Approaches to modeling converters with current programmed control

Approaches to modeling converters with current programmed control

F. J. Azcondo, Ch. Brañas, R. Casanueva University of Cantabria Department of Electronics Technology System and Automation Engineering Ave. de los Castros s/n 39005 Santander, SPAIN Email: [email protected] [email protected], [email protected] Abstract—In this paper, we present an overview of previously published approaches to dynamic modeling of current programmed converters, including basic low-frequency averaged models, as well as treatments of sampling and aliasing effects. The modeling assumptions are examined and the differences among the approaches are highlighted, with the objectives of making it easier to present these topics in power electronics courses and applying the models in practice.

I.

perturbed inductor current waveforms. It is well known that undesirable subharmonic oscillations occur for duty cycle D > 0.5 as illustrated by the waveforms in Fig. 1(a). An compensation ramp with slope Mc > 0, is added to ensure that a current perturbation in a switching period diminishes in the next period as shown in Fig. 1 (b). A simple geometrical argument leads to the following condition to prevent the subharmonic instability, M − Mc ∆I 2 = 2 0 is the inductor current slope in the dTs subinterval when the active switch is on, and M2 = V2/L > 0 is the negative of the inductor current slope in the d’Ts subinterval when the rectifier switch is on, d’ = 1−d. From (1), it follows that selecting M c > M 2 / 2 ensures stable operation of the current control loop under all steady-state operating conditions. A disadvantage of CPM control is a relatively high sensitivity to noise related to sensing the instantaneous switching current and comparing the sensed signal to the current command. To reduce the sensitivity to noise, the compensation ramp is commonly added in practical CPM designs, even when operating the converter at duty cycles less than 0.5 [5]. III.

LARGE-SIGNAL AVERAGED MODELS

The first step in a derivation of an averaged dynamic model for a CPM controlled converter is to describe a relationship between the current command ic and the average inductor current 〈iL〉Ts . In the simplest, first-order model [2, 5], it is assumed that the average inductor tracks the current command, (2) iL Ts ≈ ic , which gives an approximate reduced-order model. More accurate models [2-8] are based on large-signal descriptions that take into account the inductor current slopes and the slope of the compensation ramp. Referring to the waveforms of Fig. 2, the mid-point current values in the dTs and the d’Ts subintervals can be found as: 1 (3) i1 = i c − M c dT s − m1 dT s , 2

i 2 = i c − M c dT s −

1 m 2 d ' Ts , 2

(4)

respectively, where d’=1 − d, and d is the switch duty cycle. It is of interest to examine how various CPM modeling approaches differ in approximating the average inductor current 〈iL〉Ts using the expressions (3) and (4). For example, assuming steady-state conditions, the largesignal averaged inductor current in [3] is found using (3) only, 1 (5) iL ≈ i1 = i c − M c dT s − m1 dT s . Ts

2

In [6], however, the derivation is based on (4) only, 1 iL ≈ i 2 = i c − M c dT s − m 2 d ' T s . Ts

2

∆I1 ∆I3 ∆I2

2

2

t T

FREQUENCIES

Starting from the large-signal models presented in Section III, the corresponding low-frequency small-signal averaged models are derived by small-signal linearization, i.e. by finding how a perturbation in the current command ic relates to perturbations of the inductor current iL, the switch duty ratio d, and the voltages v1 and v2 that contribute to the inductor current slopes m1 and m2, respectively. Perturbation of the control signal is illustrated in Fig. 3. For example, the small-signal linearization of (8) yields D 2 Ts D ' 2 Ts iˆL = iˆc − M c Ts dˆ − mˆ 1 − mˆ 2 − (DM 1 − D ' M 2 )dˆ (9) 2 2

Since DM1 = D’M2 in steady state, we have D 2 Ts D ' 2 Ts iˆL = iˆc − M c Ts dˆ − mˆ 1 − mˆ 2 . 2 2

t iL

1 M c Ts

  D 2 Ts D ' 2 Ts  iˆc − iˆL − mˆ 1 − mˆ 2  .  2 2  

DT

T

ic

-M c ∆I1

m1

-m 2

IL

∆I3

∆ I2

t vrs

DTs

Ts

2Ts

DTs

Ts

2Ts

t

(b) Fig. 1: Current programmed control technique. (a) control signal with no compensation ramp. (b) with compensation ramp.

ic i c− M ct i2

i1

iL(t) −m2

m1

dTs

Ts

Fig. 2: Waveforms illustrating derivation of the large-signal averaged model equations in Section III.

