State-trajectory Control of LLC Converter Implemented by Microcontroller Chao Fei, Weiyi Feng, Fred C. Lee, Fellow, IEEE, Qiang Li, Member, IEEE Center for Power Electronics Systems Virginia Tech, Blacksburg, VA 24061 USA
[email protected] Abstract— this paper investigates how to integrate statetrajectory control functions of LLC resonant converter within a commercial low-cost microcontroller, including Simplified Optimal Trajectory Control (SOTC) for load transient, burst mode and soft start up. Simplified calculation process is proposed for SOTC to minimize digital delay. Guideline to select a proper microcontroller is presented to limit calculation delay and ADC delay within 1 switching cycle to ensure good transient performance. Optimized transient processes are proposed to eliminate large oscillation between burst mode and normal operation. Soft start-up is implemented with sensing the output voltage only. The whole control system structure is proposed to integrate all these control functions and for the first time all these control functions are integrated in one low-cost MCU. The results are demonstrated on a 130kHz 300W 380V/12V halfbridge LLC resonant converter with MCU TMS320F2808. Index Terms --- state-trajectory control, LLC resonant converter, microcontroller (MCU).
I.
INTRODUCTION
LLC resonant converter has been widely used as DC-DC converter due to its high efficiency and hold-up capability. However, the control characteristics of resonant converter are complex due to the dynamics of resonant tank. To describe and analyze the behavior of resonant tank, state-trajectory analysis and control are first employed for series resonant converter (SRC) [1]-[3]. However, LLC resonant converter has three resonant elements, which causes analysis and control of LLC converter more complex than SRC. Another way to analyze LLC converter is using small-signal transfer function. However, the small-signal transfer function of voltage-mode LLC converter varies under different input voltage and load current [4], [5]. Simplified optimal trajectory control (SOTC) for LLC resonant converter is proposed to improve the transient performance in paper [6]. SOTC senses load current and modifies pulse widths of driving signals, which works as a load current feed-forward loop and improves transient response significantly. Another way to improve transient performance is to increase the bandwidth of control loop. Average current-mode control [7], [8] and bang-bang charge control [9] for LLC converter can achieve constant dynamic performance. However, they require sensing resonant tank status or extra logic circuit to control switching instant. For LLC resonant converter, burst mode is widely used to This work was supported by the Power Management Consortium (PMC) in the Center for Power Electronics Systems, Virginia Tech.
improve light-load efficiency. There are several other papers [10]-[14] talking about different methods of burst mode, whose problem is that resonant tank can’t keep to the efficiency-optimal state trajectory. The burst-mode control with constant burst-on time and optimal switch pattern is proposed in paper [15] to solve this problem based on stateplane analysis. For start-up of resonant converter, there will be large current and voltage stress and it will take a very long time. Soft start-up for LLC resonant converter is proposed in paper [16], which can limit stress and optimize energy transfer. However, papers [6], [15], [16] solve each problem separately. In real applications, all these functions need to be realized in one single controller. Therefore, it is worthwhile to investigate how to implement all state-trajectory control functions in one controller. There are mainly three challenges to achieve this goal: firstly, complex calculation in statetrajectory control requires high-performance digital controller, such as high-end FPGA and very fast analog-to-digital converter, however, cost-effective MCU is preferred in industrial application; secondly, burst-mode control with constant burst-on time and optimal switch pattern proposed in paper [15] can’t be used directly when associated with normal operation, because there would be very large dynamic oscillation in both resonant tank and output voltage during the transient processes between burst mode and normal operation; thirdly, for given application, it is still unknown how to design the whole control system structure and then select proper MCU. As a result, further research efforts need to be spent on how to integrate these state-trajectory control functions into one low-cost MCU. The research goal of this paper is to investigate and solve the challenges in integrating all state-trajectory control functions in a cost-effective MCU. In section II, how to implement SOTC by MCU is investigated and a simplified calculation method to fit MCU’s ability is proposed. In section III, optimized transient processes are proposed to solve large dynamic oscillation problem in the transition between burst mode and normal operation. In section IV, how to implement soft start-up using the concept of state-trajectory control by MCU is investigated. In section V, guidelines to select proper MCU are presented and experimental results are demonstrated on a 130kHz 300W 380V/12V half-bridge LLC resonant converter with MCU TMS320F2808. Finally, a summary is given in section VI.
