Multi-step Simplified Optimal Trajectory Control (SOTC) for Fast Transient Response of High Frequency LLC Converters Chao Fei, Student Member, IEEE, 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 proposes Multi-step Simplified Optimal Trajectory Control (SOTC) for the high frequency LLC converters. SOTC can improve the transient performance of the LLC converters by immediately changing the pulse width of the primary driving signal during the load transient. However, the implementation of SOTC requires high-performance digital controllers. The proposed Multi-step SOTC calculates the trajectory based on only the output voltage and the load current, and settles the resonant tank within multi steps. With the proposed Multi-step SOTC, low-cost controllers can be used to control the high frequency LLC converters with the statetrajectory control concept. The number of steps in multistep SOTC is selected based on the speed of the controller and the switching frequency of the LLC converters. Experimental results are demonstrated on a 500kHz 1kW 400V/12V LLC converter. Keywords—LLC resonant converter, optimal trajectory control, transient response, digital control. I.
INTRODUCTION
The LLC resonant converters have been widely used as a DC-DC converter due to its high efficiency and hold-up capability [1][2]. However, compared with PWM converter, the control characteristics of the resonant converters are much more complex due to the dynamics of the resonant tank. It is challenging to improve the load transient response of the LLC converters. Several approaches have been tried to describe the control characteristics and improve the transient response of the resonant converters. One approach is using small-signal model of the resonant converter [3][4]. However, since the switching frequency of the resonant converters is within a relative wide range, it is hard for the small-signal model to describe the dynamics of the resonant converters of the whole operating range. For the LLC converters, the simulation-based smallsignal model has been employed, showing that the transfer function is quite complex under different operating points [5]. Recently, another approach based on the current-mode control was proposed to improve the transient response of the resonant converter [6][7][8]. The variation of the control-to-output This work was supported primarily by the Power Management Consortium (PMC) in CPES, Virginia Tech
978-1-4673-7151-3/15/$31.00 ©2015 IEEE
transfer function of the current-mode control under the different operating points is smaller than the conventional voltage-mode control due to the feedback from the resonant tank. However, the implementation of this approach is quite complex, requiring additional logic circuits and sensing circuits to control the switching instants of the primary-side switches. In addition, the stability of the current mode control for the resonant converters under the different conditions needs to be proven. Different from the small-signal approaches, the statetrajectory analysis and control can better describe and analyze the behavior of the resonant tank. It is first applied to the series resonant converters (SRC) [9][10]. The Optimal Trajectory Control (OTC) for SRC can control the resonant tank to follow the desired trajectory exactly [11]. However, the LLC converters have three resonant elements, which cause the analysis and control of the LLC converters to be more complex than the SRC. Simplified Optimal Trajectory Control (SOTC) was proposed to improve the transient response of the LLC converters [12]. The state-trajectory control for the LLC converters was also extended to the burst mode for light load efficiency [13], and soft start-up and short circuit protection [14]. All these state-trajectory control functions of the LLC converters were integrated in one microcontroller (MCU) for a 130kHz LLC converter [15]. The digital controllers are gradually taking the place of the analog controllers in the control of the LLC converters [16][17][18]. Among the digital controllers, the cost-effective MCUs are preferred in the industrial applications. With the fast development of the wideband gap devices and novel magnetic materials [19][20][21], the high frequency LLC converters are emerging in recent years due to its high power density and integrated magnetics [22][23]. Multi-step Simplified Optimal Trajectory Control (SOTC) is proposed in this paper to achieve fast load transient response for the LLC converters with the low-cost digital controllers. The proposed Multi-step SOTC consists of two major concepts: firstly, the trajectory is calculated based on only the output voltage and the load current, which simplifies the whole control scheme; secondly the resonant tank is settled within multi-step, which allows the low-cost controllers to control the high frequency LLC converters. The number of steps in multi-step SOTC is determined by the speed of the controller and the switching frequency of the LLC converters.
