71
CHAPTER 3
PSM BUCK DC-DC CONVERTER UNDER
DISCONTINUOUS CONDUCTION MODE
Discontinuous conduction mode is the operating mode in which the
inductor current reaches zero periodically. In pulse width modulated
converters under discontinuous conduction mode the inductor current rises
during ON time and when the switch is OFF the current falls and reaches zero
before the end of the cycle. There is a brief duration of time over which the
inductor current is zero and hence each switching cycle starts from zero
inductor current. While this is happening during charging cycles in a PSM
converter, the load discharges the capacitor during skipping cycle
(Angkititrakul and Hu 2008).
PSM buck converter discussed in this chapter is a DC-DC
converter with constant frequency and constant duty cycle (D). Condition for
DCM operation is discussed in section 3.1. Modeling of PSM converter and
simulation results are provided in section 3.2. Modeling and simulation
results of PSM converter operated in forced discontinuous conduction mode
due to pulse skipping are provided in sections 3.3 and 3.4. Exhibition of nonlinear phenomena is studied and the results for PSM converter under DCM
are included in sections 3.5 and for PSM converter under forced DCM in
section 3.6. Discontinuous conduction mode with switched inductor is
discussed in section
72
3.1
DISCONTINUOUS
CONDUCTION
MODE
IN
BUCK
CONVERTER
In a buck converter with constant frequency operation and with
inductance and duty cycle fixed, discontinuous conduction mode sets in
during loads when load resistance is greater than the critical resistance, Rcr
(Erickson and Maksimovic 2001).
In terms of K parameter, where K = 2L/RTs which depends on the
circuit values and the choice of fsw and Kc, the critical parameter depending
on the converter duty cycle D, discontinuous conduction results when K < Kc,
where for buck converter Kc = (1-D).
Figure 3.1
K Vs D plot showing DCM/CCM regions separated by the
borderline
As shown in the plot between K and D in Figure 3.1, for a buck
converter, if the parameter values are so chosen that K>1, DCM would never
occur. For PSM buck converter since the D value is fixed, Kc is fixed for
charging cycles. If the parameter values make K < Kc, operation will be in
DCM throughout.
73
It is possible to note that at K=Kc the average inductor current is
equal to one half of inductor current ripple as shown in Figure 3.2. Any
decrease in load current, which equals the average inductor current would
result in discontinuous conduction in converters with devices that block
negative current.
With DCM the device peak current has to be higher than that in
continuous conduction mode since the peak current has to be at least twice the
load current.
Figure 3.2
Average inductor current or the load DC current equals
half the inductor ripple
Referring to Figure 3.2 let the maximum and minimum currents are
IU and IL. Then the ripple magnitude
I
IU
IL
(3.1)
With upper and lower values fixed, the inductor current ripple is
fixed and the dc component is midway and hence is equal to I/2. Hence
I L,avg
I
2
0 .5 I U
IL
(3.2)
It is possible to express the inductor current ripple quantity as
IL
I L,avg
Vin Vo
ton
2L
I L ,avg
Vo
toff
2L
(3.3)
74
Vin Vo t on
Vo t off
(3.4)
2L
Since the load current I0 = ILavg discontinuous conduction mode
would be the result when
I0
Vin Vo t on Vo t off
2L
(3.5)
While comparing the CCM and DCM operation in converters,
output voltage in DCM is a function of load resistance, as given by
Equation (3.36), and the peak inductor and device currents are higher.
V0
Vin
where
2
1
1 4 K / D1
2
(3.6)
K=2L/RTs
Valid for K<Kcr
Since the extinction time tx is given by
tx
DVin
fV0
(3.7)
D2 can be determined from TX=D1+D2 in Seconds.
3.2
MODELLING AND SIMULATION OF PSM CONVERTER
UNDER DCM
A PSM converter is said to be operating in discontinuous
conduction mode if inductor current reaches zero and stays at zero level for a
brief period of time. Since a PSM converter operates with a constant duty
75
cycle if the condition for DCM is satisfied then in every cycle during the
charging period the inductor current would start from zero. Due to the same
reason the inductor current would remain zero throughout the skipping
period. This results in third circuit configuration alone being considered
during skipping period while modelling the converter operating under DCM.
