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Chapter 7.3
Natural and Step Responses of RLC Circuits
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Initial conditions: v 0 !( ) = v 0 +( ) , dv 0 +( )dt
=i 0 +( )C
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Initial conditions: i 0 !( ) = i 0 +( ) , di 0 +( )dt
=v 0 +( )L
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■ In dc steady state, a capacitor looks like an open circuit and an inductor looks like a short circuit.
■ The voltage across a capacitor must be a continuous function of time
■ The current flowing through an inductor must be a continuous function of time
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I. Parallel RLC Circuit: Step Response and Natural Response
II Series RLC Circuit: Step Response and Natural Response
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I. Parallel RLC Circuit: Step Response and Natural Response
iL + iR + iC = I
iL 0 +( )+ 1L
v z( )dz0
t
!"#$
%&'+vR+C
dvdt
= IiL +vR+C
dvdt
= I
iL +LRdiLdt
+ LCd 2iLdt2
= I
•
•
d 2iLdt2
+1RC
diLdt
+1LC
iL =ILC
•
•
• vL+1Rdvdt
+Cd2vdt2
= 0
d 2vdt2
+1RC
dvdt
+1LC
v = 0•
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Total Response =
{ zero-input response = natural response} + { zero-state response = forced response }
! Steady- State Response (Limit Response)
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I-1 The Natural Response of a Parallel RLC Circuit
d2vdt 2
+ 1RC
dvdt
+ 1LC
v = 0
Second-Order, Constant - Coefficient, Homogenous, Linear DE
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A Parametrization and General Solution
d2vdt 2
+ 1RC
dvdt
+ 1LC
v = 0
! = 12RC
- Neper frequency (rad/sec)
! 0 =1LC
- Resonant radian frequency (rad/sec)
! = "# 0 ! ! - damping ratio
d 2vdt2
+ 2!dvdt
+" 02v = 0
d2vdt 2
+ 2!" 0dvdt
+" 02v = 0
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Let us seek the solution in the form: v t( ) = Aest , s can be complex
Substitute into the ODE, we got an algebraic (characteristic) equation
d2vdt 2
+ 2! dvdt
+" 02v = 0 ! s2 + 2!s +" 0
2 = 0
s2 + 2! s+" 02 = 0 Characteristic Equation
s1 = !! + ! 2 !" 0
2
s2 = !! ! ! 2 !" 02
"#$
%$
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Since the ODE is linear ! linear combination of solutions remains a solution to the equation. The general solution for v t( )must be of the form
v t( ) = A1es1t + A2es2t ,
where A1 and A2 will be determined by the two initial conditions
v 0 +( ), dv 0 +( )dt
• v 0 !( ) = v 0 +( ) , dv 0 +( )dt
=i 0 +( )C
• ! circuit theory
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• •
•
•
•
s2 = !4s1 = !2
overdamped underdampeds1 = !3+ j4
s1 = !3! j4
s1 = s2 = !1critically damped
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B The Forms of the Natural Response ! = 1
2RC, ! 0 =
1LC
B1 The Overdamped Voltage Response: ! 2 >" 02
v t( ) = A1e! !! ! 2!"0
2{ }t + A2e! !+ ! 2!"0
2{ }t ! A1es1t + A2e
s2t
A1 , A2 - obtained from the initial conditions ! v 0 +( ), dv 0 +( )dt
!circuit th.