In this paper, Section II introduces the concept and the implementation of Simplified Optimal Trajectory Control (SOTC). Section III introduces the improvement for implementation of SOTC. Section IV introduces the proposed Multi-step SOTC and the derivation of the required pulse width during the load transient. Section V presents the experimental results on a 500kHz LLC converter. The conclusions are given in Section VI.
steady state, and uses SOTC to achieve fast response during the load transient. Based on the output voltage and the load current, the steady-state trajectory before and after the load transient can be calculated; then the required pulse width of the primary driving signal during the load transient can be obtained by SOTC. The control scheme of SOTC is shown in Fig. 2. Q1
II. CONCEPT AND IMPLEMENTATION OF SIMPLIFIED OPTIMAL TRAJECTORY CONTROL (SOTC) Different from previous state-trajectory control method [11], the Simplified Optimal Trajectory Control (SOTC) proposed in [12] calculates the trajectory based on only the output voltage and the load current, rather than the resonant current and/or the resonant voltage. This is because the steady-state trajectory is determined by the output voltage and the load current under certain input voltage range, regardless of the switching frequency. Fig. 1 is an example of the state-trajectory for the LLC converters operating at around the resonant frequency. The xaxis is the normalized resonant current, and the y-axis is the normalized resonant voltage. The current normalizing factor is ⁄ ⁄ , and the voltage normalizing factor is . Firstly, the centers of the trajectory are determined by the output voltage. When the high-side switch is on, the voltage across the , which is 1 , 0 in the resonant tank is state-plane. When the low-side switch is on, the voltage across the resonant tank is , which is , 0 in the stateplane. The radius of the trajectory is determined by the load current. Take switching frequency at around the resonant frequency as an example, is expressed as _
⁄
In which, expressed as:
Co
VgsQ2
Vref
Fig. 2. Control scheme of SOTC
SOTC can achieve very fast transient response since the load transient is sensed and responded directly by OTC. However, the original implementation of SOTC requires very fast digital controller and ADC, which is not applicable for the industrial application. To achieve SOTC with low-cost digital controller, paper [15] simplified the real-time calculation of SOTC for the digital controllers and proposed method to implement SOTC with lowcost MCUs. Since the digital delay is a key factor impacting the transient performance, the sampling and the calculation is allocated within one switching cycle to optimize the transient response. Fig. 3 is the control scheme of SOTC implemented by MCU. Fig. 4 is an example of sampling and calculation during the load step-up with SOTC.
Microcontroller
Q1 Lr
Q1,Q2
Q2
n :1:1 VO
Lm Cr
Co
Driver
RL
iLoad VO
2∙∆T
+
(2)
√
PI Compensator
VCO
Driver
+ _
RL
iLoad VgsQ1
is the RMS value of the resonant current,
_
VO
Q2
(1)
⁄
n:1:1 Lm
VIN
√ ·
Lr
Cr
VIN
SOTC TS
(for transient)
PI Control (for steady state)
-
Vref +
Fig. 3. Control scheme of SOTC implemented by MCU 0.3
iLrN
0.2
iLoad
R
VO
0.1
VgsQ1 VgsQ2
O2 (nVoN, 0) 0
O1 (1-nVoN, 0)
I[k]
I[k+1] ΔTUP
ΔTUP
Digital delay SOTC Fig. 4. Sampling and calculation during the load step-up with SOTC
-0.1
R -0.2 -0.3 0.2
VCrN 0.3
0.4
0.5
0.6
0.7
0.8
Fig. 1. State-trajectory of the LLC converters operating at around the resonant frequency
The Simplified Optimal Trajectory Control (SOTC) uses a linear regulator to eliminate the steady-state error during the
The state-trajectory during the load step-up with SOTC is shown in Fig. 5. Initially, the converter runs at the light load condition, corresponding to the black circle in the state-plane. After the load step-up, SOTC would increase the width the primary driving signal by ΔTUP to boost the resonant tank energy. Then the linear regulator eliminates the steady-state error and the converter is stable at the heavy load, corresponding to the red circle in the state-plane.
The real-time calculation can be replaced by the look-up table to reduce the calculation time. Take a 500kHz LLC converter as an example, the look-up table is shown in Fig. 7. It is 3demension table, with x-axis of the (k-1)th sampling, y-axis of the kth sampling, and z-axis of the ΔT.
