A generic algorithm for global steady-state characterization of POL ...

A Generic Algorithm for Global Steady-State Characterization of POL Converters Michel S. Nakhla DOE, Carleton University Ottawa, Ontario, Canada, KI S 5B6 [email protected]

Martin Plesnik Nortel Networks Nepean, Ontario, Canada, K1Y 4H7 [email protected] Abstract - k new algorithm for global steady-state characterization of switching POL (point-of-load) converters as an efficient CAD tool for design and analysis is presented. The global steady-state characterization, involving the conventional simulation of converter operation as well as stress, sensitivity and tolerance analysis, allows for a complex evaluation of converter topology. The presented algorithm i s based on efficient evaluation of peaks, ripples, average and r m s values of converter internal variables using state variables. The inherent steady-state problem is solved using a new, fast, robust and accurate method for determining converter steady-state. Computer simulation and experimental results are attached.

circuit variables vary too. Thus, it is important to know not only values of converter circuit variables at nominal operating conditions, but also, it is important to know how they vary in response to changes of operating conditions, what are their maximums and minimums, and at which operating conditions the maximums and minimums occur. A search for minimum or maximum of converter internal variables may not be a simple task.This is because the minimum and maximum values of different converter variables can occur at different and, generally unknown, operating conditions. Even further, it is important to know how both the converter performance and converter component stress are sensitive to change of device parameters. Such a complex analysis of converter steady-state response is referred to in this work, as the global steady-statecharacterization.

1. INTRODUCTlON

In generd, the problem of the global characterization as specified above can be solved using commercial general-purpose simulators, using formulas and equation commonly used for switching converter design, and using specialized switched circuit simulators. While the use of commercial simulators ([I], [2]) in VLSI designs satisfies request for many types of circuit analysis, the use of these tools in switching converters area is not straight forward and introduces many difficulties. In principle, the simulation algorithm should involve several following steps: 1. simulation of the switching operation in the time-domain at a particular operating point, 2. determination of steady-state, 3. inspection of simulated waveforms for one switching cycle including search and storage for their peaks and ripples, and computation and storage for their average and r m s values, and 4. repetition of the procedures 1 - 3 for other operating points within the specified ranges of input parameters. The disadvantages of this approach include the large amount of time consumed by the time-domain simulation (e.g. [3]), convergence issues, detection of steady-state, difficulties with repeating procedures 1 - 3 and further data processing using the set of commands available with the commercial simulators.

A thorough understanding and evaluation of converter steadystate response to changes of input voltage, load current or device parameters are crucial to the analysis of converter topology and converter design. A converter failure in steady-state, caused by excessive component stress, or by failure in meeting performance goals, results in either expensive re-design or, in the worst case, may result in project failure.

The role of software took in design and analysis of switching converters is well understood. In the case o f POL converters, the importance of this role increases further. This is not only because of the market related reasons such as a growing demand for POL type converters and the “time-to-market” factor, but also, this is due to the continuous trends in converter device integration, development of new types of high-power-density packages, and practical verification challenges when measuring, for instance, hi&, currents at high switching frequencies on shrinking POL footprints. A simulation of switching converter circuit variables such as output voltage, converter input current, inductor current in timedomain gives fundamental information about switching converter operation and provides a basic support for the converter design. Further, the thorough characterization of converter steady-state response involves investigation of their ripples, average and rms values and also power dissipated on particular devices. As converter operating conditions vary depending on input voltage, load current, device parameters, or switching frequency, the converter

0-7803-8975-1/05/$20.00 02005 IEEE.

