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P.G. Maranesi, IEEE Fellow, M. Riva, IEEE Member, F. Belloni Università degli Studi di Milano

via Celoria 16, 20133, Milan, Italy tel: (+39) 02 50317457, fax: (+39) 02 50317458

e-mail: federico.belloni@unimi.it, piero.maranesi@unimi.it, riva@unimi.it

Abstract — An analog adaptive controller of power converters based upon a switched capacitor programmable filter is described. The transfer function of the external feedback network is continuously updated by the readouts of the input variables. A rough discretization of the inputs is obtained by a small number of fast comparators that digitally program the filter by means of a look-up table. The analog output drives the error amplifier of the switching converter controller whether it is PWM or phase-shifted or resonant. The present paper refers to a DC/DC converter even though the application can be addressed to any kind of power converter. The possibility of including the circuit into IC controller drivers is open. The non-linear overall characteristics of the power cells often show large differences in the dynamic behaviour depending on the operating point when the variations of the inputs are very large. The possibility of different conduction modes makes the use of a single time-invariant feedback network extremely downgrading the frequency performances. This paper describes a possible circuit topology implementing the switched capacitor adaptive feedback as a second order external network whose transfer function, in the Z domain, can assume as many different forms as required. The potential of this solution is illustrated with a simple example concerning a PWM flyback converter spanning over the discontinuous and the continuous conduction modes. The difficult work of defining the feedback transfer functions set can be carried out by an automatic analysis and CAD tool. The control of large signal transients is also made possible.

I. INTRODUCTION Time invariant networks are generally used for the

external feedback of switched power processors. The non-linear overall behaviour of the switching cells and specifically the possibility that more than one conduction mode are involved, imply dynamic differences depending on the working point. Feedback must guarantee the stability over the whole operating range. This often causes the closed loop dynamic performances to be far from optimal. An adaptive feedback can avoid this drawback. A common analog feedback network is difficult to be made dependent on the input sources because, generally, it implies switching of feedback capacitors with unacceptable transients. For this reason, the most part of research activities on adaptive control is oriented to digital solutions [1-7]. Circuit complexity, processing delay, power dissipation, and cost have restrained until now the insertion of digital control into commercial power conditioning ICs.

* Patent Pending

An analog solution in the discrete time providing an adaptive set of transfer functions in the Z-transform domain is described hereafter. It avoids the problem of initializing switched capacitors and can be integrated more easily than digital controllers.

II. P DESCRIPTION OF THE ADAPTIVE FEEDBACK FILTER The dynamic characteristics of the switching power

cells can be suitably described in the discrete time and in the Z-transform domain when the small signal approximation is adopted [8, 9]. Although not commonly implemented, the feedback network can be profitably synthesized by means of a switched capacitor recursive filter [10]. In this case, the Nyquist frequency of the filter is naturally synchronized to the switching frequency of the power cell.

The block scheme of an nth order recursive filter and a possible implementation of a 2nd order network are reported in figures 1a and 1b.

DELAYT

DELAYT

DELAYT

+ +δ0

δ1

δ2

δn

γ0

γ1

γ2

γn

IN OUT

a)

Q

Q

vref vref

QQ

vrefvrefVout

R0a

R0b

vref vref

Q

QQ

Q

R1a R1b

R2a R2b

Ve

Rfoc

c

c

c

c

c

c

Rfoc

b) Figure 1: a) schematic representation of an nth order discrete time recursive filter and, b) 2nd order switched capacitor recursive filter