∆
300 200 100 0 -100 -200
Fig. 5. State-trajectory during the load step-up with SOTC
-300
For the SOTC implemented by MCU, there is still a maximum switching frequency limitation caused by the digital delay. Take TI’s digital controller TMS320F28027 as an example, the total digital delay is 7.2us, which means that the maximum switching frequency suitable for SOTC is 140kHz. If we want to further increase this maximum switching frequency limitation, the digital delay of SOTC must be reduced. III. IMPROVEMENT FOR IMPLEMENTATION OF SIMPLIFIED OPTIMAL TRAJECTORY CONTROL (SOTC) To reduce the digital delay of SOTC, firstly, the sampling and the calculation is re-allocated as shown in Fig. 6. The A/D for the output voltage finishes before the beginning of the present switching cycle, and then the PI calculation starts at the beginning of the present switching cycle since the PI calculation only needs the output voltage. At the same time of the PI calculation, the load current is sampled and converted, and then the SOTC calculation follows immediately after the PI calculation. With this improvement, the digital delay consists of only calculation. With digital controller TMS320F28027, the digital delay is reduced to 6.7us, and the maximum switching frequency suitable for SOTC is increased to 150kHz. A/D for iLoad
iLoad
I[k]
VO
PI Calculation SOTC Calculation (only need VO)
Fig. 6. Improve implementation of SOTC by re-allocating sampling and calculation
The second step to reduce the digital delay is to use the lookup table for the SOTC calculation instead of the real-time calculation. In the previous implementation, the equations for ΔTUP in the load step-up and ΔTDOWN in the load step-down are expressed below, which is derived in [12]:
∆
80 60
40
40
20
20 0
0
Fig. 7. Look-up table for SOTC calculation of a 500kHz LLC converter
The result from the look-up table is the same with that from the real-time calculation, but the required CPU cycles for the calculation is reduced from around 400 CPU cycles to around 240 CPU cycles. And the table consists of only 121 integers, which is around 3% RAM space of TMS320F28027. With the look-up table, the digital delay is reduced to 4us, and the maximum switching frequency suitable for SOTC is increased to 250kHz. Although the evaluations above are based on given controller, the same analogy applies to the other digital controllers. In this section, the implementation of SOTC is improved by optimizing the sampling points and using the look-up table to reduce the calculation time. With these improvements and taking TMS320F28027 as an example, the maximum switching frequency suitable for SOTC is improved from 140kHz to 250kH. But there is still a limitation if we want to further push the switching frequency with given controller.
A. Proposed Multi-step SOTC
SOTC
∆
60
IV. PROPOSED MULTI-STEP SIMPLIFIED OPTIMAL TRAJECTORY CONTROL (SOTC)
I[k+1]
VgsQ1 VgsQ2 A/D for VO
-400 80
·
(3)
·
·
(4)
Although the implementation of SOTC is optimized and improved in Section III, there is still maximum switching frequency limitation with given controller. Multi-step SOTC is proposed in this paper to further push this maximum switching frequency limitation with given controller. The concept of SOTC is to settle the resonant tank within 2step, which is the optimal way to settle 3 resonant elements: resonant inductor Lr, resonant capacitor Cr and magnetizing inductor Lm in the LLC converters. However, due to this 2-step limitation, the maximum switching frequency with given controller is also limited. If we want to further push this maximum switching frequency limitation, the 2-step solution is not suitable any more.
Instead of trying to achieve the optimal performance, Multistep SOTC can be used to settle the resonant tank within more than 2 steps as shown in Fig. 8. The number m of steps in Multi-step SOTC is determined by the speed of the digital controller and the switching frequency of the power stage. For example, 2-step SOTC with TMS320F28027 is suitable for a switching frequency of up to 250kHz; then 4-step SOTC with the same controller would be suitable for a switching frequency of up to 500kHz; the same analogy applies to the cases with more steps. Even Multi-step SOTC uses more steps to settle the resonant tank; it can still achieve the benefit of SOTC, which means there is very small oscillation in the resonant tank during the load transient. This is because Multi-step SOTC is still much faster than the linear regulator, and the resonant tank is first settled to around the final steady state; then the linear regulator takes very little effort to eliminate the steady state error. For given high frequency power stage and low-cost digital controller, the benefit of SOTC can still be achieved by using Multi-step SOTC.
I[k]
iLoad VO
I[k+1] 6 steps
1 2 3 4 5 6
VgsQ1 VgsQ2
6-step SOTC
calculation
Fig. 9. Implementation of 6-step SOTC
B. Calculation of ΔT’ in Multi-step SOTC The calculation of ΔT in SOTC has been derived in [12], which will be briefly summarized in this paper. Based on the derivation of ΔT in SOTC, the calculation for ΔT’ in Multi-step SOTC is derived in the followings. Fig. 10 is an example illustrating the load step-up. The length of AB equals to that of CD, since the ΔTs for two steps in SOTC are always the same to ensure that the magnetizing current is settled no matter during the load step-up or the load step-down. The converter is assumed to run at around the resonant frequency. The light load current is Ilight and the heavy load current is Iheavy. Then the coordinates of the point A and the point E are expressed as: . ,
:
:
0.3
(5)
. ,
(6)
iLrN
0.2
B
A
0.1
E
ΔT 0
ΔT -0.1
D
C
Light load Heavy load
-0.2
-0.3 0.2
VCrN 0.3
0.4
0.5
0.6
0.7
0.8
Fig. 10. Calculation of ΔTUP in SOTC for load step-up Fig. 8. Proposed Multi-step SOTC
Fig. 9 shows the implementation of 6-step SOTC. The output voltage and the load current are sampled every third switching cycle. After the load transient happens, the controller would have 3 switching cycles (equivalent to 6 steps) to accommodate the calculation time. Then the controller uses 6-step to settle the resonant tank. So the same controller can be applied to the higher switching frequency with Multi-step SOTC compared with the case using 2-step SOTC.
