Freewheeling Path for Inductor...
Transcript of Freewheeling Path for Inductor...
Freewheeling Path for Inductor Current
LiQ
offD onD
D
On-time Off-time
● When the on-time inductor current path is discontinued, the freewheeling path provides an alternative path for the inductor current.
● The freewheeling path is usually constructed with a diode, called the freewheeling diode.
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Inductive Switching Circuit #3
( )Lv t
1 2 25 VS SV V
S2 25 VV
( )Li t
sT
0.5 sT
A
BArea A Area B=
Closed
OpenLi
Lv
1SV 2SV
1 2: S SV = V =Case #1 50V and 25 V
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Inductive Switching Circuit #3
1 2: 3S SV = V =Case #2 50V and 0 V 1 2: 20S SV = V =Case #3 50 V and V
( )Lv t
30 V
20 V
( )Li t
20 V
30 V Area A > Area BArea A = Area B
sT
0.5 sT
sT
0.5 sT
( )Lv t
( )Li t
Closed
Open
Closed
Open
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Definition of Capacitance
( )Ci t
C
( )q t
( )Cv t
( )Definition of capacitance:( )C
q tC
v t· º
( )Definition of current: ( )Cdq t
i tdt
· =
( )( )( )
Circuit equation of capaci
( )
tanc
( )
e
CC C C
dv tdq ti t C v t i t dt
dt dt C=
·
= = ò1
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Circuit Equations for Capacitors
( ) ( )CC C C
dv ti t C v t i t dtdt C
Basic equations
1( ) ( )
1 1, ( )
( ),
SC S C C S
C SC S C
Ii t = I v t = i t t = I t = tC C C
v t Vv t =V i t = C = Ct t
Special cases
With ( ) ( ) d d
d dWith ( ) ( ) 0d d
( )Ci t
C
( )Cv t
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Current Drive of CapacitorE
nerg
y S
tora
ge a
nd T
rans
fer D
evic
e
( )Ci t
SI ( )Cv t
( )Cv t
( )Ci t
SI
SI tC
1( ) dC S
S
v t I tC
It
C
t
t
C
1( ) ( )dC Cv t i t tC
Current drive: excitation of capacitor with acurrent source
( )Ci t
C
( )Cv t
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Capacitive Switching Circuit #1
d ( )( )dC
Cv ti t C
t
SIC
Cv
Ci
0 1t
SICi
SI C Cv
1t
Closed
Open
212 CC v t1Instantaneous release of the electric energy of ( )
generation of acurrent spike that kills the switch
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Capacitive Switching Circuit #2
SI / CCv
Ci
SICi
SI Cv
ClosedOpen
Isolation diode blocks the capacitor voltage when the switch is closed.
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Charge Balance Condition
● Charge balance condition: the charge increase in one switching period should be equal to the charge decrease in that switching period.
inc decq q
Excitation of capacitor with arectangular current waveform
1 1incinc dec
dec
q I Tq q I T = I T
q I T 1 1 2 22 2
Amp-sec balance condition
● The average value of the capacitor current can be considered to be zero assuming that the averaging is performed over a sufficiently longer period than the switching period.
1I
2I1T
2T
Ci
( )Ci t
C
( )q t
( )Cq i t t
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Implications of Amp-Sec Balance Condition
● Whenever possible, a switching circuit establishes a steady-state equilibrium byadjusting the circuit variables to satisfy the amp-sec balance condition on the capacitors in the circuit.
● A circuit that violates the amp-sec balance condition can be easily devised, butthe circuit eventually destroys itself by an over-voltage condition on the capacitor.
Example
SI / CCv
Ci
SICi
SI Cv
ClosedOpen
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Capacitive Switching Circuit #3
( )Ci t
25A
25A
( )Cv t
AB
1SIC
2SIC
Closed
OpenOpen
Cv
1SI 2SI
2C Si I
1SI 2SICv
1C Si I
Cv
1SI 2SI
0Ci
Switch open
Switch closed
1 2: 25S SI = I =Case #1 A and 25 A
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( )Ci t
30A
20A
( )Cv t
( )Ci t
20A
30A
0A
( )Cv t
A
B
AB
Closed
Open
ClosedOpenOpenOpen
Capacitive Switching Circuit #3
1 2: 3S SI = I =Case #2 30 A and 0 A 1 2: 2 2S SI = I =Case #3 0 A and 0 A
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Ideal Transformer
● Definition of circuit variables
Pi Si
1 : n
Pv
Sv
( ): primary voltage ( ): primary current( ): secondary voltage ( ): secondary current
: turns ratio
P P
S Sn
v t i tv t i t
Definition of circuit equatio( ) ( )
n
( )( )S P
P S
v t n v t
i t n i t
==
·
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Circuit Equations of Ideal Transformer
0Pi 0Si 1 : n Pi Si 1 : n
dcV
Secondary sideopen Primary side open Secondaryside short Primaryside short
S dcv n V
dcV S dcv n V
● Examples
( ) ( )( ) ( )
S P
P S
v t n v ti t n i t
Pi Si
1 : n
Pv
Sv
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Practical Transformer
● Dot convention: the primary and secondary currents flowing into thewinding terminals marked ● produce a mutually additive magnetic flux.
● Lenz`s law: an electro-magnetic induction occurs in such a way that themagnetic flux produced as the outcome of the magnetic induction opposes the magnetic flux that initiated the induction process.
Pi Si
Pv
Sv
Structure and dot convention Symbol and polarity
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Modeling of Practical Transformer
1) Faraday`s law
P P
S S
N
N
c
c
c
: magnetic flux inside the core
: primary flux linkagePerfect coupling
: secondary flux linkage
Pi
PNPv SvSN
Si
cP
c
d ( ) d ( )( )
d dd ( ) d ( )
( )d d
( ) ( ) ( )( )
PP
SS S
SP PS P
S S P
t tv t N
t tt t
v t Nt t
Nv t N v t v tv t N N
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Modeling of Practical Transformer
2) Ampere`s law
c
m
Hl
: magnetic field intensity inside the core: mean magnetic path length
Pi
PNPv SvSN
Si
cH
with
P P S S c m
S c mP S
P Pc mS
P S m mP P
N i t N i t H t l
N H t li t i t
N NH t lN
i t i t i t i tN N
- =
= +
= + =
( ) ( ) ( )
( )( ) ( )
( )( ) ( ) ( ) ( )
Magnetizing current thecurrent required to create that couplesthe primary and secondary windings through
magnetic induction.
m ci t H t:( ) ( )
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Modeling of Practical Transformer
Pi
PNPv SvSN
Si
cH
Magnetizing inductance associated with and( ) ( )
( )
( ) ( )
( )( ) ( )
P P r o cP m m m r o P
m c
m m
m m
P
p
t N H t S St L i t L N
i t H
L i t t
t l l
N
l m ml
l
m m= = = = 2
Magnetizing current through primary winding( )
( ) c mm
p
H t li t
N=
Magnetic flux linkage at primary winding( ) ( ) ( )P P c P r o ct N t N H t Sl f m m= =
3) Magnetizing inductance
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Circuit Model of Practical Transformer
Pi Si
mi
mL
1 : n
Pv
Sv
S
P
NnN
Turns ratio
Sni
Voltage equation( ) ( )S Pv t nv t=
Current equation
with
( ) ( ) ( )
( ) ( )( )( )
( )( ) ( )
P S m
P r o c c mPm
m pr o P
m
c mP S
p
i t ni t i t
N H t S H t lti t
L S NN
l
H t li t ni t
N
m ml
m m
= +
= =
+
=
=
2
4) Circuit model
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