3.2.1
Modelling
To facilitate analysis and design of converters in various
applications reduced-order models and full-order models for DCM PWM
converters were reported (Cuk and Middlebrook 1977, Maksimovic and Cuk
1991, Vorperian 1990). In former, the inductor current does not appear as a
state variable due to the fact that it becomes zero every cycle and the model
predicts the low frequency behaviour correctly but the absence of inductor
current is disadvantageous in certain control techniques, which rely on
inductor current magnitude.
In the latter the inductor current is retained and they are reported to
be having improved accuracy over reduced-order models (Jian Sun 2001).
In buck converters under discontinuous conduction mode of
operation, there is an additional time interval in each cycle during which
inductor current is clamped to zero.
The converter is assumed to work in DCM and modeled using State
Space Averaging technique (Luo Ping et al 2006, Middlebrook and Cuk
1977) with inductor current included. Let for p cycles the clock pulses are
applied and for q cycles the pulses are skipped for a particular load resistance
R and input voltage Vin. The duration pT is known as charging period and the
duration qT is known as skipping period. During the charging period, in each
cycle the switch is ON for duration equal to D1T and OFF for duration equal
76
to (1-D1) T. During this period inductor current drops to zero in D2T and
hence the current is zero during the remaining (1-(D1+D2)) T. During the
skipping period the switch is OFF throughout as the pulses are not applied
and skipped.
The state space equations, assuming discontinuous conduction
mode, are obtained as:
During charging period,
x
A1 x
y
C1x
x
A2 x
y
C2 x
x
A3 x
y
C3 x
B1vin
0
t
D1T
( 3.8)
B2vin
D1T
t
D1
D2 T
(3.9)
B3vin
D1
D2 T
t
T
(3.10)
During skipping period,
x
A3 x B3 vin
y
C3 x
0 t
T
(3.11)
where,
0
A1
A3
A2
A
0
0
1
RC
0
1
C
1
L
1
RC
(3.12)
(3.13)
77
x
y
iL
vC
,
(3.14)
v0 ,
(3.15)
1
L ,
0
B1
(3.16)
B2
B3
0
C1
C2
C3
(3.17)
(3.18)
0 1
After State Space Averaging,
x
pA D1
D2 x
pA3 D3 x qA3 x
p q
pD1 Bvin
(3.19)
Defining Modulation Factor M,
M
fa
f
1
fa
f
(3.20)
p
(3.21)
p q
where,
fa
f
Actual frequency of switch
Clock frequency
Then Equation (3.19) becomes
x
1 M A D1
D2
A3 D3
MA3 x
1 M D1 Bvin
(3.22)
78
M, the modulation factor is a measure of the number of skipping.
When vin goes higher for the same V0 with constant D, M increases increasing
the number of skipped pulses to maintain the voltage. Similarly when load
decreases M increases decreasing the number of switching.
M, the modulation factor, can be obtained from
2V02 fL
Vin Vin V0 D12 R
(1 M )
(3.23)
Modulation Factor plotted as a function of Vin is shown in
Figure 3.3. As voltage increases M increases indicating increased skipping of
pulses. Modulation Factor plotted as a function of RL is shown in Figure 3.4.
As load resistance increases M increases to signify regulation through pulse
skipping.
