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Boost Switching Converter Design Equations
The boost converter is a high efficiency step-up DC/DC switching converter. The converter uses a transistor
switch, typically a MOSFET, to pulse width modulate the voltage into an inductor. Rectangular pulses of
voltage into an inductor result in a triangular current waveform. We'll derive the various equations for the
current and voltage for a boost converter and show the tradeoffs between ripple current and inductance. For
this discussion we assume that the converter is in the continuous mode, meaning that the inductor's current
never goes to zero.
First, here are some definitions:
Peak inductor current
ipk
Min inductor current
io
Ripple Current
Ripple Current Ratio to Average Current
Off Duty Cycle
Switch Off Time
Average and Load Current
RMS Current for a Triangular Wave
The relationship of voltage and current for an inductor is:
,or
For a constant rectangular pulse:
From this we can see that the current is a linear ramp, when the voltage is a constant pulse.
When the transistor switches on the current is:
, or
and when the transistor switches off the current is:
, or
Equation 1
Where VD is the voltage drop across the diode, and VTrans is the voltage drop across the transistor. Note
that the continuous/discontinuous boundary occurs when io is zero.
By equating through delta i, we can solve for Vout:
We can also solve for the duty cycle as follows,
If we neglect the voltage drops across the transistor and diode then:
So it is clear that the output voltage is related directly to the duty cycle of the pulses.
The main question when designing a converter is what sort of inductor should be used. In most designs the
input voltage, output voltage and load current are all dictated by the requirements of the design, whereas, the
Inductance and ripple current are the only free parameters. It can be seen form Equation 1, that the
inductance is inversely proportional to the ripple current. In other words, if you want to reduce the ripple,
then use a larger inductor. Thus, in practice a ripple current is decided upon which will give a reasonable
inductance.
There are tradeoffs with low and high ripple current. Large ripple current means that the peak current is ipk
greater, and the greater likelihood of saturation of the inductor, and more stress on the transistor.
So when choosing an inductor make sure that the saturation current of the inductor is greater than ipk.
Likewise, the transistor should be able to handle peak current greater than ipk. The inductor should also be
chosen such that the it can handle the appropriate rms current.
It should be noted that when there is a light load the circuit can slip into discontinuous mode, where the
inductor becomes fully discharged of it's current each cycle. When a load is reapplied the inductor needs to
recharge, and so the transistor's duty cycle increases pulling the inductor towards ground, and because of the
increased duty cycle Vout decreases when we really want it to increase. This causes an instability, which is
well known for boost converters, and not a problem with buck converters.
One way to combat this instability is to choose a large enough inductor so that the ripple current is greater
than twice the minimum load current. When this condition is met then the inductor is always in continuous
mode.
This can be expressed as follows:
For higher efficiency the diode should be an ultra fast recovery diode.
Buck Switching Converter Design Equations
The buck converter is a high efficiency step-down DC/DC switching converter. The converter uses a
transistor switch, typically a MOSFET, to pulse width modulate the voltage into an inductor. Rectangular
pulses of voltage into an inductor result in a triangular current waveform. We'll derive the various equations
for the current and voltage for a buck converter and show the tradeoffs between ripple current and
inductance. For this discussion we assume that the converter is in the continuous mode, meaning that the
inductor's current never goes to zero.
First, here are some definitions:
Peak inductor current
ipk
Min inductor current
io
Ripple Current
Ripple Current Ratio to Average Current
Duty Cycle
Switch On Time
Average and Load Current
RMS Current for a Triangular Wave
The relationship of voltage and current for an inductor is:
,or
For a constant rectangular pulse:
From this we can see that the current is a linear ramp, when the voltage is a constant pulse.
When the transistor switches on the current is:
, or
Equation 1
and when the transistor switches off the current is:
, or
Where VD is the voltage drop across the diode, and VTrans is the voltage drop across the transistor. Note
that the continuous/discontinuous boundary occurs when io is zero.
By equating through delta i, we can solve for Vout:
We can also solve for the duty cycle as follows,
If we neglect the voltage drops across the transistor and diode then:
So it is clear that the output voltage is related directly to the duty cycle of the pulses.
The main question when designing a converter is what sort of inductor should be used. In most designs the
input voltage, output voltage and load current are all dictated by the requirements of the design, whereas, the
Inductance and ripple current are the only free parameters. It can be seen form Equation 1, that the
inductance is inversely proportional to the ripple current. In other words, if you want to reduce the ripple,
then use a larger inductor. Thus, in practice a ripple current is decided upon which will give a reasonable
inductance.