II. SIMPLIFIED OPTIMAL TRAJECTORY CONTROL (SOTC) IMPLEMENTED BY MCU Fig. 1(a) shows the circuit diagram of Simplified Optimal Trajectory Control (SOTC). Fig. 1(b) shows state-trajectory during load step-up.
the sampling finished, the MCU first processes closed-loop computation based on VO to get switching period TS for next switching cycle. After that, the MCU processes SOTC computation based on I[k] and I[k+1] to get ΔTUP or ΔTDOWN. Then switching period for next switching cycle becomes TS + 2∙ΔTUP or TS - 2∙ΔTDOWN. The derivation process for equations to get ΔTUP or ΔTDOWN is presented in paper [6]. The derived final equations are shown in follows: ·
∆ ∆ (a)
(b)
Fig 1. Simplified Optimal Trajectory Control (SOTC) (a) Circuit diagram; (b) State-trajectory during load step-up
As shown in Fig.1 (b), when load step-up happens, the resonant tank is settled in 2 steps. There is almost no dynamic oscillation in output voltage. Thus PI controller takes little effort to eliminate steady-state error. This is because the whole control system senses load current and modifies pulse widths of driving signals, which works as a load current feed-forward loop. The response speed of load current feed-forward loop is not limited by bandwidth of closed-loop control, thus improving load transient response significantly. However, SOTC requires nonlinear calculation and induces digital delay in load current feed-forward loop, which has significant impact on transient response speed. Since digital delay is a key factor impacting transient performance and MCU can’t process data in parallel, it is significant to allocate sampling and calculation time and simplify calculation to fit the ability of low-cost MCU when implementing SOTC by MCU. Fig. 2 is an example of sampling and calculation during load transient for SOTC implemented by MCU with one switching cycle delay in load current feed-forward loop.
/
1
(1)
·
(2)
Where ILL is load current for light load and IHL is load current for heavy load. To implement SOTC in MCU, these equations need to be converted into digital format and simplified. In low-cost MCU, fixed-point calculation is preferred. And operations of multiplication, bit-shift, addition and subtraction take few CPU cycles to process. While division, square root and other more complex operations take a lot of CPU cycles to process. Since the digital delay in load current feed-forward loop has significant impact on transient performance, it is important to minimize the CPU cycles required to process the calculation. Let I[k+1] be the sensed load current for current switching cycle and I[k] for last switching cycle. And AD_I[k] is the digital value of load current at kth sampling. Then m is defined as the ration of AD_I[k] over I[k]. If I[k+1] > I[k], load step up calculation is employed. And the equation is simplified as below: ∆ Where
· and
_
_
(3)
are integers and defined as followings:
· ·
·
(4)
So and need to be selected properly to establish equation (4). After this, load step up calculation is converted in fixed-point calculation and take less than 100 CPU cycles to process each time. If I[k+1] < I[k], load step down calculation is employed. Since there is square root in the calculation, which takes too many CPU cycles for cost-effective MCUs, load step down calculation must be linearized and simplified. ΔTDOWN is a
Fig 2. An example of ampling and calculation for SOTC implemented by MCU during load step-up
For the example given in Fig. 2, the interrupt service routine is designed to be triggered at the beginning of each switching cycle. ADC starts sampling a short time after beginning of switching cycle to avoid switching noise. When
function of square root of range of
_
1
_
, which equals to
_
1
_
, and
is from 0 to 1 to cover all situations.
However, considering burst mode is included in the whole control system solution,
_
1
_
can be limited to a smaller
range. Assume the system enters burst mode when load is below 20%, then the range
_
1
_
of becomes from 0.2 to 1
in normal operation. After that, ΔTDOWN can be approximate to a linear function of
_
1
_
within that range. Although there
will be some error compare with exact calculation, resonant tank can still be settled to the state near new steady state during load transient and closed-loop control will eliminate the error. Fig.3 shows how to approximate ΔTDOWN as a linear function of
_
1
_
Besides SOTC calculation talked above, PWM update and control loop interrupt service routine takes about 150 CPU cycles. With simplified calculation for SOTC, total CPU cycles required to process PI control and SOTC is less than 400. Then the total delay caused by calculation and ADC can be limited within one switching cycle using a cost-effective MCU for industrial practice. To limit digital delay including calculation and sampling within n switching cycles when selecting MCU, the following equation needs to be fulfilled: .
·
.
(7)
Where Ncal. Is the number of CPU cycles required to process all the calculation; Fclock is the clock frequency of MCU; and Tperiod is switching period of power stage. Fig. 4(a) shows the simulation result of SOTC with one switching cycle delay under step load change (5A-15A). Fig. 4(b) shows the simulation result of SOTC with two switching cycles delay under same load transient condition with Fig. 4(a).