M Vs Vin
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
Vin in Volts
Figure 3.3
Modulation factor Vs Vin (Increased skipping with Vin)
79
M Vs RL
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
35
RL in Ohms
Figure 3.4 Modulation factor Vs RL( Increased skipping with RL)
The averaged model including equivalent series resistance is obtained
as below:
The state space equations, assuming discontinuous conduction
mode with rC<<R are obtained as:
During charging period,
x
A1 x B1u
y
C1 x
x
A2 x B2u
y
C2 x
x
A3 x B3u
y
C3 x
0 t
D1T
D1T
t
D1
D2 T
D1
(3.24)
D2 T
t T
(3.25)
(3.26)
80
During skipping period,
x
A3 x B3u
y
C3 x
0 t T
(3.27)
where,
A1
A2
A3
0
x
iL
vC
u
y
B1
,
(3.31)
v0
iin
0
B2
0
0
0
(3.28)
(3.30)
i0
1
L
1
L ,
0
(3.29)
vin
0
B3
A
rC
L
1
C
(3.32)
rC
L ,
1
C
(3.33)
rC
L
1
C
(3.34)
0
1
C
(3.35)
81
C1
C2
C3
x
rC
1
1
0
rC
1
0
0
3.2.2
(3.37)
0 1
(3.38)
0 0
1 M A D1
y
(3.36)
D2
x
rC D1 D2
D1
1 M
B1
B3 D1
B2
B3 D2
B3 u
1
x
0
(3.39)
(3.40)
Simulation
For simulation of the PSM DC-DC buck converter under DCM the
following parameters in Table 3.1 are considered. The load and inductance
values considered result in discontinuous conduction within each cycle.
Table 3.1 Parameters considered for PSM DC/DC converter under DCM
Parameter
vin
V0
L
12 H
C
f
R (minimum)
D1
Value
12V to 20V
5V
470 F
40KHz.
10 Ohms
0.2
Simulation of the model of the converter under DCM is carried out
with MATLAB and the circuit simulation carried out with PSIM. Simulation
results for a load current of 500mA are shown in Figure 3.5. Pulses applied to
82
the switch are shown along with clock pulses to indicate the pulses skipped.
Inductor current is discontinuous. Vin =12V and V0 is maintained at 5V.
The load current is increased from 250mA to 500mA and the
output voltage response is shown in Figure 3.6.
It is observed that the
inductor current and hence the switch current is high but does not change with
increase in load current. Here it remains at 3A at both the loads. The number
of pulses applied increases thus decreasing the number of pulses skipped.
Load Current
V0
Io(A)
6
4
2
0
Pulses Applied
0.8
0.4
IL (A)
0
Ind C urrent
3
2
1
0
C LK
0.8
0.4
0
0.0018
0.002
0.0022
0.0024
Time (s)
Time (S)
Load current, applied pulses, inductor current and clock
5.0 2
vo(V)
5.0 1
5
4.9 9
4.9 8
4.9 7
4.9 6
3
IL (A)
2 .5
2
1 .5
1
0 .5
0
0 .40 15
0.4 0 2
0.4 0 2 5
0 .4 0 3
T i m e (s)
Output voltage and Inductor current waveforms
Figure 3.5
PSM Buck converter waveforms for a load of 500mA
83
Output Voltage in V
Load current in A
Load Current
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
V0
5.2
5.1
5
4.9
4.8
0.0099
0.01
Time (s)
0.0101
(a)
vo(V)
5.02
5
4.98
IL (A)
3
2.5
2
1.5
1
0.5
0
0.4015
0.402
0.4025
0.403
Time (S)
(b)
Figure 3.6
Step increase in load and response of output voltage. Load
increased from 250mA to 500mA. (a) Load current and
output voltage (b) Output voltage and inductor current
Output voltage ripple at 250mA is less than 1% at 0.88% and at
500mA is slightly higher but less than 1% at 0.97%. At 1A the regulation is
poor and the voltage falls to 4.4V.
84
v in(V)
20
18
16
14
12
vo(V)
5.12
[0.411044 , 5.10972]
5.1
5.08
5.06
5.04
5.02
[0.409586 , 5.02012]
5
4.98
[0.409525 , 4.9719]
[0.411174 , 4.97087]
4.96
0.41
0.411
0.412
Time (S)
Figure 3.7 Step increase in load and response of output voltage
The input voltage is increased from 12 V to 20 V and the load is
maintained constant. The ripple at an input of 12V is about 0.97% and at 20V
it is about 2.5%. The inductor peak current and hence the device current is
2.5A which is around 5 times the load current.
It is found that with selection of a constant D1 the ripple in DCM
PSM Buck converter can be maintained low but increase in load may result in
regulation failure. This is due to the fact that the reservoir capacitor would
discharge to a larger extent than the charge it received in each cycle.
