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.

53

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

54

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

55

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

56

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

57

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

58

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

59

Capacitive Switching Circuit #2

SI / CCv

Ci

SICi

SI Cv

ClosedOpen

Isolation diode blocks the capacitor voltage when the switch is closed.

60

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

61

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

62

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

63

( )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

64

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

==

·

65

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

66

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

67

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

68

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:( ) ( )

69

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

70

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

71