small-signal laplace-domain analysis of uniformly-sampled pulse-width modulators

8/3/2019 Small-Signal Laplace-Domain Analysis of Uniformly-sampled Pulse-width Modulators

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2004 35lh A n n u l IE EE Powe r Elecrronics Speciulisls Conference Auchen, Germany. 2004

Small-Signal Laplace-Domain A nalysis of Uniformly-Sampled Pulse-Width Modulators

D avi d M. Van de Sype*, Koen De GussemB, A l e x P. Van den B os s che , and Jan A. M e l k e b e e k

Electrical Energy Laboratory

Department of Electrical Energy, Systems and Automation, Ghent UniversitySt-Pietersnieuwstraat 4 I , B-9000 Gent, Belgium

*Email: [email protected]

Absfracl-As the nerformance of dieital sienal nrofessors has 11. UNIFORMLY-SAMPLED PULSE-WIDTH M OD UL AT OR Sincreased rapidly dbring the last decade, there'is a growinginterest to replace th e analog controllers in low power switchingconverters by more complicated and flexible digital controlalgorithms. Compared to high power Converters, the controlloop bandwidths for converters in the lower power range aregenerally much higher. Because of this, the dynamic propertiesof the uniformly-sampled pulse-width modulators used in lo w

power applications become an important restriction for themaximum achievable bandwidth of control loops. After thediscussion of the most commonly used uniformly-sampled pulse-

width modulators, small-signal frequency- and Laplace-domainmodels for the different types ofuniformly-sampled pnlse-widthmodulators are derived theoretically. The results obtained areverified by means of experimental data retrieved from a testsetup.

1. I N TR O D U C TI O N

The pulse-width modulators embedded in modern digital

signal processors (e.g. TMS320C2XX of Texas Instruments,

ADSP2199X of Analog Devices, DSP568XX of Motorola,

etc.) operate all in a similar fashion. If we disregard quan-

tization effects, a model for the uniformly-sampled pulse-

width modulator is shown in Fig. 1. The input u( t) , a

continuous function of time, is sampled with a frequency

ws synchronously to the pulse-width modulation (Fig. I) .

The sampled input u s ( t ) s sent to a zero-order-hold circuit

(ZOH). Finally, the P W M waveform is generated by compar-

ing the output of the ZOH U&) to the value of the carrier

waveform v,(t) . a triangular waveform. Depending on the

shape and the frequency w , of the carrier waveform vc(t)

different types of uniformly-sampled pulse-width modu lators

can he obtained. While the carrier frequency wc determines

the switching frequency, the sampling frequency wg defines~. . . .

the number of samples that is taken in a switching period

T,. In commercial digital controllers two possibilities are

commonly Offered: th e

For reason s of price, control circuits for low-power switch-

ing power supplies ( < 3 kW ) are almost always implemented

using analog circuits. As the pricelperformance ratio of digitalfrequency ws is equal