Increased pulse width would result in poor ripple performance at lighter
loads. Hence constant pulse width may regulate over a short range without
considerable deterioration in performance.
85
3.3
MODELLING AND SIMULATION OF PSM CONVERTER
UNDER FORCED DCM
When conditions of a PSM converter do not favour discontinuous
conduction the inductor current is nonzero in the beginning of each switching
cycle during charging period. The inductor current reaches a peak value and
the skipping period begins after which the current drops. Now, if the number
of cycles that are skipped, are adequate then the current reaches zero and may
remain zero for a brief period causing discontinuity in inductor current. Since
in each switching cycle the current is continuous this mode may be classified
under CCM and due to discontinuity forced in inductor current it can be
treated as a converter under DCM.
3.3.1
Modelling Converter under Forced DCM
Consider a converter controlled with PSM controller that applies
pulses over duration of time and skips pulses over another duration alternately
based on the result of a condition that involves the actual and desired output
voltages. Let the duration of charging period equal pT where p is the number
of cycles applied, and the duration of skipping period equal (q + r) T where q
is the number of cycles over which the inductor current is non zero and r is
the number of cycles over which the inductor current is zero. The inductor
current waveform in Figure 3.8 includes charging cycles and skipping cycle
long enough for the inductor to dry out so that there is current discontinuity.
86
I nd Current in A
0.5
IL(A)
0.4
0.3
0.2
0.1
0
Charging
Skipping
Pulse Amplitude (V)
V 10
1
0.8
0.6
0.4
0.2
0
0.4951
0.49 52
0.4953
Time(S)
Figure 3.8 Inductor current and applied pulses
rT is a prolonged duration of time that is included to make inductor
current zero for a brief period of time to bring in discontinuous conduction. It
is to be noted that during the charging period the converter is under
continuous conduction since the condition for discontinuous conduction is not
satisfied.
The state space equations, assuming discontinuous conduction with
continuous conduction during charging period, are obtained as given below:
During charging period, for p cycles
x
A1x
y
C1 x
x
A2 x
y
C2 x
B1u
B2u
0 t
D1T
D1T
t
T
(3.41)
(3.42)
During skipping period, for q cycles with nonzero inductor current
87
x
A2 x
y
C2 x
B2u
0 t T
(3.43)
During skipping period, for r cycles with zero inductor current
x
A3 x
y
C3 x
B3u
0 t T
(3.44)
where,
0
A1
A2
0
A3
x
0
A
1
C
1
L
1
RC
0
1
RC
(3.45)
(3.46)
iL
(3.47)
vC
u
vin
(3.48)
y
v0
(3.49)
B1
1
L
0
B2
B3
0
C1
C2
C3
(3.50)
(3.51)
C
0 1
(3.52)
88
State space equations are averaged over switching period:
From Equations (3.41) and (3.42) for p cycles
x
Ax
y
Cx
D1 B1u
(3.53)
From Equation (3.43) for q cycles
x
Ax
y
Cx
(3.54)
From Equation (3.44) for r cycles
x
A3 x
y
Cx
(3.55)
This can be realised with hysteretic current limit along with PSM
control. Number of pulses can be calculated from the rise time for inductor
current to rise to peak value set for IL. Valley current is set to be zero making
the average inductor current to equal load current. Typical inductor current
waveform is shown in Figure 3.9. Here the upper current limit or threshold
was set to be 2A but there is overshoot due to delay in circuit.
Inductor Current
2.5
Current Limit
2
1.5
1
0.5
0
0.4015
0.402
0.4025
0.403
Time (s)
Figure 3.9 Typical current waveform in FDCM PSM converter
89
The PSIM model for PSM buck converter under forced DCM is
shown below in Figure 3.9. There are three sub circuits. Two sub circuits are
to limit voltage and current respectively and the third one is the skip logic.
The current limit circuit accepts inductor current as the input and produces
output HIGH when the current crosses a preset value in a way almost similar
to the one proposed by Dokania (2004) in which a device current sensor was
employed to sense the peak current. This output resets an SR flip flop to give
an output LOW to SKIP Logic which in turn would block pulses to the
switch. This results in start of the skipping cycle.
