И.2 Small-Sinul A С Mrtdeling i>fthe DCM Switch Netwoii

cuit is relatively easy to solve.

The coiitrol-to-output transfer function OJs) is found by letting = 0 in Fig. 1 1.18. Solution for v then leads to

i\\52)

with

= -;-Ц-

(11.53)

The line-to-ontpiit transfer function С 0) is found by letting <i = 0 in Fig. 11.18. One then obtains

(11.54)

with

(11.55)

Expressions for Gj, Gj,q, and to are listed in Table 11.3, for the DCM buck, boost, and buclt-boost converters with resistive loads [12,13].

The ac modeling approach described in this section is both general and useful. The transistor and diode of a DCM convertercan be simply replaced by the two-port network of Fig. 11.13(b), leading to the small-signal ac model. Alternatively, the switch network can be defined as in Fig. 11.16(a) or 11.16(b), and then modeled by the same two-port network. Fig. 11.16(c). The small-signal converter model can then be solved via conventional circuit analysis techniques, to obtain the small-signal transfer functions ofthe converter.

lible 11.3 Salient features of DCM converter stnall-signal transferfunctions

Conveiter

U>

Buck

TV \ -M

D 2-M

2-M (1 -M)RC

Boost

2V Af-l D Ш-1

m- i) c

Buck-boost

11.2.1 Exumple: Control-to-Output Frequency Response of a DCM Buost Cunverter

As a simple numerical example, let us find the small-signal control-to-output transfer function of a DCM boost converter having the following element and parameter vaiues:

K= 12fi

I = 5 pH

= 1ШкНг

(il.i6)

The output voltage is regidated to be V= 36 V, It is desired to determine Gfx) at the operating point where the load current is / = 3 A and the dc input voltage is = 24 V,

The effective resistance is found by solution of the dc equivalent circuit of Fig. 11.12(h).

Since the load current / and the input and output voltages Vand are known, the power source vaiiie P is

(11,57)

(11.58)

r = /(V-Vj) = [3 А)(36 V-24 VJ = 36W The effective resistance is therefore

V C24V)

The steady-state duty cycle Z)can now be found using Eq. (11.32):

ГЖ /~(5рН) (il.59j

V Л,Г, V (16Q)(10ps) -

The expressions given in Table 11.3 for G,yo and to of the boost converter can now be evaluated:

(36 V)

Й 2V M-1 2(36V) -л D 2M-l~ (0,25)

f < p 2M-1 2n~ 2n{My)RC

(24V)

(Збу) (24 V)

= 72V=i-37aBV

, (36V)

(24 V)

(11.60)

(36 V)

(24 V)

= 112Hz

(12i2)(47t) fi¥)

A Bode diagram of the control-to-output transfer function is constructed in Fig. 11.19. The solid lines illustrate the magnitude and phase predicted by the approximate single-pole model of Fig. 11.1Я. The dashed lines are the predictions of the more accurate model discussed in Section 11.3, which include a second pole at/j = 64 kHz and a RHP zero at f- 127 kHz, arising from the inductor dynamics. Since the switching frequency is 100 kHz, the accuracy of the inodel at these frequencies cannot be guaranteed. Nonetheless, in practice, the lagging phase asymptotes arising from the inductor dynamics can be

II G,

40dBV 20 dBV OdBV -20 dBV dBV

11.2 Snuill-Slgnal AC Modeling of the DCM Switch Network i29

ZG...

-90 -180*

-270*

10 Hz

100 Hz

10 kHz

100 kHz

Fig. 11.19 Magnilude and phase ofthe coiitrol-io-outpiit transfer funttion, DCM iioost example. Solid lines: function and its asymptotes, approximate single-pole respunst predicted by tlie model of Fig. 11, IS. Daslied lines; more accurate response that includes liigti-frcquency inductor dynamics.

observed beginning at 10 = 6.4 кН/.

11.2.2 Example: Control-to-Output Frequency Responses ofaCCIVI/DClVISEPlC

As another example, consider the SEPIC of Fig. 11.20. Acctwding to Eq. (11.34), this converter operates ill CCM if

V 1 - P s: R D RD)

(11.61)

where RJ.iy) is given by Eq. (11.33). Upon neglecting losses in the converter, one finds that the CCM conversion ratio is

Vj l-D

(11.62)

When Eqs. (11.33) and (11.62) are substituted into Eq. (11.61), the condition for operation in CCM becomes:

R< . , =4бД

(11.63)

(i-B),

The converter control-to-output frequency responses are generated using Spice ac simulations. Details of