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CHAPTER 5
Transient and Steady State
Response
(Second-Order Circuits)
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Contents
Natural response of series RLC circuit
Natural response of parallel RLC circuit
Step response of series RLC circuit
Step response of parallel RLC circuit
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What is second order?
• Circuits containing
two storage
elements.
• Second-order
circuit may have
two storage
elements of
different type or
the same type
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Initial and final values
• Combination of R, L and C
• Find v(0), i(0), dv(0)/dt, di(0)/dt, i(∞) & v(∞)
• t(0-) the time just before switching event
• t(0+) the time just after switching event
• Assume the switching event take place at t=0
• Voltage polarity across capacitor
• Current direction across inductor
• Capacitor voltage always continuous v(0+) = v(0-)
• Inductor current always continuous i(0+)=i(0-)
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Example
The switch in the figure shown has been closed for a long
time. It is open at t=0, Find:
(a) i(0+), v(0+)
(b) di(0+)/dt, dv(0+)/dt
(c) i(∞) , v(∞)
12 V
0.25 H4 Ω
0.1 F2 Ω
i
+V-
t=0
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Exercise
The switch in the figure shown was open for a long time but
closed at t=0. Determine
(a) i(0+), v(0+)
(b) di(0+)/dt, dv(0+)/dt
(c) i(∞) , v(∞)
24 V
0.4 H
1/20 F2 Ω
i
+V-
t=0
10 Ω
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The Source-Free Series RLC
• Applying KVL around the loop
𝑅𝑖 + 𝐿𝑑𝑖
𝑑𝑡+1
𝑐 −∞
𝑡
𝑖 𝑑𝑡 = 0
• Differentiate with respect to t
𝑑2𝑖
𝑑2+𝑅
𝐿
𝑑𝑖
𝑑𝑡+𝑖
𝐿𝐶= 0
• Finally,
𝑠2 +𝑅
𝐿𝑠 +
1
𝐿𝐶= 0
![Page 8: Chapter 5 Transient and steady state response(Second-Order Circuit)](https://reader034.vdocument.in/reader034/viewer/2022042501/55a93f941a28ab734e8b4582/html5/thumbnails/8.jpg)
The Source-Free Series RLC
• Roots equation
𝑠1 = −𝑅
2𝐿+
𝑅
2𝐿
2
−1
𝐿𝐶
𝑠2 = −𝑅
2𝐿−
𝑅
2𝐿
2
−1
𝐿𝐶
or
𝑠1 = −𝛼 + 𝛼2 − 𝜔02
𝑠2 = −𝛼 − 𝛼2 − 𝜔02
where
𝛼 =𝑅
2𝐿, 𝜔0 =
1
𝐿𝐶
• 𝛼 (Np/s)
• 𝜔0 (rad/s)
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The Source-Free Series RLC
Three type of solution
• If α > ω0 overdamped case
• If α = ω0 critically damped case
• If α < ω0 underdamped case
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The Source-Free Series RLC
Overdamped case (α>ω0)
• Both roots S1 and S2 are negative and real
• The response is 𝑖 𝑡 = 𝐴1𝑒
𝑠1𝑡 + 𝐴2𝑒𝑠2𝑡
![Page 11: Chapter 5 Transient and steady state response(Second-Order Circuit)](https://reader034.vdocument.in/reader034/viewer/2022042501/55a93f941a28ab734e8b4582/html5/thumbnails/11.jpg)
