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Power Electronics The Buck (Step-Down)
Converter
1
Dr. Firas Obeidat
2
Table of contents
1 • Introduction
2
• Step Down Chopper with Resistive Load
3
• Step Down Chopper with RL Load
4
• Step Down Chopper with Low Pass Filter
Dr. Firas Obeidat Faculty of Engineering Philadelphia University
3 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
Introduction
DC-DC converters are power electronic circuits that convert a DC voltage to
a different DC voltage level, often providing a regulated output.
LOAD
Vcontrol
(derived from
feedback circuit)
DC supply
(from rectifier-
filter, battery,
fuel cell etc.)
DC output
General block diagram
Applications: – Switched-mode power supply (SMPS), DC motor control,
battery chargers, subway cars, trolley buses, vehicles, etc.
4 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
Introduction
Main Types of Choppers
1- Step-down DC-DC converter.
In step down chopper output voltage is less than input
voltage.
2- Step-up DC-DC converter.
In step up chopper output voltage is more than input
voltage.
3- Buck-Boost converter (Step-down/step-up converter).
4- Cuk converter.
5 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Step down chopper as Buck
converted is used to reduce the input
voltage level at the output side.
Circuit diagram of a step down
chopper is shown in the figure.
When CH is turned ON, Vs directly
appears across the load as shown in
figure. So VO=VS.
When CH is turned OFF, Vs is
disconnected from the load. So output
voltage VO = 0.
The voltage waveform of step down
chopper
6 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter TON → It is the interval in which chopper is in ON state.
TOFF → It is the interval in which chopper is in OFF state.
VS → Source or input voltage.
VO → Output or load voltage.
T → Chopping period = TON + TOFF
F=1/T is the frequency of chopper switching or chopping frequency
Operation of Step Down Chopper with Resistive Load
When CH is ON, VO = VS When CH is OFF, VO = 0
The Average output voltage is
𝑉𝑑𝑐 = 𝑉𝑜 =1
𝑇 𝑉𝑠𝑑𝑡
𝑇𝑂𝑁
0
=𝑉𝑠𝑇𝑂𝑁𝑇
= 𝐷𝑉𝑠
𝐼𝑑𝑐 =𝑉𝑑𝑐𝑅=𝐷𝑉𝑠𝑅
7 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
The rms output voltage is
Where,
D is duty cycle = TON/T. TON can be varied from 0 to T, so 0 ≤ D ≤ 1.
The output voltage VO can be varied from 0 to VS.
𝑉𝑟𝑚𝑠 =1
𝑇 𝑉𝑠
2𝑑𝑡
𝑇𝑂𝑁
0
= 𝑉𝑠𝑇𝑂𝑁𝑇= 𝐷𝑉𝑠
The output voltage is always less than the
input voltage and hence the name step
down chopper is justified.
𝐷 =𝑇𝑂𝑁𝑇
𝑇 = 𝑇𝑂𝑁 + 𝑇𝑂𝐹𝐹
𝑃𝑜 = 𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠 =𝑉𝑟𝑚𝑠
2
𝑅= 𝐷
𝑉𝑠2
𝑅
𝐼𝑟𝑚𝑠 =𝑉𝑟𝑚𝑠𝑅=
𝐷𝑉𝑠𝑅
Step Down Chopper with Resistive Load
8 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Ripple factor (RF) can be found from
𝑅𝐹 =𝑉𝑟𝑚𝑠𝑉𝑑𝑐
2
− 1 =𝐷𝑉𝑠
2
𝐷2𝑉𝑠2 − 1 =
1
𝐷− 1 =
1 − 𝐷
𝐷
Methods of Control
1- Pulse Width Modulation
• tON is varied keeping chopping frequency ‘f’ & chopping
period ‘T’ constant.
• Output voltage is varied by varying the ON time tON
2- Variable Frequency Control
• Chopping frequency ‘f’ is varied keeping either tON or tOFF
constant.
• To obtain full output voltage range, frequency has to be
varied over a wide range.
• This method produces harmonics in the output and for large
tOFF load current may become discontinuous
V0
V
V
V0
t
ttON
tON tOFF
tOFF
T
v0
V
V
v0
t
t
tON
tON
T
T
tOFF
tOFF
Pulse Width Modulation Method
Variable Frequency Control Method
Step Down Chopper with Resistive Load
9 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Examlpe: A transistor dc chopper circuit (Buck converter) is supplied with
power form an ideal battery of 100 V. The load voltage waveform consists
of rectangular pulses of duration 1 ms in an overall cycle time of 2.5 ms.
