Bifurcations in Frequency Controlled Load Resonant DC-DC Converters Kuntal Mandal∗ , Soumitro Banerjee† , Chandan Chakraborty∗ , Mrityunjoy Chakraborty∗ ∗
† Indian
Indian Institute of Technology, Kharagpur-721302, India Institute of Science Education & Research (Kolkata), Mohanpur Campus 741252, India
Abstract—The complex behavior of load resonant converters is investigated by using a newly developed stability analysis tool. In this paper we specifically consider the series-parallel load resonant converter with L-C output filter, controlled by the frequency modulation technique. The close loop variable frequency control consists of a feedback loop and a PI controller with voltage controlled oscillator (VCO). The VCO is used to convert the error in output voltage into a change of frequency to control the voltage in close loop system. Two converter designs are considered with different combinations of L-C output filter. With the variation of the proportional gain, Neimark-Sacker and symmetry-breaking bifurcations are observed.
converters. However, it is inaccurate for discontinuous capacitor voltage mode (DCVM) due to the distortion of the circuit waveforms. A lot of work has been done to extend the averaged modeling approach from the PWM dc-dc converters to resonant converters. The drawback with this approach is that the resultant model has twice the order of the original system. The stability analysis of period-1 as well as higher
Vin
I. I NTRODUCTION The resonance based converters are capable of providing high efficiency, and compact size due to their operation with near lossless switching [1]. In many potential applications such as power sources for telecom, medical equipment, and computer systems, second- and higher order resonant topologies are used [1], [2]. Compared with the conventional secondorder resonant converters, higher-order converters of LCC and LLC-type are shown to posses more desirable characteristics. In addition, by using higher-order resonant tanks, the designer has the choice of utilizing the parasitic capacitance and inductance especially when operating at very high frequency. As a result, depending on the topology used, the parasitic reactance can turn into an asset rather than a liability. Moreover, the diversity in resonant topologies gives the designer the freedom to choose the topology that most suits the application. A large variety of control strategies have been proposed for such resonant converters, which can be divided into two categories: variable frequency control and constant frequency phase-modulated control. For conventional variable frequency control a wide range of switching frequency is required to obtain both the load and the line regulations which is not desirable from the point of view of EMI filter and transformer design. The constant frequency control technique overcomes the disadvantages of variable frequency technique, however their control has an added complexity and is applicable to fullbridge switch networks only. For largely economic reasons, frequency control technique of such converters is generally employed in the consumer electronic field since the half-bridge switch network requires fewer components. Traditionally, power electronics practitioners use the fundamental frequency technique [1] for obtaining some information about the stability and dynamic behavior of load resonant 978-1-4673-0219-7/12/$31.00 ©2012 IEEE
Vin + 2 + −
b Vin + 2
Cin S1 2 G1 Cin S1 2 G1
iL + vC-s Ls Cs
D1 a v ab
if
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Lf
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+ C f v0
Cp vC+p
RL
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Switch network Resonant tank Gate Driver
vsc
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vcon
Rectifier + K
LPF
p
+
Ki Error Amplifier
Load + Vref
Fig. 1. The variable frequency controlled half-bridge series-parallel load resonant dc-dc converter.
