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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 6, JUNE 2010
An Insight into the Switching Process of Power
MOSFETs: An Improved Analytical Losses Model
Miguel Rodrı́guez, Student Member, IEEE, Alberto Rodrı́guez, Student Member, IEEE,
Pablo Fernández Miaja, Student Member, IEEE, Diego González Lamar, Member, IEEE,
and Javier Sebastián Zúniga, Member, IEEE
Abstract—The piecewise linear model has traditionally been
used to calculate switching losses in switching mode power supplies
due to its simplicity and good performance. However, the use of the
latest low voltage power MOSFET generations and the continuously increasing range of switching frequencies have made it necessary to review this model to account for the parasitic inductances
that it does not include. This paper presents a complete analytical
switching loss model for power MOSFETs in low voltage switching
converters that includes the most relevant parasitic elements. It
clarifies the switching process, providing information about how
these parasitics, especially the inductances, determine switching
losses and hence the final converter efficiency. The analysis presented in this paper yields two different types of possible switching
situations: capacitance-limited switching and inductance-limited
switching. This paper shows that, while the piecewise linear model
may be applied in the former, the proposed model is more accurate
for the latter. Carefully-obtained experimental results, described
in detail, support the analytical results presented.
Index Terms—Losses, modeling, MOSFETs, switched mode
power supplies.
I. INTRODUCTION
HE continuously increasing range of operating frequencies
in actual power converters has caused switching losses to
greatly determine the final performance of a design. On the
one hand, switching losses can be accurately calculated using physical simulation software or circuit simulations that use
complete MOSFET models [1]; unfortunately, both methods
are quite time consuming, and the information required to obtain precise results is not always available for many designers.
Besides, as each simulation only provides valid results for the
case under study, neither general conclusions nor a clear physical insight into the switching process are easily obtained from
these tools. On the other hand, analytical switching loss models
(usually based upon certain simplifications) provide less accurate results, but enable the designer to make fast calculations,
thus allowing almost immediate comparison between different
operating conditions or between different semiconductor performances. Furthermore, analytical models also provide physical
T
Manuscript received June 23, 2009; revised November 3, 2009. Current
version published June 4, 2010. This work has been supported by the
Spanish Ministry of Science and Innovation under Grants AP2006-04777,
AP2008-03380, and project TEC-2007-66917. Recommended for publication
by Associate Editor E. Santi.
The authors are with the Power Supply Systems Group, Department of Electrical and Electronic Engineering, University of Oviedo, Gijón 33204, Spain
(e-mail: rodriguezmiguel.uo@uniovi.es; rodriguezalberto@uniovi.es; fernandezpablo.uo@uniovi.es; gonzalezdiego@uniovi.es; sebas@uniovi.es).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPEL.2010.2040852
insight into the switching process, allowing designers to acquire
in-depth knowledge of the switching loss mechanisms.
The classical analytical model, or piecewise-linear model [2],
is based on the assumption that MOSFET’s capacitances completely determine its switching behavior. However, it does not
account for parasitic inductances. The latest generation of lowvoltage power MOSFETs exhibit an extremely good switching
performance, mainly due to their low capacitances and increased
transconductances; this key fact means that parasitic inductances
limit the switching process in practice. The classical model thus
needs a review to account for the effect of these parasitics (actual values for the parasitic inductances vary from around 15 nH
in a classic TO-220 package to less than 1 n in the DirecFET
package used by International Rectifier).
Several detailed models that include such parasitics have already been proposed, including either a fully analytical treatment [3], [4] or a more empirical approach [5]. This issue has
also been addressed in [6]– [8], but applied to a current source
resonant driver. This paper presents an analytical loss model
for power MOSFETs in actual low-medium voltage (< 40 V)
switching converters that includes all the parasitics. It is obtained by solving the equivalent circuits during the switching
transitions; all the approximations used are clearly stated, and
the solutions are presented in a relatively simple form. The
model is easy to understand and provides a deep insight into the
switching process. The paper is organized as follows: first, the
motivation and basis of the model are given. Then, the model
and the procedure employed to obtain all the waveforms are
fully explained. Finally, the proposed model is compared to the
classical model and tested by means of several experiments.
II. MOTIVATION AND BASIS OF THE MODEL
Fig. 1(a) shows a synchronous buck converter. During switching transitions, the output inductance L may be considered to be
a current source, and the output capacitor C may be considered
a constant voltage source. Such assumptions yield the circuit in
Fig. 1(b), from which the output voltage has been removed as it
is in series with a current source. As the dead time control used
in synchronous buck converters causes the freewheeling diode
Df either to be conducting before the high-side MOSFET, HS,
turns on or to start conducting after HS turns off, the low-side
MOSFET, LS, does not play any role during the switching transitions, and it can be removed from the circuit. Furthermore, the
said dead time forces LS to switch in zero voltage conditions,
thus adding no extra losses. Thus, only HS has to be considered
in the switching loss analysis. After taking into account such
considerations and adding the parasitic capacitances of HS, the
0885-8993/$26.00 © 2010 IEEE
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RODRÍGUEZ et al.: INSIGHT INTO THE SWITCHING PROCESS OF POWER MOSFETS: AN IMPROVED ANALYTICAL LOSSES MODEL
Fig. 1. (a) Synchronous buck converter; (b) equivalent circuit for the switching
transition; (c) intermediate circuit after the LS MOSFET has been removed and
the parasitic capacitances of HS have been added; (d) final equivalent circuit
obtained from a rearrangement of elements in (c).