Calculate, for resistive load of 10 Ω.
(a) The duty cycle D.
(b) The average value of the output voltage Vdc.
(c) The rms value of the output voltage Vrms.
(d) The ripple factor RF.
(e) The output dc power.
𝐷 =𝑡𝑂𝑁𝑇=1𝑚𝑠𝑒𝑐
2.5𝑚𝑠𝑒𝑐= 0.4
(a)
𝑉𝑑𝑐 = 𝐷𝑉𝑠 = 0.4 × 100 = 40 V (b)
(c) 𝑉𝑟𝑚𝑠 = 𝐷𝑉𝑠 = 0.4 × 100 = 63.2 V
(d) 𝑅𝐹 =1−𝐷
𝐷 =
1−0.4
0.4= 1.225
𝑃𝑜 =𝑉𝑑𝑐
2
𝑅=402
10= 160 W (e)
Step Down Chopper with Resistive Load
10 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
When chopper is ON, supply is connected
across load. Current flows from supply to
load.
When chopper is OFF, load current
continues to flow in the same direction
through FWD due to energy stored in
inductor ‘L’.
Step Down Chopper with RL Load
Load current can be continuous or
discontinuous depending on the values of
‘L’ and duty cycle ‘D’
For a continuous current operation, load
current varies between two limits Imax and
Imin.
When current becomes equal to Imax the
chopper is turned-off and it is turned-on
when current reduces to Imin.
Outputvoltage
Outputcurrent
v0
V
i0
Imax
Imin
t
t
tON
T
tOFF
Continuouscurrent
Outputcurrent
t
Discontinuouscurrent
i0
11 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
When the switch is closed in the buck
converter, the circuit will be as shown in
the figure, the diode is reverse-biased.
Continuous Current Operation When Chopper Is ON (0 ≤t ≤
tON)
The voltage across the inductor is
𝑉𝑠 = 𝑉𝑅 + 𝑉𝐿
𝑉𝑠 = 𝑉𝑅 + 𝐿𝑑𝑖
𝑑𝑡 →
𝑑𝑖
𝑑𝑡=𝑉𝑠 − 𝑉𝑅𝐿
∆𝑖 = 𝑉𝑠 − 𝑉𝑅𝐿
𝑑𝑡
𝐷𝑇
0
=𝑉𝑠 − 𝑉𝑅𝐿
𝐷𝑇 =𝑉𝑠 − 𝑉𝑅𝐿
𝑡𝑂𝑁
Outputvoltage
Outputcurrent
v0
V
i0
Imax
Imin
t
t
tON
T
tOFF
Continuouscurrent
Outputcurrent
t
Discontinuouscurrent
i0
𝑑𝑖
𝑑𝑡=∆𝑖
𝑡𝑂𝑁=𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛
𝑡𝑂𝑁=𝑉𝑠 − 𝑉𝑅𝐿
𝑖𝑜1 = 𝐼𝑚𝑖𝑛 +𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛
𝑡𝑂𝑁𝑡 = 𝐼𝑚𝑖𝑛 +
𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛𝐷𝑇
𝑡 = 𝐼𝑚𝑖𝑛 +𝑉𝑠 − 𝑉𝑅𝐿
𝑡 From straight
line equation
(1)
(2)
Step Down Chopper with RL Load
12 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Continuous Current Operation When Chopper Is OFF (tON ≤t ≤ T)
Outputvoltage
Outputcurrent
v0
V
i0
Imax
Imin
t
t
tON
T
tOFF
