Efficient CAD tool for power electronics compensator ... - GSEP - UC3M

Efficient CAD tool for power electronics compensator design C. Martínez*, V. Valdivia*, A. Lázaro**, J. Lourido, I. Quesada*, C. Lucena*, P. Zumel** A. Barrado** * Student member, IEEE **Member, IEEE Power Electronics System Group (GSEP) Carlos III University of Madrid Avda. Universidad 30 28911 Leganés Madrid, Spain 1 Abstract -- Nowadays CAD tools are widely used in power electronics. Power circuit simulators as well as some other dedicated tools allow reducing developing times. In this paper a new and efficient CAD tool for power electronics compensator design is presented. This tool incorporates some new concepts and algorithms oriented to obtain the optimal loop performance through the compensator design. The CAD tool allows easy designing of PWM regulated converters since it provides an easy and friendly interface tool to design the compensator; both from the conventional topologies and the small signal AC analysis of power circuit simulators.

I. INTRODUCTION

S

INCE early 90´s CAD tools have been successfully applied to power electronics for analysis, design and teaching purposes. These tools can be sorted into four main groups: simulation software [1] – [6], mathematical software [7] – [8], FEA tools [9] – [10], magnetic components and thermal tools [11] - [12], vendor tools [13] and software tools for specific application design [14]. From the aforementioned summary, it is concluded that there is no any tool specifically dedicated to the design and optimization of the regulator of power converters control loop. Most of simulation software [1] - [4] provides automatic small-signal AC analysis without using averaged models. However, the compensator design and optimization process is not completely solved from a general point of view. Nevertheless, these tools are not suitable for a general case and do not provide access to the design criteria. Venable 350 Demo [14] can be considered a good approximation to a generalized compensator design program. This software uses imported data (from measurements or simulations) of the converter small signal model to tune the compensator. Only K-factor method [15] is provided as tuning strategy and only a voltage mode control (VMC) is

This work has been supported by Ministerio de Educación y Ciencia (Spain), by means of the research project SAUCE (DPI: 2009-12501).

considered. However the main idea is very useful due to the fact that the tool can be applied to a generic converter. This work starts from the limitations of compensator synthesis strategies based on direct numerical calculation and proposes some new ideas and algorithms implemented within a CAD tool. The algorithms validation has been carried out by designing a converter using the CAD tool, and comparing the behavior of the converter predicted by the CAD tool with that obtained by making use of PSIM® simulator [1]. II. LIMITATIONS OF THE CURRENT SYNTHESIS STRATEGIES AND PROPOSED NEW METHODOLOGY A. Limitations A variety of synthesis techniques can be found in the scientific literature [15] - [19]. Most of these methods allow a straightforward compensator synthesis from two numerical inputs, e.g. phase margin (PM) and crossover frequency (fcross). Although in all the aforementioned techniques, the zeroes and poles placement is direct, the selection of these two input parameters for the optimization of the loop overall performance is not always immediate. K-factor [15] provides direct pole-zero placement and determination of the resultant circuit component values. Although it is a powerful method for compensator design, the fact of the “blind” selection of fcross and PM can lead to a non optimum solution, particularly in converters with complex dynamics. B. Proposed methodology In order to overcome the abovementioned limitations, this work presents a new strategy. Its most significant features are additional optimization algorithms: “solutions map”, “Kplus method” and “manual method” (see sections 4, 5 and 6 respectively) and real-time updated plots and numerical values (Bode, polar and transient response plots) in the active windows (see Fig.1). Therefore, the designer is able to observe every parameter effect at real time, and so to select the optimized compensator design.

Fig.1: Main screen of the proposed software tool. All the plots are real-time updated under changes in the compensator design degrees of freedom or under any change in the converter parameters.