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Fig. 2. (a) Typical turn-ON waveforms in the classical model; (b) equivalent
circuit with an extra parasitic drain inductance.
circuit in Fig. 1(c) is obtained. Finally, a rearrangement of the
elements in Fig. 1(c) yields the circuit in Fig. 1(d). The same
transformations can easily be applied to boost or buck–boost
converters, yielding exactly the same final equivalent circuit; the
only difference is that the values of voltages and currents change
(e.g., in the boost converter the voltage source value is Vout and
the current source value is Iin ). A synchronous buck converter
will be used in the following sections to present the model. The
major advantage of this approach arises from a practical issue: in
a conventional buck converter, conduction losses would mask
switching losses due to the high-conduction losses caused by
the freewheeling diode Df , thus hindering any laboratory measurement aimed at testing the proposed model. However, it is
worth to comment that the analysis is still directly applicable to
a conventional buck converter: the only special concern should
be that the model would become more difficult to verify due to
the aforementioned issues.
A. Range of Application of the Classical Model
The classical model has been widely used by power supply
designers. It provides quite accurate results, especially bearing
in mind its simplicity. It also provides a simple, comprehensive
approach to the switching process. However, careful analysis
of the model shows that it is based on the assumption that the
switching transitions are strongly determined by the parasitic
capacitances. Fig. 2(a) shows the typical turn-ON waveforms
in the piecewise-linear model. First, the drain current increases
linearly until reaching its final value Iout ; then, the drain-source
voltage drops to 0 V. The procedure is easy to understand: as
the freewheeling diode of Fig. 1(d) is initially forward biased,
it forces the drain-source voltage to remain constant until the
MOSFET carries all the output current. The turn-OFF process is
very similar, but the voltage increase takes place before the current drop. This switching behavior can be referred to as capacitance limited. Fig. 2(b) shows the equivalent circuit of Fig. 1(d)
with an extra inductance in the drain terminal of the MOSFET.
This inductance limits the drain current change rate, and thus
nothing stops the drain-source voltage from changing before the
Fig. 3. (a) Simplified physical view of a power MOSFET; (b) simplified 3-D
view of the silicon die with the package.
current. In this situation, the assumptions of the classical model
are no longer valid.
B. Basis of the Proposed Model
The proposed model includes the most relevant parasitics
present in actual power MOSFETs. Fig. 3(a) shows a simplified
physical view of a MOSFET. Three intrinsic parasitic capacitances are considered in this model: the gate-source capacitance,
the gate-drain capacitance, and the drain-source capacitance. As
both the drain-source capacitance and the gate-drain capacitance
are associated with a reverse-biased p-n junction, they change
with the applied voltage. This change is modeled by the following equation:
C0
(1)
C (v) = v γ
1+
K
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 6, JUNE 2010
Fig. 4. Equivalent MOSFET circuit including internal and major external
parasitics.
beginning of each substage, the equivalent circuit with its corresponding initial conditions is stated. Capacitance values are
chosen using (1) in accordance with the initial voltage over the
capacitance at the beginning of the substage; though this procedure might yield discontinuities at the substage boundaries, it
will provide more accurate results when calculating switching
losses and the obtained waveforms will also be more meaningful
from a physical point of view. The time-domain equations of the
equivalent circuit are then obtained and solved using Laplace
transforms. Once the drain current and the internal drain-source
voltage are known, switching losses are calculated in terms of
energy by integrating the product of those waveforms, and then
the energy is multiplied by the switching frequency fsw . In certain substages, it is possible to obtain equations that allow the
energy losses to be directly found (without an integration procedure). Tables I and II summarize all the relevant equations.
As the final conditions of one substage are used as the initial
conditions for the following, and to simplify the use of Laplace
transforms, the state variables are chosen such that their initial
condition is zero in each substage. Mathematically, x being
a state variable, then x (t = 0) = 0. The initial conditions are
included in the circuits and in the corresponding equations as
independent voltage sources (for the capacitances) or current
sources (for the inductances), but are not drawn in most of the
equivalent circuits for the sake of clarity. The actual voltages
and currents in the circuits (electrical variables) are noted with
an apostrophe, while the state variables are noted without the
apostrophe. It is straightforward to find one from the other
Fig. 5. Final equivalent circuit used to obtain the proposed model, along with
the electrical variables.
x (t) = x (t) + x (0) .
where K and γ are two adjustment parameters extracted from
the capacitance versus voltage curve that can be found in the
datasheet of the MOSFET, and C0 is the 0 V capacitance value.
The gate-source capacitance remains approximately constant
regardless of the applied voltage.
The model also includes several parasitic inductances that
can appear for different reasons. Fig. 3(b) shows the silicon die
together with the MOSFET package. The pad parasitic inductances and the internal bondings between the silicon die and the
package pads contribute to the total parasitic inductance. Current
packaging technology tries to minimize such inductances by directly connecting the die with almost noninductive pads, achieving inductances as low as 0.1 nH [9]. The layout of the printed
circuit board also contributes slightly to the inductance.
Fig. 4 shows the equivalent circuit resulting from the previous considerations. The final circuit model used to analyze the
switching process is shown in Fig. 5. All the parasitic inductances in series in each terminal of the MOSFET are added and
thus represented as one single element. The gate circuit resistance Rg is made up of the gate driver equivalent resistance and
an internal resistance due to the gate contact. The internal nodes
are Gint , Dint , and Sint , while the final external nodes, which
include the effect of all the parasitic inductances are G, D, and S.
The procedure to obtain the model equations is as follows:
first, the theoretical analysis is divided into two different stages,
the turn-ON and turn-OFF transitions. Within each transition,
different substages arise; these are treated separately. At the
(2)
Furthermore, the initial time is chosen as zero at the beginning
of each substage.