Ic+îc Ic

iL+îL iL

Ts ,n

Ts ,n + 1

Fig. 3: Perturbation of the control signal

(10)

Equation (10) can be solved for the duty cycle perturbation, dˆ =

3T

CLK

0

SMALL-SIGNAL AVERAGED MODELS AT LOW

2T

(a)

which was the approach adopted in [5]. It should be noted that (5)-(8) represent different ways to approximate large-signal averaged dynamics of the inductor current in the CPM controlled converter. At low frequencies, as pointed in [8], all of the proposed approximations are nearly the same, and result in very similar predictions for the low-frequency dynamics. Lowfrequency small-signal averaged models are discussed further in the next section. IV.

∆I4

∆T 1

2

Finally, reference [4] pointed to the fact that the large-signal expression for the average inductor current should be based on averaging the transient waveform in Fig. 2 over the switching period, 1 1 iL ≈ di1 + d ' i 2 = i c − M c dT s − m1 d 2 T s − m 2 d ' 2 T s , (8) Ts

∆T1

(6)

Assuming m1d = m2d’, reference [7] combined (3) and (4) into an alternative expression: 1 (7) iL ≈ i c − M c dT s − ( m1 + m 2 ) dd ' Ts . Ts

iL

(11)

Finally, we note that the voltages v1 and v2 contributing to the slopes m1 and m2, respectively, can be expressed in terms of the converter input voltage vg and the output voltage v. Hence, the expression (11) for the duty cycle perturbation can be written as (12) dˆ = Fm (iˆc − iˆL − Fg vˆ g − Fv vˆ ) ,

where Fm is the modulator gain, and Fg, Fv, represent the input voltage and the output voltage feed-forward gains, respectively, for the CPM controller. For the model derived from (8), expressions for the gains Fg and Fv in terms of the converter parameters and the steady-state operating values can be found for all basic converters in [5]. The alternative large-signal models (5)-(7) can be used to derive the corresponding small-signal models in the same form as (12), except that the expressions for the gains Fm, Fg, and Fv become somewhat different.

v

vg

Small-signal averaged model of the switching power converter

iL

d Fm

Fg

Fv

TABLE I MODULATOR GAIN DERIVED FROM THE LARGE-SIGNAL EXPRESSIONS Fm from (5) in [3]

Fm from (6) in [6]

Fm from (7) in [7]

Fm from (8) in [4-5]

1 M    M c + 1 Ts 2   1 Fm = M    M c − 2 Ts 2   1 Fm = 1 − 2D    Mc + M2 Ts 2D   1 Fm = M cTs Fm =

The gains Fm are listed in terms of M1, M2 and Mc in Table I for the four different approaches [3-7]. It can be observed that in [6] a different Fm, Fm =1/(M1+Mc)Ts is used for the small-signal model. Nevertheless, as pointed out in Section III, the differences among the various small-signal models in predicting low-frequency dynamics are relatively small. The small-signal models described in this section are fullorder models that result in second-order transfer functions for the basic single-inductor converters such as buck, boost or buck-boost. As detailed in [5], the control-to-output transfer function Gvc = vˆ / iˆc of a CPM controlled converter modeled as shown in Fig. 4 exhibits a low-frequency pole fp1 and a high-frequency pole fp2. Experimental verifications show that the model predictions are excellent, especially at low frequencies. With an adequate compensation ramp, which is usually the case in practical designs, the model predictions are accurate in a wider range of frequencies, allowing reliable design of the voltage feedback loop. However, the small-signal continuous-time models described in this section are not able to predict perioddoubling instability of the CPM controller or additional phase lag at relatively high frequencies (approaching one half of the switching frequency) in cases when the compensation ramp is not employed or when the ramp slope is relatively low. The period-doubling instability and the additional phase lag are ascribed to sampling effects, the modeling of which is discussed in the next section.

ic Fig. 4: Low-frequency small-signal averaged model of a CPM controlled switching power converter.

V.