Adaptive Control of Power Converters by SwitchedCapacitors Filters*

978-1-4244-1668-4/08/$25.00 ©2008 IEEE

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The transfer function of the circuit in b) is

ca

cac

fo

cb

cb

cbc

acfo

ca

cfo

ca

cfo

cb

cfo

cb

cfo

cb

cfo

ca

cfo

YzYRz

YzYzYYR

RR

zRR

z

RR

zRR

zRR

RR

zH

21

221

20

0

21

2

21

2

0

0

)(++

++=

++

++

= (1)

with cij

cij RY 1=

By varying the resistors cijR or their corresponding

admittances cijY , the DC gain, the zeros and the poles of

( )zH can be modified at will. With the aim of proposing an adaptive filter, and just

for giving an example of realization, each resistor of the circuit can be replaced by a programmable operational transconductance amplifier (OTA), easily integrable on a chip. An example of digital programmable OTA is shown in figure 2 [11]. Many other solutions are proposed in the literature, f.i. [12-15].

The values of the input variables, f.i. input voltage and output current in the case of DC/DC voltage converters, can drive the OTAs.

The pairing of input variable readouts with the programmable filter commands can be done by a look-up table. The discretization of the inputs can be simply obtained by a few fast comparators.

Large differences of dynamic behaviours may depend on the status of the input variables, mainly when a variation of the inductor current conduction mode appears. Other programmable switched capacitor filter schemes are reported in the literature, f.i. [16-22], with emphasis on various aspects of the integration.

VDD

VSS

B0

VPOL

B1B2

B3

VR

VZ

Figure 2: digitally programmable OTA

Figure 3 shows the block scheme of an adaptive

controller based upon programmable switched capacitor filter.

LookupTable

Vin

Iout

Vout

MOSFET

n1

n2

k DigitallyProgrammableSwitched Capacitors Filter

Analog Voltageto Pulse Width,

Converter

or Phase Shift,or Frequency

FlashComparators

n1 Levels

FlashComparators

n2 Levels

to

Drivers

Figure 3: switched capacitors adaptive controller for DC/DC converter

The operating ranges of the input variables inV and

outI are subdivided respectively in 1n and 2n discretization levels by means of fast comparators. Each of the resulting 21 nnn ⋅= subsets can be associated to a specific transfer function of the external feedback that optimizes the dynamics within the selected operating area. In most cases 126 ≤≤ n provides a significant improvement due to adaptive control. The look-up table drives the digital programmable filter according to the running readouts of the inputs. The input variable readings can be updated period after period or on the base of averages in a few periods depending on specific characteristics of load and input voltage sources. The commutation of the filter only involves the switching of the programmable OTAs and has no impact on the capacitor voltages in the delay chain. Hence, no initialization or settling time effect of the analog memories occurs. The OTAs switching commands can be adjusted to avoid spikes at the input of the error amplifier. In correspondence to large signal transients, dedicated up/down fast recovery paths can be activated bypassing all of the capacitor voltages in the delay chain.

The hard and tedious work of defining a transfer function set covering all the subsets within the operating range has to be delegated to an automatic analysis and computer aided control design software. The suggested tool is FREDOMSIM, which adopts a state space discrete time model, as described in the literature with many proofs of accuracy in the case of fixed frequency PWM converters [9, 23, 24]. The adaptive control technique described here is applicable also to variable frequency converters.

III. P A SIMPLE EXPLANATORY APPLICATION The power cell of a 60W flyback converter switching at

68=sf kHz is show in figure 4.

Vin

M

CL

n

D

Ds

s

zs

magD

C IC CL

n=8/63

o1 o2f

f load

C =1nFsL =530uHmag

C =1000uFo1

C =1000uFo2

C =100uFf

L =2.2uHf

DCV =136V-184Vin

I =0A - 5Aload

V =12Vout

M: STP9Nk70ZFPD: STPS2H100

Figure 4: schematic of the flyback converter ST Microelectronics EVALSTSR30-60W

The working area involves the DCM and the CCM

operating modes and it was divided in 6 subsets ( 21 =n , 32 =n ). The frequency response forecasts were obtained by FREDOMSIM in correspondence to 6 operating points each of them belonging to one subset. Two predictions of control-to-output voltage frequency responses, )2( fjGco π , are shown in figure 5 together with relevant measurements of gain and phase. In 5a, the inductor current is continuous; in 5b, it is discontinuous. The dynamic differences in the two cases are quite strong.