The ultimate goal of this paper is to obtain relatively simple and handy analytical expressions of voltage and current
waveforms that enable the designer to easily calculate switching losses using a conventional spreadsheet. A simple model
for MOSFET behavior must be used to achieve such objective.
Thus, the MOSFET is considered to be a resistance, an open
circuit or a dependent current source in all the analytical calculations presented in this paper. The behavior as a dependent
current source (controlled by the voltage between Sint and Gint )
is described by (3).
(3)
ichannel = gm vgs
− VTH
gm and VTH being the MOSFET transconductance and the
threshold voltage, respectively. Obviously, modeling MOSFET
behavior using (3) provides results with limited accuracy. However, its use provides a physical insight into the switching process that would be very difficult to obtain if more complicated
expressions were used.
III. TURN-ON TRANSITION
The equivalent circuit at the beginning of the turn-ON transition is shown in Fig. 6. As the freewheeling diode is forward
biased, the voltage Vin is applied over the MOSFET equivalent circuit and its parasitics. A detailed analysis of this circuit
leads to the four different substages described in the following
sections.
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RODRÍGUEZ et al.: INSIGHT INTO THE SWITCHING PROCESS OF POWER MOSFETS: AN IMPROVED ANALYTICAL LOSSES MODEL
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TABLE I
SUMMARY OF LOSS CALCULATIONS DURING TURN-ON
A. Substage I
During substage I, the gate-source equivalent capacitance is
charged until it reaches VTH . As the MOSFET is open circuited,
approximately no drain current flows in the circuit and the drainsource voltage remains constant. Thus, only the gate circuit must
be considered. The equivalent circuit is shown in Fig. 7. The
initial conditions for the electrical variables are:
ig (0) = 0
(10)
vgs
(0) = 0 .
(11)
From the circuit in Fig. 7, the following equations are
obtained:
dig
Ls Ld
+ ig Rg + vgs
(12)
Vg = Lg +
Ls + Ld dt
ig = (Cgs + Cgd )
dvgs
.
dt
(13)
Applying Laplace transforms to (12) and (13) and solving for
vg s yields
vgs (s) =
Vg
1
2
s Leq Ceq s + Rg Ceq s + 1
Leq and Ceq now being
Ls Ld
Leq = Lg +
Ls + Ld
(14)
Ceq = (Cgs + Cgd ) .
The inverse Laplace transform of (14) is shown in Table I as
(4). The current ig can be obtained from (12) and (13), following
the same procedure. This current, as shown in Table I as (5), is
used to calculate the initial current for the following substage.
As the MOSFET is not activated, there is no power loss during
this period, except for the losses in the gate drive circuit that are
included as a separate term in the final loss calculations.
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 6, JUNE 2010
TABLE II
SUMMARY OF LOSS CALCULATIONS DURING TURN-OFF
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RODRÍGUEZ et al.: INSIGHT INTO THE SWITCHING PROCESS OF POWER MOSFETS: AN IMPROVED ANALYTICAL LOSSES MODEL
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A total of seven differential equations can be obtained from
the circuit in Fig. 8, one for each state variable in the circuit
(yielding six equations) and (3) being the seventh. This set of
differential equations can be reduced to four by eliminating three
state variables: the current through the source inductance is , the
channel current ichannel , and the voltage across the gate-drain
capacitance vdg . The remaining set of equations is
Vg − Vth = (ig + Ig 1 ) Rg
did
dig
dig
+ vgs + Ls
+
dt
dt
dt
did
did
dig
+ Ls
+
0 = vds + Ld
dt
dt
dt
dvds
dvgs
dvgs
= ig + Ig 1 + Cgd
−
Cg s
dt
dt
dt
dvds
dvgs
dvds
−
+ gm vgs .
id = Cgd
+ Cds
dt
dt
dt
+ Lg
Fig. 6.
Equivalent circuit at the beginning of the turn-ON transition.
Fig. 7.
Equivalent circuit during substage I.
Fig. 8.
Equivalent circuit during substage II.
(29)
(30)
(31)
(32)
As this set of equations is, in general, coupled, a simple analytical solution cannot be easily found. However, considering
the typical values of the capacitances and transconductances in
modern low-voltage power MOSFETs, it can be concluded that
the channel current will increase very fast, rapidly discharging the drain-source capacitance. The parasitic inductances stop
the drain current from increasing, thus allowing Cds to be discharged by the channel current. The drain current id and its
derivatives can be neglected.
B. Substage II
When vgs reaches VTH , the MOSFET channel starts conducting. Equation (3) then applies, and the channel current is directly
proportional to Vgs − VTH . Fig. 8 shows the equivalent circuit
for this substage.
The initial conditions are:
with Ig
1
(0) = VTH
vgs
(24)
vgd
(0) = Vin − VTH
(25)
vds
(0) = Vin
(26)
id
(0) = 0
(27)
ig
is
(0) =
(0) = Ig
1
being the initial current in the gate circuit.
(28)
Equation (34) is no longer needed, and the other three equations are uncoupled. Thus, the remaining set of equations can
be solved to find voltages and currents in a simple form. Using
Laplace transforms, vgs (s) is obtained as
vgs (s) =
·
1
s
Vg − Vth + Ig 1 Leq s
s2 (Leq Ceq ) + s (Rg Ceq + (M − 1) (Leq /Rg )) + M
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(37)
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 6, JUNE 2010
Fig. 9.
Equivalent circuit during substage III.
with M , Leq and Ceq now being
M = 1 + Rg gm
Cgd
Cgd + Cds
Leq = Lg + Ls
Ceq = Cgs +
Cgd Cds
.
