Summary for CH#8
Natural and Step Responses of RLC Circuits
•
The characteristic equation of the following RLC circuits,
is:
s 2 + 2α s + ωo 2 = 0
where ,
α=
1
( parallel )
2RC
α=
R
(series )
2L
ωo =
1
(for parallel & series )
LC
s1,2 = −α ± α 2 − ωo 2
•
Depending on the values of α 2 & ωo 2 the natural and step response of the series &
parallel RLC circuits can be classified as follow:
Type
Overdamped
Roots (S1,2)
Real, distinct
Condition
α 2 > ωo 2
Underdamped
Complex
α 2 < ωo 2
Critically damped
Real, repeated
α 2 = ωo 2
•
For natural response:
Type
Overdamped
Equations
x (t ) = A1e s1t + A 2e s 2t
Underdamped
x (t ) = (B l cos ωd t + B 2 sin ωd t )e −αt
Initial Conditions
x (0) = A1 + A 2 ;
dx (0) = i c (0) = A s + A s
1 1
2 2
dt
C
x (0) = B 1 ;
dx (0) = i c (0) = −α B + ω B ,
d
1
2
dt
C
where : ωd = ω0 2 − α 2 (W arning !!)
Critically damped
•
x (t ) = (D1t + D 2 )e −αt
x (0) = D 2
dx (0) = i c (0) = D − α D
1
2
dt
C
For step response: ( See examples 8.6-8.10)
Type
Overdamped
Equations
x (t ) = X f + A '1 e s1t + A '2 e s 2t
Underdamped
x (t ) = X f + (B 'l cos ωd t + B '2 sin ωd t
Critically damped
x (t ) = X f + (D '1 t + D '2 )e −αt
Initial Conditions
x (0) = X f + A '1 + A '2 ;
dx (0) = i c (0) = A ' s + A ' s
1 1
2 2
dt
C
x (0) = X f + B '1 ;
dx (0) = i c (0) = −α B ' + ω B ' ,
1
2
d
dt
C
x (0) = X f + D '2
dx (0) = i c (0) = D ' − α D '
1
2
dt
C
Two Integrator Amplifier: (see example 8.13)
Applying KCL at the inverting terminals result in the following:
dv o
1
=−
v g __________________________________1
dt
R1C 1
dv o
1
=−
v o1
dt
R 2C 2
Differentiating
⇔
d 2v o
1 dv o 1
=−
_________ 2
2
dt
R 2C 2 dt
From 1& 2 :
d 2v o
1
1
=
vg
2
dt
R1C 1 R 2C 2
•
Two Integrating Amplifiers with Feedback Resistors:
The reason for adding the feedback resistors is the fact that the op amp in the integrating
amplifier saturates because of the feedback capacitor’s accumulation of charge. A resistor is
placed in parallel with each feedback capacitor (C1 and C2) to overcome this problem.
vg
d 2v o  1 1  dv o  1 
+ + 
+
v o =
2
dt
R aC 1R bC 2
 τ 1 τ 2  dt  τ 1τ 2 
The characteristic equation :
1 1
1
=0
s 2 +  + s +
 τ 1 τ 2  τ 1τ 2
The roots :
s1 = −
1
τ1
; s2 = −
1
τ2
Example 8.14 illustrates the analysis of the step response of two cascaded integrating
amplifiers when the feedback capacitors are shunted with feedback resistors.