Chapter 6. Inductance, Capacitance, etc.

Inductor
An inductor is an electrical component that opposes any change in electrical
current based on the magnetic field. The source of the magnetic field is the
charges in motion.

Using the passive sign convention, we have:

di
v=L
v in volts dt
L in henrys
i in amps + v −
t in seconds
i L
Inductance in Picture
Electrical Relationship for Inductor

di
v=L
dt
di
vdt = L dt = Ldi
dt
1
di = vdt
L
i ( t1 ) 1 t1
∫i (t0 ) di = L ∫t0 v(t )dt
1 t1
i (t1 ) − i (t0 ) = ∫ v(t )dt
L t0
1 t1
i (t1 ) = ∫ v(τ )dτ + i (t0 )
L t0
 1 t 
p (t ) = v (t ) ⋅ i (t ) = v (t ) ⋅  ∫ v (τ ) d τ + i (t 0 ) 
 L t0 
dw dw(t )
= p⇒∫ dt = ∫ p (t ) dt = ∫ v(t ) ⋅ i (t ) dt
dt dt
dw(t )  di 
∫ dt dt = ∫  L dt  ⋅ i(t ) dt
 
∫ dw = ∫ Li di
i (t )
w(t ) − w(t0 ) = ∫i (t ) Li di
0

i (t )
w(t ) = L ∫i (t ) i di + w(t0 )
0

2 i (t )
i
w(t ) = L + w(t0 )
2 i ( t0 )

w(t ) =
2
[
L 2
]
i (t ) − i 2 (t0 ) + w(t0 )

L 2
if w(t0 ) = 0 and i (t0 ) = 0, then w(t ) = i (t ).
2
 Example
for t≤0
i=0
i = A sin wt for 0 ≤ wt ≤ 2π
i=0 for wt ≥ 2π

+
i L v
−

Find v(t), p(t), w(t).

 i(t)
A
i = A sin wt
wt
π /2 π 3π / 2 2π
v(t )
di
v=L LAw
dt
v = ( LAw) cos wt
wt
π /2 π 3π / 2 2π
p = vi
p (t )
p = LAw cos wt ⋅ A sin wt
p = LA w cos wt ⋅ sin wt
2 LA 2 w
2
LA 2 w
p= sin 2 wt v
2
π wt
t
t LA 2 w π /2 3π / 2 2π
w = ∫ p (t ) dt = ∫ ⋅ sin 2 wt dt
0 2
0
w(t )
Set τ = 2 wt ⇒ dτ = 2 w dt
2
LA w 1 2 wt
LA 2

2 w ∫0
w= ⋅ ⋅ sin τ dτ 2
2
2 wt wt
=−
LA 2
cos τ =
LA 2
(1 − cos 2 wt ) π /2 π 3π / 2 2π
4 0
4
 Capacitor
A capacitor is an energy storage element that stores the energy in the electric field. The electric
field results from the separation of charges (voltage).

Using the passive sign convention, we have:

i in amps
+ v −
dv
C in farads i =C
v in volts dt
t in seconds
i
C
Capacitors in Picture
 Electrical Relationship for Capacitor

p (t ) = v(t ) ⋅ i (t )
dv(t ) 1 t 
i (t ) = C p (t ) =  ∫ i (τ ) dτ + v(t0 )  ⋅ i (t )
dt  C t0 
i (t) dt = C dv
v ( t1 ) 1 t1 dw(t ) dv(t )
∫v (t0 ) C ∫t0 i(t ) dt
dv = p(t) =
dt
= v(t ) ⋅ C
dt
1 t1  dw(t )   dv(t ) 
v(t1 ) − v(t0 ) = ∫ i (t ) dt  dt = Cv(t ) dt
C t0  dt   dt 
1 t
v(t ) = ∫ i (τ ) dτ + v(t0 ) ∫ dw = ∫ Cv dv
C t0
Cv 2 (t )
w(t ) =
2
 Example

i=0 t=0 dv
i =C
dt
i = Bt 0 ≤ t ≤ 10
i dt = C dv
i=0 t ≥ 10 1
dv = i dt
C
+ v(t ) 1 t
i C v
∫v(0) C ∫0 i dτ
dv =

− 1 t
v(t) = ∫ i dτ + v(0)
C0
v(0) = 0
1 t
v(t) = ∫ Bτ dτ
Find v(t), p(t), w(t). C0
1 Bt2
v(t) =
C 2
 10B
i(t )
i (t ) = Bt
5 10 t

1 B100
C 2 1 Bt 2
v(t ) v(t ) =
C 2
5 10 t
1 Bt 2
1 B 21000 p (t ) = v(t ) ⋅ i (t ) = ⋅ Bt
C 2
C 2
p(t ) 1 B 2t 3
=
C 2
5 10 t
dw
= p (t ) ⇒ dw = p (t ) dt
1 B 210,000 dt
C 8 t
1 B 2τ 3
t

w(t ) w(t ) = ∫ p (τ ) dτ = ∫ dτ
0
C 0
2
5 10 t =
B2 t 4 1
= Cv(t ) 2
C 8 2
 Series and Parallel Combinations
Inductors in series/parallel combine in manners analogous to resistors.
Š for inductors in series i i2 in
1
Leq = L1 + L2 + Κ + Ln
i (t ) = i1 (t ) = i2 (t ) = Κ = in (t ) L1 L2 Ln

Š for inductors in parallel i

1 1 1 1 i
= + +Κ + i1 i2 in
Leq L1 L2 Ln
i (t ) = i1 (t ) + i2 (t ) + Κ + in (t ) L1 L2 Ln

Capacitors in series/parallel combine in the opposite manners.

Š for capacitors in series i1 i2 in
1 1 1 1
= + +Κ +
Ceq C1 C2 Cn
C1 C2 Cn
i (t ) = i1 (t ) = i2 (t ) = Κ = in (t )
Š for capacitors in parallel i
Ceq = C1 + C2 + Κ + Cn i
i1 i2 in
i (t ) = i1 (t ) + i2 (t ) + Κ + in (t )
C1 C2 Cn