Unformatted text preview: Fig. 13.2.
In t he network in Fig. 13.2, we replace t he i nductor b y a c urrent source o f c urrent X l
a nd t he c apacitor by a voltage source of voltage X 2, as shown i n Fig. 13.3. T he r esulting
network consists o f five resistors, two voltage sources, a nd o ne current source. We can
determine t he voltage v L across t he i nductor and t he c urrent i c t hrough t he c apacitor by
u sing t he principle o f s uperposition. This s tep c an be accomplished by inspection. For
example, V L h as t hree c omponents arising from three sources. To compute t he c omponent
d ue t o f , we a ssume t hat X l = 0 ( open circuit) a nd X 2 = 0 ( short circuit). Under these
conditions, all of t he network t o t he r ight of t he 2 0 r esistor is opened, a nd t he c omponent
o f V L due t o f is t he voltage across t he 2 0 resistor. T his v oltage is clearly ~f· Similarly,
t o find t he c omponent of V L due t o X l, we s hort f a nd X 2. T he source X l sees a n equivalent
resistor of 1 0 across it, a nd hence V L =  Xl. C ontinuing t he process, we find t hat t he
c omponent of V L d ue t o X 2 is  X2. Hence VL .
I
= X l ="2 f  Xl  X2 (13.12a) Using the same procedure, we find
(13.12b) I n S ec. 6.6 w e s aw t hat a g iven t ransfer f unction c an b e r ealized i n s everal w ays.
C onsequently, w e s hould b e a ble t o o btain d ifferent s tatespace d escriptions o f t he
s ame s ystem b y u sing d ifferent r ealizations. T his a ssertion will b e c larified b y t he
f ollowing e xample.
• E xample 1 3.4
D etermine t he s tatespace description of a s ystem specified by t he t ransfer function
Hs _
( )  s3 2 s + 10
+ 8s 2 + 19s + 12 (13.15a) (13.15b)
!
2
~
= _3_ ___ + _3_ s +1 8 +3 s +4 (13.15c) tThis procedure requires modification if the system contains allcapacitor voltage source tie sets
or allinductor current source cut sets. In the case of allcapacitor voltage source tie sets, all
capacitor voltages cannot be independent. One capacitor voltage can be expressed in terms of
the remaining capacitor voltages and the voltage source(s) in that tie set. Consequently, one of
the capacitor voltages should not be nsed as a state variable, and that capacitor should not be
replaced by a voltage source. Similarly, in allinductor current source tie sets, one inductor should
not be replaced by a current source. If there are aIIcapacitor tie sets or allinductor cut sets only,
no further complications occur. In allcapacitorvoltage source tie sets and/or a11inductorcurrent
source cut sets, we have additional difficulties in t hat the terms involving derivatives of the input
may occur. Thls problem can be solved by redefining the state variables. The final s tate variables
will not be capacitor voltages and inductor currents. a
792 13 S tateSpace A nalysis 13.2 A S ystematic P rocedure for D etermining S tate E quations 7 93 Using t he p rocedure developed in Sec. 6.6, we s hall realize H{s) i n Eq. (13.15) w ith
four different realizations: (i) t he c ontroller canonical form [Eq. (13.15a)], (ii) t he o bs...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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