Cgd + Cds
The inverse Laplace transform of (37) is shown in Table I as
(6). The drain-source voltage state variable can also be obtained
by applying Laplace transforms to vds
gm
vds (s) = 2
s (Cgd + Cds )
·
(Vg − Vth + Ig 1 Leq s) (s(Cgd /gm ) − 1)
. (38)
s2 (Leq Ceq ) + s (Rg Ceq + (M − 1) (Leq /Rg )) + M
The inverse Laplace transform of (38) is shown in Table I as
(7). The power loss during this substage has to be found by integrating the product ichannel vds . The end of this substage takes
place when the drain-source voltage reaches approximately 0 V.
If the input voltage were higher than a few tens of volts, Vds
would not reach 0 V, but would tend to reach an intermediate
voltage. As the present analysis is limited to input voltages
below approximately 40 V, this situation is not considered and
Vds will drop to 0 V in most practical situations.
C. Substage III
After the drain-source voltage reaches approximately 0 V,
the MOSFET starts behaving as a resistor. Fig. 9 shows the
equivalent circuit. The initial condition is
id (0) ≈ 0 .
(39)
Thus, the drain current is approximately determined by the
input voltage source and the parasitic inductances
id ≈
Vin
t.
Ld + Ls
(40)
Once again, the state variable id and the actual drain current id
are the same. As this substage lasts for a short period of time and
Rds, ON is usually very low, switching losses can be neglected.
This substage finishes when the drain current reaches the output
current.
D. Substage IV
This substage, which is usually called the reverse recovery
period, has been extensively analyzed in [3]. When the drain
current reaches the output current, the reverse recovery period
Fig. 10. Equivalent circuits during substage IV. The initial current through the
inductances has been drawn as an independent current source: (a) equivalent
circuit at the beginning of substage IV; (b) equivalent circuit after an appropriate
rearrangement of elements in (a); (c) equivalent circuit when the freewheeling
diode becomes capable of blocking voltage, i.e., when the drain current has
reached its maximum value Ip e a k ; (d) equivalent circuit after an appropriate
rearrangement of elements in (c).
begins. Fig. 10(a) shows the equivalent circuit at the beginning
of this period. The freewheeling diode is unable to block reverse
voltage until a certain amount of charge has been removed from
its junction. Thus, the drain current increases until it reaches a
certain peak current Ip eak ; at this moment, the charge has been
removed from the diode junction and it becomes capable of
blocking reverse voltage. Fig. 10(c) shows the equivalent circuit
at this moment, CD being the equivalent freewheeling diode
capacitance. A resonant period then begins, at the end of which
the capacitance is charged to Vin and the current through the
inductance is Iout once again.
Fig. 11 shows the definition of the reverse recovery charge Qrr
and the reverse recovery time trr , under common test conditions.
The reverse recovery charge is the total charge that has to be
provided to the diode junction until the voltage across it reaches
its steady-state value and the current through the diode drops to
0 A. The charge provided to the equivalent diode capacitance
during the resonant period is hence included in the term Qrr .
This is the major difference between the present analysis and
previously presented results.
The energy stored in the circuit at the beginning of substage
IV is
1
2
(41)
EL = (Ls + Ld ) Iout
2
EC D = 0 .
(42)
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RODRÍGUEZ et al.: INSIGHT INTO THE SWITCHING PROCESS OF POWER MOSFETS: AN IMPROVED ANALYTICAL LOSSES MODEL
1633
As CD depends on the applied voltage, (47) can be rewritten as
1
Eloss = Vin Qrr − QD Vin
2
(48)
with QD being the total charge supplied to the diode junction;
this charge can be calculated integrating the capacitance versus
voltage graph.
Noteworthily, significant reverse recovery parameters are
very difficult to calculate; for instance, Qrr strongly depends
on Iout and on the rate of change of the current, see, e.g., [10].
If the freewheeling diode is a Schottky diode, which suffers
almost no reverse recovery effects, then (47) becomes
Qrr
Eloss
= Vin (Vin CSchottky )
1
1
− CSchottky Vin2 = CSchottky Vin2
2
2
(49)
with CSchottky being the Schottky diode equivalent capacitance.
Fig. 11. (a) Test circuit for reverse recovery charge and reverse recovery time.
At time t0 , when the forward current through the diode is IF , switch S 1 opens
and switch S 2 closes; (b) current through the diode.
When the current reaches Ip eak , the energy stored in the
inductance is
EL ,p eak =
1
(Ls + Ld ) Ip2 eak .
2
(43)
The energy provided by the voltage source during said period
is
EV i n = Vin Qrr,1
(44)
with Qrr,1 being the part of Qrr that is removed from the diode
before the current reaches Ip eak . Thus, the energy loss is
Eloss,1 = Vin Qrr,1 −
1
2
(Ls + Ld ) Ip2 eak − Iout
.
2
(45)
During the resonant period, the energy stored in the induc2
tance changes from 12 (Ls + Ld )Ip2 eak to 12 (Ls + Ld )Iout
, so the
net energy provided to the inductance during the whole substage
is zero. The energy loss during the resonant period is
Eloss,2
1
= Vin Qrr,2 − CD Vin2
2
1
+ (Ls + Ld ) Ip2 eak − I02
2
(46)
with Qrr,2 being the part of Qrr that is needed to charge the
equivalent parasitic diode capacitance once it is able to block
reverse voltage. The second term in (46) stands for the final
energy stored in the diode parasitic capacitance. The total energy
loss is thus given by (47)
Eloss = Eloss,1 + Eloss,2 = Vin (Qrr,1 + Qrr,2 )
1
1
− CD Vin2 = Vin Qrr − CD Vin2 .