signal processors has decreased rapidly durin g the last decade,

the interest for digi tal control of switching nower s u ~n li es n

the swi tching frequency wc fo r single-update-mode modulators

(Figs. an d 3, Or the frequency is twice as high as

the switching frequency in double-update-mode modulatorsY I I ,

the low nnwer range ha% ernwn 111 121. ~ ~..=... . L , , L _ , .r - - ~ ~ - - ~ ~

~~~.

When applying digital control to a switching power supply,

the different switches in the supply are often controlled by

a digital or uniformly-sampled pulse-width modulator. Conse-

quently, the dynam ics of a digitally co ntrolled switching power

supply are influenced by two nonlinear effects: quantization

effects and modulation effects. As the effects of quantization

in digital control of switching power supplies have been

addressed before [3], 141, this study focuses on modulation

effects. The dynamic behavior of switching power supplies

with digital pulse-width modulators in the feedback chainis inherently different from those with analog pulsewidth

modulators. Hence, the derivation of new models is required.

Though exact discrete-time models exist for converters [SI,

their use in control is limited because of the presence of

(Fig. 4). A further subdivision can be made depending onthe waveform of th e carrier: an isosceles-triangular-carrier

or a sawtooth-carrier. When the carrier U, is an isosceles-triangular-carrier waveform, two single-update-mode mcd ula-

tors and one double-update-mode modulator can be identified.

Th e single-update-mode mod ulato rs with isosceles-triangular-

carrier are the symmetric-on-time modu lator Fig. 3(a) and the

symmetric-off-time modu lator Fig. 3(b). As for the douhle-

update-mode mod ulato rs only the modulator w ith isosceles-

triangular-carrier is commercially available, this modulatorwill be referred to as the double-update-mode modulator

Fig. 4. For modulators with a sawtooth carrier only single-

update-mode modulators are commonly available: the end-of-on-time m odulator Fig. 2(a) and the begin-of-on-time modu-Intnr Gin ?th\.U1 I .b. "\U,

matrix exponentials and other highly nonlinear vector func-

tions. App roximations of the discrete-time models such as [6],

may present usable expressions for control hut don't provide

the sam e "feel" for converter and modulator behavior that

exists for converters controlled with analog circuitry. To meet

these requirements a small-signal Laplace-domain analysis

is performed yielding both frequency- and Laplace-domainmodels for the different types of modulators. The derived

models d o not only provide important quantitative results but

also provide the designer with the necessary insight.

111. L A P L A C E - D O M A I NN A L Y S I S

A. Single-Update-Mode odulators

To derive the frequency- and Laplace-domain models of

the single-update-mode modulators, the waveforms of the

general single-update-mode modulator are used. This general

single-update-mode modulator has as carrier waveform ue , a

triangular waveshape determined by the period T, nd the ratio

a. Th e lat ter is the duration of the falling edge of the triangle

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2004 35th Annual IEEE Power Elecrronics Specialists Conference Aachen. Germany, 2004

Fig. I. A general uniformly-sampled pulse-width

modulator

I U

t

on Il e y I

of f -t

I U I U

(4 (b)

Fig. 2.on-time

The single-update-mode sawtooth-carrier modulato n. (a): end-of-on-time, (b): begin-of-

I I

(a) (b)

Fiz. 3. The single-upda te-made triangular-carrier madula ton. (a): symmetric-o n-time, (b): sym meu ic- Fig. 4. The double-update-mode triangular-

of-t ime

relative to the period T, (Fig. 5 ) . Choosing cy equal to 0, 1 / 2

and 1 allows to obtain the waveforms for the end-of-on-time

modulator, the symmetric-on-time modulator and the begin-

canier modulator

of-on-time modulator respectively. Hence, three out of four

different single-update-mode modulators can be analyzed in a

unified way.

The input of the modulator u(t) s separated into a steady-

state part U an d a small excursion to this steady-state i i ( t ) ,or

(1 )

