Elementy L i C w obwodach prądu przemiennego

sinusoidalnego

I n d u k c y j n o ś ć L

Założenia:

L

= const, R = C = 0 oraz i I

t

=

m

sin

ω

Φ

Φ

i

zi

R

zI

R

t

t

=

=

=

µ

µ

ω

ω

m

m

sin

sin

i

u

L

e

L

⎟

⎠

⎞

⎜

⎝

⎛

π

−

=

−

=

−

=

−

=

2

sin

cos

d

d

d

d

m

m

t

L

I

t

L

I

t

i

L

t

z

e

i

L

ω

ω

ω

ω

Φ

⎟

⎠

⎞

⎜

⎝

⎛

π

−

=

2

sin

m

t

E

e

L

L

ω

u

e

u

e

L

L

+

= ⇒ = −

0

⎟

⎠

⎞

⎜

⎝

⎛

π

+

=

⎟

⎠

⎞

⎜

⎝

⎛

π

+

=

=

2

sin

2

sin

cos

m

m

m

t

U

t

L

I

t

L

I

u

ω

ω

ω

ω

ω

t

u, i

2

π

i

e

L

u

L

0

π

π

/2

I

U LI

=

ω

E

L

ϕ π

= + /2

U

I

L

U

I L

m

m

2

2

=

⇒

=

ω

ω

X

L

L

=

ω

U

IX

I

U

X

L

L

=

⇒ =

X

L

fL

L

=

=

ω

2

π

[X

L

]

= [

ω

] [L]

= Ω = (1 s)

−1

⋅1 H =(1 s)

−1

⋅ (Ω⋅s)

t

UI

t

t

I

U

ui

p

ω

ω

ω

2

sin

2

sin

sin

m

m

=

⎟

⎠

⎞

⎜

⎝

⎛

π

+

=

=

t

u, i, p

π

i

p

2

π

u

L

0

π

/2

+

+

−

−

∫

=

T

T

t

p

A

0

d

A

P

A

T

T

T

= ⇒ =

=

0

0

∫

∫

∫

=

=

=

=

=

4

/

0

0

m

2

m

4

/

0

4

/

m

2

d

d

d

d

d

T

I

T

T

W

LI

i

Li

t

i

t

i

L

t

ui

A

⎟

⎠

⎞

⎜

⎝

⎛

π

=

∠

=

2

,I

U

UI

Q

[Q]

= var

A

b

= Qt

[A

b

]

= var⋅s

P o j e m n o ś ć C

Założenia:

C

= const, R = L = 0 oraz u U

t

=

m

sin

ω

t

u

C

i

u

C

q

t

i

q

d

d

d

d

d

d

=

⇓

=

∧

=

i

u

C

u

C

i

C

u

t

CU

t

t

CU

t

=

=

=

d

d

d(sin

d

m

m

ω

ω

ω

)

cos

⎟

⎠

⎞

⎜

⎝

⎛

π

+

=

⎟

⎠

⎞

⎜

⎝

⎛

π

−

−

=

2

sin

2

sin

cos

t

t

t

ω

ω

ω

⎟

⎠

⎞

⎜

⎝

⎛

π

+

=

⎟

⎠

⎞

⎜

⎝

⎛

π

+

=

2

sin

2

sin

m

m

t

I

t

CU

i

ω

ω

ω

Na kondensatorze prąd wyprzedza napięcie o

π/2 (90°).

t

u, i

2

π

i

u

C

0

π

π

/2

I

ϕ π

= − /2

U I C

=

ω

/

I

U

C

I

CU

m

m

2

2

=

⇒ =

ω

ω

X

C

C

= 1

ω

I

U

X

U

IX

C

C

=

⇒

=

X

C

fC

C

=

=

1

1

2

ω

π

[X

C

]

= [

ω

]

−1

[C]

−1

= Ω = 1s:1F = 1s:(1C:1V) = 1V:1A

t

UI

t

t

I

U

ui

p

ω

ω

ω

2

sin

2

sin

sin

m

m

=

⎟

⎠

⎞

⎜

⎝

⎛

π

+

=

=

t

u, i, p

π

i

p

2

π

u

C

0

π

/2

+

+

−

−

∫

=

T

T

t

p

A

0

d

A

P

A

T

T

T

= ⇒ =

=

0

0

∫

∫

∫

=

=

=

=

=

4

/

0

0

e

2

m

4

/

0

4

/

m

2

d

d

d

d

d

T

U

T

T

W

CU

u

Cu

t

t

u

uC

t

ui

A

⎟

⎠

⎞

⎜

⎝

⎛

π

=

∠

=

2

,

c

I

U

UI

Q

Q

CU

c

=

ω

2

[Q

c

]

= var

A

b

= Q

c

t oraz [A

b

]

= var⋅s