2
2
(47)
E. Example Waveforms
Fig. 12 shows a set of example waveforms provided by the
model, using typical MOSFET parameters. Fig. 12(a) shows
the drain to source voltage and the channel current, as well
as the instantaneous power in the channel, during the first three
substages (the reverse recovery substage has not been included).
It can be seen that, as long as the reverse recovery process is
not taken into account, significant power loss only takes place
during substage II, which lasts for a relatively short period. Thus,
the model suggests that, when an inductance-limited switching
takes place, there are small turn-ON losses. Furthermore, as
stated in [11], there is no need to include the rather arbitrary
output capacitance loss term, 12 Cds Vin2 . Fig. 12(b) shows the
gate to source voltage; it can be seen that the Miller effect takes
place during substage II, and that it is predicted by the analytical
model. The gate to source voltage is essentially equal to that of
the classical model and to the waveforms found in conventional
MOSFET datasheet.
IV. TURN-OFF TRANSITION
The equivalent circuit at the beginning of the turn-OFF transition is shown in Fig. 13. As the freewheeling diode is reverse
biased, the current Iout circulates through the MOSFET and the
parasitic inductances. A detailed analysis of this circuit leads to
the five different substages described in the following sections.
A. Substage I
The gate circuit discharges the gate-source equivalent capacitance. Fig. 14 shows the equivalent circuit during this period.
The initial conditions are
(0) = Vg
vgs
(50)
ig (0) = 0 .
(51)
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 6, JUNE 2010
Fig. 12. (a) Turn-ON example waveform (the reverse recovery period is not
included). The sudden change in the channel current is a consequence of the
approximations that were made during the analysis; (b) gate to source voltage
example waveform. The Miller effect takes place during substage II and causes
the actual channel current waveform.
Fig. 14.
Equivalent circuit during substage I.
Fig. 15.
Equivalent circuit during substage II.
The inverse Laplace transform of (54) is shown in Table II
as (15). The current ig can be obtained from (52) and (53), following the same procedure. This current, as shown in Table II
as (16), is used to calculate the initial current for the following substage. Even though the MOSFET is carrying the output
current, this period lasts for a short time and the losses can be
neglected.
Substage I finishes at a time t1 , when the MOSFET enters into
the linear region, where (3) applies. Thus, t1 can be calculated
as
vgs (t1 ) = −Vg + VTH +
Iout
.
gm
(55)
B. Substage II
Fig. 13.
Equivalent circuit at the beginning of the turn-OFF transition.
From the circuit in Fig. 14, the following equations can be
easily obtained:
dig
+ ig Rg + vgs
−Vg = (Lg + Ls )
dt
dvgs
.
ig = (Cgs + Cgd )
dt
Fig. 15 shows the equivalent circuit during this period. The
channel current suddenly decreases, thus causing the excess
current Iout − ichannel charging the MOSFET capacitances. The
initial conditions are
(0) = VTH +
vgs
(52)
Iout
vdg
(0) = − VTH +
gm
(53)
(0) ≈ 0
vds
Using Laplace transforms, vgs is found
Vg
1
vgs (s) = −
s Leq Ceq s2 + Rg Ceq s + 1
ig (0) = −Ig
(54)
id
Ceq = Cgs + Cgd .
with Ig
2
(56)
(57)
(58)
(59)
2
(0) = Iout
is (0) = Iout − Ig
Leq and Ceq now being
Leq = Lg + Ls
Iout
gm
(60)
2
being the initial current in the gate circuit.
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(61)
RODRÍGUEZ et al.: INSIGHT INTO THE SWITCHING PROCESS OF POWER MOSFETS: AN IMPROVED ANALYTICAL LOSSES MODEL
1635
The circuit equations are the following:
dig
0 = (ig + Ig 2 ) Rg + Lg
dt
diL s
Iout
− vgs − VTH +
+ Ls
gm
dt
vds = vgs + vgd
ig + Ig
2
(62)
(63)
+ Cgs
dvgs
dvgd
= Cgd
dt
dt
dvds
dvgd
= ichannel + Cgd
dt
dt
ig + is = 0 .
Iout − Cds
(64)
(65)
(66)
These equations can be solved directly using Laplace transforms
to find vgs
vgs (s) = −
·
1
s
sLeq Ig 2 + (Vth + (Iout /gm ))
Leq Ceq s2 + (Rg Ceq + (M − 1) (Leq /Rg )) s + M
(67)
M = 1 + Rg gm
Ceq
t→∞
Cds Cgd
= Cgs +
Cds + Cgd
s→0
C. Substage III
has reached Vin and the
At the beginning of this substage, vds
freewheeling diode becomes forward biased. Fig. 16(a) shows
the equivalent circuit during this period. The initial conditions
are
(0) = Iout
(68)
is (0) ≈ Iout
(69)
vds
(0) = Vin .
Iout
gm
Iout (M − 1) − VTH gm
.
(72)
M
In this substage, the initial conditions have been represented
by independent voltage and current sources in the circuit in
Fig. 16(a); instead of solving the circuit equations, a rearrangement of these sources easily yields the circuit in Fig. 16(c).
Fig. 16(c) shows that the final equivalent circuit is a simple
parallel resonant circuit. As the current and voltage waveforms
in this circuit are oscillating, it seems clear that the proposed
approximation is valid as long as the substage lasts for a time
shorter than half a resonant period. vds is easily found from
Fig. 16(c)
=
Applying the inverse Laplace transform to (67) yields (17)
in Table II. The results are very similar to those obtained in
Section III-B. The drain-source voltage can be obtained from
(62)–(66) using the same procedure. The result is shown in
Table II as (18). This period ends when the drain-source voltage
reaches Vin . The channel current could reach 0 A before that
moment; such a situation will be addressed in Section IV-F.
ichannel (0) = Ich,∞
t→∞
= lim svgs + VTH +
Cgd
.