periodic orbit of any resonant converters is difficult due to the fact that the number of the topological modes or subsystems within a period can vary dramatically depending on the power circuit topologies and control techniques. Due to the lack of a generic method, the complex behaviors of the resonant dc-dc converters remain largely unexplored unlike the basic dc-dc converters [3]. The demand of high bandwidth performance necessitates the detailed numerical analysis of the complex behavior of this class of converters. Utilizing the periodicity matrix and Poincar´e map, Nagy and Dranga detected the first instability in one member of a double channel resonant dc-dc converter family [4]. However, this method cannot be applied for all resonant converters and all high-periodic orbits due to the fact that the complexity of the method increases with the number of subsystems because the switching conditions implicitly appeared in the final expression. Computationally efficient and generic algorithms are, therefore, sought to accurately determine the boundaries between different operating modes and to predict the mechanisms of instability of the desired operation under parameter variation. Once we have an understanding of the pathways by which an orbit can lose stability, it is possible to use that knowledge to devise methods of avoiding (or delaying) the onset of these instabilities
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[5]. Based on the “fundamental solution matrix” [5] where each switching is treated separately, we have developed an algorithm for the stability analysis of the periodic orbit [6] of such a complex system. In our earlier work, we have reported the subharmonic, quasiperiodic [6], symmetry-breaking [7], and chaotic operation [8] in the fixed frequency phase-shift modulated (PSM) series-parallel resonant converter (SPRC). The aim of this paper is to extend the work to the frequency controlled SPRC which is a nonautonomous but not timeperiodic system. II. C ONVERTER C ONFIGURATION AND O PERATION The system shown in Fig. 1 and its operation have been extensively discussed in the literature [9]. Like all other resonant converters it uses a resonant tank circuit which consists of three reactive elements: a series resonant inductor L𝑠 , a series resonant capacitor C𝑠 , and a parallel resonant capacitor frequency C𝑝 . The√resulting resonant tank will have a resonant √ f𝑟 = 1/ 𝐿𝑠 𝐶𝑠 and characteristic impedance Z0 = 𝐿𝑠 /𝐶𝑠 . The typical waveform of the studied system operating above resonance (switching frequency 𝑓𝑠 > 𝑓𝑟 ) is presented in Fig. 2. This converter possesses six topological circuit configurations and two operation modes: I and II. Mode I operation (which is called continuous capacitor voltage mode (CCVM) results for low 𝑄𝑠 (𝑍0 /𝑅𝐿 ) or lower load, and the topological sequence is given as 𝑀4 −𝑀3 −𝑀5 −𝑀6 (see Fig. 2). In the topological sequence 𝑀4 − 𝑀8 − 𝑀3 − 𝑀5 − 𝑀9 − 𝑀6 (operation mode II) the circuit will provide poor gain and higher load. In operation mode II, there is an additional subsystem where parallel resonant capacitor voltage is zero (i.e. DCVM) in each half cycle. G1 G1 Vin 2
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Assuming ideal components in the circuit (shown in Fig. 1), the frequency controlled SPRC can be described by the following differential equations with discontinuous right hand sides 1 1 dvCs iL 𝑑𝑖𝐿 = [−𝑣𝐶𝑠 − 𝑣𝐶𝑝 + 𝑉in sign(sin(2𝜋fs t))], = 𝑑𝑡 𝐿𝑠 2 dt Cs 𝑑𝑣𝐶𝑝 1 1 𝑑𝑖𝑓 = = [𝑖𝐿 − 𝑖𝑓 sign(v𝐶𝑝 )], [abs(v𝐶𝑝 ) − 𝑣0 ] 𝑑𝑡 𝐶𝑝 𝑑𝑡 𝐿𝑓 1 𝑣0 𝑑𝜌 𝑑𝑣0 = = 𝐾𝑖 [𝑣0 − 𝑉ref ] [𝑖𝑓 − ], 𝑑𝑡 𝐶𝑓 𝑅𝐿 𝑑𝑡 From a “hybrid system” point of view, the system studied here consists of three switching surfaces and six subsystems when operating above resonance and with continuous filter inductor current. = The equations can be expressed in matrix form as 𝑑x [ ]𝑇𝑑𝑡 Ax + Bu where, x = 𝑖𝐿 𝑣𝐶𝑠 𝑣𝐶𝑝 𝑖𝑓 𝑣0 𝜌 = [ ]𝑇 ]𝑇 [ 𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 , u = 𝑉𝑖𝑛 𝑉𝑟𝑒𝑓 . Here 𝑇 denotes transpose. Circuit symmetry implies an useful relationship among the matrices appearing in the different subsystems: B3 = B4 = B8 , B5 = B6 = B9 , B6 = W 𝑐 B3 , A4 = A6 , A3 = A5 , A8 = A9 , A3 = W 𝑐 A4 W 𝑐 , A8 = W 𝑠 A3 W 𝑠 , W 𝑐 W 𝑐 = I. Here, I is the identity matrix and the periodicity matrix ⎡ ⎡ ⎤ ⎤ −1 0 0 0 0 0 −1 0 0 0 0 0 ⎢ 0 −1 0 0 0 0 ⎥ ⎢ 0 −1 0 0 0 0 ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ 0 0 −1 0 0 0 ⎥ ⎢0 0 0 0 0 0⎥ ⎢ ⎢ ⎥ ⎥ W𝑐 = ⎢ ⎥, W 𝑠 = ⎢ 0 0 0 1 0 0 ⎥ ⎢0 0 0 1 0 0⎥ ⎢ ⎥ ⎣0 0 0 0 1 0⎦ ⎣0 0 0 0 1 0⎦ 0 0 0 0 0 1 0 0 0 0 0 1 and
Ts
A. Control of Output Voltage
∫ The control voltage vcon = K𝑝 (v0 -Vref ) + K𝑖 (v0 -Vref )𝑑𝑡 is fed to the input of a voltage controlled oscillator (VCO). The VCO output is a periodic signal at twice the switching frequency. The toggle flip-flop outputs the symmetrical square wave signal to control the Mosfets through gate drivers. The output frequency of an ideal VCO is given by =
III. M ATHEMATICAL M ODEL
D7 D8
Fig. 2. The typical waveforms of steady-state period-1 for variable frequency controlled SPRC operating above resonance in CCVM.