Since the length of AB equals that of CD, the point B is approximately in the middle of the point A and the point E. And during AB, the magnetizing current ILm can be treated as approximately constant, expressed as: ·
(7)
And the resonant capacitor voltage is charged by the magnetizing current ILm during AB. Then ΔTUP can be derived: ∆
·
·
/
(8)
The ΔT’ in Multi-step SOTC for the load step-up is derived iteratively based on the conclusion from SOTC. Firstly, the ΔT’s in the multi steps are always the same since this ensures the magnetizing current are balanced and PWM cannot be updated every switching cycle in the digital controller due to the limited controller speed. Take 4-step SOTC for the load step-up as an example, shown in Fig. 11. 0.3
iLrN
.
·
.
.
(14)
The coordinates of the point A and the point E are expressed as: . ,
:
. ,
:
ΔT’
(15)
(16)
0.2
With equations (14) – (16), VCN is expressed as: ·
0.1
.
⁄
(17)
0
Since ΔAMO and ΔCNO are similar triangles, then:
Light load -0.1
.
Heavy load -0.2
-0.3 0.2
Load = IX
ΔT’
IAN is y-coordinate of the point A, which is expressed in equation (15), so the resonant current at the point C is expressed as:
VCrN
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 11. Calculation of ΔTUP_4’ in 4-step SOTC for load step-up
In 4-step SOTC, assume the first 2-step settles the resonant tank from the Ilight equivalent trajectory to the IX equivalent trajectory, then ΔTUP_4’ can be expressed as ·
∆
/
(9)
_
The next 2-step will settle the resonant tank from the IX equivalent trajectory to the Iheavy equivalent trajectory. So anther expression of ΔTUP_4’ is: ·
∆
/
_
·
/
_
(11)
·
By iteration, ΔTUP_m’ of m-step SOTC for the load step-up can be derived as below: ·
·
/
∆ 0.3
0.1
. .
·
(20)
Since the converter runs at around the resonant frequency, so the ΔTDOWN of SOTC for the load step-down is:
Fig. 12 is an example illustrating the load step-down. The angle of α1 equals to that of α2, since the ΔTs for the two steps in SOTC are always the same to ensure that the magnetizing current is settled. The length of AO equals to that of BO, and the length of CO equals to that of DO. So ΔABO and ΔCDO are similar triangles, then:
/
(19)
/
0.2
_
·
·
(12)
∆
/
Since the duration from the point B to the point C is a negligible time compared with the duration from the point A to the point B on the heavy load trajectory. In the time from the point A to the point B, the magnetizing inductor is clamped by the output voltage. So duration from the point A to the point B is approximately expressed as:
(10)
The expression of ΔTUP_4’ can be derived by combining equation (9) and equation (10) as below: ∆
(18)
.
(21)
iLrN B
M
0
-0.1
α2
E
A
C
α1 N
Light load
D
Heavy load
-0.2
. .