v 0 +( ) = A1 + A2dv 0 +( )dt
= s1A1 + s2A2
!"#
$#
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The two initial conditions: need v 0 +( ) and dv 0 +( )dt
v 0 +( ) - the initial voltage on the capacitor ! V0
Cdv 0 +( )dt
= iC 0 +( ) ! dv 0 +( )dt
= 1CiC 0 +( )
KVL at 0 + : iC 0 +( ) = !V0R! I0
dv 0 +( )dt
= ! V0CR
! I0C
- the initial voltage change on the capacitor
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B2 The Underdamped Voltage Response: ! 02 >" 2 ! = 1
2RC, ! 0 =
1LC
s1 = !! + ! 2 !" 02 = !! + j " 0
2 !! 2
s2 = !! ! ! 2 !" 02 = !! ! j " 0
2 !! 2 ! s2 = s1"
! d = ! 0
2 !" 2 - damped radian frequency (rad/sec)
v = B1e!"t cos # dt( )+ B2e!"t sin # dt( )
• The response has the oscillatory damped character if ! > 0 • If ! = 0 (R!" ) ! max. oscillations with ! d !! 0
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B1 , B2 - obtained from the initial conditions ! v 0 +( ), dv 0 +( )dt
v 0 +( ) = B1dv 0 +( )dt
= !!B1 +" dB2
"#$
%$
! v t( ) = B1e!!t cos " dt( )+ B2e!!t sin " dt( )
• v 0 !( ) = v 0 +( ) , dv 0 +( )dt
=i 0 +( )C
• ! circuit theory
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Euler’s Formula Euler, L. Introductio in Analysin Infinitorum, Vol. 1. Bosquet, Lucerne, Switzerland, 1748
The Master of Us All
e jx = cos x( )+ j sin x( )
e! jx = cos x( )! j sin x( )
cos x( ) = ejx + e! jx
2 , sin x( ) = e
jx ! e! jx
2 j
e j! +1= 0
How many mathematicians does it take to change a light bulb?
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B3 The Critically Damped Voltage Response: ! 02 =" 2 ! 0 =
1LC
! = 12RC
s1 = !! + ! 2 !" 02 = !! !
s2 = !! ! ! 2 !" 02 = !! R = 1
2LC
v t( ) = D1te!"t + D2e!"t
D1 , D2 - obtained from the initial conditions ! v 0 +( ), dv 0 +( )dt
! circuit th.
v 0 +( ) = D2
dv 0 +( )dt
= D1 !!D2
"#$
%$
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• •
•
•
•
s2 = !4s1 = !2
overdamped underdampeds1 = !3+ j4
s1 = !3! j4
s1 = s2 = !1critically damped
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Examples of Second-Order Linear Homogenous Differential Equations with Constant Coefficients
1. y 2( ) +10y 1( ) + 21y = 0
Characteristic Equation: s2 +10s + 21= 0
! s2 +10s + 21= s + 3( ) s + 7( ) = 0 ! s1 = !3 , s2 = !7
Solution: y t( ) = C1e!3t +C2e!7t
2. y 2( ) + 7y 1( ) = 0 ! s2 + 7s = 0 ! s1 = 0 , s2 = !7 ,
Solution: y t( ) = C1 +C2e!7t
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3. y 2( ) + 4y 1( ) + 5y = 0 ! s2 + 4s + 5 = 0
! s1,2 =!4 ± 42 ! 4 " 5
2 ! s1 = !2 + j , s2 = !2 ! j
Solution: y t( ) = C1e!2t cos t( )+C2e!2t sin t( )
4. y 2( ) + 4y = 0 ! s2 + 4 = 0 ! s1,2 = ±2 j
Solution: y t( ) = C1 cos 2t( )+C2 sin 2t( )
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5. y 2( ) + 8y 1( ) +16y = 0 ! s2 + 8s +16 = 0 ! s + 4( )2 = 0
! s1 = s2 = !4
Solution: y t( ) = C1e!4 t +C2te!4 t
6. y 2( ) = 0 ! s2 = 0 ! s1 = s2 = 0
Solution: y t( ) = C1e0!t +C2te0!t = C1 +C2t
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I-2 The Step Response of a Parallel RLC Circuit
d2iLdt 2
+ 1RC
diLdt
+ 1LC
iL =ILC
iL = iLnatural + I
!
d2iLdt 2
+ 2! diLdt
+" 02iL = 0
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• iL = I + A1es1t + A2e
s2t - overdamped
• iL = I + B1e!!t cos " dt( )+ B2e!!t sin " dt( ) - underdamped
• iL = I + D1te
!!t + D2e!!t - critically damped
The required constants can be found from iL 0 +( ) , diL 0 +( )dt
• iL 0 !( ) = iL 0 +( ) , diL 0 +( )dt
=v 0 +( )L
• ! circuit theory
! v 0 +( ) = v 0 !( )
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Example
I = 24mA
1. Finding the Overdamped Step (DC) Response of a Parallel RLC Circuit R = 400!
the initial energy stored in the circuit is zero ! iL 0 +( ) = 0 , vC 0 +( ) = 0
! = 12RC
, ! 0 =1LC
?