III. AVAILABLE TOPOLOGIES AND OPERATING POINT The CAD tool includes some conventional topologies such as Buck, Boost, Buck-Boost, Flyback and Forward converters. All of them present both voltage and current control loops. Additionally, a powerful characteristic is provided: the CAD tool is able to easily import any transfer function obtained from the AC sweep analysis provided in some software simulators. In this way, the designer is able to design the control loop of power stages with complex dynamics without deriving its small signal model. Thus, the developing time of the converter control can be reduced significantly. When designing a DC/DC converter, the static operating point of the converter is a key aspect to be taken into account by the designer. Thus, in the data entry window itself (see Fig. 2), the conduction mode and the inductance current values (maximum, minimum and average) are calculated from the input data. Hence, the operating point of the converter, which is a key aspect in control loop design, is calculated and displayed On Fig. 2, the data entry and the static operating point of a VMC Buck converter is shown. This converter will serve as an example to analyze the most important features of the CAD tool.

Fig. 2: Data entry screen and static operating point of the converter. Example shown: Buck converter (VMC).

IV. SOLUTIONS MAP Once the converter parameters have been defined and the sensor type has been selected, the next step in the designing process is the selection of the compensator type. The CAD tool provides four different kinds of compensators such as "Single pole", PI compensator, Type 2 compensator and Type 3 compensator. The schematic and the Bode diagram of each of them (only module is represented) can be found on Fig. 3.

a)

VO (dB ) Vin

20dB/dec -20dB/dec

-20dB/dec

Vin

Ref

VO

fZ1 fZ2

+

b)

VO (dB ) Vin

fP1 fP2

log(f)

-20dB/dec 0dB/dec

Vin

Ref

-20dB/dec

VO

fZ

+

c)

VO (dB ) Vin

Vin

Ref

fP

log(f)

-20dB/dec 0dB/dec

VO

+

fZ VO (dB) Vin

d)

log(f)

0dB/dec -20dB/dec

Vin

Ref

PM max_ T 3 = 4·atan( K max ) − 90 + P( f )

(2)

PM max_T 2 = 2·atan( K max ) + P( f )

(3)

Fig. 4, Fig. 5 and Fig. 6 show the solutions map for a Type 3, Type 2 and PI compensators respectively, for the example case of a VMC Buck converter. In all cases, the lower boundary corresponds to a simple integrator. Within the valid area (white color between the two curves) each set of fcross and PM corresponds to a stable solution, so a stable design can be obtained at the first attempt. Afterwards the designer is able to optimize the overall performance of the control loop, for instance using the K-plus method explained in detail in section VI. fsw

VO

+

fP

log(f)

PM min = 90 + P ( f )

(1)

where P(f) is the phase of the open loop transfer function without compensator. The upper limit is given by the maximum phase boost provided by each kind of compensator. If the K-factor method [15] is selected, for any fcross and PM and fixing K to its maximum value (Kmax), the upper limit of the solutions map for a Type 3 compensator (PMmax_T3) and for a Type 2 compensator (PMmax_T2) can be obtained by the rearranging of the K-factor equations as is shown in (2) and (3) respectively.

Crossover Frequency (Hz) Fig. 4: Solutions map of a VMC Buck converter with a Type 3 compensator.

fsw

Phase Margin (degrees)

At this point, every component comprising the control loop has been chosen (power stage, sensor and regulator type). Next, the parameters of the compensator have to be designed. One of the key issues during this task is the appropriate selection of fcross and PM. In order to ease the first attempt when designing a control loop, the “solutions map” feature has been developed. Based on the selected power stage, sensor and type of regulator, the solutions map provides a “safe operating area” of the different combinations of fcross and PM that lead to stable systems. Thus, this map minimizes the number of trial and error iterations associated to this process. The boundaries, which determine the valid area, represent the maximum and minimum phase margin that can be achieved for any kind of compensator. In terms of frequency, the range is delimited by the switching frequency (fsw). The simple integrator is a particular case of any compensator, therefore it provides the lower phase margin limit (PMmin) by adding 90 degrees to the phase of the open loop transfer function without compensator (power stage, sensor and modulator). PMmin is represented in expression (1).

Phase Margin (degrees)

Fig. 3: Compensator allowed by the CAD tool. a) Type 3 compensator. b) Type 2 compensator. c) PI compensator. d) "Single pole".

Crossover Frequency (Hz) Fig. 5: Solutions map of a VMC Buck converter with a Type 2 compensator.