Continuouscurrent
Outputcurrent
t
Discontinuouscurrent
i0
0 = 𝑉𝑅 + 𝑉𝐿
0 = 𝑉𝑅 + 𝐿𝑑𝑖
𝑑𝑡 →
𝑑𝑖
𝑑𝑡= −
𝑉𝑅𝐿
∆𝑖 = −𝑉𝑅𝐿𝑑𝑡
𝑡𝑂𝐹𝐹
0
= −𝑉𝑅𝐿𝑡𝑂𝐹𝐹
𝑑𝑖
𝑑𝑡=∆𝑖
𝑡𝑂𝐹𝐹=𝐼𝑚𝑖𝑛 − 𝐼𝑚𝑎𝑥𝑡𝑂𝐹𝐹
= −𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛𝑡𝑂𝐹𝐹
= −𝑉𝑅𝐿
(3)
𝑖𝑜2 = 𝐼𝑚𝑎𝑥 +𝐼𝑚𝑖𝑛 − 𝐼𝑚𝑎𝑥𝑡𝑂𝐹𝐹
𝑡 − 𝑡𝑂𝑁 = 𝐼𝑚𝑎𝑥 −𝑉𝑅𝐿(𝑡 − 𝑡𝑂𝑁)
From straight line equation
(4)
Step Down Chopper with RL Load
13 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
𝑉𝑠 − 𝑉𝑅𝐿
𝑡𝑂𝑁 −𝑉𝑅𝐿𝑡𝑂𝐹𝐹 = 0
𝑉𝑠 − 𝑉𝑅𝑉𝑅
=𝑡𝑂𝐹𝐹𝑡𝑂𝑁
𝑉𝑠𝑉𝑅− 1 =
𝑡𝑂𝐹𝐹𝑡𝑂𝑁
𝑉𝑠𝑉𝑅=𝑡𝑂𝐹𝐹𝑡𝑂𝑁
+ 1
𝑉𝑠𝑉𝑅=𝑡𝑂𝐹𝐹 + 𝑡𝑂𝑁𝑡𝑂𝑁
=𝑇
𝑡𝑂𝑁 𝑉𝑅 = 𝐷𝑉𝑠
From equation (1)
∆𝑖 =𝑉𝑠 − 𝐷𝑉𝑠𝐿
𝐷𝑇 =𝑉𝑠 1 − 𝐷 𝐷
𝐿𝑓
since
𝑓 =1
𝑇
𝐷 =𝑡𝑂𝑁𝑇
Step Down Chopper with RL Load
Steady-state operation requires that the inductor current at the end of the
switching cycle be the same as that at the beginning, meaning that the net change
in inductor current over one period is zero. This requires
14 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
At steady state operation, the average inductor current must be the same as
the average current in the load resistor.
𝐼𝐿 = 𝐼𝑅 =𝑉𝑅𝑅
The maximum and minimum values of the inductor current are computed as
𝐼𝑚𝑎𝑥 = 𝐼𝐿 +∆𝑖
2
𝐼𝑚𝑎𝑥 = 𝐼𝐿 +𝑉𝑠 1 − 𝐷 𝐷
2𝐿𝑓= 𝐼𝐿 +
𝑉𝑅 1 − 𝐷
2𝐿𝑓
𝐼𝑚𝑖𝑛 = 𝐼𝐿 −∆𝑖
2
𝐼𝑚𝑖𝑛 = 𝐼𝐿 −𝑉𝑠 1 − 𝐷 𝐷
2𝐿𝑓= 𝐼𝐿 −
𝑉𝑅 1 − 𝐷
2𝐿𝑓
The average dc output voltage and current can found as
𝑉𝑑𝑐 = 𝐷𝑉𝑠 𝐼𝑑𝑐 ≅𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛
2
Step Down Chopper with RL Load
15 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Examlpe: A dc chopper has a resistive load of 20Ω and input voltage
VS=220V. When chopper is ON, its voltage drop is 1.5 volts and chopping
frequency is 10 kHz. If the duty cycle is 80%, determine the average output
voltage and the chopper on time.
𝑉𝑑𝑐 = 𝐷𝑉𝑠 =𝑡𝑂𝑁𝑇
𝑉𝑠 − 𝑉𝐶𝐻 = 0.8 220 − 1.5 = 174.8 V
𝑉𝑠 = 220V
𝑇 =1
𝑓=
1
10 × 10−3= 0.1m 𝑠𝑒𝑐
𝑡𝑂𝑁 = 𝐷𝑇 = 0.8 × 0.1 × 10−3 = 80μ 𝑠𝑒𝑐
𝐷 =𝑡𝑂𝑁𝑇= 0.8
Step Down Chopper with RL Load
16 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Examlpe: A Chopper circuit is operating at a frequency of 2 kHz on a 460
V supply. If the load voltage is 350 volts, calculate the conduction period of
the thyristor in each cycle.