There are tradeoffs with low and high ripple current. Large ripple current means that the peak current is ipk
greater, and the greater likelihood of saturation of the inductor, and more stress on the transistor.
So when choosing an inductor make sure that the saturation current of the inductor is greater than ipk.
Likewise, the transistor should be able to handle peak current greater than ipk. The inductor should also be
chosen such that the it can handle the appropriate rms current.
For higher efficiency the diode should be an ultra fast recovery diode.
Inverting Switching Converter Design Equations
The inverting converter is a high efficiency DC/DC switching converter. The converter uses a transistor
switch, typically a MOSFET, to pulse width modulate the voltage into an inductor. Rectangular pulses of
voltage into an inductor result in a triangular current waveform. We'll derive the various equations for the
current and voltage for a boost converter and show the tradeoffs between ripple current and inductance. For
this discussion we assume that the converter is in the continuous mode, meaning that the inductor's current
never goes to zero.
First, here are some definitions:
Peak inductor current
ipk
Min inductor current
io
Ripple Current
Ripple Current Ratio to Average Current
Off Duty Cycle
Switch Off Time
Average and Load Current
RMS Current for a Triangular Wave
The relationship of voltage and current for an inductor is:
,or
For a constant rectangular pulse:
From this we can see that the current is a linear ramp, when the voltage is a constant pulse.
When the transistor switches on the current is:
, or
and when the transistor switches off the current is:
, or
Equation 1
Where VD is the voltage drop across the diode, and VTrans is the voltage drop across the transistor. Note
that the continuous/discontinuous boundary occurs when io is zero.
By equating through delta i, we can solve for Vout:
We can also solve for the duty cycle as follows,
If we neglect the voltage drops across the transistor and diode then:
So it is clear that the output voltage is related directly to the duty cycle of the pulses.
The main question when designing a converter is what sort of inductor should be used. In most designs the
input voltage, output voltage and load current are all dictated by the requirements of the design, whereas, the
Inductance and ripple current are the only free parameters. It can be seen form Equation 1, that the
inductance is inversely proportional to the ripple current. In other words, if you want to reduce the ripple,
then use a larger inductor. Thus, in practice a ripple current is decided upon which will give a reasonable
inductance.
There are tradeoffs with low and high ripple current. Large ripple current means that the peak current is ipk
greater, and the greater likelihood of saturation of the inductor, and more stress on the transistor.
So when choosing an inductor make sure that the saturation current of the inductor is greater than ipk.
Likewise, the transistor should be able to handle peak current greater than ipk. The inductor should also be
chosen such that the it can handle the appropriate rms current.
This can be expressed as follows:
For higher efficiency the diode should be an ultra fast recovery diode.
Boost Power Factor Correction
Maximum Peak to Peak Ripple Current in the Boost PFC Inductor:
Because the switching frequency is usually more than three orders of magnitude higher than the line
frequency, for a single switching cycle the inductor can be considered as having a triangle AC switching
current with the instantaneous line current as a DC bias. After making this approximation, the following
equations can be developed:
In general, for an inductor the following equation will apply:
(15-1)
When the switch is ON, the current in the boost inductor will increase from Il to Ih:
(15-2)
Where Vi is the instantaneous input voltage.
When the switch is OFF, the current in the boost inductor will decrease from Ih to Il:
(15-3)
We will eliminate now D between equations (15-2) and (15-3):
(15-4)
To find at what instantaneous input voltage the peak to peak ripple current will have a maximum, we need to
analyze the function:
(15-5)
From the above equation, we see that:
a) If the instantaneous input voltage can reach half of the boost voltage, the peak to peak ripple current has a
maximum in this point. This maximum is:
(15-6)
The maximum instantaneous RMS value will be:
(15-7)
b) If the instantaneous input voltage cannot reach half of the boost voltage, the peak to peak ripple current
has a maximum when the instantaneous input line reach its peak.
In a normal design, 85-276Vac, or 176-276Vac, or 85-138Vac, the instantaneous input voltage would always
reach a value equal with half of the boost voltage. The only situation when the instantaneous input voltage
will not reach half of the boost voltage, is in a 85-136Vac design with 400Vdc boost voltage witch is unusual.
This particular case will not be discussed further.
