Transfer functions of DCM converter
1 ˆ ( s) j2 ( R // r2 )d sC ˆ ˆ ( s ) j2 d ( s ) v 1 1 s( R // r2 )C ( R // r2 ) sC ( R // r2 )
control-to-output Gvd ( s)
Small-signal ac model of DCM buck-boost converter
Small-signal ac models of the DCM buck and boost converters
Simplification of DCM small-signal model
v2(t) Ts
Let L short circuit C open circuit
I1

f i (V1 , V2 )
I2


Re (d1 )
+
vL
L
vg() t Ts
-


C R
V2

R
V

-
-
Converter input power:
Converter output power:
i2(t) Ts
v2(t) Ts
Re (d1 )
+
vL
L


C R
v(t ) Ts
-
Solution of averaged model: steady state
fi ( v1 Ts , v2 Ts )
i1 (t ) Ts v1(t) Ts
i2(t) Ts
+
vg (t )
v1 (t )
Ts
Ts
+
i1(t) Ts
i2 (t )
+
Ts
Re (d )
C
R
v(t) Ts
-
Steady-state model: DCM buck, boost
Let L short circuit C open circuit
i1(t) Ts Re(d)
+
L
+
v1 (t )
Buck
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
i1 (t ) Ts
v1 (t ) Ts Re (d (t ))
f1 v1 (t ) Ts , v2 (t ) Ts , d (t )


Expand in three-dimensional Taylor series at the quiescent operating point:
1 vL (t ) Ts Ts

t Ts
t
1 vL dt Ts

t Ts
t
L
di L dt [i t Ts i (t )] dt Ts
Solve for d2:
Average switch network port voltages
Average v1(t) waveform:
1. A two-port lossless network 2. Input port obeys Ohm’s Law 3. Power entering input port is transferred to output port
The loss-free resistor (LFR)
Review of last week study
df1 v1 ,V2 , D df1 V1 , v2 , D ˆ ˆ1 (t ) ˆ2 (t ) I1 i1 (t ) f1 V1 , V2 , D v v dv1 dv2 v V v V
1 1 2
2
df1 V1 ,V2 , d ˆ d (t ) ...... dd d D
ˆ( s ) v line-to-output Gvg ( s ) ˆg ( s ) v

ˆ ( s ) 0 d
g 2 ( R // r2 ) 1 s ( R // r2 )C
Stage 1
Switch is on and diode is off
Inductor current increase linearly
Stage 2
Switch is off and diode is on
Inductor transfers energy to output The stage is ended once inductor current reduce to zero.
d1 d 2 d3 1
use:
Similar analysis for v2(t) waveform leads to
Average switch network port currents
Average the i1(t) waveform:
The integral q1 is the area under the i1(t) waveform during first subinterval. Use triangle area formula:
Input port
i1 (t ) Ts v1 (t ) Ts Re (d (t )) f1 v1 (t ) Ts , v2 (t ) Ts , d (t )


Output port
i2 (t ) Ts
Similarly
Re d (t ) v2 (t ) Ts
Input port: Averaged equivalent circuit
where
Output port: Averaged equivalent circuit
Power balance in lossless two-port networks
In a lossless two-port network without internal energy storage: instantaneous input power is equal to instantaneous output power.
DCM buck, boost model
i1(t) Ts Re(d)
Buck
L
+
vg (t )
+ v1 (t ) +
Ts
Ts
i2 (t )
+
Ts
v2 (t )
Ts
C
R
, v2 Ts )
v (t )
Ts
-
fi ( v1
-
Ts
+
Boost
L
v2 ( t ) -
Ts
fi ( v1
Ts
, v2 Ts )
vL (t )
Ts
0
ˆL (t ) 0 v
Buck, boost, and buck-boost converter models all simplify to
DCM buck, boost, and buck-boost converters exhibit a single-pole system
-
Conversion ratio of DCM converter
Small-signal ac modeling of the DCM switch network
Perturb and linearize:
We get
Linearization by Taylor series
Given the nonlinear equation
2 v1 (t ) T s
f 2 v1 (t ) Ts , v2 (t ) Ts , d (t )


DC terms Small-signal ac linearization
Small-signal DCM switch model parameters
Table Small-signal DCM switch model parameters
Stage 3
Both Switch and diode are off Capacitor output energy to load
DCM waveforms
Peak inductor current:
d2(t)= ?
Average inductor voltage:
In DCM, the diode turns off when the inductor current reaches zero. Hence, i(0) = i(Ts) = 0, and the average inductor voltage is zero. This is true even during transients.
Equate and solve:
Steady state input to output ratio
For the buck-boost converter, we have
Eliminate Re:
Averaged models of DCM converters