This input is sampled at the sample rate T, (=Tc) ielding the

waveform u,(t). If we use an ideal sampler instead of a real

sampler, the small-signal output of the ideal sampler becomes

(2 )

This sampled input is passed on to th e ZOH (see also Fig. I )

before it is compared to the general triangular carrier. The

output of the modulator y(t) ca n also be separated into a

steady-state portion Y ( t ) the response to U) and a small

excursion to the steady-state G(t) ,or

u(t)= U +q t ) .

T ( t ) (U- U)*(+

Y ( t ) = Y ( t )+w. (3 )

If the pulses of P ( t ) are sufficiently small (this can beguaranteed by choosing G ( t ) sufficiently small), y^(t)may be

approximated as a series of impulses, in which the separate

impulses have the same surface as their corresponding pulses

and where the impulses are positioned at the flanks of the

steady-state output Y ( t ) see Fig. 5 , compare the two lower

curves).

If we consider the response of the modulator to a small-

signal impulse in the sampled input at time zero

G * ( t )= 6 ( t ) , (4 )ig. 5. Th e key waveforms for a general single-update-mode modulator

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2004351hAnnual IEEE PowerElectronics Specialisrs Conference

Fig. 6 .

modulator

The black diagram of a idealized genenl single-update-mode

the small-signal outpu t of the modulator can he approximated

by

F ( t )x T,ab(t- To)+T,(l - a)J(t- (T, -TI)), (5 )

with TO nd TI chosen as in Fig. 5. As the small-signal input

is a series of impulses

+m

G*(t) = E(nT,)b(t-

nT,), (6 )n=--m

the small-signal output can be written in the Laplace-domain

by using (4), (5) and (6)

This equa tion can be schematically represented by the diagram

of Fig. 6. (Equation (7) is used in [7 ] to derive the z-domain

models for digitally controlled converters.) H ence, the general

single-update-mode modulator behaves, in small signal, as

a device that responds to an impulse at its input with two

delayed impulses at its output. The impulses at the output are

positioned at the switching edges of the steady-state output

Y( t ) f the modulator (see also Fig. 5 ) .

Taking into account tkat the Laplace-transform of the sam-pled small-signal input U * ( s ) an be expressed as a functioncf the Laplace-transform of the continuous small-signal input

(8)- 1 +-

a'(s) =- G ( s - j k w * ) ,

W ) 81

k = - m

the small-signal output of the general single-update-mode

modulator finally becomes

(9 )Hence, the frequency spectrum of the small-signal output ofthe modulator contains an inf ini te number of images of the

frequency spectrum of the input with as Nyquist frequen cy half

the sampling frequency, or w,/2. f the frequency spectrum

of the small-signal input G ( t ) contains only frequencies lower

than the Nyquist frequency and if we consider only frequencies

lower that the Nyquist frequency in the output, the latter can

be approximated by

If the proper values for a, To an d TI are chosen, equation

( I O ) provides the frequency- and Laplace-domain models

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Aachen. Germany. 2004

IZnT, (Zn+l)T, t

Fig. 7 . The key waveforms for the double-update-mode modulator

for the different modulators. For a modulator with a carrier

waveform v J t ) limited between 0 an d 1, the average value of

the input U is equal to the average duty-ratio D. he model

for the end-of-on-time modu lator is derived from ( I O ) by using

cu=O,T~=(l-D)T, ,or

For a = 1, TO (1- D)T, the Laplace-domain model for the

begin-of-on-time modu lator becom es

(12)

The model for the symmetric-on-time modulator is obtained

by substituting a=1/2 an d To=T1=(1- ) / 2 ) in (IO)

s ( l - D ] T ,GPWM(S)e -

Th e Laplace-domain model of the symmetric-off-time modula-tor can't be derived directly from ( I O) . From a similar analysis

the following model is obtained

B. The Double-Update-Mode Modulator

For the double-update-mode modulator the series of im-

pulses P ( t ) 6) at the input of the modulator with sampling

period T, s divided into two subseries with sampling period



2004 35rh Annual IEEE P ow er Elecrronics Specialists Conference Aachen, Germany, 2004

