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First Order Circuit
• Capacitors and inductors
• RC and RL circuits
RC and RL circuits (first order circuits)
Circuits containing no independent sources
•
Excitation from stored energy
•
‘source-free’ circuits
•
Natural response
Circuits containing independent sources
•
DC source (voltage or current source)
•
Sources are modeled by step functions
•
Step response
•
Forced response
Complete response = Natural response + forced response
RC circuit – step response
t=0
R
Vs
+
vc

Taking KCL,
C
Objective of analysis: to find expression for vc(t)
for t >0 , i.e. to get the voltage response of the
circuit to a step change in voltage source OR
simply to get a step response
dv c v c  Vs

0
dt
R
t dt
dv c


v c (0) v  V  o RC
c
s
vc (t)
For vc(0) = Vo,
For vc(0) = 0,
v  Vs 
dv c
 c
dt
RC


ln
t

dv c
dt

v c  Vs  RC
v c ( t )  Vs
1

t
v c (0)  Vs
RC
v c (t )  Vs  Vo  Vs e


v c ( t )  Vs (1  e )

1
t
RC
, where  = RC = time constant
RC circuit – step response
vc(t)
Vs
0.632Vs

2
3
4
t
5
Vs -- is the final value i.e. the capacitor voltage as t  
In practice vc(t) considered to reach final value after 5
When t = , the voltage will reach 63.2%
of its final value
t

v c ( t )  Vs (1  e  )
RL circuit – step response
t=0
Objective of analysis: to find expression for iL(t)
for t >0 , i.e. to get the current response of the
circuit to a step change in voltage source OR
simply to get a step response
R
Vs
+
vL

iL(t)
di
 Vs  iLR  L L  0
dt
Taking KCL,

iL ( t )
iL

V 
di L
R
   iL  s 
dt
L
R

t
di L
R
   dt
V 
L
(0) 
0
 iL  s 
R


ln
iL ( t ) 
di L
R
  dt
  Vs 
L
 iL  
R

Vs
R  R t
V
L
iL (0)  s
R
V 
V   t
iL (t )  s  Io  s e L
R 
R
R
For iL(0) = Io,
t
For iL(0) = 0,

V
iL (t )  s (1  e  )
R
, where  = L/R = time constant
RL circuit – step response
iL(t)
Vs/R
0.632(Vs/R)

2
3
4
5
t
(Vs/ R) -- is the final value i.e. the inductor current as t  
In practice iL(t) considered to reach final value after 5
t
When t = , the current will reach
63.2%
of its final value

V
iL (t ) 
s
R
(1  e  )
The complete response
•
The combination of natural and step (or forced) responses
•
For RC circuit, the complete response is:

t


t

v c (t )  Vo e  Vs (1  e )
Natural response:
• Response due to initial energy
stored in capacitor
• Vo is the initial value, i.e. vc(0)
Forced response:
• Response due to the present
of the source
• Vs is the final value i.e vc()
Note: this is what we obtained when we solved the step
response with initial energy (or initial voltage) at t =0
The complete response
•
Complete response is also can be written as the combination of
steady state and transient responses:
v c (t )  Vs  ( Vo  Vs )e
Steady state response:

t

Transient response:
• Response that exist long after the
excitation is applied
• Response that eventually
decays to zero as t  
• For DC excitation, this is the term
in the complete response that
does not change with time
• For DC excitation, this is the
term in the complete response
that changes with time
• This is the final value, (i.e. vc())
• Vo is the initial value (i.e. vc(0))
and Vs is the final value (i.e. vc())
The complete response
Complete response of an RL circuit can be written as:

t

t

Vs
iL (t )  Io e  (1  e  )
R
Vs
Vs
iL (t ) 
 (Io  )e
R
R

t

(natural response) + (forced response)
(steady state response) + (transient response)
The General Solution
In general, the response to all variables (voltage or current) in RC or RL
circuit can be written as:
x(t )  x()  x(t o )  x()e

t  t o 

•
x(t) can be v(t) or i(t) for any branch of the RC or RL circuit
•
x() – final value of x(t) (long after to)
•
x(to) – initial value of x(t) – for continuous variables, x(to+) = x(to-)
For to = 0, the equation becomes :
x(t )  x()  x(0)  x()e

t