(13)
In equation (13), VXN means the normalized resonant voltage at the point X. With VBN = VCN and VDN = VEN, then:
-0.3 0.2
VCrN 0.3
0.4
0.5
0.6
0.7
0.8
Fig. 12. Calculation of ΔTDOWN in SOTC for load step-down
The ΔT’ in Multi-step SOTC for the load step-down is derived iteratively based on the conclusion from SOTC. Firstly, the ΔT’s in the multi steps are always the same since this ensures the magnetizing current are balanced and the PWM cannot be updated every switching cycle in the digital controller due to the limited controller speed. Take 4-step SOTC for the load step-down as an example, shown in Fig. 13. 0.3
iLrN
0.2
V. EXPERIMENTAL RESULTS The Multi-step SOTC is verified on a 500kHz LLC converter. The 500kHz LLC converter is designed based on Matrix Transformer for LLC Resonant Converters [24], which was originally a 1MHz 1kW 390V/12V unregulated DCX with GaN devices. Here the LLC converter is a 500kHz 1kW 390V/12V regulated DC/DC converter with Si devices. The schematic of the LLC converter with the matrix transformer and SOTC implemented by MCU is shown in Fig. 14. In addition to Multi-step SOTC for the load transient, the soft start-up [25], burst mode for the light load efficiency and the adaptive SR driving are also integrated in the controller. Lr1 4:1:1
400V
0.1
*
*
12V
SR1
*
VO
SR2
0
Lm1 *
Heavy load
SR3
*
Light load
-0.1
*
Lr
4:1:1 4:1:1
Cr
SR4 SR5
*
Lr2
*
-0.2
-0.3 0.2
*
VCrN 0.3
0.4
0.5
0.6
0.7
4:1:1
0.8
In 4-step SOTC, it is assumed that the first 2-step settles the resonant tank from the Iheavy equivalent trajectory to the IX equivalent trajectory, then ΔTDOWN_4’ can be expressed as _
(22)
The next 2-step will settle the resonant tank from the IX equivalent trajectory to the Ilight equivalent trajectory. So anther expression of ΔTDOWN_4’ is: ∆
_
(23)
Expression of ΔTDOWN_4’ can be derived by combining equation (22) and equation (23) as below: ∆
_
(24)
By iteration, ΔTDOWN_m’ of m-step SOTC for the load stepdown can be derived as below: ∆
_
SR7
*
Fig. 13. Calculation of ΔTDOWN_4’ in 4-step SOTC for the load step-down
∆
SR6
Lm2
Load = IX
2∙∆T
Q1,Q2 Driver
+ +
iLoad
SOTC
TS
-
PI Control
Microcontroller
Vref +
Fig. 14. Schematics of the LLC converter with matrix transformer and SOTC implemented by MCU
The parameters of the 500kHz LLC converter is: Lr = 4.5uH, Cr = 22nF, Lm = 22uH. The hardware of the 500kHz LLC converter is shown in Fig. 15, with the efficiency curve shown in Fig. 16. To control the 500kHz LLC converter, a 60MHz MCU (TMS320F28027) is selected, which is popular in the industrial application for power supplies of the servers and telecom. However, in the industrial application, the switching frequency of power converter is normally below 130kHz.
Primary devices
Transformer1
Transformer2
(25)
In this section, the adaptive Multi-step SOTC is proposed so that SOTC can be applied to the high frequency LLC converters with the low-cost digital controllers. The number of steps in Multi-step SOTC is selected based on the speed of the controllers and the switching frequency of the power stage. Furthermore, the required ΔT’ for Multi-step SOTC is derived based on the conclusion from SOTC.
SR8
MCU
SRs
SRs
Fig. 15. The 500kHz LLC converter hardware
efficiency
VI. CONCLUSIONS
96% 94% 92% 90% 88% 86% 84% 82% 80% 78%
Power(W) 0
200
400
600
800
1000
Fig. 16. Efficiency curve of the 500kHz LLC converter
The experimental results of the load step-up from 40A to 80A and the load step-down from 80A to 40A are shown in Fig. 17 and Fig. 18 respectively. The output capacitor is totally 5mF ceramic capacitor (X5R), whose capacitance drops to around 10% at 12V DC bias. So the actual output capacitance is only around 500uF. The overshoot for both the load step-up and the load step-down are within 600mV (5% VO), and there is almost no oscillation in the resonant tank during the load transient.
In this paper, the Simplified Optimal Trajectory Control (SOTC) is investigated and improved to push the maximum switching frequency limitation for given controller, and SOTC can be implemented by the low-cost controllers to achieve the fast load transient response. Then the Multi-step Simplified Optimal Trajectory Control (SOTC) is proposed to further push the maximum switching frequency limitation, which calculates the trajectory based on only the output voltage and the load current, and settles the resonant tank within multi steps. With Multi-step SOTC, the low-cost controllers can be used to control the high frequency LLC converters with the state-trajectory control. The number of steps in Multi-step SOTC is determined by the speed of the controllers and the switching frequency of the LLC converters. Experimental results are demonstrated on a 500kHz 1kW 400V/12V LLC converter with 60MHz MCU (TMS320F28027). Fast load transient response is achieved by using 6-step SOTC.
REFERENCES [1]
[2]
iLr (6A/div)
SOTC [3]
iLoad (40A/div)
Load transient (≈10us)
[4]
VO (500mV/div)
[5]
600mV (5% VO )
50us/div
Fig. 17. Multi-step SOTC for load step-up (from 40A to 80A)
[6]
[7]
[8]
[9]
iLr (6A/div) iLoad (40A/div)
SOTC [10]
Load transient (≈10us)
[11]
600mV (5% VO )
VO (500mV/div) 50us/div Fig. 18. Multi-step SOTC for load step-down (from 80A to 40A)
[12]
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