! ! 2 >" 02
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(a) iL 0 +( ) , diL 0 +( )dt
I = 24mA
the initial energy stored = 0 ! iL 0 +( ) = 0
! vC 0 +( ) = 0 and vC 0 +( ) = v 0 +( ) = L diL 0 +( )dt
! diL 0 +( )dt
= 0
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(b) Characteristic Equation ! = 12RC
, ! 0 =1LC
I = 24mA
• ! 02 = 1
LC = 1012
25 ! 25 = 16 !108
• ! = 12RC
= 109
2 ! 400 ! 25= 5 !104 ! ! 2 = 25 !108
Since ! 2 >" 02 ! overdamped
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s1 = !! + ! 2 !" 02 = -5 !104+ 3!104 = -20,000
s2 = !! ! ! 2 !" 02 = 5 !104 - 3!104 = -80,000
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(c) iL t( ) iL t( ) = I + A1es1t + A2es2t
iL 0 +( ) = I + A1 + A2 = 0 diL 0 +( )
dt= s1A1 + s2A2 = 0
! A1 = !32mA , A2 = 8mA
iL t( ) = 24 ! 32e!20,000t + 8e!80,000t mA, t ! 0
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2. Finding the Underdamped Step Response of a Parallel RLC Circuit ! 0 =
1LC
R = 625! iL 0 +( ) = 0 , vC 0 +( ) = 0 ! = 12RC
! 0
2 = 16 !108 , ! 2 = 10.24 !108 ! ! 2 <" 02 ! underdamped
s1 = !! + j " 02 !! 2 = !3.2 "104 + j2.4 "104
s2 = !! ! j " 02 !! 2 = !3.2 "104 ! j2.4 "104
iL = I + B1e!!t cos " dt( )+ B2e!!t sin " dt( )
! d = ! 02 !" 2 = 24,000 rad/s
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iL t( ) = I + B1e!!t cos " dt( )+ B2e!!t sin " dt( ) iL 0 +( ) = I + B1 = 0
diL 0( )dt
=! dB2 !"B1 = 0
! B1 = !24mA , B2 = !32mA
iL t( ) = 24 ! 24e!32,000t cos 24,000t( )! 32e!32,000t sin 24,000t( ) mA, t ! 0
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3. Finding the Critically Damped Step Response of a Parallel RLC Circuit ! = 1
2RC , ! 0 =
1LC
R = 500! iL 0 +( ) = 0 , vC 0 +( ) = 0
! 02 = 16 !108 , ! 2 = 16 !108 ! ! 2 =" 0
2 critically damped
iL t( ) = I + D1te!!t + D2e!!t !
iL 0 +( ) = I + D2 = 0
diL 0( )dt
= D1 !!D2 = 0 ! D1 = !960,000mA / s , D2 = !24mA
iL t( ) = 24 ! 960,000te!40,000t ! 24e!40,000t mA, t ! 0
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4. Comparing the Three-Step Response Forms
iL !( ) = 24
•
tod90% =130µs
tcd90% = 97µs
tud90% = 74µs
•
•
•
••
tud90% < tcd
90% < tod90%
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On the basis of the results obtained, which response would you specify in a design that puts a premium on reaching 90% of the final value of the output in the shortest time ?
The underdamped response reaches 90% of the final value in the fastest time, so it is the desired response type when speed is the most important design specification.
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Which response would you specify in a design that must ensure that the final value of the current is never exceeded? From the plot, you can see that the underdamped response overshoots the final value of current, whereas neither the critically damped nor the overdamped response produces currents in excess of 24 mA. Although specifying either of the latter two responses would meet the design specification, it is best to use the overdamped response. It would be impractical to require a design to achieve the exact component values that ensure a critically damped response.