1 ⎡ π P ( f cross ) ⎞ ⎤ ⎛ (5) tan ⎢atan (α ) − ⎜ PM − − ⎟⎥ 4 2 ⎝ ⎠ ⎣ ⎦ The additional design degree of freedom provided by Kplus method can be used as follows: if “α” is set to be lower than K, higher gain at low frequencies but less attenuation at fsw are achieved (see Fig. 8). On the contrary, if “α” is set higher than K, the control loop has less gain at low frequency but more attenuation at fsw (see Fig. 8). It should be remarked that the phase margin is the same in all cases.

fsw

Phase Margin (degrees)

β=

Kplus with α < K: Higher Gain at Low frequencies 40

Fig. 6: Solutions map of a VMC Buck converter with a PI compensator.

fc

fsw

K-factor 20

A new optimization method for compensator synthesis, hereinafter Kplus, is presented along this epigraph. The Kplus method is based on the K-factor method [15], indeed the inputs are the same (fcross and PM). However, unlike K-factor method [15], fcross is no longer the geometric mean of the zeroes and the poles frequencies. The Kplus method provides an additional design degree of freedom with respect to the conventional K-factor [15], since the new method places the zeroes frequency fz a factor “α” below fcross (fz = fcross/α) and the poles a factor “β” above fcross (fp = fcross•β) (see Fig. 3). Therefore, “α” is set from fcross and PM and “β” is automatically calculated. The definition of “β” depends on the type of compensator: (4) defines “β” for a Type 2 compensator and (5) defines “β” for a Type 3 compensator. PM is the desired phase margin and P(fcross) is the phase of the open loop without regulator at fcross.

T (s)

0 dB

α

α

fcross β

fP1, fP2

log(f)

f cross ⋅ β

Fig. 7 Basis of the Kplus method: example of a Type 3 compensator design. R(s) is the regulator transfer function. T(s) is the open loop transfer function. Both fZ1 and fZ2 are the designed zeroes frequency and fP1, fP2 are the designed poles frequency once zeroes have been fixed.

β=

Kplus with α < K: Lower attenuation at fsw

0

Kplus with α > K: Lower Gain at Low frequencies

− 20

Kplus with α > K: More attenuation at fsw − 40 100

3

1×10

4

1×10

5

1×10

6

1×10

Frequency (Hz)

Fig. 8: Comparative Bode plots of K-factor vs Kplus.

Therefore, the Kplus method can be used to improve the overall performance of the control loop in those cases where a slightly larger high frequency ripple could be admitted at the input of the PWM modulator. Once the zeroes and poles frequencies by means of Kplus method have been determined, the regulator components can be easily obtained. VI. MANUAL METHOD

R(s)

fZ1, fZ2 f cross

T(dB)

V. K-PLUS METHOD

1 π ⎡ ⎛ ⎞⎤ tan ⎢atan (α ) − ⎜ PM − − P ( f cross ) ⎟⎥ 2 ⎝ ⎠⎦ ⎣

(4)

This method places poles and zeroes independently from each other. It is used to optimize the results obtained from the K-factor [15] and Kplus methods or when these automatic methods do not provide a valid solution. The designer has two ways to place zeroes and poles: • Introduce the exact frequencies into a dialog box. • Drag the poles and zeroes directly on the open loop transfer function Bode plot (see Fig. 9). Thus, it can be observed how this placement affects the system stability.

The developed algorithm starts obtaining the so-called transfer functions called “output impedance” ( vˆO / iˆO ), audiosusceptibility ( vˆO / vˆin ) and the transfer function of the output voltage with respect to the reference ( vˆO / vˆ ref ) [19], [20], where vˆO is the output voltage disturbance, iˆO is the output current disturbance, fP1 fP1, fP2

Manual fP2

K-factor

vˆin

is the input voltage

disturbance and vˆ ref is the reference voltage disturbance. Next, these transfer functions are represented in a LinearTime-Invariant Canonical-State-Space-Model form [21]. A key aspect for real time transient plot is the automatic axis adjustment. Therefore, the time interval of the transient response (total time, tt) to be represented is automatically chosen as a function of the magnitude of the lower frequency pole. The time interval between calculated points (time step, ts) is automatically chosen as a function of the magnitude of the higher frequency pole. Finally, the transient response is calculated using tt and ts. VIII. ANALOG AND DIGITAL COMPENSATOR

Fig. 9: Open loop transfer functions obtained with the K-factor (blue trace) and manual (red trace) methods.