The second subcircuit accepts input from both the output voltage
sensor and inductor current sensor. When inductor current reaches zero AND
if voltage is below the reference value the circuit produces an out put HIGH
that set the SR flip flop which in turn would make the SKIP Logic to release
pulses resulting in start of the charging cycle.
Thus in every cycle it is ensured that the inductor dries out
resulting in discontinuous conduction.
FDCM with Hysteretic current limit
L1
A
A
C1
Resr
RL
T2
Q R
Q
T2 skip
logic
If iL>Ipeak
S
T2
If iL<Iv
&
v0<Vref
Figure 3.10 Forced discontinuous conduction in PSM buck converter
90
Number of pulses applied till the inductor current reaches Ipeak
equals p and the number of pulses skipped till the current reaches zero equals
q and the pulses will be skipped till the voltage is less than Vref.
3.4
OBSERVATIONS WITH VARIATION IN INPUT VOLTAGE
Following parameters are considered for simulation that is carried
out with PSIM. Parameters correspond to continuous conduction mode.
Table 3.2 Parameters considered for simulation of forced discontinuous
conduction mode
Parameter
Value
12V to 20V
5V
vin
V0
L
156 H
C
470 F
40KHz.
5 Ohms
0.6
f
R (minimum)
D1
Input Voltag e in V
20
18
16
14
12
10
Ou tput Voltage in V
5 .2
5.15
5 .1
5.05
5
4.95
4 .9
4.85
Ind Cu rrent i n A
2 .5
2
1 .5
1
0 .5
0
0 .403 2
0.4034
0.4036
0.4038
0 .4 04
0.4 042
0.4044
0 .4 046
Time (s)
Figure 3.11 Waveforms of Vin, v0 and iL in PSM forced DCM
discontinuous conduction with Ripple – 6%
91
Change in input voltage does not affect the ripple as observed and
the output voltage is regulated over the input range.
3.5
BIFURCATION AND CHAOS IN PSM CONVERTER
UNDER DCM
When the converter operates under DCM the inductor current goes
down to zero before the end of each switching cycle and the inductor is reset.
The current rises from zero in each switching cycle and hence disturbances in
the previous switching cycle do not have any effect on the next switching
cycle (Middlebrook 1988 and Teuvo Suntio 2006).
A simple buck converter having two independent storage elements
is a second order system. Inductor current is zero at the start of each
switching period when the converter operates in DCM.
iL ( nT )
0
n
(3.56)
where T is the switching period and n is an integer.
Thus with the inductor current no longer a dynamic variable, the
converter becomes a first-order system with the capacitor voltage serving as
the only state variable.
The dc-dc converter is operating in discontinuous conduction mode
with operating frequency fixed and the switch and diode are assumed ideal,
with fsw and T the switching frequency and time period, for a
two-dimensional buck converter. The operating condition switches from S1
with switch ON to S2 with switch OFF and at the end to S3 after inductor is
dried out once a clock cycle. Hence every cycle begins from S3 and the
switch goes to ON state so that the operating condition switches to S1 or
92
remains in S3 depending on the constraint equation v(nT)>vref. It is also to be
noted that the duty cycle is fixed and hence at the beginning of every cycle
based on the constraint the next operating state is S1 or retaining the state as
S3 the next pulse is skipped.
The sampled data model of the converter with input constant
operating in discontinuous conduction mode is a first-order iterative map
given by:
vn
f vn , d n
1
(3.57)
The discrete-time map for the converter is (Fang and Abed 1998)
2
vn
d n Vin Vin
vn
vn
1
vn
(3.58)
where
vn
vc tn
(3.59)
T
1
CR
1 T
2 CR
2
(3.60)
T2
2 LC
(3.61)
The constraint equation is
dn
0
dn
D1
if v nT
if
v nT
(3.62)
vref
v ref
(3.63)
93
The bifurcation diagram is as shown in Figures 3.12 and 3.13.