MODELING OF SAMPLING EFFECTS

In a switched-mode power converter, sampling is performed by the modulator: the value of the duty cycle is updated once per switching period. To account for the highfrequency effects due to sampling in the CPM modulator, while retaining the simplicity of a continuous-time model, the following modeling approach was first introduced in [6], and then applied in a different manner in [7]: (a) Since the objective is to investigate dynamic responses only at high frequencies, assume that the voltage perturbations are negligibly small; hence, vˆg ≈ 0 , vˆ ≈ 0 , mˆ 1 ≈ 0 , mˆ 2 ≈ 0 , and the high-frequency asymptote of the duty-cycle to inductor current transfer function Gid(s) becomes iˆ (s ) M 1 + M 2 (13) Gid ( s ) = L = s dˆ ( s )

(b) Based on the assumption (a), derive a sampled-data (discrete-time) model for the sampled inductor current dynamics. (c) From the sampled-data model (b), derive continuoustime dynamics of the inductor current by means of a zeroorder-hold (ZOH). (d) Approximate the inductor-current dynamics from (c) using a continuous-time transfer function. (e) Incorporate the transfer function from (d) into the low-frequency small-signal averaged model (such as the model shown in Fig. 4). To demonstrate part (b) of the approach, let us derive the discrete-time control to inductor current transfer function using the waveforms of Fig. 1(b): I L (0) + M 1dTs = I c (Ts ) − M c dTs I L (0) + iˆL (0) + M 1 D + dˆ Ts = I c (Ts ) + iˆc (Ts ) − M c D + dˆ Ts

(

)

iˆL (0) + M1dˆTs = iˆc (Ts ) − M c dˆTs

(

)

(14)

I c (Ts ) − M c dTs − M 2 (1 − d )Ts = I L (Ts ) ˆ I c (Ts ) + ic (Ts ) − M c D + dˆ Ts − M 2 1 − D + dˆ Ts = I L (Ts ) + iˆL (Ts ) (15) iˆ (T ) − M dˆT + M dˆT = iˆ (T ) .

(

c

)

s

( (

c

s

2

))

s

L

s

Combining (14) and (15) we have iˆ (T ) − iˆL (0 ) . (16) dˆT s = c s M1 + M c and (17) iˆc (Ts )(1 + α ) − iˆL (0)α = iˆL (Ts ) , where a is defined as [6]: M − Mc . (18) α= 2 M1 + M c From (18), we have the discrete-time inductor dynamics: (19) iˆL [k ] = −αiˆL [k − 1] + (1 + α )iˆc [k ] . The Z-transform of (19) yields the desired discrete-time transfer function iˆL ( z ) z , (20) = (1 + α ) (z + α ) iˆc ( z ) which completes step (b) of the modeling approach. It can be observed that both [6] and [7] use the same discrete-time transfer function (20) in the derivations. References [6] and [7] also use the same method to go from the discrete-time transfer function to a continuous-time transfer function: z = e sTs to obtain the sampled-Laplace domain representation, and a ZOH with the transfer function 1 − e − sTs sT s

Fig. 7(a). Taking into account (13), the transfer function He(s), 1 sTs ≈ 1− sTs 2 /π e −1

obtained from the model in Fig. 6(a). In [7], the sampling effect at high frequencies is incorporated by modifying the current loop gain (at high frequencies), Tc(s) = FmGid, into a current loop gain that includes sampling effects, T’c(s) so that the transfer function iˆL (s ) T ' (s) = c ˆic (s ) 1 + Tc' ( s )

d Fm

vg

Fg

 s   s    +    ωs / 2   ωs / 2 

1 2/π

He (s )

îL

(a)

s

d 1 s 1+ ω

2

p

(23) 2

 s   s    +    ωs / 2   ωs / 2  Substituting (23) in (22) yields (24) iˆL (s ) 1 ≈ 2 ˆic (s )  s  π 2  s  +  1+  − 1 2 1+ α  ωs 2  ωs 2  From (24), the condition to prevent subharmonic oscillation at half the switching frequency is also derived, 2 (1 + α ) > 1 , which is the same as (1). The final step (e) in the model derivation, i.e., incorporating the high-frequency inductor dynamics into the lowfrequency small-signal averaged model is accomplished differently in [6] and [7]. In [6], the sampling action of the CPM modulator is represented by a sampling gain He(s) inserted as shown in 1+

v

Fv

+− îc

s

e − sTs ≈

−− +

(21)

s

1 2/π

(27)

from the model (see Fig. 7) matches the result (24).

to obtain a continuous-time Laplace representation of the inductor current perturbations from the samples: 1 − e − sT iˆL (s ) e sT (22) ≈ (1 + α ) sT (e + α ) sTs iˆc (s ) To accomplish step (d), i.e., to approximate (22) using a rational continuous-time transfer function, both [6] and [7] make use of the Pade approximation: 1−

2

 s   s  (25)   +    ωs / 2   ωs / 2  is found by matching (at high frequencies) the result (24) to the transfer function iˆL (s ) FmGid (26) = iˆc (s ) 1 + FmGid H e H e ( s) =

Fm vg

Fg

−− + +− îc

Fv

v

îL

(b) Fig.6: Modification of the CPM small-signal model in Fig. 4 to include the sampling effects according to: (a) [6], and (b) [7].