Fig. 9 shows the open loop transfer function of a VMC Buck converter with a Type 3 compensator obtained with the K-factor method [15] (blue trace) and the one obtained with the manual method (red trace). Using the manual method, one of the poles of the Type 3 compensator has been moved to a higher frequency. The new compensator obtained with this method provides more phase margin but less attenuation at switching frequency. It is remarkable that every changes can be observed in real time. VII. TRANSIENT RESPONSE Transient response specifications, such as setting time and overshoot are usually critical specifications when designing the compensator of a power converter. Therefore, providing a real time view of the converter transient response may greatly help the designer while designing the control loop. This feature reduces dramatically the designing time, because it is not necessary to simulate the transient response of the system each time any parameter of the control loop is modified.

The presented CAD tool allows obtaining both analog and digital compensators. The digital compensator has been obtained from the continuous-time compensator by means of discretization through either the bilinear transform or the Tustin method. The effects of the time delays are included once the compensator has been obtained. If the time delay implies a phase delay too high, the continuous compensator should be redesigned with more phase margin. On the other hand, the rounding of the coefficients which appears in fixed-point systems has been taken into account. From the number of available bits to represent data in digital system, the coefficients are rounded and the new discrete compensator with rounding coefficients is analyzed. IX. PARAMETRIC SWEEP An important feature that presents the proposed new CAD tool is the parametric sweep. By means of this characteristic, once the compensator has been obtained, the power stage and the compensator values can be changed. Therefore the designer is able to observe every effect in the control loop. In this parametric sweep the compensator obtained can be modified or not depending on the choice of the designer. In this way, a sensitivity analysis can be made by the designer.

a) 3.3002

3.3

Vo(V)

This section has two purposes. The first one involves comparing the optimized design using the new method of synthesis (Kplus) ) with respect to the design obtained from the K-factor method, and the second one seeks to compare the results obtained from PSIM® [1] with the results obtained from the proposed CAD tool. The comparison is made in terms of both the frequency response (Fig. 10) and the output-current step transient response (Fig. 11 and Fig. 12). For these purposes it has been implemented the abovementioned VMC Buck converter with a Type 3 compensator. Fig. 10 shows the open-loop Bode plots computed with the CAD tool and those obtained using PSIM® [1] (only the magnitude is shown). It should be noticed that the new optimization method (Kplus) achieves more gain at low frequencies but slightly less attenuation above fcross than the K-factor method [15].

3.2998

PSIM®: Kplus 3.2996

3.2994

0.0148

0.0154

0.0156

0.0158

0.016

0.0158

0.016

b) 3.3002

3.3

3.2998

PSIM®: K-factor 3.2996

Kplus 20

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T(dB)

0.0152

f cross

a) 40

0.015

0.0152

0.0154

0.0156

Time (ms)

0

K-factor

Fig. 11: Output voltage transient response under an output current step obtained with PSIM® [1].a) Kplus. b) K-factor method [15].

-20

a)

PSIM® -40 100

b)

0.015

Time (ms)

Vo(V)

X. SIMULATIONS RESULTS

1K

Frequency (Hz)

10K

100K

40

Kplus

Vo(V)

30 20

CAD tool: Kplus

T(dB)

10

K-factor

-10 Time (ms)

-20 -30

b)

CAD tool

-40 1K

Frequency (Hz)

10K

100K

Fig. 10: Open loop transfer function Bode plots. a) PSIM® [1]. b) CAD tool.

Fig. 11 and Fig. 12 show, respectively, both the simulated and the computed with the CAD tool transient responses under an output current step. Due to the higher low frequency gain achieved, the response of the compensator designed using Kplus presents less overshoot than the designed using the K-factor. Both the transient and frequency responses obtained with PSIM® [1] and the CAD tool fit accurately.

