Chapter-6-update
Download
Report
Transcript Chapter-6-update
L
6.1 The Inductor
i
Is a passive element ( can not generate energy)
v
However the inductor is capable of storing energy in the magnetic field which can be released
Represented graphically as a coiled wire
Symbolized by the letter L
Measured in henrys (H)
The voltage drop across the inductor the inductor and the current in the direction
of voltage drop are related as follows
di
v=L
dt
Example 6.1
0
i = 5t
10te
i
v
100 mH
t<0
t>0
(b) At what instant of time is the current maximum ?
0
i = 5t
10te
0.736
0.2
Since the maximum will occur were the derivative of the current is zero
di
=0
dt
di
= 10( 5te 5t e 5t )
dt
= 10e5t (1 5t) = 0
As
(1 5t) 0
t
1
s 0.2 s
5
t<0
t>0
(c) Find the voltage across the terminals of the inductor ?
v
i
0
i = 5t
10te
100 mH
t<0
t>0
di
v=L
dt
=
(0.1)10( 5te
=
0
V
5t
e
5t
t<0
(d) Sketch the voltage across the terminals
of the inductor ?
)
=
e
5t
(1 5t) V
t>0
(e) Are the voltage and the current at a maximum at the same time ?
Since the voltage across the inductor is
proportional to di/dt not i
100 mH
v
i
the voltage maximum is not at the same time as
current maximum
0
i = 5t
10te
0.736
t<0
t>0
0.2
0
v = 5t
e (1 5t)
t<0
t>0
(f) At what instant of time does the voltage change polarity ?
100 mH
v
i
At t = 0.2 s which corresponds to the
moment when di/dt is passing through
zero changing sign
0
i = 5t
10te
0.736
t<0
t>0
0.2
0
v = 5t
e (1 5t)
t<0
t>0
(g) Is there ever an instantaneous change in voltage across the inductor ? If so at what time ?
Yes, At t = 0 s
100 mH
v
i
Note that the voltage can change instantaneously
across the terminals of an inductor
0
i = 5t
10te
0.736
t<0
t>0
0.2
0
v = 5t
e (1 5t)
t<0
t>0
Current in an Inductor in Terms of the Voltage Across the Inductor
L
v
Since
di
v=L
dt
Multiplying both side by the differential
i
di
v dt = L
dt
dt
di
v dt = L dt
dt
vdt = Ldi
Integrating both side with respect to the differentials
i(t )
L
i(t 0 )
t
dx = v d
t0
di
OR
and
dt
dt
Ldi = vdt
we have
Note we used x and the variables of integration, were
become limits on the integrals
( x and become dummy variables )
t
L i(t ) - i(t 0 ) = v d
t0
t
1
i(t ) =
v d i(t 0 )
L
t0
i
and
t
Example 6.2 The voltage pulse applied to the 100mH inductor. Assume
v(t)
(a) Sketch the voltage v(t) ?
i
i = 0 for
0
v (t ) = 10t
20te
t<0
t>0
t ≤0
v(t)
i(t)
0
v (t ) = 10t
20te
t<0
t>0
(b) Find the inductor current i(t) ?
1
i(t) =
0.1
t
20 e 10 d 0
= 2 1 10te 10t e 10t A t > 0
0
(c) Sketch the current i(t) ?
Note the inductor current i(t) approaches a constant vlue of 2 A as t increases.
Power and Energy in the Inductor
t
di
v=L
dt
L
1
i(t ) =
v d i(t 0 )
L
t0
v
i
di
p = vi = L
dt
i = Li di
dt
dw
=
dt
Integrating both side with respect to the differentials
w
0
i
dx = L y dy
0
1 2
w = Li
2
dw = Li di
dw
and di were w=
0
if
i= 0
L
The Inductor
i
v
t
di
v=L
dt
1
i(t ) =
v d i(t 0 )
L
1
w = Li 2
2
di
p = Li
dt
t0
i
L
The Capacitor
v
dv
i =C
dt
1
v(t ) =
C
t
i d v (t
t0
0
)
1
w = Cv 2
2
p = Cv
dv
dt
L
v
i
L
v
i
di
v=L
dt
dv
i =C
dt
t
1
i(t ) =
v d i(t 0 )
L
t0
1
w = Li 2
2
di
p = Li
dt
v(t ) =
1
C
t
i d v (t 0 )
t0
1 2
w = Cv
2
dv
p = Cv
dt
Inductors in Series
1 1
1
Lequ L1 L 2
1
Ln
Therefore inductor combine like resistor
5.1.1 Capacitors in Series
5.1.1 Capacitors in Parallel
Therefore capacitor combine like conductor
Example 5.1
A voltage source is applied to a 5-F capacitor as
shown.
Sketch the capacitor current and the stored energy as a
function of time.
dv t
i t C
dt
dv t
5
dt
1
2
w t C v t
2
1
5v s2 t
2
Example 5.2
A current source is applied to a 5-F capacitor.
Sketch the capacitor voltage as a function of time.
t
1
v t i s d
5
The capacitor voltage is related to
area Under current source.
For example area at t = 1 s is
Area of Triangle is (1)(10)=10
V(1s)=10/5= 2V.
For example area at t=3 s is
10+10+5 = 25 hence
V(3s)=25/5= 5V.
Example 5.3
A current source is applied to a 5-H inductor as shown. Sketch
the voltage across the inductor versus time.
di t
v t L
dt
1 2
w t Li t
2
Example 5.4
A voltage source is applied to a 5-H inductor as
shown. Sketch the inductor current versus time.
1
i t
L
t
v d
1
i t
L
t
v d