The other solution for the global characterization problem is to use formulas which are used in design practice. The formulas can provide relatively efficient way for computing values of converter internal variables. The problems of computational convergence is not relevant because the formulas are derived for converter steady-state. Results can be obtained very quickly even for many combinations of input parameters and in many operat-

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ing points. Conversely, tesults obtained using these formulas have an approximative or an “envelope” character, the formulas are often not know and necd to be derived for each new converter circuit. Typically, dificulties with deriving the formulas are introduced by the implementation of large amounts of parasitic device parameters, by the accurate determination of switching instances (which need to be done also using formulas), and by the complexity of the converter circuit. In comparison to the use of formulas and commercial simulators, the specialized switching circuit simulators based on fast time-domain methods or accelerated steady-state methods introduce more options for the global steady-state characterization problem. Although the overall procedure of characterization may involve all steps 1 - 4 as specified above, the advantages of the analytical methods are in the short time needed for determining converter steady-state, satisfactory accuracy and relative programming freedom. The performance of the specified switching converter simulators was demonstrated in several works ([4] through 161). However, the authors concentrate, first of all, on effectiveness of time-domain simulation of a few internal converter variables (usually output voltage and output inductor current) at particular operating point defined by the input voltage and load current.

In this work, a new ancl efficient method for the global steadystate characterization of switching converters is presented. Peak, ripple, average and rms values of converter circuit variables are inspected using the PRAR functions. The PRAR fmctions ace assigned to converter circuit variables using state variables. The PRAR functions are further evaluated over a range of operating conditions including input voltage and load current, and over a range of device parameter changes. To achieve a maximum simulation speed, high level switching device models are used. The inherent steady-state problem is solved using a new, fast, robust and accurate algorithm for determining converter steady-state, Implementation examples include characterization of buck converter with synchronous rectifiers.

which are important from design stand point and are object of analysis:

(3) The elements of the output vectoryj can be, for instance, gate-tosource voltages of synchronous rectifier transistors, drain-tosource voltage of switching transistors, magnetizing currents in transformer and inductor windings, and currents through sensing resistors. The analytical steady-state solution for the converter model above in time-domain is as follows:

where X,(O) are values of state vector at the beginning of i-fh topology within switching cycle, and for i applies

\. .

i = -2

~

n

~f T 2

~

~

~

for T 1 + T 2 + T n - 1 I r < T

where TI,T,, ... T,,., are time intervals associated with duration of converter topologies and T i s the converter switching period. The PRAR functions can be defined using equation (4)as follows: I . Maximum peak function

11. MILESTONES OF ALGORITHM FOR GLOBAL STEADY-STATE CHARACTERIZATION OF SWITCHING POWER CONVERTERS

Let’s assume a piece-wise linear model of switching converter in state-space specified tiy a set of linear differential equations as

follows: i

=

A i l + bju

(7) 4. Average

function

(1)

where the index i varies from 1 to n depending on converter topology, Ai, bi, ciTand ,di are system matrices and vectors, and U is the input vector. The output equation (2) can be expanded such that it includes (besides the converter output) those converter circuit variables

1.

T

where functions y , throughy, are elements of vectory, p~ through P k are input parameters such as input voltage, load current, capac-

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T

itance and inductance of converter output filter, winding resistance, and ESR of output capacitor.

0 Begin

The following comments should be made at this point: 1. To compute the converter steady-state response y in (4) and to evaluate the PRAR hnctions ((5) through (9)), only the values o f time intervals related to duration of the converter topologies Ti are needed to be found. Than, the values of state vector at the beginning of switching topology q(0) can be calculated using Newton shooting method.

U Enter device parameters

#

Update state-spaced e l

2. Evaluation of P U R functions can be done both numerically and analytically. When choosing the analytical computation, the derivation of final forms of the average and rms PRAR functions will involve integration of a sum of n functions defined in (4). In case of minimum and maximum peak PRAR functions, the algorithm will include search for local minimum or maximum of the set of n linear functions defined in (4).