Cgd + Cds
id
using (67) and the final value theorem
Ich,∞ = lim ichannel = lim gm Vgs (t)
with M , Leq , and Ceq being the same as in Section III-B
Leq = Ls + Lg
Fig. 16. (a) Equivalent circuit during substage III; (b) intermediate circuit
obtained by changing the position of the current sources; (c) final equivalent
circuit during substage III.
(70)
(71)
(t) = Vin + (Iout − Ich,∞ ) Ld ω0 sin (ω0 t)
vds
with ω0 being
ω0 = 1
(Ls + Ld ) Cds
.
(74)
This substage ends when the drain current reaches the channel current Ich,∞ . At this moment, the drain-source voltage has
reached its peak voltage Vds,p eak . This value can easily be calculated once t3 is known.
(t3 ) .
Vds, p eak = vds
As the drain inductance impedes a sudden change in the drain
current, the drain-source voltage continues increasing. It is assumed that the gate voltage remains constant during this period,
and as does the channel current. The latter can be approximated
(73)
(75)
D. Substage IV
During this period, the drain current continues decreasing
toward zero, forced by the voltage difference that exists between
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 6, JUNE 2010
Fig. 17. (a) Equivalent circuit during substage IV; (b) equivalent circuit for
the resonant turn-OFF stage (substage V).
Vds, p eak and Vin . As the current that charged Cds to Vds, p eak
was the difference between the drain and the channel current,
there will not be a high amount of current to change Vds during
this period. Thus, Vds is considered to remain approximately
equal to Vds, p eak during this substage. In order to simplify the
analysis, the channel current is assumed to decrease as if it
were controlled by the voltage source Vin , instead of by the
gate-source voltage. The power loss can thus be obtained as
Ploss = Vds ichannel dt ≈ Vds, p eak ichannel dt .
(76)
Fig. 17(a) shows the equivalent circuit. Thus, the channel
current can be approximated as
Vds, p eak − Vin
t.
ichannel (t) = Ich,∞ −
Ls + Ld
(77)
E. Substage V
A resonant stage begins when the channel current reaches
0 A; the MOSFET stops conducting current and becomes an
open circuit. The gate circuit continues discharging Cgs and
Cgd more or less independently from the rest of the circuit.
At the same time, the resonant circuit formed by Ls + Ld and
Cds starts oscillating until it reaches its steady-state value. The
equivalent circuit is shown in Fig. 17(b). R stands for the equivalent resistance that dumps the oscillations, which is mainly
made up of the circuit tracks high-frequency resistance and the
parasitic inductance and capacitance high-frequency equivalent
resistances.
The loss calculations during this substage are identical to
those in Section III-D. Once again, losses can be calculated by
considering the energy stored at the beginning and the end of
the substage. The energy stored at the beginning is
EL = 0
EC ,i =
(78)
1
Cds (Vds, p eak )2 .
2
(79)
The energy stored at the end of the period is
EL = 0
EC ,f =
1
Cds Vin2 .
2
(80)
(81)
Fig. 18.
Turn-OFF example waveform.
The energy provided by the voltage source is
EV in = Vin Cds (Vin − Vds, p eak )
(82)
an equation that shows that some energy has been returned to
the input source. Finally, the energy loss is
Eloss = EV in − (EC ,f − EC ,i )
1
= Cds Vp2eak + Vin2 − Cds Vp eak Vin .
(83)
2
The latter expression is the same as the one obtained in [3].
F. Substage III: Drain Current Drops to Zero Before vds
reaches Vin
As previously stated in Section IV-B, the channel current
could drop to 0 A during substage II, thus causing a sudden end
of said substage. This is likely to happen if Iout is relatively
small. In that case, the MOSFET becomes open circuited and
the analysis is straightforward; switching losses during the rest
of the turn-OFF transition can be approximated by considering
that all the energy stored in the parasitic inductances is lost
during the resonant period
1
2
(Ls + Ld ) Iout
.
(84)
2
Equation (84) neglects the voltage difference between Vds and
Vin at the moment when this substage begins.
However, as the switching losses given by (84) are noticeably
smaller than in the normal switching transition, this case is of
minor relevance.
Eloss ≈
G. Example Waveforms
Fig. 18 shows a set of example waveforms provided by the
model, using typical MOSFET parameters. As the gate to source
voltage provided by the model is again essentially equal to what
can be found in a conventional MOSFET datasheet, it has not
been included in this section. It can be seen in Fig. 18 that
a noticeable power loss occurs during the turn-OFF transition,
especially in comparison with the turn-ON. Thus, the model
suggests that turn-OFF switching losses are much higher than
turn-ON switching losses.
This fact can be easily explained taking into consideration
the effect of the parasitic inductances: these limit the current
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RODRÍGUEZ et al.: INSIGHT INTO THE SWITCHING PROCESS OF POWER MOSFETS: AN IMPROVED ANALYTICAL LOSSES MODEL
Fig. 20.
Fig. 19. Comparison between the losses provided by the proposed model and
the classical piecewise–linear model [2].
change rate during turn-ON, thus inducing an almost zerocurrent-switching transition. Conversely, during the turn-OFF
transition, the inductances lengthen the duration of the process,
thereby causing an increase in the losses.
1637
LM3152 demo board.
and its effect should be taken into account. Both conditions are
stated in (86) and (87):
Iout <<
tturn−on Vin
⇒ capacitance limited
2Ld
(86)
Iout >>
tturn−on Vin
⇒ inductance limited.
