Download Aim is to obtain to obtain state equations in the following form

Higher Order Circuits – How To Obtain State Equations?
Consider a circuit with capacitor, inductors, n-terminal resistors and
independent sources.
Aim is to obtain to obtain state equations in the following form:
x  Ax  Bu
y  Cx  Du ,
x  Rn state variables
y  R r output variables
u  R p input variables
Method:
x(0)  x0
- some capacitor voltages and some inductor
currents (mostly all)
- currents and voltages under consideration
- values for independent sources
Step 1: Draw the circuit graph and choose an appropriate tree:
a-) Choose voltage sources as twigs, if tree is not completed
continue with (b).
b-) Choose capacitors as twigs (you may have to exclude some capacitors), if
tree is not completed continue with (c).
c-) Choose current-controlled edges of resistors as twigs, if
tree is not completed continue with (d).
d-) choose inductors as twigs.
Step 2: Determine the state variables:
capacitor voltages in twigs and inductor currents in chords.
Write element equations for state variables:
C
dvc (t )
 ic (t )
dt
L
diL (t )
 vL (t )
dt
Step 3: Calculate state equations:
Write linearly independent current equations:
KCL’s for fundamental cut-sets
Write linearly independent voltage equations:
KVL’s for fundamental loops
Using these equations obtain currents of capacitors in twigs and
voltages of inductors in chords in terms of state variables and
inputs.
Which elements are these?
An Example
Obtain the state equations for the following circuit.
L
R
+
e(t)
C
j(t)
An Example
Write state equations for the following circuit.
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
Solutions of 2. Order Differential Equations
 x1   a11
 x   a
 2   21
a12   x1   b1 
  u,



a22   x2  b2 
x  Ax  Bu ,
x(t0 )  x0
x(t0 )  x0
Solution can be obtained as the sum of the solution of the homogeneous
equation and the particular solution:
xT (t) = xh (t) + x par (t)
x  Ax,
Homogeneous equation:
xh (t )  Se
t
x(t0 )  x0
 S1  t
  e
 S2 
Set  ASet
S  AS
I  AS  0
detI  A  0
exponential solution as a guess
What should we find now?
S=0 is of course a solution, but can we
find a nonzero solution too?
2  a  b  0
Characteristic Equation
Roots of the characteristic polynomial:
1, 2
Also need to find S : eigenvectors
1I  AS1  0
2 I  AS2  0
S1  c1V1
S2  c2V2
eigenvalues
An element of
Solve this to find the eigenvector for
Solve this to find the eigenvector for

1t
xh (t )  e V1
1
2

 c1 
 c1 
e V2    M (t )  
c2 
c2 
2t
M (t )
Particular Solution: (should be found by quess method)
é x (t) ù
1par
ê
ú
x par (t) =
ê x2 (t) ú
par
ë
û
Total solution: x(t) = M (t)C + x par (t)
x(t0 ) = M(t0 )C + x par (t0 )
C = M (t0 )-1 éë x(t0 ) - x par (t0 )ùû
x(t) = M (t)M -1 (t0 )éë x(t0 ) - x par (t0 )ùû + x par (t)
x(t) = M (t)M -1 (t0 )éë x(t0 ) - x par (t0 )ùû + x par (t)
 (t )
State Transition
Matrix
x(t) = F(t)x(t0 ) + x par (t) - F(t)x par (t0 )
x(t) = F(t)x(t0 ) + x par (t) - F(t)x par (t0 )
zero-input
solution
zero-state
solution
An Example
Solve the state equations for the following circuit.
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
An Example
Write state equations for the following circuit!
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
An Example
Draw the voltage of the capacitor for different initial values!
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
A constant solution of a dynamical system: Equilibrium
x (t )  Ax(t )
x (t )  f ( x(t ))
0  Axd
0  f ( xd )
How many
equilibria are
there?
What happens near the equlibrium?
Definition: Lyapunov stability
Let xd be an equilibrium of the system given by x (t )  f ( x(t )) .
The equilibrium is Lyapunov stable if for every   0 there exists a
 ( )  0 such that
x(0)  xd   ( )

x(t )  xd  
t  0.
A Lyapunov stable equilibrium xd is asymptotically stable if there
exists a   0 such that x(0)  xd    lim x(t )  xd  0 .
t 
Zero-input solution:
eigenvalues
x zi (t )  e 1 (t to ) S1c1  e 2 (t to ) S 2 c2  .....e n (t to ) S n cn
eigenvectors
How do eigenvalues and eigenvectors affect the solution?
.............................................................................................................
Solutions of Linear Systems
S. Haykin, “Neural Networks- A Comprehensive Foundation”2nd Edition, Prentice Hall, 1999,New Jersey.
What is common in all these systems?