The result obtained in [7] is that the sampling effect can

be incorporated by adding a single-pole transfer function to the model, as shown in Fig. 6(b). The pole frequency is

îc + -

îe

Tc(s) or T’c(s)

(28)

|ZOH(jω)|/Ts

2πf s  2  ωp = π − 1 4 1+ α 

1

îL

0.8 0.6 0.4 0.2 0

0.5

1

1.5

2

2.5

2

2.5

3

f/fs

The high-frequency CPM model extensions shown in Fig. 6 can also be incorporated into the model of reference [5]. Appendix A gives a simulation example based on the converter and CPM models described in [5], including large-signal and small-signal simulation results. The effect of the high-frequency extension based on the model in Fig. 6(b) is also shown.

-π/2

−π

A current programmed converter is a nonlinear, timevarying system. It is important to realize that the results of Sections IV and V give approximate transfer functions based on a single-frequency sine-wave perturbation and narrow-band measurement tuned to the perturbation frequency. Taking into account the time-varying nature of the system and the sampling, it has been pointed out that the picture presented by the small-signal models of Section III and IV is not complete, especially when the perturbation frequency approaches or exceeds one half of the switching frequency [8-11]. The Laplace transform of the output signal of an ideal sampler is j 2 nπ  1 ∞ ˆ (29) iˆc* = ∑ ic  s + T  Ts n = −∞  s  In the derivations of (22) and (24), the references [6, 7] take into account only one component (n=1) of (29), 1 (30) iˆc* = iˆc Ts As the perturbation frequency increases the responses at the sideband frequencies are more prominent, and produce more significant distortion in iL which is not taken into account in the models of Section V. Replicas are found at nfs ± fm, where fs =1/Ts is the switching (or sampling) frequency, fm the perturbation frequency, and n an integer.

0

0.5

1

1.5

3

f/fs

ALIASING EFFECTS

Fig.8. ZOH responses in the frequency domain

Once the fundamental response is obtained with the models of Section V, the zero-order hold transfer function in the frequency domain, sin (ωTs ) , (31) ZOH ( jω ) = e − jωT / 2Ts ωTs can be used to evaluate the aliasing effect by locating the frequencies where the replicas take place, and then obtaining the relative amplitude compared to the fundamental. For example, Figs. 9 and 10 illustrate the distortion caused by the replicas in the case of a lowfrequency perturbation, and a high frequency perturbation, respectively. It is clear that the aliasing effects can be significant and should be taken into account when interpreting predictions of the models described in Section V. s

1

|ZOH(jω)|/Ts

VI.

Arg{ZOH(jω)} (rad)

0

Fig.7: Current loop gain. (a) Considering that the averaged functions are continuous (b) considering the sampling effect

0.8 0.6 0.4 0.2 0

fm/fs

0.5

1

1.5

2

2.5

3

f/fs

Fig.9. Fundamental response and replicas to a low frequency perturbation

0.6

of a perturbation at side band frequencies which are multiples of the switching frequency. The effect of the perturbation takes place not only at the perturbation frequency but also at sideband frequencies around the switching frequency. This effect is not considered when the Laplace transform of the sampler output is simplified.

0.4

ACKNOWLEDGMENT

|ZOH(jω)|/Ts

1 0.8

0.2 0

0.5 f

m/fs

1

1.5

2

2.5

3

f/fs

This work of F. J. Azcondo, Ch. Brañas, and R. Casanueva is sponsored by the Spanish Government through the project CICYT TEC 2004-02607/MIC "Power systems for discharge lamps and electrical discharge machining."

Fig.10 Fundamental response and replicas to a high frequency perturbation

REFERENCES [1]

VII.