+m

1+ )e-*DT* (-l)kO(s - jku,) (22)Ik=-m

U

Note that, though the sampling frequency for the double-

update modulator is twice as high as the switching frequency,

the Nyquist frequency for this modulator is half the switching

frequency w,/2.

Under the assumption that the frequency spectrum of the

input .^(t) ontains only frequencies below the Nyquist fre-

quency, and that we are only interested in the frequency

content of the output in the same frequency range, the small-

. (15) signal output of the double-update-mode modulator can be

approximated by

- T,

Fig. 8. The block diagram of the idealized double-update-mode modulator

T, (=2T,), (see Fig. 7):

+m

.^;(t )=

.^f(t) = 1 ^(t)6(t- nT, - Ts)

G(t)b(t- nT,)n=-m+mI -=-m

).

(23)

(1-D)T, + e-sDT,I

G ~ ~ ~ ( S )( e -The response of the modulator to these separate subseries is

different. If we consider an impulse i n the first subseries at c, ~ i ~ ~ ~ ~ if ~ e s u l r stime zero

.^;(t) 6 ( t ) ,The Laplace-domain models for the single-update-mode and

double-update-mode modulators are summarized i n Table I.I 6 )

domain models are also shown i n Table I.mpulse (Fig. 7)

= Tab@ 1 - D ) T , ) . (17)

An impulse in the second subseries

Ci;(t)= 6 ( t ) (18)

yields approximately the output

c 2 ( t ) = TJ( t- DT , ) . (19)

By considering the above, the output of the douhle-update-

mode modulator can be expressed in the Laplace domain as

B(s)= i q s ) +A(s)

= T~ e - * ( ' - ~ ) ~ * c ~ ( s )e-SDT*G;(s)). (20)

Hence, for small signals the dynam ic behavior of the double-

update-mode modulator can he represented schematically by

Fig. 8.As both subseries, q ( t ) nd G ( t ) 15), are obtained by

sampling the same continuous small-signal input .^(t), heirLaplace transforms are related [SI:

By substituting these Laplace transforms in (20), an expression

similar to (9) is obtained for the double-update-mode modu-

Whereas the models for the end-of-on-time modulator have

been published before and are in agreement with the results

of [9], [ IO ] an d [ I I ] , the models for the other modulatorsare new expressions. The frequency-domain models for thesawtooth-carrier modulators (end-of-on-time and kgi n-o f-o n-

time, Table I) show that these modulators behave as a pure

delay that is depend ent on the average duty-ratio D. his delaycan he regarded upon as caused by the delay between the

taking of the sample and the response of the modulator to

this sample: either D T , for the end-of-on-time modulator or

(1-D)T, for the begin-of-on-time modulator (T ,=T,).

Conversely, the single-update-mode isosceles-triangular-

carrier modulators (symmetric-on-time, symmetric-off-time,

Table I) have a gain that is dependent on the frequency and

the average duty-ratio D, hile the phase-shift represents

a delay of half a sampling-period. The delay of TB/Z an

be intuitively understood. After all, the response F(t) of a

symm etric modulator to a new sample at the input consists of

two parts, a part before and a part past half of the switching

period (see Fig. 5 for a = 4).Hence, on average, the response

to a sample occurs in the center between the two parts or with

a delay of half a switching period.

The frequency domain model for the double-update-mode

(Table I) shows again a dependency on the average duty-ratio

D and the frequency of the input w . Moreover, only the gain

varies with the average duty-ratio D, hile the delay is fixed at

half a sampling frequency T,/2. This delay can be explained as

follows. The waveform of the double-update-mode modulator(Fig. 7) can also be constructed by applying one sample to a

hegin-of-on-time modulator and the other to a end-of-on-time

modulator. As a consequence, the delay to the samples i&(t)

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2004 35th Annual IEEE Power Elecrronics Specialists Conference Aachen, Germany . 2w4

TABLE I

T H E RRQUENCY- AND L A P L A C E - U O M A I N0DRI.S FOR UNIFOKMLY.SAMI'LED PUI.SF.-WII)TH MODULATORS

end-of-on-time p D T . 1 L ( - j w D T , )

begin-of-on-time e--s( l -D)T. 1 L ( - j w ( 1 - D ) T , )

.-s I f +&+) II f c cos(E!p)+)ymmetric-on-time I