𝑉𝑑𝑐 = 𝐷𝑉𝑠 =𝑡𝑂𝑁𝑇𝑉𝑠
𝑉𝑠 = 460V
Chopping period
𝑇 =1
𝑓=
1
2 × 10−3= 0.5m 𝑠𝑒𝑐
𝑡𝑂𝑁 =𝑇𝑉𝑑𝑐𝑉𝑠
=0.5 × 10−3 × 350
460= 0.38m 𝑠𝑒𝑐
Step Down Chopper with RL Load
17 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Step Down Chopper with Low Pass Filter
This converter is used if the
objective is to produce an output
that is purely DC.
If the low-pass filter is ideal, the
output voltage is the average of the
input voltage to the filter.
Analysis for the Switch Closed
When the switch is closed in the buck
converter circuit of fig. a, the diode is
reverse-biased and fig. b is an
equivalent circuit. The voltage across
the inductor is
18 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Step Down Chopper with Low Pass Filter
Analysis for the Switch Closed
Since the derivative of the current
is a positive constant, the current
increases linearly. The change in
current while the switch is closed is
computed by modifying the
preceding equation.
(1)
(∆𝑖𝐿)𝑐𝑙𝑜𝑠𝑒𝑑= 𝑉𝑠 − 𝑉𝑜𝐿
𝐷𝑇
0
𝑑𝑡 =𝑉𝑠 − 𝑉𝑜𝐿
𝐷𝑇
or
19 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Step Down Chopper with Low Pass Filter
Analysis for the Switch Opened
When the switch is open, the diode
becomes forward-biased to carry the
inductor current and the equivalent
circuit of fig. c applies. The voltage
across the inductor when the switch
is open is
The derivative of current in the inductor is a negative constant, and the
current decreases linearly. The change in inductor current when the switch is
open is
(2) (∆𝑖𝐿)𝑜𝑝𝑒𝑛𝑒𝑑=
−𝑉𝑜𝐿
(1−𝐷)𝑇
0
𝑑𝑡 =−𝑉𝑜𝐿(1 − 𝐷)𝑇 or
20 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Step Down Chopper with Low Pass Filter
Steady-state operation requires that the
inductor current at the end of the switching
cycle be the same as that at the beginning,
meaning that the net change in inductor
current over one period is zero. This
requires
Using equations 1&2
The average inductor current must be the
same as the average current in the load
resistor, since the average capacitor current
must be zero for steady-state operation:
t
t
t
t
t
t
t
0
Vo
Ic
lV
Di
Si
Li
bev
onT offT
sT
oI
in oV V
oV
Q
Vo
21 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Step Down Chopper with Low Pass Filter
The maximum and minimum values of the inductor current are computed as
Since Imin=0 is the boundary between continuous and discontinuous current,
The minimum combination of inductance and switching frequency for
continuous current in the buck converter is
22 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Step Down Chopper with Low Pass Filter
where Lmin is the minimum inductance required for continuous current. In
practice, a value of inductance greater than Lmin is desirable to ensure
continuous current.
Since the converter components are assumed to be ideal, the power supplied by
the source must be the same as the power absorbed by the load resistor.
This relationship is similar to the voltage-current relationship
for a transformer in AC applications. Therefore, the buck
converter circuit is equivalent to a DC transformer.
In the preceding analysis, the capacitor was assumed to be very large to keep
the output voltage constant. In practice, the output voltage cannot be kept
perfectly constant with a finite capacitance. The variation in output voltage, or
ripple, is computed from the voltage-current relationship of the capacitor. The
current in the capacitor is
23 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Step Down Chopper with Low Pass Filter
While the capacitor current is positive, the
capacitor is charging. From the definition of
capacitance,
The change in charge ∆Q is the area of the
triangle above the time axis
Substitute (∆iL)open in the above equation yields
∆Vo is the peak-to-peak ripple voltage at the output
The required capacitance in terms of specified voltage ripple:
24 Dr. Firas Obeidat Faculty of Engineering Philadelphia University
The Buck (Step-Down) Converter
Step Down Chopper with Low Pass Filter
Examlpe: buck dc-dc converter with Low Pass Filter has the following
parameters:
Assuming ideal components, calculate (a) the output voltage Vo, (b) the
maximum and minimum inductor current, and (c) the output voltage
ripple.
(a)
(b)
(c)
The average inductor current is 1 A, and ∆iL=1.5 A.
25