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101 102 103 104 105-20

0

20

40

60M

agni

tude

[dB

]Control to Output Voltage

101 102 103 104 105-300

-200

-100

0

Frequency [Hz]

Pha

se [d

eg]

Experimental DataFREDOMSIM Forecast

a)

101 102 103 104 105-20

0

20

40

60

Mag

nitu

de [d

B]

Control to Output Voltage

101 102 103 104 105-300

-200

-100

0

Frequency [Hz]

Pha

se [d

eg]

Experimental DataFREDOMSIM Forecast

b) Figure 5: control-to-output dependencies: a) at VVin 136= ,

AIload 5= (CCM) and, b) at VVin 184= , AIload 5.0= (DCM)

The control-to-output transfer function in the Z-domain

( cTsez ⋅= , sfT sc μ706.14/1 == ) at VVin 136= ,

AIload 5= is:

0.4959) 0.7469z-0.9872)(z-0.8414)(z-(z0.2899)0.8965)(z-15)(z-0.23152(z-)( 2 +

+=zGa

co (2)

and at VVin 184= , AIload 5.0= is:

0.05743)0.7659)(z-0.9918)(z-(z0.9596)-1.6337z(z)(

+=zGb

co (3)

The following optimizing transfer functions of the

external feedback networks have been chosen respectively:

0.719)-0.9984)(z-(z0.3755)-0.9875)(z-0.023401(z)( =zH a

ext (4)

0.9984)-0.9603)(z-(z0.9917)-0.77)(z-0.10047(z)( =zH b

ext (5)

In both cases, the compensation networks allow

reaching a DC gain close to 100 and insert a dominant pole at a lower frequency than that of the power cell that is cancelled by a new zero. Another zero is placed close to the second pole of the power cell and finally a high frequency pole is added. The results on the phase margin can be appreciated in the loop gain and phase forecasts shown in figure 6 for the two operating points considered in this example.

100 101 102 103 104-50

0

50

Mag

nitu

de [d

B]

Loop Gain

100 101 102 103 104-300

-200

-100

0

Frequency [Hz]

Pha

se [d

eg]

a)

100 101 102 103 104-50

0

50

Mag

nitu

de [d

B]

Loop Gain

100 101 102 103 104-300

-200

-100

0

Frequency [Hz]

Pha

se [d

eg]

b) Figure 6: loop gain dependencies: a) at VVin 136= ,

AIload 5= (CCM) and, b) at VVin 184= , AIload 5.0= (DCM)

The use of a single time-invariant network up to

guaranteeing stability over the whole operating area would lower the bandwidth very much.

IV. CONCLUSIONS An analog approach to adaptive feedback for IC

controller drivers devoted to switching power converters has been based on programmable switched capacitor networks. This solution is alternative to digital adaptive control, more simple and free from processing delay. The possibility of synthesizing the feedback network in the discrete time and in the Z-transform domain copes well with exact modelling of the power cells and takes profit from the availability of automatic dynamic analysis and CAD tools. Such facilities make the design easier, even though a set of models of the power cell and of corresponding transfer functions of the feedback network have to be identified. The capacity of promptly facing the large signal transients is an additional merit. The control procedure is based upon a set of small signal linear models in the discrete time, a step forward as compared with only one averaged model in the continuous time considered for usual dynamic optimization. Complete guarantee of overall stability is approached but not reached as it implies non-linear analysis and Lyapunov criteria. The work is supported by an application example, dynamic simulations and some experimental results. For explanatory simplicity, the order of the circuit in the given example has been limited to the second. It can be easily increased according to the dynamics of the considered switching cell and taking into account possible requirements due to filters and loads. The implementation of a circuit suited to monolithic integration is in progress.

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