Download Inductance and Capacitance

Chapter 4
Inductance and Capacitance
Mdm Shahadah Ahmad
Inductance and Capacitance
• Inductor
• Relationship between voltage,
current, power and energy
• Capacitor
• Relationship between voltage,
current, power and energy
• Series-parallel combinations for
inductance and capacitance
Inductor
Inductor concept
• An inductor consist of a coil of conducting
wire.
• Inductance, L is the property whereby an
inductor exhibits opposition to the change
of current flowing through it, measured in
henrys (H).
Inductance
• Inductance, L
N A
L

2
L = inductance in henrys (H).
N = number of turns
µ = core permeability
A = cross-sectional area (m2)
ℓ = length (m)
Inductance and Capacitance
• Inductor
• Relationship between voltage,
current, power and energy
• Capacitor
• Relationship between voltage,
current, power and energy
• Series-parallel combinations for
inductance and capacitance
Relationship between voltage,
current, power and energy
Inductor
symbol
Inductor
Voltage
di(t )
v(t )  L
dt
Inductor
current
1 t
i (t )   v( ) d  i (t0 )
L t0
di
p  vi  Li
dt
1 t

 v   v dt  it 0 
 L t0

Power
t
w(t )  w(t0 )   p( ) d
t0
1 2
1 2
 Li (t )  Li (t0 )
2
2
• Assuming that energy is zero at time
t=t0, then inductor energy is:
1
w(t ) 
Li (t )
2
2
Inductance and Capacitance
• Inductor
• Relationship between voltage,
current, power and energy
• Capacitor
• Relationship between voltage,
current, power and energy
• Series-parallel combinations for
inductance and capacitance
CAPACITOR
Capacitor physical concept:
• A capacitor consists of two
conducting plates separated by an
insulator (or dielectric).
• Capacitance, C is the ratio of the
charge on one plate of a capacitor to
the voltage difference between the
two plates, measured in farads (F).
• The amount of charge stored,
represented by q, is directly proportional
to the applied voltage v,
q Cv
q = cas dalam coulomb (C)
C = kapasitans dalam farad (F)
v = voltan dalam volt (V)
• Capacitance, C:
eA
C
d
C = Capacitance in farads (F)
e = permittivity of dielectric material
between the plates (C2/N∙m2)
A = surface area of each plates (m2)
d = distance between the plates (m)
Inductance and Capacitance
• Inductor
• Relationship between voltage,
current, power and energy
• Capacitor
• Relationship between voltage,
current, power and energy
• Series-parallel combinations for
inductance and capacitance
Relationship between
voltage, current, power and
energy
• Capacitor symbol
Capacitor
current
dv( t )
i( t )  C
dt
1 t
v(t )   i ( )d  v(t0 )
C t0
Capacitor voltage
Power:
p(t )  v(t )  i (t )
 dv(t ) 
 v(t )   C

dt 

• Energy stored in a capacitor from time t to t0:
t
w(t )  w(t0 )   p( )d
t0
dv( )
  v( ) C
d
t0
d
t
 C
v (t )
v (t0 )
v( )dv( )
v (t )
1
2
 C v( )
2
v (t 0 )
1
1
2
2
 C v(t )  C v(t0 )
2
2
• Capacitor is not discharge at t=-∞,
therefore the voltage is zero.
1
2
w(t )  C v(t )
2
Energy capacitor
Inductance and Capacitance
• Inductor
• Relationship between voltage,
current, power and energy
• Capacitor
• Relationship between voltage,
current, power and energy
• Series-parallel combinations for
inductance and capacitance
Series and parallel
capacitors
• The equivalent capacitance, Ceq of N
parallel-connected capacitors is the sum
of the individual capacitances.

v

i1
i2
iN
C1
C2
CN
• Using KCL,
dv
I n  Cn
dt
I  I1  I 2  .........  I N
dv
dv
dv
I  C1  C2
 ...........  C N
dt
dt
dt
dv
 C1  C2  .......  C N 
dt
N
dv

 dv
   Cn 
 Ceq
dt
 n 1  dt
• Equivalent circuit for the parallel
capacitor,

N
Ceq   Cn
n 1
is
v

Ceq
• The equivalent capacitance, Ceq of N
series-connected capacitors is the
reciprocal of the sum of the reciprocals
of the individual capacitances.
i
Vs


C1
C2
 V1 
 V2 

VN

CN
• Using KCL,
1
Vn 
Cn

t

i ( ) d
V  V1  V2  ...........  VN
 1
1
1
V   
 ........ 
CN
 C1 C2
N
1 t

i ( )d


C
n 1
n
1

Ceq

t

i ( )d
 t
  i ( )d


• Equivalent circuit for the series
capacitor,
i
N
1
1

Ceq n 1 Cn
Vs


Ceq
Series and parallel
inductors
• The equivalent inductance, Leq of N
series-connected inductors is the sum of
the individual inductances.
i
L1
 V1
Vs


L2

 V2


VN

LN
• Using KVL,
di
Vn  Ln
dt
V  V1  V2  .........  VN
di
di
di
V  V1  L2  ...........  LN
dt
dt
dt
di
 L1  L2  .......  LN 
dt
N
di

 di
   Ln   Leq
dt
 n 1  dt
• Equivalent circuit for the series
inductor,
i
N
Leq   Ln
n 1
Vs


Leq
• The equivalent inductance, Leq of N
parallel-connected inductors is the
reciprocal of the sum of the reciprocals
of the individual capacitances.
is

i1
i2
iN
V
L1
L2
LN

• Using KVL,
I  I1  I 2  ...........  I N
1 1
1
I     ........ 
LN
 L1 L2
N
1 t

v( )d


L
n 1 n
1

Leq

t

v( )d
 t
  v( )d


1 t
In 
v ( ) d

Ln 
• Equivalent circuit for the parallel
inductor,
N
1
1

i
Leq n 1 Ln s

V

Leq
Question 1
Obtain the total of capacitance.
4F
CT
2F
3F
12F
1F
Answer
• Short circuit, then:
CT  3F
Question 2
•
Voltage stored in a 10µF capacitor is
shown in figure below. Obtain the graph
for current of the capacitor.
Answer
• Capacitor voltage:
10
v(t ) 
t
6
5  10
0
0  t  5 saat
t  5 saat
dv(t )
• current: i (t )  C
dt
 10 
 10 10 
6 
 5  10 
 20 A
0  t  5 saat
6
• Thus:
Question 3
• Determine the voltage across a 2 µF
capacitor if the current through it is
i(t )  6e
3000t
mA
Assume that initial capacitor voltage is zero
Answer
• Capacitor voltage:
1 t
v(t )   i (t )dt  v(0)
C 0
v ( 0)  0
1 t
v(t )   i (t )dt  v(0)
C 0
t
1
 3000t
3

6
e

10
dt
 6 0
2  10
3  10 3000t

e
 3000
3
 (1  e
 3000t
)V
t
0