Vo(V)

100

CAD tool: K-factor

Time (ms)

Fig. 12: Output voltage transient response under an output current step obtained with the CAD tool. a) Kplus. b) K-factor method [15].

XI. COMMUNICATION PSIM® - CAD TOOL It has been developed a bidirectional communication between PSIM® [1] and the CAD tool. It allows importing the small signal AC analysis obtained with the simulator and after finishing the design process, exporting the resultant compensator to PSIM® [1]. The designer can choose to export the data into a .txt file or directly to export the components to the simulator (s-domain and z-domain). This communication reduces significantly the converter closed loop simulation.

[8]

JMAG. POWERSYS Solutions. [on line]. [Consulted: May, 6, 2010]

[9]

MAXWELL. Ansoft high-performance EDA software. [on line]. [Consulted: May, 6, 2010]

[10]

PEXPRT. Ansoft high-performance EDA software. [on line]. [Consulted: May, 6, 2010]

[11]

GECKOHEAT. Gecko Research [on line]. [Consulted: May, 6, 2010]

[12]

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[13]

POWERESIM. Powersim Software AS. [on [Consulted: May, 6, 2010]

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VENABLE 350 DEMO. Venable Corporation. [on [Consulted: May, 6, 2010]

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[15]

VENABLE, H. D. "The k-factor: A New mathematical Tool for Stability, Analysis, and Synthesis". Proceeding of. Powercon 10, San Diego, CA, March 22-24, 1983.

[16]

O'DWYER, A. “Handbook of PI and PID controller tuning rules”. Imperial College Press, 2006. ISBN 978-1-84816-242-6.

[17]

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[18]

ROSSETTO, L., SPIAZZI, G. “Design Considerations on Current-Mode and Voltage-Mode Control Methods for Half-Bridge Converters” in Proc. of IEEE App. Power Elect. Conf. (APEC’97). ISBN 0-7803-37042.

[19]

ERICKSON, R. W. “Fundamentals Of Power Electronics 2nd edition", Springer, 2001. ISBN 0-7923-7270-0.

[20]

KAZIMIERCZUK, M. “Pulse-Width Modulated DC-DC Power Converters”, Wiley, 1997. ISBN 978-0412105418.

[21]

K. Ogata, “Modern Control Engineering, 4th edition”, Prentice Hall, 2008. ISBN 978-8131703113.

ACKNOWLEDGMENT This work has been supported by Ministerio de Educación y Ciencia (Spain), by means of the research project SAUCE (DPI: 2009-12501). XII. CONCLUSIONS CAD tools for power electronics state-of-the-art presents a gap in the compensator design and optimization. In this work a new philosophy for control loop optimization have been proposed based on complete and real-time loop shaping according to user´s commands. All this features have been developed in a new CAD tool and some new optimization algorithms have been also included. Some predefined small signal models have been provided and also external data can be imported to the proposed CAD tool. This feature provides the flexibility to design an optimized control loop for almost any system specially in those with complex dynamics. Within the CAD tool it has been included the converter transient responses under an output current step, reference voltage step and input voltage step. This characteristic allows the designer to observe the dynamic performance without a simulation. All features presented within the CAD tool allows to reduce the designing process of the converter control loop. Therefore the total developing time is reduced significantly. XIII. REFERENCES [1]

PSIM Simulation Software. POWERSYS Solutions. [on [Consulted: May, 6, 2010]

line].

[2]

SIMPLORER. Ansoft high-performance EDA software. [on line]. [Consulted: May, 6, 2010]

[3]

SIMPLIS. Simetrix Technologies Ltd. [on line]. [Consulted: May, 6, 2010]

[4]

PLECS. Plexim inc. [on line]. [Consulted: May, 6, 2010]

[5]

PSPICE A/D. Cadence Design Systems, Inc. [on line] [Consulted: May, 6, 2010]

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MATHCAD. Parametric Technology Corporation. [on [Consulted: May, 6, 2010]

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MATLAB. The Mathworks, Inc. [on line] [Consulted: May, 6, 2010]

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