! 2 =" 02 !"! R = 1
2LC
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5. Finding Step Response of a Parallel RLC Circuit with Initial Stored Energy
R = 500! iL 0 +( ) = 29mA , vC 0 +( ) = 50V
I = 24mA
! 0
2 = 16 !108 , ! 2 = 16 !108 ! ! 2 =" 02 ! critically damped
iL t( ) = I + D1te!!t + D2e!!t
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iL t( ) = I + D1te!!t + D2e
!!t
iL 0 +( ) = I + D2 = 29mA
diL 0( )dt
= D1 !!D2 ! LdiL 0 +( )dt
= vC 0 +( ) = 50V
! diL 0 +( )dt
= 2000A / s
iL 0 +( ) = I + D2 = 29mA ! D2 = 5mA
diL 0( )dt
= D1 !!D2 = 2000 ! D1 = 2.2 !106mA / s
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iL t( ) = 24 + 2.2 !106 te"40,000t + 5e"40,000t mA, t ! 0
Also
v t( ) = L diL t( )dt
= !2.2 "106 te!40,000t + 50e!40,000t V, t ! 0
! vC 0 +( ) = 50V
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iL t( ) , v t( )
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II Series RLC Circuit: Step Response and Natural Response
KVL: V = Ri + L didt
+ vC and i = C dvCdt
d2vCdt 2
+ RLdvCdt
+ 1LC
vC = VLC
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• ! parallel =12RC
• ! = R2L
- Neper frequency ! rad/sec
! 0 =1LC
- Resonant radian frequency!rad/sec
! d = ! 02 !" 2 - damped radian frequency
d 2vCdt 2
+ 2! dvCdt
+" 02vC = V
LC
s2 + 2!s +" 02 = 0 ! s1,s2
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• vC =V + A1es1t + A2e
s2t - overdamped
• vC =V + B1e!!t cos " dt( )+ B2e!!t sin " dt( ) - underdamped
• vC =V + D1te
!!t + D2e!!t - critically damped
The required constants can be found from
vC 0 +( ) , dvC 0 +( )dt
! circuit theory
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Example E7. 17 Find i0 t( ) and v0 t( )
v0 t( ) =18i0 t( ) +12
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1. t > 0
KVL: !4 + 11/ 36
i0 z( )dz + vC 0 ±( )0
t
"#$%
&'(+ 2 di0 t( )
dt+18i0 t( )+12 = 0
! d 2i0dt 2
+ 9 di0dt
+18i0 = 0
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d 2i0dt 2
+ 9 di0dt
+18i0 = 0
Ch. Eq: s2 + 9s +18 = 0 ! s1 = !3 , s2 = !6 ! overdamped
! i0 t( ) = A1e!3t + A2e!6t
Need i0 0 +( ) and di0 0 +( )dt
to determine A1,A2
We must go back to t = 0 !
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2. t < 0
vC 0 !( ) = 0
iL 0 !( ) = i0 0 !( ) = 24 !126 +18
= 0.5A
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3. t = 0 +
iL 0 +( ) = iL 0 !( ) = i0 0 +( ) = 0.5A ; vC 0 !( ) = vC 0 +( ) = 0
KVL: !4 ! vC 0 +( )=0
!"# $# + vL 0 +( )+18 i0 0 +( )=0.5!"# +12 = 0 ! vL 0 +( ) = !17 V
Also vL 0 +( ) = L di0 0 +( )dt
! di0 0 +( )dt
= vL 0 +( ) / L = !172
!
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4. A1 , A2 from i0 0 +( ) = 0.5A and di0 0 +( )dt
= !172
i0 t( ) = A1e!3t + A2e!6t
i0 0 +( ) = A1 + A2 , di0 0 +( )dt
= !3A1 ! 6A2
A1 + A2 = 0.5
!3A1 ! 6A2 = !172
"#$
%$ ! A1 = !11/ 6 , A2 = 14 / 6
i0 t( ) = !116e!3t + 14
6e!6t , t > 0
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i0 t( ) = !116e!3t + 14
6e!6t , t > 0
iL 0 !( ) = 0.5A i0 !( ) = 0
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Examples: Tutorial Chapter 7