Similar analysis for i2(t) waveform， we got
i2 (t) Ts
d12 (t)Ts 2L

v1(t)
2 Ts
v2 (t) Ts

1

v1 (t )
2 Ts
Re (d1) v2 (t) Ts

fi ( v1 Ts , v2
Ts )
Stage 2
Switch is off and diode is on
Inductor transfers energy to output The stage is ended once inductor current reduce to zero.
Stage 3
Both Switch and diode are off Capacitor output energy to load
v2(t) Ts


C R v(t) Ts

-
Solution of averaged model: steady state
vg(t)Ts
i1(t) Ts fi ( v1 Ts , v2 Ts ) i2(t) Ts


பைடு நூலகம்
v1(t) Ts -
DCM waveforms
Peak inductor current:
d2(t)= ?
1
vL (t) Ts Ts
t Ts t
vLdt

1 Ts
tTs L di dt L [i
t
dt
Ts
t Ts
i(t)]
In DCM, the diode turns off when the inductor current reaches zero. Hence, i(0)
Average v1(t) waveform:
d1 d2 d3 1
use:
Similar analysis for v2(t) waveform leads to
Average switch network port currents
Average i1(t) waveform:
The integral q1 is the area under the i1(t) waveform during first subinterval. Use triangle area formula:
= i(Ts) = 0, and the average inductor voltage is zero. This is true even during
transients.
vL (t) Ts 0
Average inductor voltage:
Solve for d2:
Average switch network port voltages
DTS

2V R
Then DCM mode
DCM
Output to input ratio of Buck converter
In DCM
Derivation of DCM averaged switch model: buck-boost example
Define switch terminal quantities v1, i1, v2, i2, as shown
• ,
Let us find the averaged quantities for operation in DCM, and determine
it1hTes ,rev1laTtsio, ni2sTbs ,etvw2 eTsen
them
Stage 1
Switch is on and diode is off Inductor current increase linearly
In a lossless two-port network without internal energy storage: instantaneous input power is equal to instantaneous output power.
1. A two-port lossless network 2. Input port obeys Ohm’s Law 3. Output is (nonlinear) dependent current source 4. Power entering input port is transferred to output port
We need equivalent circuits that model both the steady-state and small signal ac models of converters operating in DCM.
CCM
Buck converter
IL Io
(Vg V ) L
Input port: Averaged equivalent circuit
where
The loss-free resistor (LFR)
Output port: Averaged equivalent circuit
Power balance in lossless two-port networks
Averaged switch model: DCM buck-boost example
Original circuit
Averaged model
vg(t)Ts
i1(t) Ts fi ( v1 Ts , v2 Ts ) i2(t) Ts


v1(t) Ts -
Re (d1) + vL L
Chapter 2 Modeling of DCM DC/DC Converter
Characteristics at the CCM/DCM boundary
•All converters may operate in DCM at light load •Steady-state output voltage becomes strongly load-dependent •Dynamics in DCM mode is different to CCM mode