RMS Ripple Current in the Boost PFC Inductor:
Re-write the formula (15-4) function of time:
The RMS value of this current will be:
Boost PFC Inductor Calculation:
We will try now to determine a formula for calculating the boost inductor having as input data: PFC output
power, PFC boost voltage, Minimum AC input voltage, PFC switching frequency and Maximum boost ripple
current as a percentage from the peak Input AC current:
(15-8)
Maximum Boost PFC Inductor Peak Current:
Analyzing the above equation, the conclusion will be that the inductor peak current will occur at the
minimum AC line voltage (V rms ), and when the instantaneous line voltage reaches its peak. Therefore, the
maximum peak current in the boost inductor will be:
Now, going back to the equations (15-2) and (15-3) and eliminating ripple current, we will have the
following equation for the Boost duty cycle:
(15-9)
MOSFET Switching: Part I - Turn-ON, Hard Switching by Constantin Darius
Livescu
This article is trying to make sense out of confusing information regarding the behavior of a MOSFET during
switching sequences, in numerous technical articles.
We are not attempting to explain the physics behind a MOSFET structure. For those interested to find more
about a MOSFET structure, we recommend the SGS-Thomson technical articles mentioned in the references.
The purpose of the article is to present a power supply design engineer with facts that will help design a
MOSFET driving circuit, calculate the estimated losses for critical events, predict the efficiency of a power
supply, estimate the junction temperature for critical components and various stresses, and ultimately,
helping make decision to optimize a design.
The MOSFET switching events are analyzed for an inductive load, diode clamping circuit, the only one that
applies to a switching power supply. The datasheet information or technical articles regarding resistive loads
have little or no relevance to switching a MOSFET in a switch mode power supply. Also the article is
considering only 500V/600V MOSFETs, most relevant and for switch mode power supplies.
Capacitors reference designators are the same as in SGS-Thomson articles. Check the references if a more
detailed explanation of their significance is necessary.
Below are the waveforms, mostly self-explanatory related with a MOSFET switching on (inductive load,
diode clamping, hard switching):
And now the comments:


Gate voltage has a sharp rise at t1 due mainly to the MOSFET source inductance.
Internal MOSFET channel will carry also the Coss discharge current.


At t3 the drain voltage will reach Vx, usually around 25V for 500V MOSFETS. After this time
Vds*Idrain loss should be considered part of the conduction loss.
Q3, gate charge associated with drain voltage reaching Vx, is much smaller then Q3+Q4, commonly
specified in a MOSFET datasheet.
Common errors and misconceptions:
Error: Drain voltage will decreased linearly to zero during t2 - t4 period, when gate voltage is constant (gate
plateau voltage).
Reality: Drain voltage will decrease much faster reaching Vx voltage, during t1 - t2 period. Calculating the
switching loss associated with this period of time, considering that the drain voltage will decrease linearly for
the entire "plateau" period, will results in huge errors.
Error: Drain voltage will reach zero at t2b (the end of the diode reverse recovery period).
Reality: Drain voltage will decrease much faster reaching Vx voltage, during t1 - t2 period. Calculating the
switching loss associated with this period of time, considering that the drain voltage will decrease linearly for
the entire "plateau" period, will results in huge errors.
Error: MOSFET capacitances cannot be used to determine switching behavior, you need gate charges
values.
Reality: MOSFET capacitances, if fully characterized, can fully explain (together with other parameters) the
switching behavior, without the need for gate charges.
Error: MOSFET datasheets give enough information to characterize the switching behavior of a MOSFET
in your application.
Reality: Most MOSFET datasheets, for reasons not discussed in this article, are not giving enough
information to be able determine the switching behavior in a typical application for which the part was
intended to be used. Time characteristics (turn-on delay time, rise time, turn-off delay time, fall time) are
measured in conditions that bear no resemblance with a typical application of the device, so, the best for a
design engineer would be to disregard them altogether. An attempt to use them to "guestimate" the real
useful data, it is time consuming and usually not done by engineers.
A notable exception: Motorola datasheets, similar with the one mention in the references for MTW20N50E,
are providing the information that really matters in determining the switching behavior. To the best of our
knowledge, current at the time this article was last updated, Motorola datasheets were the only ones
providing all following critically needed characteristics: Vx voltage, gate charge needed for drain voltage to
drop from Vdd to Vx, high voltage capacitance variation in a readable chart, low voltage capacitance
variation in a readable chart, Ciss and Crss variation at low voltage, zero Vgs and Vds, zero Vds and Vgs
zero to 10V, internal source inductance.