Figure 3.12 Bifurcation diagram for PSM DC/DC buck converter under
DCM
Figure 3.13 Bifurcation diagram for PSM DC/DC buck converter under
DCM-enlarged view
94
3.6
BIFURCATION AND CHAOS IN PSM DC-DC VMC –FDCM
The input voltage is varied from 10V to 35V. There is bifurcation
but the tendency to be chaotic is brought down every time the operating state
becomes S3 with the state variable iL becomes zero. The system reduces to
first order as the inductor is reset and hence the chaos does not set in as seen.
In Figure 3.14 the number of charging pulses repeat making fa constant. In
Figure 3.15 the waveforms of v0 and iL repeat in a 4-3 sequence as marked
both indicating periodicity.
Load Current in A
Ind current in A
Waveforms showing no chaos
Input voltage is 12V and output voltage is 5V
2.5
2
1.5
1
0.5
0
Output Voltage in V
5.2
5.15
5.1
5.05
5
4.95
4.9
4.85
0.402
0.4025
0.403
0.4035
0.404
0.4045
Time (s)
Figure 3.14 Inductor current and output voltage with Vin = 12V for
PSM buck converter under forced DCM with Ripple – 6%
Ind C urrent in A
Loa d C urrent in A
3
2.5
2
1.5
1
0.5
0
Output Voltage in V
W av efo rm s show ing no c haos
Input Volatge is 35V a nd output voltage is 5V
5.4
5.3
5.2
5.1
5
4.9
0.40 1
0.40 2
0.403
0.404
Time (s)
Figure 3.15 Inductor current and output voltage with Vin = 35V for
PSM buck converter under forced DCM with Ripple – 6%
95
Phase plane trajectory between iL, the inductor current and vC the
capacitor voltage is shown below in Figure 3.16 indicating no chaos at 12V.
Ind Cu rre nt in A
Phase Plane Trajectory PSM under Forced DCM
iL Vs vC
Vin = 12V
2 .5
2
1 .5
1
0 .5
iL becoming zero showing Discontinuous conduction
0
4.9 6
4.9 8
5
5 .0 2
5 .0 4
5 .06
5 .08
5 .1
vC
Figure 3.16 Phase plane trajectory iL Vs vC showing discontinuous
current and no Chaos at Vin = 12V for PSM converter
under forced DCM
Phase plane trajectory between iL, the inductor current and vC the
capacitor voltage is shown below in Figure 3.17 indicating no chaos at 35V.
vC
Phase Plane trajectory for PSM under Forced DCM
vC in V and iL in A
5.22
5.2
5.18
5.16
5.14
5.12
5.1
5.08
5.06
iL becoming zero showing discontinuous conduction
0
0.5
1
1.5
iL
2
2.5
3
Figure 3.17 Phase plane trajectory iL Vs vC showing discontinuous
current and no Chaos at Vin = 35V for PSM converter under
forced DCM
96
Ripple is at 6% due to high inductor current ripple with ESR not
negligible.
3.7
CONCLUSION
Since the inductor current becomes zero every cycle the order of
the system reduces by one. In a typical CCM buck converter there are two
poles and system may become unstable when feedback is applied whereas the
DCM has one pole and the system is stable with feedback. A small inductor
implies a larger ripple and perhaps a DCM operation. Device peak current is
significantly higher. The system now reacts quicker since a smaller inductor
offers less opposition to current changes. However, with change in load or
supply voltage the ripple goes up or the converter fails to regulate if the duty
cycle is inadequate. The ripple current being important, if high results in
higher conduction losses due to resistive paths like RDS,on etc.: DCM
operation brings larger conduction losses compared to CCM if operation is
over a wide range.
With a selection of a constant duty ratio, it is found that, the ripple
in DCM PSM Buck converter can be maintained low at a particular load but
would not regulate over the entire range and may result in regulation failure
due to inadequate charge transfer per cycle. Adequate charge transfer is
ensured in the case of forced DCM operation with hysteretic current control,
but the output voltage ripple is slightly higher. Increased pulse width to
ensure adequate charge transfer also results in poor ripple performance. This
is due to the ESR of the capacitor considered being not negligible. Low ESR
ceramic capacitors that are used for output filtering (John Betten and Dave
Strasser 2002), result in lower ripple and offer a solution to minimisation of
ripple in converters that are inherently stable.