inverter with fixed switching frequency, realizing high-performance closed-loop control.

The purpose of the study in this paper was to achieve a high-performance closed-loop control for DCM inverter. Therefore, a linear first-order mathematical model is firstly proposed, and on the basis of it, the closed-loop control method, named DCC, is constructed. The parameter design of the control system is also completed.

3. DCM Operation Analysis

A half-bridge circuit, which is the basic unit of inverters, is shown in Figure 1. It is taken as a research example in this study for analyzing the DCM operation. As shown in Figure 1, the inverter circuit consists of two power devices Q₁ and Q₂, a filter inductor L, and a filter capacitor C. The DC bus voltage V_dc is separated by two capacitors C_dc1 and C_dc2. The connection point between L and C is denoted as N_a.

3.1. DCM

Assume that the inductor current i_L and the output inductor voltage v_o are positive; then, the steady-state waveforms for one switching cycle are as illustrated in Figure 3. The voltages of C_dc1 and C_dc2 are equal. The switching frequency is high enough, so both v_o and V_dc are constant within one switching cycle. The steady-state waveforms for the switching frequency that is much higher than the grid frequency is analyzed in the following.

In one switching cycle, there are three intervals denoted as Intervals I–III.

• Interval I t₀~t₁: Switch Q₁ is turned on at t₀; Q₁ is ZCS because i_L is zero at t₀, as shown in Figure 3. In this interval, the DC capacitor C_dc1 charges the filter capacitor C and supports the load current through the inductor L. The inductor voltage is expressed as v_L = V_dc/2−v_o > 0; thus, i_L increases.
• Interval II t₁~t₂: Switch Q₁ is turned off at t₁, and the current i_L commutates to D₂. The inductor voltage is V_L = −V_dc/2−v_o < 0, and i_L decreases to zero at t₂.
• Interval III t₂~T_s: In this interval, switches Q₁ and Q₂ keep off. The filter inductor L and the parasitic capacitance of Q₁ and Q₂ form a resonance circuit, and i_L shows a resonant waveform, but its average value is close to zero. The filter inductor L is equivalent to the open circuit, and the load is powered by the filter capacitor C until the next switching cycle starts.

3.2. Circuit Equations

In Intervals I and II, when the inductor current and capacitor voltage are used as state variables, the state equations are given as follows:

$$\left\{\begin{array}{l}{v}_{PWM}=L\left(d{i}_L/dt\right)+v_o,\\ i_L=i_o+C\left(d{v}_o/dt\right).\end{array}\right.$$

(1)

In Interval III, when establishing the average model, i_L can be considered to be zero, and the state equation in this interval is given by

$$i_L=i_o+i_C=0.$$

(2)

According to Equations (1) and (2), the linear average state model cannot be derived, which is the reason why the closed-loop controllers for DCM inverter cannot be easily designed.

4. Direct Charge Model and Controller Design

It should be noted that, for node N_a, Equation (3) can be applied to the whole switching period, from which the charge equation can be derived as Equation (4). Charges Q_C and Q_O flow through the capacitor and load in one switching cycle, respectively. The Q_L denotes charge output by the inverter bridge, which flow through the inductor. Therefore, the charge quantity Q_L can be used as a control variable to establish the average model.

$$i_L=i_o+i_C.$$

(3)

$$Q_L={\displaystyle \underset{k{T}_s}{\overset{(k+1)T_s}{\int }}{i}_{L}dt}={\displaystyle \underset{k{T}_s}{\overset{(k+1)T_s}{\int }}{i}_{o}dt}+{\displaystyle \underset{k{T}_s}{\overset{(k+1)T_s}{\int }}{i}_{C}dt}=Q_C+Q_O.$$

(4)

4.1. Charge Quantity Model

Equation (4) implies that within a switching period T_s, the amount of charge quantity Q_L is equal to the sum of the charge quantities Q_C and Q_O. Therefore, Equation (4) can be expressed as

$$Q_L(k)=Q_C(k)+Q_O(k)=C\left[v_o(k)-v_o(k-1)\right]+i_o(k)T_s.$$

(5)

Using the Z-transformer, Equation (5) can be expressed as

$$V_o(z)=\left[Q_L(z)-T_s{I}_o(z)\right]/\left[C(1-z^{-1})\right].$$

(6)

Equation (6) can be used as a model for the analysis and controller design of a DCM inverter, as shown in Figure 4. It is a linear first-order model, and in this paper, it is called the charge quantity model (CQM). The model uses charge quantity Q_L as the input control variable and the capacitor voltage V_o as the state and output variables. The load current I_o is regarded as a disturbance.