1

I I I Determine ana store

+-

3. Sensitivity of peak, ripple, average and rms converter variables to change of input parameters can be computed numerically with a minimum additional effort as follows:

I

All parametemp!?

t

wherel; stands for i-th PRAR function, pjo.stands forj-th parameter and A means an increment from the nominal point pj0 Power losses estimates can be computed as shown, for instance, in [ 121 through [ 161 using average and rms voltages and currents determined by the PRAR functions. A generic flow diagram for the global characterization of switching converter shown in Fig. 1 includes k nested loops, one for each input parameter p! through p b If the input parameter causes a change in the converter state-space model (e.g. a change of load current or output inductor inductance causes changes to the characteristic matrices Ai), the converter model has to be updated accordingly. After the steady-state is found, time intervals and state vectotsXJ0) are stored, and PRAR functions are evaluated. The procedure of setting a new operating point, determining the steady-state, and evaluating PRAR functions repeats over all the predefined ranges of the input parameters p I through Pk. The outputs from the algorithm are waveforms of converter selected circuit variables, curves, surfaces or multidimensional matrices of values of P U R functions, devices power losses, converter overall efficiency and parametric sensitivities. Note that the concept Of the shown in may be used when evaluating the PRAR functions over the switching fiefor component tolerance analysis.

i

All parametmph?

I Plot yj

Fig. I. Flow diagram for the global characterization.

111. A NEW METHOD FOR COMPUTATION SWITCHING CONVERTER STEADY-STATE

The speed, accuracy and convergence ofthe steady-state algadthmwhen determining output vectory in (4) is critical. The o f the majority steady-state algorithms ([7] through [lo]) comprises an initial guess of state vector, and includes one-cycle time-domain simulation, which is further followed by a state vector correction using a Newton’s iteration. The steady-state method proposed in this work uses the timeinterval concept which will be explained in following. Let’s consider a piece-wise linear model o f switching converter operating

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with three topology changes per switching cycle. Using the Newton’s shooting method, the state vector at the beginning of the switching cycle at steady4ate is

where: K I.

=

A [ ’ ( e A i T i - f ) b .I u

for i=1,2,3.

To determine the converter steady-state, e.g. to determine the state vectorXl(0) in (1 l), the only values for time intervals TI, T2 and T3 in (11) and (12) are unknown. Those can be computed from the “time-interval e.quations” obtained for average output voltage Yossand average inductor current IL,. Computing the average quantities Y,, and IL,, from definition or using the statespace averaging principle, the “time-interval equations” are found in following form:

in steady-state can be replaced by the selection of converter average output voltage as for one o f the “time-interval equation”. This way, the steady-state method becomes independent on converter control scheme and convergence issues caused by the feedback operation (e.g. [lo]) are eliminated. Also, system description is reduced to minimum. Selection of output inductor current for the “time-interval equation” is optional and, in principle, any average quantity (e.g. average output capacitor voltage) can be chosen. Second, the average inductor current can be expressed as it is used for the reduced-order averaged model 1Lss=ipd2 in 1181 and [19], or, as it is used for the full-order averaged model IL,=ipk(T~+Tz)/2T in 1191 without impact on final results obtained for the steady-state. This is because the difference in expression for average inductor current in reduced-order and 111order models is compensated by differences in system descriptions. This statement is in agreement with the results obtained for reduced-order and full-order models in the frequency domain ([19]). As it was demonstrated in [19], the gain and phase for both models for low frequencies, and, therefore, also for the DC state, are identical. Finally, the risk of Newton-Raphson algorithm failure when computing time intervals is the only potential convergence issue of the proposed steady-state method. However, the associated risk of algorithm convergence failure is minimum because the initial guess for the time intervals can be made close to the solution, thus the performance of the Newton-Raphson algorithm near the solution is excellent (e.g. [6]).

In summary, in comparison with the published methods, the major advantages of the proposed method are: elimination convergence problems, elimination of time delays caused by onecycle time-domain simulation, independence on converter control scheme, and minimizing system description. Thus, the steady-state is determined in a fast and accurate manner.

1v.EXPERIMENTAL VERIFICATION where c , vector ~ ~ selects the inductor current from the state vector X.For simplicity reasons, the vectors d l , d2 and d3 are set to nil vectors.