CONCLUSIONS

This paper presents an overview of previously published approaches to dynamic modeling of current programmed (CPM) converters [3-8]. The modeling assumptions are examined and the differences among the approaches are highlighted, with the objectives of making it easier to present these topics in power electronics courses and applying the models in practice. A starting point in CPM model derivations is an approximate large-signal averaged model that relates the current command, the average inductor current, the slopes of the inductor current, and the compensation ramp slope. In Section III, four different approaches ([3], [4,5], [6] and [7]) are described and compared. A basic low-frequency small-signal averaged model is derived by linearization of the large-signal averaged model as discussed in Section IV. It is pointed out that this small-signal model is quite appropriate for practical CPM designs in most cases. Approaches to including sampling effects and improving high-frequency model predictions based on the approaches described in [6] and [7] are discussed in Section V. Finally, we point out to the importance of aliasing effects [8-11] in understanding a complete picture of converter dynamic models. A paper to motivate a research work on current programmed converters has been presented by comparing the analysis given in different contributions. The inductor current average function is approached in different works by reconstructing it from the sampled average values with a zeroth-order hold. The small signal model from which [7] starts is the same as the accurate model of [5] section 12.3. The small signal model, from which [6] starts, does not take into account equation (3), therefore the influence of perturbations on m1 is neglected. Despite the differences in the low-frequency model between [6] and [5,7] the analyses of the sampling effect in [6] and [7] are equivalent, and the resulting modifications of the current-programmed ac models can be easily introduced into the models of [5]. As a general summary it can be said that [6] and [7] carry out the same analysis of the sampling effect while this point is not included in [5]. On the other hand [8] and [10,11] complete the analysis performed in [7] highlighting the consequence

C. Deisch, “Simple switching control method changes power converter into a current source,” IEEE PESC, 1978 Record, pp.300306. [2] S. Hsu, A. Brown, L. Rensink, R. D. Middlebrook, “Modeling and analysis of switching dc-to-dc converters in constant-frequency current programmed mode,” IEEE PESC, 1979 Record, pp.284-301. [3] R. D. Middlebrook, “Topics in multiple-loop regulators and currentmode programming,” IEEE PESC, 1985 Record, pp.716-732. [4] G.C. Verghese, C.A. Bruzos, K.N. Mahabir, “Averaged and sampleddata models for current mode control: a re-examination,” Proc. of the Annual IEEE Power Electronics Specialists Conference, 1989. PESC '89. vol.1 pp.:484 - 491 June 1989. [5] R. W. Erickson, D. Maksimovic, Fundamentals of Power Electronics, 2nd Edition, Chapter 12 and Appendix B.3,” Kluwer Academic Publishers, 2001. [6] R. B. Ridley, “A New, Continuous-Time Model for Current-Mode Control,” IEEE Trans. on Power Electronics. Vol. 7 No. 2, April 1991. pp. 271-280. [7] F. Dong Tan and R.D. Middlebrook, “Unified and Measurements of Current-Programmed Converters,” Proc. of the PESC’93, pp 380-387. [8] D. J. Perreault, G. C. Verghese, “Time-varying effects and averaging issues in models for current-mode control,” IEEE Transactions on Power Electronics. Vol. No. 3. May 1997. pp 453-461. [9] V. A. Caliskan, G.C. Verghese, A.M. Stankovic, “Multi-frequency averaging of DC/DC converters,” Proc. of the IEEE Workshop on Computers in Power Electronics, 1996. pp. 113-119 Aug. 1996. [10] G. C. Verghese, V. J. Thottuvelil, “Aliasing effects in PWM power converters,” Proc. of the Annual IEEE Power Electronics Specialists Conference, 1999. PESC '99. vol.2 pp.:1043 - 1049 July 1999. [11] V. J. Thottuvelil, G. C. Verghese, “Simulation-based exploration of aliasing effects in PWM power converters,” 6th Workshop on Computers in Power Electronics, pp. 177-183. July 1998.