MOSFET Switching: Part II - Turn-ON, Soft Switching by Constantin Darius
Livescu
This article is trying to make sense out of confusing information regarding the behavior of a MOSFET during
switching sequences, in numerous technical articles.
We are not attempting to explain the physics behind a MOSFET structure. The purpose of the article is to
present a power supply design engineer with facts that will help design a MOSFET driving circuit, calculate
the estimated losses for critical events, predict the efficiency of a power supply, estimate the junction
temperature for critical components and various stresses, and ultimately, helping make decision to optimize a
design.
The MOSFET switching events are analyzed for an inductive load, diode clamping circuit, the only one that
applies to a switching power supply. The datasheet information or technical articles regarding resistive loads
have little or no relevance to switching a MOSFET in a switch mode power supply. Also the article is
considering only 500V/600V MOSFETs, most relevant and for switch mode power supplies.
Below are the waveforms, mostly self-explanatory related with a MOSFET switching on (inductive load,
diode clamping, soft switching):
And now the comments:




Gate voltage has a sharp rise at t1s due mainly to the MOSFET starting to discharge the Coss and
MOSFET source inductance .
At t2s the drain voltage will reach Vx, usually around 25V for 500V MOSFETS.
Q3, gate charge associated with drain voltage reaching Vx, is much smaller then Q3+Q4, commonly
specified in a MOSFET datasheet.
Notice the change of events order compared with hard switching! In soft switching circuits first Vds
will drop to a low level before the drain current will begin to rise, having as consequence a dramatic
decrease in switching losses!
MOSFET Switching: Part III - Turn-OFF, Hard Switching by Constantin Darius Livescu
This article is trying to make sense out of confusing information regarding the behavior of a MOSFET during
switching sequences, in numerous technical articles.
We are not attempting to explain the physics behind a MOSFET structure. For those interested to find more
about a MOSFET structure, we recommend the SGS-Thomson technical articles mentioned in the references.
The purpose of the article is to present a power supply design engineer with facts that will help design a
MOSFET driving circuit, calculate the estimated losses for critical events, predict the efficiency of a power
supply, estimate the junction temperature for critical components and various stresses, and ultimately,
helping make decision to optimize a design.
Below are the waveforms, mostly self-explanatory related with a MOSFET switching off (inductive load,
diode clamping, hard switching):
And now the comments:


During t3r to t2r period, the Coss will charge with a current depending of the dV/dt associated with
this period. This will decrease the actual current going through internal MOSFET channel, and
therefore reducing the switching loss associated with this time interval.
Q3, gate charge associated with drain voltage reaching Vx, is much smaller then Q3+Q4, commonly
specified in a MOSFET datasheet.
MOSFET Switching: Part IV - Turn-OFF, Soft Switching by Constantin Darius Livescu
This article is trying to make sense out of confusing information regarding the behavior of a MOSFET during
switching sequences, in numerous technical articles.
We are not attempting to explain the physics behind a MOSFET structure. For those interested to find more
about a MOSFET structure, we recommend the SGS-Thomson technical articles mentioned in the references.
The purpose of the article is to present a power supply design engineer with facts that will help design a
MOSFET driving circuit, calculate the estimated losses for critical events, predict the efficiency of a power
supply, estimate the junction temperature for critical components and various stresses, and ultimately,
helping make decision to optimize a design.
Below are the waveforms, mostly self-explanatory related with a MOSFET switching off (inductive load,
diode clamping, soft switching, typical of a power supply topology):
Other Considerations:

SMPS Power Supplies is using the above described correct theory regarding MOSFET switching to
accurately calculate the switching losses in PFC hard switching and soft switching topologies.
Combined with our accurate models for diodes (with voltage drop, reverse recovery time and reverse
recovery current being functions of operating temperature, forward current, dI/dt), the design
spreadsheets (ADH2450Des__.xls, ADH8100Des__.xls) are the most accurate design tools for
designing and predicting the performances of a switching power supply.
References:



Study of a Model for Power MOSFET Gate-Charge, SGS-Thomson Microelectronics, Power MOS
Devices Data book, 1st Edition - June 1988.
Gate Charge Characteristics Lead to Easy Drive Design for Power MOSFET Circuits, by M. Melito
and F. Portuese, SGS-Thomson Microelectronics, Designers' Guide to Power Products, Application
Manual, 2nd Edition - June 1993.
MTW20N50E, Designer's Data Sheet, Motorola - REV 4, 1996.