Fig 3. approximate ΔTDOWN as a linear function of
_ _
With approximated linear curve, the expression of ΔTDOWN can be simplified as below:
∆
0.675 · ·
(5)
For the case of 130kHz switching frequency with a 100MHz MCU, the equation can be further simplified as below:
∆
_
_ _
(a)
(6)
With equation (6), load step down calculation is converted into fixed-point calculation and takes less than 200 CPU cycles to process each time. Another way to decrease CPU cycles required for SOTC calculation is using look-up tables. All data are pre-calculated and stored in MCU. For load step up calculation, the table is two-dimension and relatively small since ΔTUP is a function of (IHL - ILL). For load step down calculation, if ΔTDOWN is stored as a function of
, the table is two-dimension but stores
floating-point data, which is not suitable for cost-effective MCU; if ΔTDOWN is stored as a function of both IHL and ILL, the table would be fixed-point but three-dimension, which would take a lot of space and many CPU cycles to loop up required data. After all, real-time calculation using the simplified method talked above can achieve good performance with little time to process when implementing by a costeffective MCU.
(b) Fig 4. Simulation result of SOTC with delay (a) one switching cycle delay (VO overshoot 50mV); (b) two switching cycles delay (VO overshoot 75mV)
It is shown clearly in Fig. 4 that less delay mean better load transient performance. However, for digital control, the delay is discrete so there is no need to use the best controller to minimize the delay. For industrial application, the switching frequency of power stage is around 100kHz and delay can be limited within 1 switching cycle using simplified calculation proposed above when implementing by a cost-effective MCU. III. OPTIMIZED TRANSIENT PROCESSES BETWEEN BURST MODE AND NORMAL OPERATION Burst mode in LLC resonant converter is employed to improve light load efficiency. However, when burst mode is implemented by MCU and combined with normal operation,
the transient processes between burst mode and normal operation have not been investigated and optimized. Section III first investigates transient performance between the burst-mode proposed in paper [15] and normal operation with analysis from perspective of state-trajectory, and then proposes optimized processes for normal operation to burst mode using simplified one-step SOTC calculation and burst mode to normal operation using combination of burst mode pulse pattern and SOTC, thus simplifying complexity of program and calculation burden of MCU. A. Transient process from burst mode to normal operation Fig. 5 shows waveform of transient process from burst mode to normal operation. It is shown clearly in Fig. 5 that there is dynamic oscillation when the circuit enters normal operation. And PI control takes a lot of cycles to reach new steady state. This is because the energy in resonant tank extends to very large if the circuit changes to normal operation directly with switching frequency equals to resonant frequency.
During burst off time, the state locates at point t0. When the controller senses that the load current become larger than burst mode limit, it starts normal operation with an initial switching frequency equals to resonant frequency. From t0 to t1, the energy in resonant tank already extends to much larger than needed. And from t1 to t2, the resonant tank follows a trajectory which is corresponding to a load current much larger than IH. Since PI control is very slow, it takes a long time to reach new steady state with large dynamic oscillation. To solve this problem, an optimized transient process from burst mode to normal operation is proposed. Firstly the circuit works at burst mode; when load current becomes larger than burst mode limit, the controller will generate first 2 pulses of burst mode 3-pulse pattern, as shown in Fig. 7(a) from t0 to t3. Then the state is settled to iOPT equivalent trajectory (iOPT is the efficiency-optimal load current). After that, the controller can use SOTC to settle resonant tank from iOPT equivalent trajectory to iH equivalent trajectory, as shown in Fig. 7(a) from t2 to t4. Fig. 7(b) is the corresponding state trajectory.
(a)
Fig 5. Transient process from burst mode to normal operation
Fig. 6 analyzed how this happen from the perspective of state-trajectory. The dash line is the state-trajectory during burst on time. The Green circle the steady state trajectory of IH = 10A.
(b) Fig 7. Optimized transient process from burst mode to normal oeperation (a) time-domin waveform; (b) corresponding state trajectory
After this, the controller can use frequency control with PI compensator and SOTC to ensure the circuit works at steady state. With the proposed method, the whole transient process does not involve much calculation burden for MCU, and there is no dynamic oscillation. Fig 6. State-trajectory illustrating how energy in resonant tank extends to much larger than needs
B. Transient process from normal operation to burst mode
It is shown in Fig. 8 that when the controller stops normal operation and starts burst mode, there will by dynamic oscillations in the first several burst on time cycles, during which the state can’t fix to iOPT equivalent trajectory.