2Ld
(87)
A comprehensive differentiation of these two cases can also
be found in [12].
V. COMPARISON WITH THE CLASSICAL PIECEWISE-LINEAR
MODEL
VI. EXPERIMENTAL RESULTS
Fig. 19 shows a comparison between the results provided by
the classical model (extracted from [2]) and the proposed model.
It is apparent that the classical model yields higher losses during
turn-ON than the proposed model, whereas during turn-OFF the
opposite happens and the classical model yields smaller losses
than the proposed model.
However, there is a “compensation effect” between these two
errors that leads to the final losses not being very different from
one model to another. This coincidence means that the classical
model is still very useful, but only when the switching process is
not inductance limited: in that case it yields inaccurate results,
as is shown in Section VI. The proposed model clearly indicates
that turn-OFF losses are much higher than turn-ON losses in the
inductance-limited case.
From a designer’s point of view, it is interesting to establish an
approximate boundary between the region where the classical
model can be reliably applied (capacitance-limited switching)
and the region when the proposed model should be used because the effect of parasitic inductances cannot be neglected
(inductance-limited switching). The classical model states that
the turn-ON time is
Qswitching Rg
tturn−on =
(85)
Vth + (Iout /gm )
To experimentally test the proposed model, several laboratory measurements were conducted. First, the efficiency of a
synchronous buck converter was measured and compared with
the data provided by the proposed model. A National Semiconductor LM3152 demo board switching at 500 kHz (see Fig. 20)
was used. As not enough reverse recovery data are given in
the datasheet of the MOSFETs (RJK0305), a Schottky diode
was connected in parallel with the low-side MOSFET. Thus,
(49) can be used to estimate the losses during substage IV of the
turn-ON transition. Conduction losses of both MOSFETs and the
diode were fully taken into account and calculated using conventional equations (e.g., in the case of one of the MOSFETs,
the expression i2rm s Rds,ON was used, irm s being the root mean
square current through the MOSFET); power losses in input and
output capacitances and in the control circuitry were also considered, as well as the increase of the MOSFETs on-resistance
with operating temperature.
The results are shown in Fig. 21(a) and (b). It can be seen that
the model provides accurate results and that the trend followed
by experimental data is clearly reproduced by the model. Switching losses account for approximately 25% of the total amount
of losses in this experiment. There is a certain amount of uncertainty in the efficiency results provided by the model mainly
caused by the uncertainty in the current ripple of the demo board
output inductance and by the reverse recovery losses that may
still be existing in the circuit.
A second experiment was carried out to minimize these uncertainties. A synchronous buck converter switching at 1.3 MHz
was built using RJK0305 MOSFETs and an LM2727 controller.
Several changes were made with respect to the previously described experiment. First, the switching frequency was increased
to give more weight to switching losses against other loss terms.
Second, a very small inductor (≈ 300 nH) was used to obtain a
with Qswitching being the total switching charge that has to
be transferred into the gate-drain capacitance during the Miller
effect period, the value of which is usually given in the datasheet.
Thus, the current increase through Ld during the estimated turnON time can be compared with the output current Iout ; if the
latter is much smaller than the former, Ld will not be able to limit
the current rate of change and the switching will be capacitance
limited. On the contrary, if Iout is much bigger than the current
increase through Ld , then Ld will limit the current rate of change
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 6, JUNE 2010
Fig. 22. Prototype with the small coreless inductance and different positions
of decoupling capacitors in the layout used to achieve the extra drain inductance.
Fig. 21. (a) Experimental and model results (V in = 6 V); (b) experimental
and model results (V in = 12 V).
very high current ripple. The effect of reverse recovery losses
was thus diminished, because the inductor current at the turnON instant was much smaller than the current at the turn-OFF.
Moreover, the high current ripple allowed us to have less average output current while maintaining a high turn-OFF current,
thus achieving lower conduction losses and giving more weight
to switching losses. As core losses were also another source of
uncertainty, a coreless inductor was used. The output voltage
ripple was kept small by incrementing the output capacitance
(≈ 150 µF) and by using several low-equivalent series resistance capacitors. Third, the current through the inductor was
measured using a current transformer built with a small toroid.
Fig. 22 shows the prototype. Switching losses were more than
30% of total losses with this setup.
Two different sets of measurements were made: first, the efficiency was measured and compared to the results provided by
the classical model and the proposed model; second, a small
drain inductance was added to the circuit and the efficiency was
measured again. Fig. 22 shows how the inductance was achieved
in practice: the decoupling capacitors were separated about
1 cm from their previous position. The current loop that causes
the inductance was thus made longer and the inductance was
increased. The layout was measured by a precision impedance
analyzer before and after the change. The measurements yielded
an extra inductance of approximately 10–15 nH.
Fig. 23. (a) Comparison between the experimental data and the results provided by the models with decoupling capacitors in position 1 (see Fig. 22);
(b) comparison between the experimental data and the results provided by the
models with decoupling capacitors in position 2.
Fig. 23 shows a comparison between the experimental results
obtained and the results provided by the classical and the proposed model. Fig. 23(a) demonstrates that both the classical and
the proposed model can approximately reproduce the measured
efficiency when decoupling capacitances are in position 1; i.e.,
when the parasitic inductance is minimized. An error of nearly
one point can be appreciated in the results: as the exact values of
most of the parameters are not precisely known, it is very difficult to achieve more accuracy using datasheet values, regardless
of the model being used.