Further, it applies T3

=

T- T I - T2

In this section, evaluation results for proposed steady-state method for converters operating in two and three topologies within switching cycle are presented. Also, the results for global steady-state characterization for buck converter with synchronous rectifiers are shown.

(1 5 )

Hence the average 0utpu.t voltage Y , i s known a priori and, the average inductor current can be computed using expression JLss=ipJ2 ([ 181 and [ 1911, the equations (13) (1 5 ) can be solved for the time intervals TI,T, and T3. The problem of matrix inversion in expressions (13) and (14) can be solved using the Newton-Raphson iterative algorithm.

-

Several important notes should be made at this point. First, it is not necessary to implement converter feedback since its function

A. Steudy-StateMethod

Verification

The performance of proposed steady-state method was evaluated using topology of buck converter with synchronous rectifiers operating in CCM mode and using boost topology converter operating in DCM mode. Parameters for the buck converter were set as follows: L=6uW, RL=25mOhm, C=9OuF,R,=IOmOhm and Fsw=300kHz. For the boost converters, the following device parameters were chosen: L=2.7uH, RL=50mOhm, C=33uF, R,=lOmOhm and FS,,,=200kHz.Table 1 and Table 2 demonstrate

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the speed and accuracy in determining converter steady-state at nine operating points specified by the input voltage and load current.

voltage and load current, only a single solution exists. Also, the risk of Newton iterations to fail is minimum since both surfaces Errl(Tl.Td and Err,(r,,Td are monotonous and shallow in wide neighborhood of solution.

Table 3: Average output voltage Y,, and iteration number Nat nine operating points for boost converter operating in DCM.

I,,

14:

I

,

I

I

b;.,=5.5V

3.3000

5

3

0.5

3.3000

3.3000

3.3000

3.3000 I

I

3.3000

1

1

3.3000

1

L x

N

Ym,

N

YO$*

N

5,,=3,0V

4.9997

4

4.9999

4

4.9999

4

v,=3.3v

4.9997

4

4.9999

4

4.9999

5

V,=3.6V

4.9997

4

4.9999

4

5.0000

5

,.... - ' /

..... ... ~

Table 2: Average output voltage ,Y

...

.,,.. j :.... ../... .:. ,... .. .:. ' .i .... '... .. ; ,,.... ':'

I .

and iteration numberNat nine operating

points for boost converter operating in DCM.

06

E. Converter Global Steady-State Charucterization ~

~

~

~

~

It is important to emphasize that the steady-state will be always determined with an error resulting from the use of state-space averaging principle, and therefore, further increase of iteration numbers does not necessarily lead to the steady-state error reduction. However, this error is relatively small. Particularly, in the case of boost converter, the output voltage relative error was 0.52% at worst case. If a higher accuracy is requested, a steadystate correction step can be implemented. This correction can be done either by an adjustment of output voltage Yo, in equation (13) by a value proportional to the output voltage steady-state mor, or by a time interval correction using linear approximation of error functions Emj(Tl,Td and Err2(Tl,T2, at neighborhood of solution point. Table 3 shows improvements obtained after the correction was implemented. Figure 2 shows the error functions Err, and Err2 which were calculated using two-dimensional Jacobian functions. As it is shown in this figure, for given input

The topology of buck converter with synchronous rectifiers (Fig.3) was selected for the global steady-state characterization. Device parameters were following: L=2SuH, inductor core TDK PC34, RL=15mOhm, Rg=30mOhm, C,=47uF and Co=2x33uF ceramic, Rceq=8mOhm, controller and MOS FETs used were ISL6522 and FDS6898A respectively, and switching frequency was set to 330kHz.

Fig.3. Buck converter with synchronous rectifiers.

In total, 25 converter internal variables were chosen for the steady-state characterization. The range of input voltage and load

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current was from 3.9V to 5.5V and Rom 0.5A to 5A respectively. The nominal input voltagc: was 5V and nominal load current was set to 1A. Results for the output voltage ripple (Fig.4 through Fig.9), inductor peak current and peak current sensitivity to changes of inductor inductance (Fig.8 and Fig.9), inductor core peak magnetic flux density (Fig. IO), and conversion efficiency (Fig.11 and Fig.12) are shown. The evaluation o f 25 P U R functions at 200 operating points using numerical approach took 6sec (MatlabS, Windows XP, 2.6GHz Athlon AMD).

