Inductance, Capacitance, Etc

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: v in volts L in henrys i in amps t in seconds

v=L

+ v

i

L

−

di dt

Inductance in Picture

Electrical Relationship for Inductor v=L

di dt

vdt = L

di dt = Ldi dt

1 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 di =

1 t  p (t ) = v (t ) ⋅ i (t ) = v (t ) ⋅  ∫ v (τ ) d τ + i (t 0 )   L t0  dw dw(t ) dt = ∫ p (t ) dt = ∫ v(t ) ⋅ i (t ) dt = p⇒∫ 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

w(t ) = L w(t ) = if

2 i (t )

i 2

+ w(t0 )

i ( t0 )

[

]

L 2 i (t ) − i 2 (t0 ) + w(t0 ) 2

w(t0 ) = 0

and

i (t0 ) = 0, then w(t ) =

L 2 i (t ). 2

Example i=0 i = A sin wt i=0

i

for

t≤0

for

0 ≤ wt ≤ 2π

for

wt ≥ 2π

L

+ v −

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

i(t) A

i = A sin wt

di v=L dt v = ( LAw) cos wt p = vi p = LAw cos wt ⋅ A sin wt p = LA w cos wt ⋅ sin wt 2

LA 2 w p= sin 2 wt 2 t

w = ∫ p (t ) dt = ∫

t

0

0

LA 2 w ⋅ sin 2 wt dt 2

Set τ = 2 wt ⇒ dτ = 2 w dt w=

2

LA w 1 ⋅ ⋅ 2 2 w ∫0

LA 2 =− cos τ 4

2 wt

2 wt

0

sin τ dτ

LA 2 = (1 − cos 2 wt ) 4

v(t )

wt

π /2

π

3π / 2

2π

π /2

π

3π / 2

2π

π /2

π

3π / 2

2π

wt

π /2

π

3π / 2

2π

wt

LAw

p (t )

wt

LA 2 w 2

v

w(t ) LA 2 2

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 C in farads v in volts t in seconds

+

i

v − C

dv i =C dt

Capacitors in Picture

Electrical Relationship for Capacitor p (t ) = v(t ) ⋅ i (t )

dv(t ) i (t ) = C dt i (t) dt = C dv v ( t1 ) 1 t1 dv = ∫v (t0 ) C ∫t0 i(t ) dt 1 t1 v(t1 ) − v(t0 ) = ∫ i (t ) dt C t0 1 t v(t ) = ∫ i (τ ) dτ + v(t0 ) C t0

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

∫ dw = ∫ Cv dv Cv 2 (t ) w(t ) = 2

Example i=0

t=0

i = Bt

0 ≤ t ≤ 10

i=0

t ≥ 10

dv dt i dt = C dv

i =C

+

i

C

v −

v(0) = 0

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

1 dv = i dt C v(t ) 1 t = dv ∫v(0) C ∫0 i dτ 1 t v(t) = ∫ i dτ + v(0) C0 1 t v(t) = ∫ Bτ dτ C0 1 Bt2 v(t) = C 2

i(t )

10B

i (t ) = Bt 5

10

t

1 B100 C 2

v(t )

1 Bt 2 v(t ) = C 2 5

10

t

1 B 21000 C 2

p(t ) 5

10

t

1 B 210,000 C 8

w(t ) 5

10

t

1 Bt 2 p (t ) = v(t ) ⋅ i (t ) = ⋅ Bt C 2 1 B 2t 3 = C 2 dw = p (t ) ⇒ dw = p (t ) dt dt t t 1 B 2τ 3 w(t ) = ∫ p (τ ) dτ = ∫ dτ C 2 0 0 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 i2 i

Š

Leq = L1 + L2 + Κ + Ln

1

i (t ) = i1 (t ) = i2 (t ) = Κ = in (t )

L1

Ln

L2

i

for inductors in parallel

i

1 1 1 1 = + +Κ + Leq L1 L2 Ln

i1

i (t ) = i1 (t ) + i2 (t ) + Κ + in (t )

L1

Capacitors in series/parallel combine in the opposite manners. Š for capacitors in series i1

1 1 1 1 = + +Κ + Ceq C1 C2 Cn

C1

i (t ) = i1 (t ) = i2 (t ) = Κ = in (t ) Š

in

i2

i (t ) = i1 (t ) + i2 (t ) + Κ + in (t )

Ln

L2

i2

in

C2

Cn

i

for capacitors in parallel

Ceq = C1 + C2 + Κ + Cn

in

i

i2

i1

C1

C2

in Cn