4.2. Duty Cycle Calculation

In practice, the inverter is controlled by the duty cycle d instead of Q_L. Therefore, the duty cycle calculation is presented in this section.

In Figure 3, the charge quantity Q_L in one switching period is shown as the shaded area, and it can be realized by the control of duty cycle d. According to the relationship between the inductor voltage and the duty cycle, the inductor current ripple within a switching cycle and the corresponding charge Q_L can be calculated. The value of Q_L can be obtained as

$$Q_L=(T_{on}+T_d)\Delta i_L/2.$$

(7)

In Equation (7), T_on = dT_s and T_d = d’T_s, as illustrated in Figure 3. Therefore, the inductor current ripple within a switching cycle is given by

$$\Delta i_L=(V_{dc}/2-v_o)d{T}_s/L.$$

(8)

From Equation (8), the falling time of the inductor current T_d is given by

$$T_d=d^{\prime }{T}_s=\left(V_{dc}/2-v_o\right)d{T}_s/\left(V_{dc}/2+v_o\right).$$

(9)

The inductor current i_L decreases to zero before the next switching cycle, and the total duty-ratio d_all of the Intervals I and II is calculated as

$$d_{all}=\left(T_d+d{T}_s\right)/T_s=V_{dc}d/\left(V_{dc}/2+v_o\right).$$

(10)

Therefore, Equation (10) can be rewritten as

$$Q_L=\left(V_{dc}/2-v_o\right)V_{dc}{d}^2{T}_s^2/\left[2L\left(V_{dc}/2+v_o\right)\right],$$

(11)

On the basis of the required charge quantity Q_L, the duty cycle d can be determined as Equation (12):

$$d=\sqrt{2L{f}_s^2\left(V_{dc}/2+v_o\right)Q_L/\left[V_{dc}\left(V_{dc}/2-v_o\right)\right]}.$$

(12)

According to Equation (12), when Q_L is required to output from the bridge, the calculated duty circle d should be applied to switches.

Many factors can affect the realization accuracy of the charge quantity Q_L, such as the inductance deviation or a DC bus voltage ripple. However, the influence of these disturbances can be suppressed by the closed-loop controller.

4.3. Controller Design

On the basis of the CQM, the closed-loop controllers for the DCM operation can be easily designed. This control method is denoted as the DCC method. The PID controller, PR controller, and voltage and current dual-loop controller can all be used as a controller. Due to the fixed switching frequency, the proposed DCC can be effectively implemented into digital controllers, such as a digital signal processor (DSP). Meanwhile, the proposed DCC does not require high-precision current sampling.

Due to the first-order structure of the CQM model, the simplest controller, which is a proportional controller, can be used. In order to achieve a better response-ability to the step load, the load-current feedforward control is adopted. Considering the one-beat delay, the control method is designed in z domain. The block diagram of the closed-loop control is shown in Figure 5, where the proportional controller is defined by G_p (z) = k_p. The one-beat delay caused by the DSP is taken into consideration in the control loop. Although this method cannot be a strong contribution, as shown in [21], it is a good way to check the performance of the mathematic model.

The closed-loop transfer function can be expressed as

$$v_o\left(z\right)=G_v\left(z\right)\cdot v_r\left(z\right)+G_i\left(z\right)\cdot I_o\left(z\right),$$

(13)

where

$$G_v\left(z\right)=k_p/\left[Cz+\left(k_p-C\right)\right],$$

(14)

$$G_i\left(z\right)=-\left(T_{s}z-k_f\right)/\left[Cz+\left(k_p-C\right)\right].$$

(15)

The characteristic equation of Equation (14) is

Cz + k_p − C = 0.

(16)

The pole of both G_v(z) and G_i(z) are

$$z_p=1-k_P/C.$$

Therefore, when k_p increases from 0 to +∞, the z_p decrease from 1 to −∞, as the blue line given in Figure 6. To ensure the system stable, z_p should be inside the unit circle, which means 0 < k_p < C.

Considering the transfer function G_i(z), the zero is

$$z_0=k_f/T_s.$$

z₀ is proportional to k_f, so it increases as k_f increases. Considering the need of achieving the anti-interference ability to the load-current, the amplitude response at the power frequency of the transfer function G_i(z) needs to be close to 0. Therefore, the zero z₀ is set to 1 and k_f is set as T_s.

Therefore, the parameter of a proportional controller should be 0 < k_p < C to ensure the magnitude of pole is |z_p| = 1 and k_p/C < 1, which means the system is stable.

Furthermore, according to Figure 5, the parameter of load-current feedforward control k_f should be similar with T_s to eliminate the influence of the load current as much as possible.

