Zwarty koniec linii
i (k) = G u (k) - G u (k-m) - i (k-m)
1 f 1 f 2 2
u (k) = u (k-m) = 0
2 2
i (k) = G u (k) - i (k-m)
1 f 1 2
i (k) = - G u (k) - i (k-m)
2 f 1 1
i (k-m) = - G u (k-2m) - i (k-2m)
2 f 1 1
i (k) = G u (k) + G u (k-2m) + i (k-2m)
1 f 1 f 1 1
Otwarty koniec linii
i (k) = G u (k) - G u (k-m) - i (k-m)
1 f 1 f 2 2
i (k) = i (k-m) = 0
2 2
i (k) = G u (k) - G u (k-m)
1 f 1 f 2
i (k) = G u (k) - G u (k-m) - i (k-m) = 0
2 f 2 f 1 1
G u (k) = G u (k-m) + i (k-m)
f 2 f 1 1
G u (k-m) = G u (k-2m) + i (k-2m)
f 2 f 1 1
i (k) = G u (k) - G u (k-2m) - i (k-2m)
1 f 1 f 1 1
RL niejawna metoda Eulera
u(k) = u (k) + u (k)
R L
1
( )
u (k) = G i k
R
R
1
( ) ( ))
u (k) = G (i k - jL k - 1
L
L
1 1
( ) ( ) ( ))
u(k) = G i k + G (i k - jL k - 1
R L
1 1 1
( ) ( ) ( )
u(k) = G i k + G i k  G jL k - 1
R L L
1
( ) ( )
u(k) = G 1 i k  G jL k - 1
GL
R L
( ) ( )
i(k) = GRGLu k + GRjL k - 1
GRRL T
G = G +GL = L+LT
R
(k-1) (k-1) (k-1)
j(k-1) = GRjL = jL = .. = Li
1 T
GR+GL R(R+L) L+RT
RC niejawna metoda Eulera
u(k) = u (k) + u (k)
R C
1
( )
u (k) = G i k
R
R
1
( ) ( ))
u (k) = G (i k - jC k - 1
C
C
1 1
( ) ( ) ( ))
u(k) = G i k + G (i k - jC k - 1
R C
1 1 1
( ) ( ) ( )
u(k) = G i k + G i k  G jC k - 1
R C C
1
( ) ( )
u(k) = G 1 i k - G jC k - 1
GC
R C
GRRC C
G = G +GC = T+RC
R
(k-1) (k-1) (k-1)
j(k-1) = GRjC = jC = Tj =
1 C
GR+GC R(R+T) T+RC
T
= - Gu(k - 1)
T+RC
RL jawna metoda Eulera
i(k) = i(k-1) + T uL(k)
L
( )
uL k = u(k) - uR(k)
( )
uR k = Ri(k)
i(k) = i(k-1) + T(u(k) - Ri(k))
L
i(k) = i(k-1) + Tu(k) - RTi(k)
L L
( ) ( )
(1-RT)i k = i k - 1 + T u(k)
L L
T L
i(k) = L+RT u(k) + L+RT i(k-1)
i(k) = Gu(k) + j(k-1)
L T
