Page 787 - The Mechatronics Handbook

0066_Frame_C24 Page 35 Thursday, January 10, 2002 3:45 PM

Op.Amp.
i (t) R 1
R 3 v (t) R1
+ + v (t)
-
−
v (t)
i v (t) C 3 v (t)
C3 2 R
C 1 o
FIGURE 24.14 Electronic circuit.

And for the right half, we have

di R1 t()  R 1 + R 2 1
---------------- = – ----------------- i R1 t() + -----------------v_ t() (24.204)
dt  C 1 R 1 R 2  C 1 R 1 R 2
v o t() = – R 1 i R1 t() + v_ t() (24.205)

The operational amplifier, in voltage follower connection, ensures that v + (t) = v − (t), so we can combine
the state space models given in Eqs. (24.202)–(24.205):

dv () 1
t
C3
------------------ – ------------ 0 1
dt = R C v C3 t() + ------------
R C v i t()
3 3
di () 1 R + R i R1 t() 3 3 (24.206)
t
R1
1
---------------- ------------------- – ------------------- 2 0
dt C R R C R R
1 1 2 1 1 2
v C3 t()
v o t() = [ 1 – R 1 ] (24.207)
i R1 t()
The observability matrix is given by
1 – R 1
[
ΓΓ Γ Γ C, A] = C = R + R (24.208)
c
1
1
1
CA −------------ – ------------ ------------------ 2
R C 3 R C 1 R C 1
3
2
2
To determine the complete observability, or otherwise, we need to compute the matrix determinant
1
(
[
det ΓΓ ΓΓ C, A]) = ------------------ –( R 1 C 1 + R 3 C 3 ) (24.209)
c
R 3 C 3 C 1
from where we conclude that the model system is completely observable if and only if, R 1 C 1 ≠ R 3 C , 3
which is the same condition we obtained in Example 24.13.
Applying Laplace transform to Eqs. (24.204)–(24.203) we obtain the transfer function from V i (s) to V o (s):
1
1
V o s() V + s() V o s() s + ------------ ------------
R C
R C
⋅
------------ = ------------- ------------- = ----------------------- ------------------ (24.210)
1
3
3
1
R +
V i s() V i s() V − s() s + ------------------ s + ------------
1
R
2
1
R R C 1 R C 3
3
2
1
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