After this, the controller can use constant burst on time control. With the proposed method, the whole transient process does not involve much calculation burden for MCU, and the performance is quite good. Below are approximate equations for one-step SOTC. Equation (8) is for the case IH < Iinitial, and the extra one step ∆ . Equation (9) is for the case IH > Iinitial, would be and the extra one step would be
∆ ∆
Fig 8. Transient process from normal operation to burst mode
To solve this problem, an optimized transient process from normal operation to burst mode is proposed. Firstly the circuit works at normal operation; when load current becomes smaller than burst mode limit, the controller will generate extra one pulse, as shown in Fig. 9(a) from t0 to t1, to settle resonant tank within one step. Then after t2, the state stays at burst mode initial point. Fig. 9(b) is the corresponding state trajectory. This extra one pulse is calculated using the concept of one-step SOTC.
2·
·
1
∆
. /
·
(8) (9)
Where Iinitial is burst mode initial point equivalent trajectory, which may vary for different power stage. These two equations can be simplified using the same concept referred in section II. IV. SOFT START-UP IMPELMENTED BY MCU During normal operation, LLC resonant converter operates near resonant frequency. However, during start-up process, the output voltage needs to be built up from zero to reference voltage and switching frequency changes from very high switching frequency to resonant frequency. Some commercial controllers can achieve fast start-up but there is larger stress during start-up process. Other commercial controllers may restrict stress but cannot achieve fast start-up process. The soft start-up proposed in paper [16] can achieve fast start-up while at the same time restricting stress based on state-trajectory analysis. However, the whole start-up process involves 3 stages and the state-trajectory calculation of each stage is quite complex, thus not suitable for MCU. To solve this problem, section IV first investigates what is the problem if using state-trajectory control directly. Then a method of implementing soft start-up with sensing output voltage is illustrated, which is suitable for cost-effective MCU.
(a)
(b) Fig 9. Optimized transient process from normal operation to burst mode (a) time-domin waveform; (b) corresponding state trajectory
Fig. 10(a) illustrates state trajectory for start-up Stage 1 if state trajectory control is employed directly by sensing state variables and using real-time calculation. The resonant frequency of this example is 130kHz and ADC delay in the MCU is 1us. At the beginning, the states locate at (0, 0). The controller keeps sensing iLr and vCr for calculation of state trajectory control. When the states touch current limiting band +IMAXN, the controller turns off Q1 and turns on Q2 after 1us ADC delay. When the states touch current limiting band – ILmN, the controller turns off Q2 and turns on Q1 after 1us ADC delay. At this time, the states are already very far away from destination (0.5, 0) of Stage 1 with very large stress in the two pulses. Since the state variables change very fast in one switching cycle, state trajectory control by sensing state variables and using real-time calculation is not applicable for cost-effective MCU.
To solve this problem, section IV illustrates a method of implementing start-up by MCU with sensing output voltage only. Fig. 10(b) is the state trajectory and gate signal of soft start-up Stage 1 implemented by MCU. Instead of sensing state variables to determine switching instant, each switch period is pre-calculated using equation (10) and stored in MCU. (10)
When VO reaches 12V, start-up ends and PI control takes over. With this start-up solution, the MCU has little calculation burden and almost achieves optimal performance. V. EXPERIMENAL RESULTS The whole control system is implemented by MCU TMS320F2808. The main interrupt service routine is triggered at the beginning of each switching cycle. Fig. 12 is the interrupt service program flowchart.
When the system is reset, MCU generates the precalculated pulses consecutively, thus the state will reach the destination region within the current limiting band.
(a)
(b)
Fig 12. Interrupt sevice program flowchart
Fig 10. State trajectory control for start-up Stage 1 (1) by sensing state variables (2) pre-calculated and stored switching pulses
Where fS = f(VO) is the pre-calculated relationship between switching frequency and output voltage for start-up Stage 2. Vlimit is the threshold voltage for start-up Stage 2. ΔV in Stage 3 is used to gradually decrease switching frequency by comparing VO and Vref. Both Vlimit and ΔV need to be adjusted for different design.
Then in Stage 2, relationship between fS and VO is precalculated to limit state variables in the given band (detailed calculation in paper [16]), then piece-wise linearized and stored in MCU. Each switching cycle, the MCU sense VO, and update fS for next switching cycle based on the pre-calculated relationship between fS and VO. When output voltage reaches certain limit (in this case it’s 9.5V), Stage 2 ends. In stage 3, the MCU decreases fS gradually, thus the converter will reach steady state gradually. Fig. 11 is the gate signal during soft start-up implemented by MCU.