j(k-1) = L+RT i(k-1) G=L+RT
RC jawna metoda Eulera
u (k) = u (k-1) + 5ØGÜ i(k)
C C
5Ø6Ü
u (k) = u(k)  u (k) = u(k)  Ri(k)
C R
u(k)  Ri(k) = u(k) + u(k-1)  Ri(k-1)
Ri(k) + 5ØGÜ 5ØVÜ(5ØXÜ) = u(k)  u(k-1) + Ri(k-1)
5Ø6Ü
i(k)(R + 5ØGÜ) = u(k)  u(k-1) + Ri(k-1)
5Ø6Ü
5Ø6Ü 5Ø6Ü 5Ø6Ü5ØEÜ
i(k) = 5ØGÜ+5ØEÜ5Ø6Ü 5ØbÜ(5ØXÜ) - 5ØGÜ+5ØEÜ5Ø6Ü 5ØbÜ(5ØXÜ - 1) + 5ØGÜ+5ØEÜ5Ø6Ü 5ØVÜ(5ØXÜ - 1)
i(k) = Gu(k) + j(k-1)
5Ø6Ü 5Ø6Ü 5Ø6Ü5ØEÜ
G = 5ØGÜ+5ØEÜ5Ø6Ü j(k-1) = - 5ØbÜ(5ØXÜ - 1) + 5ØGÜ+5ØEÜ5Ø6Ü 5ØVÜ(5ØXÜ - 1)
5ØGÜ+5ØEÜ5Ø6Ü
RL metoda trapezów
5ØGÜ
i(k) = i(k-1) + 25Ø?Üu (k)
L
u (k) = u(k)  u (k)
L R
u (k) = Ri(k)
R
5ØGÜ
i(k) = i(k-1) + 25Ø?Ü(u (k) + u (k-1))
L L
5ØGÜ
i(k) = i(k-1) + 25Ø?Ü(u(k)  Ri(k) + u(k-1)  Ri(k-1))
5ØGÜ 5ØGÜ
i(k) = i(k-1) + 25Ø?Üu(k) - 5ØGÜ5ØEÜi(k) + 25Ø?Üu(k-1) - 5ØGÜ5ØEÜi(k-1)
25Ø?Ü 25Ø?Ü
5ØGÜ 5ØGÜ
i(k)(1+5ØGÜ5ØEÜ) = i(k-1) + 25Ø?Üu(k) + 25Ø?Üu(k-1) - 5ØGÜ5ØEÜi(k-1)
25Ø?Ü 25Ø?Ü
5ØGÜ 5ØGÜ
i(k) = i(k-1)(25Ø?Ü-5ØEÜ5ØGÜ) + 25Ø?Ü+5ØEÜ5ØGÜu(k) + 25Ø?Ü+5ØEÜ5ØGÜu(k-1)
25Ø?Ü+5ØEÜ5ØGÜ
i(k) = Gu(k) + j(k-1)
5ØGÜ
G = 25Ø?Ü+5ØEÜ5ØGÜ j(k-1) = Gu(k-1) + (25Ø?Ü-5ØEÜ5ØGÜ)i(k-1)
25Ø?Ü+5ØEÜ5ØGÜ
RC metoda trapezów
5ØGÜ
u (k) = u (k-1) + 25Ø6Ü(i (k) + i (k-1))
C C C C
u (k) = u(k)  u (k) = u(k)  Ri(k)
C R
5ØGÜ
u(k)  Ri(k) = u(k-1)  Ri(k-1) + 25Ø6Ü(i (k)+ i (k-1))
C C
5ØGÜ 5ØGÜ
( )
Ri(k) + 25Ø6Ü 5ØVÜ 5ØXÜ = u(k) - u(k-1) + Ri(k-1) - 25Ø6Ü i(k-1)
5ØGÜ 5ØGÜ
i(k)(R + 25Ø6Ü) = u(k)  u(k-1) + Ri(k-1) - 25Ø6Ü i(k-1)
25Ø6Ü 25Ø6Ü
i(k) = 25ØEÜ5Ø6Ü+5ØGÜu(k) - 25ØEÜ5Ø6Ü+5ØGÜu(k-1) + (25ØEÜ5Ø6Ü-5ØGÜ) 5ØVÜ(5ØXÜ - 1)
25ØEÜ5Ø6Ü+5ØGÜ
i(k) = Gu(k) + j(k-1)
25Ø6Ü 25Ø6Ü
G = 25ØEÜ5Ø6Ü+5ØGÜ j(k-1) = - 25ØEÜ5Ø6Ü+5ØGÜu(k-1) + (25ØEÜ5Ø6Ü-5ØGÜ) 5ØVÜ(5ØXÜ - 1)
25ØEÜ5Ø6Ü+5ØGÜ