State Space Approach to Solving RLC circuits
[Pages:14]State Space Approach to Solving RLC circuits
Eytan Modiano
Eytan Modiano Slide 1
Learning Objectives
? Analysis of basic circuit with capacitors and inductors, no inputs, using
state-space methods
? Identify the states of the system ? Model the system using state vector representation ? Obtain the state equations
? Solve a system of first order homogeneous differential equations using
state-space method
? Identify the exponential solution ? Obtain the characteristic equation of the system ? Obtain the natural response of the system using eigen-values and vectors ? Solve for the complete solution using initial conditions
Eytan Modiano Slide 2
Second order RC circuits
e1
R1
+ i1
v1
C1
-
e3
R3
R2
e2
i2 +
C2
v2
-
R1 = R2= R3 = 1 C1 = C2= 1F
Eytan Modiano Slide 3
!!!!!!!i1
=
C1
dv1 dt
!!!!!!i2
=
C2
dv2 dt
Node equations :
e1 :!!!i1 + (e1 ! e3 ) / R1 = 0 e2 :!!!!(e2 ! e3 ) / R2 +!i2 = 0 e3 :!!!(e3 ! e1) / R1 + (e3 ! e2 ) / R2 + e3 / R3 = 0
State of RLC circuits
? Voltages across capacitors ~ v(t) ? Currents through the inductors ~ i(t) ? Capacitors and inductors store energy
? Memory in stored energy ? State at time t depends on the state of the system prior to time t ? Need initial conditions to solve for the system state at future times
E.g, given state at time 0, can obtain the system state at timest > 0 State at time 0 ~ v1(0), v2(0), etc.
Eytan Modiano Slide 4
State equations for RLC circuits
? We want to obtain state equations of the form: x"!(t) = f (x"(t))
? Where f is a linear function of the states
? In our example,
x(t
)
=
!"#vv12
(t ) $ (t )%&
,!!and!we!need!to!find,
d dt
v1(t) =
d f1(v1(t), v2 (t))!!and !! dt
v2 (t) =
f2 (v1(t ), v2 (t ))
Eytan Modiano Slide 5
Obtaining the state equations
We!have,
d dt
v1(t )
=
i1(t)!!and !! d
C1
dt
v2 (t )
=
i2 (t ) C2
? So we need to find i1(t) and i2(t) in terms of v1(t) and v2(t)
? Solve RLC circuit for i1(t) and i2(t) using the node or loop method
? We will use node method in our examples
e1 :!!!i1 + (e1 ! e3 ) / R1 = 0 e2 :!!!!(e2 ! e3 ) / R2 +!i2 = 0 e3 :!!!(e3 ! e1) / R1 + (e3 ! e2 ) / R2 + e3 / R3 = 0
?
Note that e2, e3
the
equations
at
e1
and
e2
give
us
i1
and
i2
directly
in
terms
of
e1,
? Also note that v1 = e1 and v2 = e2
e = e +3 e ? Equation at e3 gives e3 in terms of e1 and e2
12 3
Eytan Modiano Slide 6
Obtaining the state equations...
? We now have,
dv1 dt
=
i1
=
!2 3 v1
+
1 3
v2
;!!
dv2 dt
=
i2
=
1 3 v1
!
2 3 v2
"#$vv!!12
% &'
=
"!2 / 3 #$ 1 / 3
1/ !2
3% / 3&'
" #$
v1 v2
% &'
? Guessing an exponential solution to the above ODE's we get,
Eytan Modiano Slide 7
v1(t ) = E1est ,!v2 (t ) = E2est
E1sest
+
2 E1e st 3
!
E2est 3
= 0 " E1(s + 2 / 3) ! E2
/3= 0
E2 sest
!
E1est 3
+
2 E2 e st 3
= 0 " !E1
/ 3 + E2 (s + 2 / 3) = 0
The non-trivial solution
"s + 2 / 3 #$ !1 / 3
!1 / s+2
3% / 3&'
" #$
E1 E2
% &'
=
0
? The above equations have a non-trivial (non-zero) solution if
equations are linearly dependent. From linear algebra we know
this implies:
"s + 2 / 3 det #$ !1 / 3
!1 / s+2
3% / 3&'
=
0
(
(s
+
2
/
3)2
!
1
/
9
=
0
s2
+
4 3
s
+1/
3
=
0
(
s1
=
!1,! s2
=
!1 /
3
Eytan Modiano Slide 8
s1
=
!1
(
v1(t )
=
E s1 1
e!
t
,!!
v2
(t
)
=
E s1 2
e!
t
s2
=
!1 /
3(
v1(t )
=
E s2 1
e!
t
/
3
,!!
v2
(t
)
=
E e s2 !t /3 2
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- rlc and series rlc circuits pitt
- lecture 6 parallel resonance and quality factor transmit
- rlc circuit response and analysis using state
- the rlc circuit transient response series rlc circuit
- chapter 8 natural and step responses of rlc circuits
- state space approach to solving rlc circuits
- ee101 rlc circuits with dc sources
- characteristics equations overdamped underdamped and
- chapter 21 rlc circuits
- parallel rlc second order systems
Related searches
- philosophical approach to life
- best approach to problem solving
- aristotelian approach to ethics
- approach to learning activities
- trait approach to leadership pdf
- holistic approach to lupus
- topical approach to lifespan development
- human behavior approach to management
- social justice approach to ethics
- biological approach to personality
- philosophical approach to teaching
- philosophical approach to multicultural education