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
3
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.
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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.
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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
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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
= 𝑉𝑠𝑇𝑂𝑁
𝑇 = 𝐷𝑉𝑠
𝐼𝑑𝑐 = 𝑉𝑑𝑐
𝑅 = 𝐷𝑉𝑠 𝑅
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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 𝑉𝑠2𝑑𝑡
𝑇𝑂𝑁
0
= 𝑉𝑠 𝑇𝑂𝑁
𝑇 = 𝐷𝑉𝑠
The output voltage is always less than the input voltage and hence the name step down chopper is justified.
𝐷 = 𝑇𝑂𝑁
𝑇 𝑇 = 𝑇𝑂𝑁 + 𝑇𝑂𝐹𝐹
𝑃𝑜 = 𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠 = 𝑉𝑟𝑚𝑠2
𝑅 = 𝐷𝑉𝑠2 𝑅
𝐼𝑟𝑚𝑠 = 𝑉𝑟𝑚𝑠
𝑅 = 𝐷𝑉𝑠 𝑅
Step Down Chopper with Resistive Load
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The Buck (Step-Down) Converter
Ripple factor (RF) can be found from
𝑅𝐹 = 𝑉𝑟𝑚𝑠 𝑉𝑑𝑐
2
− 1 = 𝐷𝑉𝑠2
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
t tON
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
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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 = 𝑡𝑂𝑁
𝑇 = 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−𝐷𝐷 = 1−0.40.4 = 1.225
𝑃𝑜 = 𝑉𝑑𝑐2
𝑅 = 402
10 = 160 W (e)
Step Down Chopper with Resistive Load
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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.
Output voltage
Output current v0
V
i0 Imax
Imin
t
t tON
T tOFF
Continuous current
Output current
t Discontinuous
current i0
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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 ≤ t
ON)
The voltage across the inductor is 𝑉𝑠 = 𝑉𝑅 + 𝑉𝐿
𝑉𝑠 = 𝑉𝑅 + 𝐿 𝑑𝑖
𝑑𝑡 → 𝑑𝑖
𝑑𝑡 = 𝑉𝑠 − 𝑉𝑅 𝐿
∆𝑖 = 𝑉𝑠 − 𝑉𝑅 𝐿 𝑑𝑡
𝐷𝑇
0
= 𝑉𝑠 − 𝑉𝑅
𝐿 𝐷𝑇 = 𝑉𝑠 − 𝑉𝑅 𝐿 𝑡𝑂𝑁
Output voltage
Output current v0
V
i0 Imax
Imin
t
t tON
T tOFF
Continuous current
Output current
t Discontinuous
current i0
𝑑𝑖
𝑑𝑡 = ∆𝑖
𝑡𝑂𝑁 = 𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛
𝑡𝑂𝑁 = 𝑉𝑠 − 𝑉𝑅 𝐿 𝑖𝑜1 = 𝐼𝑚𝑖𝑛 + 𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛
𝑡𝑂𝑁 𝑡 = 𝐼𝑚𝑖𝑛 + 𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛
𝐷𝑇 𝑡 = 𝐼𝑚𝑖𝑛 + 𝑉𝑠 − 𝑉𝑅 𝐿 𝑡
From straight line equation
(1)
(2)
Step Down Chopper with RL Load
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The Buck (Step-Down) Converter
Continuous Current Operation When Chopper Is OFF ( t
ON≤t ≤ T )
Output voltage
Output current v0
V
i0 Imax
Imin
t
t tON
T tOFF
Continuous current
Output current
t Discontinuous
current i0
0 = 𝑉𝑅 + 𝑉𝐿 0 = 𝑉𝑅 + 𝐿 𝑑𝑖
𝑑𝑡 → 𝑑𝑖
𝑑𝑡 = −𝑉𝑅 𝐿
∆𝑖 = −𝑉𝑅 𝐿 𝑑𝑡
𝑡𝑂𝐹𝐹
0
= −𝑉𝑅
𝐿 𝑡𝑂𝐹𝐹 𝑑𝑖
𝑑𝑡 = ∆𝑖
𝑡𝑂𝐹𝐹 = 𝐼𝑚𝑖𝑛 − 𝐼𝑚𝑎𝑥
𝑡𝑂𝐹𝐹 = −𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛
𝑡𝑂𝐹𝐹 = −𝑉𝑅 𝐿 (3)
𝑖𝑜2 = 𝐼𝑚𝑎𝑥 + 𝐼𝑚𝑖𝑛 − 𝐼𝑚𝑎𝑥
𝑡𝑂𝐹𝐹 𝑡 − 𝑡𝑂𝑁 = 𝐼𝑚𝑎𝑥 − 𝑉𝑅
𝐿 (𝑡 − 𝑡𝑂𝑁) From straight line equation
(4)
Step Down Chopper with RL Load
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The Buck (Step-Down) Converter
𝑉𝑠 − 𝑉𝑅
𝐿 𝑡𝑂𝑁 − 𝑉𝑅
𝐿 𝑡𝑂𝐹𝐹 = 0 𝑉𝑠 − 𝑉𝑅
𝑉𝑅 = 𝑡𝑂𝐹𝐹 𝑡𝑂𝑁 𝑉𝑠
𝑉𝑅 − 1 = 𝑡𝑂𝐹𝐹
𝑡𝑂𝑁 𝑉𝑠
𝑉𝑅 = 𝑡𝑂𝐹𝐹
𝑡𝑂𝑁 + 1 𝑉𝑠
𝑉𝑅 = 𝑡𝑂𝐹𝐹 + 𝑡𝑂𝑁
𝑡𝑂𝑁 = 𝑇
𝑡𝑂𝑁 𝑉𝑅 = 𝐷𝑉𝑠
From equation (1)
∆𝑖 = 𝑉𝑠 − 𝐷𝑉𝑠
𝐿 𝐷𝑇 = 𝑉𝑠 1 − 𝐷 𝐷 𝐿𝑓
since
𝑓 = 1 𝑇 𝐷 = 𝑡𝑂𝑁
𝑇
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
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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 = 𝑉𝑅 𝑅
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
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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 = 𝑡𝑂𝑁
𝑇 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 = 𝑡𝑂𝑁
𝑇 = 0.8
Step Down Chopper with RL Load
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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 = 𝑡𝑂𝑁 𝑇 𝑉𝑠 𝑉𝑠 = 460V
Chopping period 𝑇 = 1
𝑓 = 1
2 × 10−3 = 0.5m 𝑠𝑒𝑐
𝑡𝑂𝑁 = 𝑇𝑉𝑑𝑐
𝑉𝑠 = 0.5 × 10−3 × 350
460 = 0.38m 𝑠𝑒𝑐
Step Down Chopper with RL Load
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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
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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 − 𝑉𝑜 𝐿
𝐷𝑇
0
𝑑𝑡 = 𝑉𝑠 − 𝑉𝑜 𝐿 𝐷𝑇
or
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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= −𝑉𝑜 𝐿
(1−𝐷)𝑇
0
𝑑𝑡 = −𝑉𝑜
𝐿 (1 − 𝐷)𝑇 or
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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
Vl
iD
iS
iL
vbe
Ton Toff Ts
Io
in o
V V Vo
Q
Vo
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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
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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
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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:
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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.
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