APPENDIX A. SIMULATION EXAMPLE Step 1. Averaged model For the buck converter of Fig. 11 (a) with duty cycle, d=0.6, and switching frequency fs=200kHz, the averaged model is obtained by replacing the switches section by its averaged equivalent, as is shown in fig. 11 (b). A transient simulation can be used to compare the switched and averaged models and to find the inductor current and capacitor voltage waveforms. Step 2. Parameters of the current programmed control For the circuit of Fig. 11 considering ideal switches and steady state conditions, the output voltage, Vo = Vg D=7.16V the average and peak values of the inductor current are

Ts=V/R=716mA and I Lpeak = iL

+ (Vg − V ) 2 L =914mA,

Ts

respectively. The control parameter of the current programmed control model, Ic, that correspond to D=0.6 is calculated as follows: ILpeak=Ic-McDTs (see Fig. 5). Using a compensating slope parameter Va=McTs=0.6 that corresponds to Mc=0.12A/µs, then Ic=1.274A. iL

L

RL

35uH

0.05

Vg

R1

C1

D2

100u 10

12V R2

U7

0 1 2

MD MS DC

12Vdc

Vg

DA VD

L

3 4

IC = 0.72

RL

35uH V6

0.05

CCM_DCM5

V

C1 R

100u

V2

10

IC = 7.2

1Vac 0.6Vdc

+

0

V

(a)

-

F

S1

U8

0 1 2

12Vdc

47

V2

Laplace part that is used to introduce the block (1/1+ωp) that results from the sampling effect [7], as described in Section V.

MD MS DC

Vg

L

3 4

DA VD

IC = 0.72 D

E

35uH

R4 0.05 C3

R6

100u

CCM_DCM5

IC = 7.2

V9

0

Vg

U4

0 1 2

12V

(a)

MD MS DC

DA VD

3 4

L

RL

35uH

0.05

V8

+

R

C2

100u 10

CCM_DCM5

D

+ -

H E2 + + - E

F

-

0

0.6Vdc

E

U9 CONTROL CURRENT 1

E

3 4

2 D

CPM

0

V

V6

L = 35u

0 1 2

H2

E

1.274Vdc

0

1Vac

V

10

FS = 2E5 VA = 0.6 L = 35E-6 RF = 1

0

(b) 0

U4

C

(b) Fig.11. Converter example: (a) switched circuit (b) averaged model

0 1 2

Vg 12V

L

MD MS DC

3 4

DA VD

IC = 0.72 A

B

35uH

RL 0.05

C2 R

100u

CCM_DCM5

V

10

IC = 7.2

L = 35u C Vg

0 1 2

12V

U4 MD MS DC

DA VD

L

3 4

A

RL

B

35uH

0.05

CCM_DCM5

C2

R3

100u

10

+ V

0

V5

-

L = 35u

V3 A

1.274Vdc

1Vac

+ -

B

0

V3 A

1.274Vdc

0 1 2

H1 + -

B

0

C

H

B

0

E1 + + - E

CONTROL CURRENT 1

2 D

3 4

B

FS = 2E5

0

C B

E1

+ -

CONTROL CURRENT 1

2 D

3 4

B 493340 493340 + s

CPM

+ -

FS = 2E5

E

0

L = 35E-6

VA = 0.6

(c) Fig.14. Small signal simulation circuits to obtain the control to output voltage transfer function (a) direct duty cycle control (b) current programmed control (c) current programmed control considering the sampling effect.

CPM

0

H

0

U6

U6 0 1 2

H1

RF = 1

L = 35E-6 VA = 0.6

Fig.12. Converter example with current programmed control

40 1.0A

(a)

20

iL

ILpeak Ts

0.8A

-40 0d

0.6A

DB(V(R1:2))

DB(V(R3:1))

(b)

DB(V(R6:1))

(a)

(b)

-90d

(c)

-180d 0.4A 5.980ms I(L1)

(c)

0 -20

5.982ms I(L2)

5.984ms

5.986ms

5.988ms

5.990ms

5.992ms

5.994ms

5.996ms

5.998ms

6.000ms

Time

Fig.13. Transient simulation of the steady state inductor current: for the switched and the averaged circuit model.

Step 3 Small signal simulation The ac analysis is performed by connecting ac sources to the control voltages of the large-signal circuit models shown in Figs. 11(b) and 12 and the operation points are fixed using the initial conditions values in inductors and capacitors. Sampling effect is taken into account following Fig. 6 (b). For the example case, M1=0.137A/µs, M2=0.206A/µs and Mc=0.12A/µs. Using (18) α = 0.33 . With (27) ωp=2π78.52kHz. The PSpice ABM library provides a

10Hz 30Hz 100Hz 300Hz P(V(R1:2)) P(V(R3:1)) P(V(R6:1))

1.0kHz

3.0kHz

10kHz

30kHz

100kHz

Frequency

Fig.15. Small signal simulation results. Control to output voltage transfer function with direct duty cycle control, current programmed control and current programmed control considering the sampling effect.