Element Equations
for 2-terminal elements
f R (v, i , t )  0
v
i
fC (v, q, t )  0
q
i  q
v  
f m ( , q, t )  0
memristor
inductor
capacitor
resistor
f L ( , i, t )  0
Ø
Resistor: An element that has an algebraic equation between v and i.
Inductor: An element that has an algebraic equation between Ø and i.
Capacitor: An element that has an algebraic equation between v and q.
Memristor: An element that has an algebraic equation between Ø and q.
2-Terminal Capacitors and Inductors
linear and time-invariant
Capacitor
Inductor
q (t )  C  v(t )
 (t )  L  i (t )
dv(t )
i (t )  C 
dt
di (t )
v(t )  L 
dt
In order to model nonlinear and or time-varying capacitors and
inductors (q ) and ( ) are used:
t
q(t ) ˆ  i ( )d

t
[C]
 (t ) ˆ  v( )d [Wb]

L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
Capacitor
f C ( q, v )  0
charge-controlled
v  vˆ( q )
voltage-controlled
q  qˆ (v)
If qˆ (v ) is
differentiable
dq
d (qˆ (v)) dv
ˆ i 
dt
dv dt
dv C (v) ˆ dqˆ (v)
i  C (v )
dv
dt
Inductor
f L ( , i)  0
flux-controlled
i  iˆ( )
current-controlled
  ˆ(i )
ˆ
If  (i ) is
differentiable
d
d (ˆ(i )) di
ˆ v 
dt
di dt
dˆ(i )
di
v  L(i )
L(i ) ˆ
di
dt
Capacitor
Linear and time-invariant
Inductor
Nonlinear and time-invariant
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
Capacitor
Linear and time-varying
Inductor
q(t )  C (t )v(t )
 (t )  L(t )i(t )
dv(t ) dC (t )
i (t )  C (t )

v(t )
dt
dt
v(t )  L(t )
q(t )  (2  sin t )v(t )
C (t )  2  sin t
 (t )  (2  sin t )i(t )
L(t )  2  sin t
di (t )
dv(t )
v
(
t
)

(
2

sin
t
)
 (cos t )i (t )
i (t )  (2  sin t )
 (cos t )v(t )
dt
dt
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
di (t ) dL(t )

i (t )
dt
dt
First-Order Linear Circuits
RTH iC  vC  vTH EE+KVL
dvC
iC  C
 CvC
dt
vC
vTH

vC  

RTH C RTH C
GN vL  iL  iN
di
vL  L L  LiL
dt
i
i
iL   L  N
GN L GN L
EE+KCL
State Equations, Kalman (1960)
x  ax  bu,
x(0)  x0
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
Solutions of First-Order Differential Equations
x  ax
dx
 ax
dt
x(0)  x0
x (t )

C
t
dx
  adt
x
0
ln x(t) - lnC = at
x(t)
ln
= at
C
x(t )  Ceat
t
x (0)  Cea 0
x(t )  eat x0
x  ax  bu
x (0)  x0
Assumption: x(t )  S (t )e at
d
( S (t )e at )  a ( S (t )e at )  bu(t)
dt
dS(t) at
at
aS (t )e 
e  aS (t )e at  bu(t)
dt
t
dS(t) at
e  bu(t)
dt
t
dS(t)
 e  at bu(t)
dt
Assumption: x(t )  S (t )e
S (t )   e  a bu()d  S (0)
0
t
x (t )  S (0)e at   e a ( t  ) bu ( ) d ,
at
t
0
0
0
x (0)  S (0)e a 0   e a ( t  ) bu ( ) d ,
0
t
x (t )  e at x (0)   e a ( t  ) bu ( ) d
0
t
x (t )  e x (0)   e
at
a ( t  )
bu ( ) d
0
zero-input
response
zero-state
response
t
Solutions of First-Order Differential Equations
x  ax  bu
x (0)  x0
t
x (t )  e at x (0)   e a ( t  ) bu ( ) d
0
zero-input
response
zero-state
response
If a<0 then z.i. response and probably some part of the z.s. Response
converge to zero as time goes to infinity. This part of the solution is
called transient solution (geçici çözüm). The remaining part in z.s.
solution (which does not converge to zero) is called the steady state solution
t
(kalıcı çözüm).
x (t )  e at x (0)   e a ( t  ) bu ( ) d
0
transient
solution
steady
solution
First-Order Linear Circuits
x  ax  bu,
x(0)  x0
t
x(t )  e x(0)   e a ( t  )bu ( )d
at
0
RTH
vC t   vC 0   e
VTH

t
RTH C
 vTH  vTH  e
vC t   vC 0   vTH  e
RTH iC  vC  vTH


t
RTH C
t
RTH C
transient
solution
dvC
 CvC
dt
vC
vTH
vC = +
RTH C RTH C
iC  C
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
 vTH
steady
solution
An Example
Find the transient dynamic behaviour of the voltage buffer!
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York
An Example
L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York