. . . .. . . . . .. . . . . .. .. . . . ,.................... ..... ... .. .....,... ..,... ...,......I.+."'") 1 : ~ ~ ~ ~ :. . . . . .

. . . . . ,. . . . . ,. . . . .. . . . . .

Fig. 9. The experimentally measured in- and output of an end-of-an-time Fig. IO.

modularorThe small-signal input E ( t ) and output & ( t ) after filtering

and averaging, together with the filtered output y ( t ) for the end-of-on-timemodulator.

is ( l - D ) T a ,while the delay to the samples &(t) is DT,, ron average T.1 2 .

Note that the Nyquist frequency for both single-update-

mode and double-update-mode modulators is equal to half the

switching frequency T c / 2 for single-update-mode modulators

T,=Td.Hence, if these models are used to predict the close d-loop stability of a digitally controlled converter, their validity

is limited to frequencies below half the switching frequency.

1V. EXPERIMENTALESULTS

Experimental verification of the theoretical results obtained

was performed using the digital pulse-width modulators of

the ADMC401 of Analog Devices. For this purpose a non-

synchronous sine wave with a small amplitude, programmable

frequency and offset is supplied to the analogue-to-digitalconverter (ADC ) of the DSP (Fig. 9); the ADC is synchronized

with the PWM. The sample obtained by the ADC is passed

on to the pulse-width modulator. Both the signal at the input

of the ADC and the signal at the output of the modulator are

provided to two matched first order low pass filters (with a

time constant of 33 ps ) before they are offered to the inputs

of a digital oscilloscope. The attenuation and phase shift dueto the low pass filtering is the same for both signals. As the

applied sine wave is non-synchronous, the averaging function

of the oscilloscope allows to clearly visualize the fun damental

&(t)of th e PWM output (Fig. IO). The experimental resultsin this section are all obtained by comparing th e amplitude

and phase shift of the two signals on the oscilloscope.

The switching frequency w,/27r of the modulator was set

to 51 kHz, corresponding to a switching period T, of 19.6 ps.

Fig. I I shows the theoretical delay introduced by an end-of-

on-time modulator as a solid line; experimental results are

added for comparison. As the gain of this modulator is always

unity, no plot is added. nevertheless a good correspon dence to

experiments was noticed. A similar experiment was performed

to check the validity of the transfer function for the begin-of-

on-time modulator, the results are exhibited i n Fig. 12. The

comparison between theoretical results and the experimental

measurements show the validity of the approach.As for the isosceles-triangular-carrier mod ulato rs only the

gain of the frequency-domain model is variable with D, hase

plots have been omitted. The gain for the symmetric-on-time

modulator, the symmetric-off-time and the double-update-

mode modulator are shown in Fig. 13, Fig. 14 and Fig. 15,

respectively. Note the subtle difference between Figs. 13 and

14: in Fig. 13 the lowest curve corresponds to D=0.95

while in Fig. 14 this curve matches D=0.05. gain, a good

correspondence exists between the experimental data and the

theoretical results.

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2004 351h Annual IEEE Power Elecrronics Specialisrs Conference

0 0.2 0.4 0.6 0.8 1

Aachen. Cer mn y, 2004

D 1.1-

Fig. 11 . Experimental measurement of the transpan delay of the end-of-on-time modulator as a function of the input frequency w and the averageduty-ratio D

D [ . I+

Fig. 12. Experimental measurement of the transpan delay of the begin-of-on-time madulatar as a function of the input frequency w and the averageduty-ratio D

w / 2 r [ kHz]+

Fig. 13. Experimental measurement of the gain of the symmehc-on-timemodulator as a function of the input frequency w and the average duty-ratio

D

V. CONCLUSI ON

Th e increasing performance, the good flexibility and the

decreasing price of digital control circuitry have entailed arenewed inkrest in the digital control of switching power

supplies. A commonly encountered part of such a digital con-

trol circuit is the uniformly-sampled pulse-width modulator.

