Second Order Circuits
Table 1: Natural Response of Parallel RLC
Revised 4/13/11
Circuit
Diff-Eq
d 2v
dv
 2
 02v  0
2
dt
dt
where  
1
2RC
1
LC
and 0 
Damping
Over: 20 < 2
Under: 20 > 2
Form of Soln
v(t )  A1 e s1t  A2 e s2t
v(t )  e t ( B1 cos d t  B2 sin d t )
Roots
s1, 2      2  0
Boundary
Conditions
2
𝑠1,2 = −𝛼 ± 𝑗𝜔𝑑
Critical: 20 = 2
, d

v(t )  ( A2  A1t ) e t
s1, 2   
02   2
𝑣(0 +) = 𝐴1 + 𝐴2
𝑣(0 +) = 𝐵1
𝑣(0 +) = 𝐴2
𝑑𝑣(0 +) 𝑖𝐶 (0 +)
=
= 𝑠1 𝐴1 + 𝑠2 𝐴2
𝑑𝑡
𝐶
𝑑𝑣(0 +) 𝑖𝐶 (0 +)
=
= −𝐵1 + 𝑑 𝐵2
𝑑𝑡
𝐶
𝑑𝑣(0 +) 𝑖𝐶 (0 +)
=
= 𝐴1 − 𝐴2
𝑑𝑡
𝐶
Table 2: Step Response of Parallel RLC
Circuit
Diff-Eq
Damping
Form of Soln
Roots
Boundary
Conditions
2
d 2i 1 di di v I s dv
 02v  0


 2
dt 2 LC dt
dt 2 RC dt LC
Over: 20 < 2
𝑖(𝑡) = 𝐼𝑠 + 𝐴1 𝑒 𝑠1 𝑡 + 𝐴2 𝑒 𝑠2 𝑡
s1, 2      2  0
2
where  
1
2RC
and 0 
1
LC
Under: 20 > 2
Critical: 20 = 2
𝑖(𝑡) = 𝐼𝑠 +
𝑒 −∝𝑡 (𝐵1 cos 𝜔𝑑 𝑡 + 𝐵2 sin 𝜔𝑑 𝑡)
𝑖(𝑡) = 𝐼𝑠 + 𝐴1 𝑡 𝑒 −∝𝑡 + 𝐴2 𝑒 −∝𝑡
𝑠1,2 = −𝛼 ± 𝑗𝜔𝑑
, d

02   2
s1, 2   
𝑖(0 +) = 𝐼𝑠 + 𝐴1 + 𝐴2
𝑖(0 +) = 𝐼𝑠 + 𝐵1
𝑖(0 +) = 𝐼𝑠 + 𝐴2
𝑑𝑖(0 +) 𝑣𝐿 (0 +)
=
= 𝑠1 𝐴1 + 𝑠2 𝐴2
𝑑𝑡
𝐿
𝑑𝑖(0 +) 𝑣𝐿 (0 +)
=
= −𝐵1 + 𝑑 𝐵2
𝑑𝑡
𝐿
𝑑𝑖(0 +) 𝑣𝐿 (0 +)
=
= 𝐴1 − 𝐴2
𝑑𝑡
𝐿
Second Order Circuits
Table 3: Natural Response of Series RLC
Circuit
Diff-Eq
Damping
Form of Soln
Roots
Boundary
Conditions
d 2i
di
 2  02i  0
2
dt
dt
Over: 20 < 2
𝑖(𝑡) = 𝐴1 𝑒 𝑠1 𝑡 + 𝐴2 𝑒 𝑠2 𝑡
s1, 2      2  0
2

R
2L
1
LC
and 0 
Under: 20 > 2
Critical: 20 = 2
𝑖(𝑡) =
𝑒 −∝𝑡 (𝐵1 cos 𝜔𝑑 𝑡 + 𝐵2 sin 𝜔𝑑 𝑡)
𝑖(𝑡) = 𝐴1 𝑡 𝑒 −∝𝑡 + 𝐴2 𝑒 −∝𝑡
𝑠1,2 = −𝛼 ± 𝑗𝜔𝑑
, d
02   2

s1, 2   
𝑖(0 +) = 𝐴1 + 𝐴2
𝑖(0 +) = 𝐵1
𝑖(0 +) = 𝐴2
𝑑𝑖(0 +)
= 𝑠1 𝐴1 + 𝑠2 𝐴2
𝑑𝑡
𝑑𝑖(0 +)
= −𝐵1 + 𝑑 𝐵2
𝑑𝑡
𝑑𝑖(0 +)
= 𝐴1 − 𝐴2
𝑑𝑡
Table 4: Step Response of Series RLC
Circuit
Diff-Eq
vs
d 2 v R dv v



dt 2 L dt LC LC
Damping
Over: 20 < 2
Form of Soln
𝑣(𝑡) = 𝑉𝑠 + 𝐴1 𝑒 𝑠1 𝑡 + 𝐴2 𝑒 𝑠2 𝑡
Roots
s1, 2      2  0
Boundary
Conditions
2

R
2L
1
LC
and 0 
Under: 20 > 2
Critical: 20 = 2
𝑣(𝑡) = 𝑉𝑠 +
𝑒 −∝𝑡 (𝐵1 cos 𝜔𝑑 𝑡 + 𝐵2 sin 𝜔𝑑 𝑡)
𝑣(𝑡) = 𝑉𝑠 + 𝐴1 𝑡 𝑒 −∝𝑡 + 𝐴2 𝑒 −∝𝑡
𝑠1,2 = −𝛼 ± 𝑗𝜔𝑑
, d

02   2
s1, 2   
𝑣(0 +) = 𝑉𝑠 + 𝐴1 + 𝐴2
𝑣(0 +) = 𝑉𝑠 + 𝐵1
𝑣(0 +) = 𝑉𝑠 + 𝐴2
𝑑𝑣(0 +)
= 𝑠1 𝐴1 + 𝑠2 𝐴2
𝑑𝑡
𝑑𝑣(0 +)
= −𝐵1 + 𝑑 𝐵2
𝑑𝑡
𝑑𝑣(0 +)
= 𝐴1 − 𝐴2
𝑑𝑡