Fig 11. Gate signal during soft start-up implemented by MCU
When the MCU is reset, the State equals “Start-up” and the MCU starts to generate pre-calculated switching pulses, which is start-up Stage 1. After these pulses ends, the MCU senses VO and uses frequency control based on the precalculated relationship fS = f(VO) in Stage 2. When output voltage reaches certain limit Vlimit, Stage 2 ends. In Stage 3, the reference voltage starts from Vlimit and increases gradually until 12V, and start-up ends. Then the State equals “Normal operation” and the MCU sense both output voltage and load current. Output voltage is processed for closed-loop control. And load current is processed for SOTC. If load current is below burst mode limit, the State equals “Burst” and the MCU enters burst mode.In burst mode, when the output voltage reaches Vref, the
MCU generates burst mode 3 pulses. When the load current is larger than burst mode limit, the MCU enters normal operation. In both normal operation and burst mode, when the output voltage is below Vlimit, the MCU enter start-up to achieve short-circuit protection. The whole control system solution only requires sensing 2 variables: output voltage and output current for all conditions. 1 ePWM module is used for primary side switches. One internal interrupt connected to ePWM is used for interrupt service program. The whole control system takes less than 4K 16 flash to store the program and data. Besides these requirements, digital delay in transient response can be limited within one switching cycle using equation (7) referred in section II. Then a proper MCU can be selected to fulfill these requirements. Since all MCUs specified for switching converter have ePWM module, more than 1 ADC channels and internal interrupt, requirements for CPU, ADC speed, and flash are limitations when selecting MCU. Table I lists CPU, ADC and flash specifications of some TI’s controllers.
(b) Fig 13. SOTC for load transient (a) step up from 5A to 15A; (b) step down from 15A to 5A
Fig. 14 is the experiment result of optimized transient processes between burst mode and normal operation. Fig. 14(a) is from burst mode (2A) to normal operation (10A). Fig. 14(b) is from normal operation (10A) to burst mode (2A). The resonant tank is settled immediately in both cases.
TABLE I CPU and ADC specification of some TI’s Controllers Controllers
CPU frequency
ADC Speed
Flash ( 16)
TMS320F2802x
40-60MHz(32-bit)
2 - 4.6MSPS
8K - 32K
TMS320F280x
60-100MHz(32-bit)
3.75-12.5MSPS
16K - 128K
TMS320F2833x
100-150MHz(32-bit)
12.5MSPS
64K - 256K
For industrial practice (switching frequency is around 100kHz), TMS320F28027 is capable of implementing all these control functions.
(a)
Then this control system solution is demonstrated on TI’s demo board TMDSHVRESLLCKIT, which is a 130kHz 300W 380V/12V half-bridge LLC resonant converter with the following circuit parameters: Lr = 55uH, Cr = 24nF, Lm = 280uH, CO = 1.32mF. Fig. 13 is the experiment result of SOTC for load transient. Fig. 13(a) is load step up from 5A to 15A, and Fig. 13(b) is load step down from 15A to 5A. There is almost no dynamic oscillation in resonant tank during load transient.
(b) Fig 14. optimized transient processes (a) from burst mode (2A) to normal operation (10A); (b) from normal operation (10A) to burst mode (2A)
Fig. 15 is the experiment result of soft start-up implemented by MCU. The whole start-up process takes less than 1.2ms. Current stress is only 20% larger than full load condition. AC voltage stress is only 30% larger than full load condition. (a)
[6]
[7]
[8]
[9]
[10] Fig 15. Soft start-up implemented by MCU
VI. SUMMARY AND CONCLUSION How to integrate SOTC for load transient, optimal burst mode and soft start-up within cost-effective MCU is investigated and provided in this paper. To implement SOTC for load transient, complex calculation process is simplified to fit MCU’s calculation ability. It is also shown that digital delay can be limited within one switching cycle for industrial practice. To combine burst mode with normal operation, optimized transient processes are necessary. Combination of burst mode first 2-pusles and SOTC is employed for the transient process from burst mode to normal operation. And one-step SOTC is employed for the transient process from normal operation to burst mode. Both these two methods involve little calculation burden for MCU. For soft start-up, using the concept of state-trajectory control directly by sensing state variables and real-time calculation is not suitable for MCU. One method by pre-calculating switching pulses and relationship between switching frequency and output voltage is provided, which requires only sensing output voltage and little real-time calculation for whole start-up process. The whole control system requires only two ADC channels, one interrupt service routine, one ePWM module and less than 4K 16 flash to store the program and data. REFERENCES [1]
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