In modern microcontrollers different types of digital pulse-

width modulators can he categorized, depending on the ratio

between the sampling-frequency and the switching-frequency,

and the shape of the carrier waveform. Four types of single-

update-mode modulator and one type of double-update-mode

modulator can he distinguished. The single-update-mode mod-

ulators are the symmetric-on-time modulator, symmetric-off-

time modulator, the end-of-on-time modulator and the begin-

of-on-time modulator. As there is in most cases only one pulse-

width modulator using a double update mode, it is simply

called the double-update-mode modulator.

Of the 5 different types of these pulse-width modulatorsdynamic models only exist for the end-of-on-time modulator.

To fill this void the small-signal Laplace-domain a nalys is for

the different types of uniformly-sampled pulse-width modula-

0 5 10 15 20 25

w / 2 r [kHz]+

Fig. 14. Experimental measurement o f the p i i n of the symmetric-off-timemodulator as a function of the input frequency w and the average duty-ratio

D

w / 2 r [ kHz]+

Fig. 15. Experimentd meaurement of the gain of the double-update-modemodulator as a function of the input frequency w and the average duty-ratioD

tors is performed. A s a result of this analysis both frequency-

domain models and Laplace-domain models for the differentmodulators are derived. The general conclusions are that

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2004 351h A n n u a l lEEEPower El e c t ro n i c s SpecialistsConfe rence Ao c h e n . Ge rm a n y , 2004

modulators with a sawtooth carrier behave dynamically as a

pure transportation d elay depend ent on th e average value of th e

duty-ratio, while the isosceles-triangulacarrier modulators

have a fixed delay of half a sampling period with a gain

that varies with the applied frequency and the average value

of the duty-ratio. Moreover, for both single-upd ate-mode anddouble-update-mode modulators the Nyquist frequency is half

the switching frequency. The obtained models are verified

by experim ent using the pulse-width modu lators on-board the

ADMC401 of Analog Devices.

R E F E R E N C E S

[ I ] - Power supply ‘Digital Control’-Real or Vinual,” RA P session. lRrh

Ann. Appl. Power Eleatmn. Con$ and Erp. (APECZW3). Miami Beach.

USA, Feb. 9-13, 2003.[2 ] - Special issue on digital control in power electronics,” IEEE T m x

Power Elccrmn.. Vol. 18. No. I. Parr IIof Tw o Pam, Jan. 2W3.

[3] D.A. Grant. M. Stevens, and J.A. Hauldswonh. “The effen ofword length on the harmonic content of microprocessor-based PWMwaveform generators,” IEEE Tmns. Id . Applic, Vol. 21, No. I,

Janf feb . 1985, pp. 218-225.

[4] A.V. Peterchev. S.R. Sanders. and J.A. Houldswonh. “Quantizationresolution and limit cycling in digitally controlled PWM conveners:’Pmceedings lEEE 32nd Annual Power Elect,: Spec. Con$, Vancouver.Canada, 2001. on cd-tom.

[SI G.C . Vergehese. M.E. Elbuluk, and J.G. assakian, ” A general approachto sampled-data modeling for power electronic circuits,” IEEE TransPower Elecrmn.. Val. I. Apr. 198 6, pp. 76-89.

[6 ] V R a j a se k m n , 1.Sun. nd B.S. Heck. “Bilinear discrete-tim e modelingfor enhanced stability prediction and digital control design,” Trms.Power Electron., Vol. 18. No . I . Jan. 2003. pp. 381-389.

[7] D.M. Van de Sype, K. De Gussemb. A.P. Van den Bossche. J.A. Melke-beek, “Small-signal z-domain analysis of digitally controlled convenen:’

Pmc. of the 35th Ann. Power Electmn. Spec. Con$. Aachen. Germany.2W4, on cd-rom.

[8] G.F. Franklin, J.D. Powell, and M. Workman, “Digital control ofdyna mic systems:’ Add iror Wesley Longtnnn, California, USA. 3rdedition. 1998.

191 G.W. Wester. and R.D. Middlebrook. “Low-freauencv characterization,of switched dc-d c conveners:’ IE EE Trms.Aerosp. and Elecrr Syst..Vol. 9. No . 3, May 1973, pp. 376385.D.M. Mitchell, “Pulsewidth modulator phase-shift,”lEEE Tms.Aemsp.

and E l e m Syst., Vol. 16. No. 3, May 1980, pp. 272-278.R. Tymenk i, “Frequency analysis of time-interval-modulated switchednetworks,” Tmns. Power Electron.. Vol. 6, No. 2, Apr. 1991. pp. 287-295.

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