Fig. 23(b) shows the results when the decoupling capacitors
are in position 2. It is apparent that, as the classical model does
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RODRÍGUEZ et al.: INSIGHT INTO THE SWITCHING PROCESS OF POWER MOSFETS: AN IMPROVED ANALYTICAL LOSSES MODEL
Fig. 24.
Fig. 25.
1639
Breakdown of total losses provided by the model.
Detailed breakdown of total losses (Io u t, average = 7 A).
not take into account any inductance, its results were noticeably different from the measured data. In contrast, the proposed
model was able to reproduce the experimental efficiency.
Fig. 24 shows the breakdown of losses provided by the proposed and the classical model. These values were used to obtain
the results, as shown in Fig. 23. The influence of each contribution can be clearly appreciated: as stated, the contribution
of switching losses to the total losses was above 30%. Fig. 25
shows the detailed contribution of each loss term when the average output current was 7 A. Yet again, the contribution of
switching losses to the total losses can be clearly noticed, along
with the higher contribution of the turn-OFF term in all operating
conditions. Furthermore, the extra drain inductance (the capacitance in position 2) only increased the turn-OFF losses, while
the rest of the terms remained constant.
VII. CONCLUSION
This paper has presented a new detailed switching model for
power MOSFETs, which allows switching losses to be calculated. It has been shown that although the classical model is
very useful and provides relatively accurate results, it may not
precise enough in actual synchronous buck converter designs
if their switching process is inductance limited. The proposed
model takes into account the major circuit parasitics, especially
inductances, providing analytical solutions for the waveforms
and enabling the designer to create a simple spreadsheet to
estimate switching losses. All the equations required for loss
calculations are summarized in Tables I and II.
Furthermore, the model provides a clear physical overview
of the switching process and shows the effect that the different
parasitics of the transistor and the circuit have on this process.
For instance, it shows that the turn-ON and the turn-OFF transitions are quite different in nature. This paper has shown, via
carefully planned experimental measurements, that the model
provides quite accurate results. It has also demonstrated that the
classical model, though very useful and accurate, has certain application limits; the proposed model is intended to provide better
results than the classical model in such situations. The conditions in which each model should be used have also been stated,
clearly differentiating between capacitance-limited switching
and inductance-limited switching.
The experimental results have also shown that it is very difficult to accurately predict the efficiency, regardless of the model
being used, mainly due to the uncertainty that exists when determining the value of most of the parameters. For instance,
both the proposed and the classical model are very sensitive to
the MOSFET threshold voltage and the transconductance. As
the operating conditions in which these parameters are given in
the datasheet may not match actual working conditions, and also
given that each parameter has its own variation range within
each MOSFET, a certain deviation in the model predictions
is almost unavoidable. Furthermore, the values of the parasitic capacitances change with the voltage applied across them,
while the parasitic inductances change with the frequency of
operation.
It is however worth pointing out that, even though the use of
analytical models cannot provide results as accurate as, e.g., a
physical simulation software, a relatively precise prediction can
be made and tested by carefully planned setups, as this paper
has shown.
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 6, JUNE 2010
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[14] R. Eriksson and D. Maksimovic, Fundamentals of Power Electronics.
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Miguel Rodrı́guez (S’06) was born in Gijón, Spain,
in 1982. He received the M.S. degree in telecommunication engineering from the Universidad de Oviedo,
Spain in 2006, where he is currently working toward
the Ph.D. degree in the Department of Electrical and
Electronic Engineering (granted by the Spanish Ministry of Science and Education under the Formación
de Profesorado Universitario Program).
His research interests include dc/dc conversion,
high-frequency power conversion, and power-supply
systems for RF amplifiers.
Alberto Rodrı́guez (S’07) was born in Oviedo,
Spain, in 1981. He received the M.S. degree in
telecommunication engineering in 2006 from the
University of Oviedo, Gijón, Spain, where he is currently working toward the Ph.D. degree.
In 2006, he was a Telecommunications Engineer
with the Government of the Principality of Asturias
and an Assistant Professor with the Department of
Electrical Engineering, University of Oviedo. Since
2007, he has been working in University of Oviedo at
full time. His current research interests include bidirectional dc–dc power converters.
Pablo Fernández Miaja (S’07) was born in Oviedo,
Spain, in 1984. He received the M.S. degree in
telecommunication engineering in 2007 from the
University of Oviedo, Gijón, Spain, where e is currently working toward the Ph.D. degree.
His work is funded by the Spanish Ministry of
Science and Technology to study dc–dc converters
to feed RF power amplifiers. His current research
interests include dc–dc converters, digital control of
switching power supplies and techniques to increase
the efficiency of RF power amplifiers like envelope
tracking.
Diego González Lamar (M’05) was born in
Zaragoza, Spain, in 1974. He received the M.Sc. and
Ph.D. degrees in electrical engineering from the Universidad de Oviedo, Gijón, Spain, in 2003 and 2008,
respectively.
In 2003, he was a Research Engineer at the University of Oviedo, where he has been an Assistant Professor since September 2005. His current research
interests include switching-mode power supplies,
converter modeling, and power-factor-correction
converters.
Javier Sebastián Zúniga (M’´87) was born in
Madrid, Spain, in 1958. He received the M.Sc. degree
from the Polytechnic University of Madrid, Madrid,
Spain, and the Ph.D. degree from the University of
Oviedo, Gijón, Spain, in 1981 and 1985, respectively.
He was an Assistant Professor and an Associate Professor at both the Polytechnic University of
Madrid and at the University of Oviedo. Since 1992,
he has been with the University of Oviedo, where he
is currently a Professor. His research interests include
switching-mode power supplies, modeling of dc-todc converters, low-output voltage dc-to-dc converters, and high-power-factor
rectifiers.
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