𝑓𝑠
consequently causes the shift from the center frequency. The VCO block generates a signal whose frequency shifts from the center frequency and maintains a linear relationship with the input signal of the VCO, vcon (𝑡). The VCO output may be given as vvco = 𝐴vco sin(2𝜋𝑓𝑠 𝑡).
⎤ ⎡ 1 ⎤ 0 0 0 − 𝐿1𝑠 − 𝐿1𝑠 0 0 2𝐿 𝑠 ⎢1 0 0 0 0 0⎥ ⎢ 0 0 ⎥ ⎥ ⎢𝐶𝑠 ⎢ ⎥ 1 ⎢1 ⎢ 0 0 𝐶𝑝 0 0⎥ 0 ⎥ ⎥ ⎢𝐶𝑝 0 ⎢ ⎥ ! = A4=⎢ ⎥, B − 𝐿1𝑓 0 ⎥ 3 ⎢ 0 − 𝐿1𝑓 0 0 0 ⎥ ⎢0 ⎢ ⎥ ⎥ ⎢ ⎣ 0 0 ⎦ ⎣0 0 0 𝐶1𝑓 − 𝐶𝑓1𝑅𝐿 0 ⎦ 0 −𝐾𝑖 0 0 0 0 0 𝐾𝑖 In each half cycle of CCVM operation, there are two kinds of switchings: (i) those determined by zero voltage across the parallel resonant capacitor and (ii) those at the zero crossing of the VCO output signal (see Fig. 3(a)): ⎡
(
𝑓0 + 𝐾vco 𝑣con (𝑡)
where, vcon (𝑡) is the input to the VCO. If vcon (𝑡) = 0, the VCO operates at a set frequency 𝑓0 , called the center frequency. The input sensitivity Kvco scales the vcon (𝑡) and
ℎ2 : t − 𝑎𝑏𝑠
𝑛 f0 + Kvco vcon
)
ℎ1 : x3 = 0,
(1)
= 0, 𝑓 𝑜𝑟 𝑛 = 1, 2, 3...
(2)
There may be another switching (Fig. 3(b)) at the end of crossing-sliding of the parallel resonant capacitor voltage in
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Ts
1st half cycle T1 M4
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M3
x3
2nd half cycle T1
Th M5
where, the normal to the switching surface ℎ2 : 𝐾 (𝐾𝑝 E+D) n𝑇2 = [𝑓0 +𝐾vco 𝐾𝑝 (E x(𝑡vco , [ ] 1 − )−𝑉ref [ )+𝐾vco D x(𝑡1 −])]2∂ℎ 2 E = 0 0 0 0 1 0 , D = 0 0 0 0 0 1 , ∂𝑡 ∣𝑡=𝑡1 = 1. When the period-1 orbit loses stability, i.e., beyond the first instability, a higher periodic stable orbit may occur; but the symmetry between the half cycles will break. Whenever the Poincar´e section is encountered, the extra term within the second bracket in the equation (6) is multiplied.
Th
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V. D ESIGN OF THE C ONVERTER
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(b) Fig. 3. Time sequences of subsystems in period-1 steady state for (a) CCVM (b) DCVM. In the switching surface ℎ𝑖𝑗 , the first and second subcripts indicate switching number and half cycle number respectively.
the DCVM operation: ℎ3 : x4 − 𝑎𝑏𝑠(x1 ) = 0.
(3)
IV. S TABILITY ANALYSIS OF THE PERIODIC ORBIT 1st half cycle
x3>0
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2nd half cycle M5
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For the given specifications in Table-1, the operating region is chosen based on the design method given in [9]. The design values given in Table-2 are chosen for above-resonance worstcase loading conditions (i.e., for maximum load current with minimum input voltage). We also want the filter inductor current to be in continuous conduction mode from the light load (𝑅𝐿 = 22.5 Ω) to the full load (𝑅𝐿 = 4.5 Ω). To reduce the rectifier diode switching loss and EMI noise level the maximum filter inductor ripple current is set to ±10%. The converter with these design values operates entirely in the CCVM. Table-1: Specifications Input voltage, V𝑖𝑛 30 𝑉 ± 20% (24 𝑉 − 36 𝑉 ) Output voltage, v0 15 𝑉 ± 1% Filter inductor ripple ±10% Output power, P0 10 − 50 𝑊 (4.5 Ω − 22.5 Ω) Resonant frequency, f𝑟 40 kHz
M6
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