5. LC Filter Design

In order to ensure that the inverter can work under the DCM for all switching periods, it is necessary to choose the LC filter parameters and switching frequency properly. Therefore, this section provides a design reference for DCM inverters.

The basic parameters of the inverter design include the DC voltage V_dc, the RMS value of the output AC voltage V_orms, the rated power P_N, the peak value of inductor current I_L, and the switching frequency f_s. Furthermore, on the basis of the proposed model, the maximum value of d_all = d + d’, which is denoted as D_max, was set as a design parameter. First of all, the inverter working in the DCM operation mode must satisfy the following condition d + d’ ≤ 1. Considering the step load and other disturbances, D_max should be smaller than one, and it was set to 0.95 in this paper.

According to Equation (10), the maximum of Q_L is limited by the available duty cycle, which can be defined as Q_Llim:

$$Q_L\le Q_{Llim}=\left(V_{dc}/2-v_o\right)\left(V_{dc}/2+v_o\right)D_{max}^2/\left(2L{f}_s^2{V}_{dc}\right).$$

(17)

A well-designed inductance should make sure that the output charge ability Q_Llim is greater than the charge spends on the load in any switching period. In the case of full-resistive load, Q_Llim reaches its minimum at the peak value of the AC output voltage v_o. At the same time, the charge spent on load also reaches its maximum. Therefore, the value of the inductance L is designed to the charge transfer capability at the peak value of the AC output voltage under the rated power of the resistive load, as shown in Equation (18).

$$Q_{\mathit{Llim},\mathit{max}}=\left(V_{dc}^2-8V_{orms}^2\right)D_{max}^2/\left(8L{f}_s^2{V}_{dc}\right)\ge Q_o=I_o{T}_s$$

(18)

Thus, the limitation of inductance L is expressed as

$$L\le L_{max}=\left(V_{dc}^2-8V_{orms}^2\right)V_{orms}{D}_{max}^2/\left(8\sqrt{2}{f}_s{V}_{dc}{P}_N\right).$$

(19)

The high frequency components of the inductor current i_L flow through the filter capacitor C and cause high-frequency ripple in the output voltage. The value of the filter capacitor should be designed as Equation (20) to effectively reduce the ripple of the output voltage ΔV_pp.

$$C\ge C_{min}=I_{Lpeak}/\left(4f_s\Delta V_{pp}\right).$$

(20)

where I_Lpeak denotes the peak value of the inductor current.

In DCM, the inductance of the filter inductor can be greatly reduced. However, due to the large ripple of the inductor current, it is necessary to increase the capacitance of the filter capacitor to absorb the harmonic current of the switching frequency. However, this will not significantly increase the volume of the capacitor. On the other hand, because the direct charge control method has a good control effect, it can effectively reduce the low harmonic of the output voltage. Therefore, the proposed method will not cause significant harmonic impact on the close equipment.

6. Simulation and Experimental Results

The simulation results verified the results given in Section 5. The simulation and experimental parameters are given in

Table 1. Simulation and experimental parameters.

Input voltage	400 Vdc	Output voltage	110 Vrms
Switching frequency	10 kHz	Output power	1 kW
Dmax	0.95
ILpeak	30 A	ΔVpp	10 V
Lmax	126.04 μH	Cmin	75 μF
Filter inductor L	120 μH	Filter capacitor C	80 μF
kp	4 × 10−5	kf	1 × 10−4
Switch	IPW65R048CFDA (650 V/40 A)

.

6.1. Simulation Results

According to Equation (14), the controller parameter was set as k_p = 0.5C, and k_f = T_s was adopted to suppress the load current disturbance effectively.

The bode diagram can be given as Figure 7 and shows that the system is stable.

The output current i_o, the duty cycle d, the total duty cycle d_all, the current stress Δi_L, the average value of inductor current i_Lavg, and the RMS value of the inductor current i_Lrms for a different power factor angle ϕ of the load are presented in Figure 8. As shown in Figure 8, when the power factor angle was ϕ = 0, both the duty cycle and the inductor current were the largest. Therefore, if the designed inverter can work well under the resistance load, it can also work well under other conditions.

6.2. Experimental Results

The proposed CQM and DCC methods were verified by the experiment using a 1 kW prototype, as shown in Figure 9. The experimental parameters are shown in Table 1. The Si MOSFETs were used as switching devices to take advantage of low turn-off losses and low conduction losses. The controller used in the prototype was based on a DSP TMS320F28335.

The waveforms of the inverter with no-load and under a resistive load are shown in Figure 10 and Figure 11, respectively, where it can be seen that the system could stably output the sine wave, and its THD was 2.77% at P_o = 1 kW.