-

.......i

Fig.7. Changes of output voltage peak-to-peak ripple in response to the variation o f output filter parameters.

.......

..i

........

..... ..... .

......

: ..

........ ................. ......... . ....... .......... .......... . .......... . ......... . . ..... 1 ; : :: : i : : .. : .: : : . . ....... ......i...... .I........ ........ ........ ................. . . .........

time(us) Fig.4. Output voltage ripple at nominal operating point as obtained by simulation. Fig.8. Changes o f inductor peak current in response to the variation of input voltage and load current.

J- .

.Lc

. . . . .i by measure-

Fig.9. Changes of peak inductor current sensitivity to change o f inductor inductance in response to the variations of input voltage and load current.

5

5

Fig.6. Changes of output voltage peak-to-peak ripple in response to the variation of input voltage and load current.

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Fig.10.Changes of inductor core peak magnetic flux density in response to the variations of input voltage and load current.

[3] E.S.Lee and T.GWilson Js. “Electrical Design Inspection of ELectronic Power Supplies via Time-domain Circuit Simulation”, IEEE COMPEL,pp.29-43,1993.

-

[4] P.Pejovic and D.Maximovic, “PETS A Simulation Tool for Power Eleceon-

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[SI L.Egiziano, N.Femia, and GSpagnuolo, “PESC: a Power Electronics CircuitOriented Simulator”, IEEE I K O N , pp.749-754, 1997.

Gw

.. .. :

81

.....................

0-

:

:. :. .: .. . .. .

.. .. i / j . .. .. I ! .: :. : : j ..........j .......... i..........1 .......... :.......... :.................... / ; : : : : .. ! : j ; j , :. :. : : : : .j .........j.............,. . . . .... . . . . ... . . . .. .. . .. .. .. .. . . . .. . . . i : .. .. .. ., i ’ : . . . . .i

[6] D.Li, R.Tymerski and T,Ninomiya, “PESC-Power Electronics Circuit Simulator”, IEEECOMPEL, pp.159-165, 2000.

:

I

.

.

[7] R.C.Wong, “Accelerated convergence to the steady-state solution of closedloop regulated switching-mode systems as obtained through simulation”, IEEE Power Speciulirt Confmence, f 987 Record, pp.682-692.

~

[8] J.Vlach, D.GBedrosian, “An Accelerated Steady-State Method for Networks with Intemally ContmHed Switches”, IEEE Trunructiom on Cimuilr and Systems-I: Fundamental Theory md Applications, vo1.39, n0.7, pp.520-530, July 1992. [9] D.LI, and R.Tymerski, “A Comparison of Steady-State Methods for Power Electronic Circuits“, IEEE PESC’98 Record, pp. 1084-1090.

.....:.__

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5

Fig. 12. Converter efficiency as obtainedbysimulation in response to the variation of input voltage and load current.

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V. CONCLUSION A new algorithm for global steady-state characterization of switching POL converters as a new and efficient CAD tool for design and analysis has been proposed. In addition to the conventional simulation of converter operation, the global steady-state characterization includes investigation of peaks, ripples, and average and rms values of converter internal variables as they vary in response to changes of operating conditions and component parameters. This analysis is based on evaluation of P U R functions expressed using state variables. The inherent steadystate problem is solved using a new, fast, robust and accurate method for determining converter steady-state. The performance of the steady-state method in terms of accuracy, speed and convergence has been demonstrated using several relevant examples.

[ 161 R.W.Erickson, “Fundamentals of Power Electronics”, Chapman and Hall,

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[20] M.Plesnik, and M.S.NakhIa, “Gtobd Characterization of Switching Power Converters”, IEEE COMPEL. pp.141-147, 2004.

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