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r 2
i 2
⋅ v 2
− i 1
i 2
r = ⋅ 1
i 1
⋅ v 1
− i 1
i 2
r 2
i 2
⋅ v 2
v = m
r m
− i 1
i 2
r = ⋅ 1
i 1
⋅ v 1
i 2
v m
v 2
r 2
i = 1
v m
v 1
r 1
C
C
dv 1
dt
dv 2
dt
i 1
i 2
C
dv 2
dt
⋅ i 2
C =
dv 1
dt
⋅ i 1
III. Generate the Needed Equations
Determine v 1
and v 2
as a function of time using eigenanalysis.
II.Goals
v 2
v0 := 20 1
r
1
i
1
v
1
v
2
r
m
r
2
i
+^2
v
m
C
1
C
2
r m
r :=7. 2
r :=17. 1
C :=11.
C := 2
The parameters for the following circuit are:
I. Define the Situation
ORIGIN := 1
C:\JLAW\CLASSES\S05PSSII page 1 of 5 January 24, 2005
r 1
r m
i 1
⋅ r m
i 2
= − r m
i 1
⋅ r 2
r m
i 2
r 1
r m
r m
r m
r 2
r m
i 1
i 2
v 1
v 2
F
r 1
r m
r m
r m
r 2
r m
:= F
F
i 1
i 2
v 1
v 2
i 1
i 2
F
1 −
v 1
v 2
C
C
dv 1
dt
dv 2
dt
⋅ F
− 1 −
v 1
v 2
C
C
C
:= C
C
dv 1
dt
dv 2
dt
⋅ F
− 1 −
v 1
v 2
dv 1
dt
dv 2
dt
C
− 1 − F
− 1 ⋅
v 1
v 2
C:\JLAW\CLASSES\S05PSSII page 2 of 5 January 24, 2005
0 50 100 150 200 250 300 350
0
5
10
15
20
25
z 1
( )t
z 2
( )t
t
z 2
( )t z 2
e
λ 2
⋅t
z := ⋅ 1
( )t z 1
e
λ 1
⋅t
λ 1
z
z0 ψ =
v 1
v 2
dz
dt
= Λ ⋅z
Λ φ =
− 1
:= A ⋅φ
V. Transform to an Orthogonal Space
ψ 2
ψ =( −0.742 0.7) 2
ψ
T
2
T
Left Eigenvector
Associated with (^) λ 1
ψ 1
ψ =( 0.833 0.588) 1
ψ
T
1
T
Left Eigenvector
Associated with (^) λ 1
The program nomalizes the right eigenvectors such that the product of
the left and right eigenvectors associated with the same eigenvalue is
one.
ψ φ⋅
C:\JLAW\CLASSES\S05PSSII page 4 of 5 January 24, 2005
VI. Solution
d 11
φ 1 1, ψ 1 1, ⋅ v 1
⋅ φ 1 1, ψ 1 2, ⋅ v 2
:= + ⋅ d 11
d 12
φ 1 2, ψ 2 1, ⋅ v 1
⋅ φ 1 2, ψ 2 2, ⋅ v 2
:= + ⋅ d 12
d 21
φ 2 1, ψ 1 1, ⋅ v 1
⋅ φ 2 1, ψ 1 2, ⋅ v 2
:= + ⋅ d 21
d 22
φ 2 2, ψ 2 1, ⋅ v 1
⋅ φ 2 2, ψ 2 2, ⋅ v 2
:= + ⋅ d 22
x 1
( )t d 11
e
λ 1
⋅t
⋅ d 12
e
λ 2
⋅t
:= + ⋅ x 2
( )t d 21
e
λ 1
⋅t
⋅ d 22
e
λ 2
⋅t
x 1
x 2
0 50 100 150 200 250 300 350
0
5
10
15
20
x 1
( )t
x 2
( )t
t
C:\JLAW\CLASSES\S05PSSII page 5 of 5 January 24, 2005