The zoomed-in waveforms of Figure 11 at different output voltage values, including v_o = 0 and 150 V, are illustrated in Figure 12. The drain-source voltage of active switch V_Q1 and inductor current i_L for DCM operation is illustrated in Figure 12, where the output voltage v_o and output current io are also given as a reference. As shown in Figure 12, the active switch was always under ZCS turn-on. What is more, the oscillating waveforms on i_L and V_Q1 were due to the resonance caused by the inductor and the parasitic capacitance of the switching devices in the corresponding interval in which both switches were turned off.

As shown in Figure 13, when the step load was applied, the voltage sag was 25.5 V, and the corresponding drop time was 0.4 ms, which was about four switching periods. In Figure 13, it can be seen that the proposed DCC method had good dynamic performance while using the proportional feedback controller plus the proportional load current feedforward controller. It should be noted that PID control, PR control, and other control methods can also be easily adopted in the proposed method if higher performance is required, but due to the limitation on the paper length, this is not discussed in this paper. Furthermore, although the inductor current maximum value of the inverter under no load and under the resistive load were similar, the duty cycles were different, as given in Equation (12). Under very small Q_L and d, the conduction losses in the no-load situation were much smaller than those in the full-load situation.

The losses of the inverter consists of inductor loss and switching device loss. For the soft-switching turn-on being realized by operating in DCM, the switching device losses include the turn-off losses and the conduction losses. The losses can be calculated by

$$P_{loss}=P_L+P_{con,MOSFET}+P_{con,diode}+P_{swi,off}$$

(21)

$$\left\{\begin{array}{l}{P}_L=R_L{I}^2\\ P_{con,MOSFET}=R_{on}{I}^{2}d\\ P_{con,diode}=\left(V_{D}I+R_D{I}^2\right)d\\ E_{swi,off}=(k_{off1}{I}^2+k_{off2}I+k_{off3})V_{swi}\end{array}\right.,$$

(22)

where R_L is the total equivalent resistance of inductor, d denotes the duty ratio of the current passing through a power device, R_on represents on-state equivalent resistance of MOSFET, V_D represents on-state voltage drop of the body diode at zero current, R_D represents its on-state equivalent resistance, and k_off1–k_off3 are the corresponding coefficient of turn-off losses. According to the datasheet, the parameters using for losses calculation are listed in

Table 2. Parameters used for loss calculation.

RL (W)	0.15	Vswi(V)	400
Rdson (W)	0.043	koff1	1.03 × 10−7
VD (V)	0.28	koff2	1.01 × 10−5
RD (W)	0.005	koff3	2.53 × 10−6

.

Taking the positive half period as an example, the current goes through MOSFET when Q₁ is turned on, and goes through the body diode of Q₂ when Q₁ is turned off and current is still not zero.

The efficiency of the experimental prototype, including the DC capacitors, half-bridge leg, and LC filter, was tested by the power analyzer. The comparison of the theoretical efficiency curve and experimental efficiency curve is given in Figure 14, where it can be seen that the peak efficiency of measured efficiency was above 98%. What is more, the efficiency curve obtained by the theoretical calculation was similar to the experimental efficiency curve and slightly higher than the experimental efficiency measured by the experiment. The error was mainly caused by the inconsistency between the operating temperature and parasitic parameters of the switch tube under the experimental conditions and the test conditions set in the data sheet.

In particular, the DCC could achieve high efficiency at a low load. The efficiency of the inverter can be further improved by optimization of the inductor design.

7. Conclusions

This paper proposes a CQM model for the inverter under the DCM operation. The proposed model uses the charge quantity output by the inverter bridge as a control variable to realize a linear model that has a simple first-order structure. This model is convenient for the simple realization of digital closed-loop control in a DCM inverter with a fixed switching frequency. The combination of the proportional controller and load current feedforward controller can provide good performances regarding the dynamic response and steady-state sine wave output.

Since the fixed frequency control is realized, the control system can be easily designed. Benefiting from the closed-loop control, the proposed DCM inverter is immune to the change in hardware parameters and does not require high-precision current sampling. The DCC can achieve high efficiency at a low load.

The key points of this paper’s contribution are summarized as follows:

• This paper proposed a mathematical model based on charge quantity. The model has a simple first-order linear structure and it is easy to design the closed-loop control.
• On the basis of the charge quantity model, a direct charge control method was proposed, and the performance was verified by the experiments using a 1 kW prototype, and a high-quality output voltage waveform with a peak efficiency of above 98% was achieved.

The proposed control method for the DCM provides a good solution for low AC power supply applications.