The Source-Free Series RLC
Critically damped case (α= ω0)
• Roots
𝑠1 = 𝑠2 = −𝛼 = −𝑅
2𝐿• The response is
𝑖 𝑡 = (𝐴2+𝐴1𝑡)𝑒−𝛼𝑡
![Page 12: Chapter 5 Transient and steady state response(Second-Order Circuit)](https://reader034.vdocument.in/reader034/viewer/2022042501/55a93f941a28ab734e8b4582/html5/thumbnails/12.jpg)
The Source-Free Series RLC
Underdamped case(α<ω0)
• Roots
𝑠1 = −𝛼 + − 𝜔02 − 𝛼2 = −𝛼 +j𝜔𝑑
𝑠2 = −𝛼 − − 𝜔02 − 𝛼2 = −𝛼-j𝜔𝑑
where 𝜔𝑑 = 𝜔02 − 𝛼2
• The response is 𝑖 𝑡 = 𝑒−𝛼𝑡(𝐵1 cos𝜔𝑑𝑡 + 𝐵2 sin𝜔𝑑𝑡)
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Example
Find i(t) for t > 0
+
v(t)
-
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Exercise
Find i(t) in the circuit below. Assume that the
circuit has reached steady state at t=0-
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Source Free Parallel RLC Circuits
• Initial inductor current and
initial voltage capacitor
𝑖 0 = 𝐼0 =1
𝐿 ∞
0
𝑣 𝑡 𝑑𝑡
𝑣 0 = 𝑉0• Applying KCL
𝑣
𝑅+1
𝐿 −∞
𝑡
𝑣𝑑𝑡 + 𝐶𝑑𝑣
𝑑𝑡= 0
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Source Free Parallel RLC Circuits
• Derivatives with respect t and diving by C
𝑑2𝑣
𝑑𝑡2+1
𝑅𝐶
𝑑𝑣
𝑑𝑡+1
𝐿𝐶𝑣 = 0
or 𝑠2 +1
𝑅𝐶𝑠 +
1
𝐿𝐶
• Roots of the characteristics equation are
𝑠1,2 = −1
2𝑅𝐶±
1
2𝑅𝐶
2
−1
𝐿𝐶
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Source Free Parallel RLC Circuits
or 𝑠1,2 = −𝛼 ± 𝛼2 −𝜔02
where 𝛼 =1
2𝑅𝐶, 𝜔0 =
1
𝐿𝐶
• 𝛼 (Np/s)
• 𝜔0 (rad/s)
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The Source-Free Parallel RLC
Three type of solution
• If α > ω0 overdamped case
• If α = ω0 critically damped case
• If α < ω0 underdamped case
![Page 19: Chapter 5 Transient and steady state response(Second-Order Circuit)](https://reader034.vdocument.in/reader034/viewer/2022042501/55a93f941a28ab734e8b4582/html5/thumbnails/19.jpg)
The Source-Free Parallel RLC
Overdamped case (α>ω0)
• Both roots S1 and S2 are negative and real
• The response is
𝑣 𝑡 = 𝐴1𝑒𝑠1𝑡 + 𝐴2𝑒
𝑠2𝑡
![Page 20: Chapter 5 Transient and steady state response(Second-Order Circuit)](https://reader034.vdocument.in/reader034/viewer/2022042501/55a93f941a28ab734e8b4582/html5/thumbnails/20.jpg)
The Source-Free Parallel RLC
Critically damped case (α= ω0)
• The roots are real and equal so the response is
𝑣 𝑡 = (𝐴1+𝐴2𝑡)𝑒−𝛼𝑡
![Page 21: Chapter 5 Transient and steady state response(Second-Order Circuit)](https://reader034.vdocument.in/reader034/viewer/2022042501/55a93f941a28ab734e8b4582/html5/thumbnails/21.jpg)
The Source-Free Parallel RLC
Underdamped case(α<ω0)
• Roots
𝑠1,2 = −𝛼 ± j𝜔𝑑
where 𝜔𝑑 = 𝜔02 − 𝛼2
• The response is
𝑣 𝑡 = 𝑒−𝛼𝑡(𝐴1 cos𝜔𝑑𝑡 + 𝐴2 sin𝜔𝑑𝑡)
![Page 22: Chapter 5 Transient and steady state response(Second-Order Circuit)](https://reader034.vdocument.in/reader034/viewer/2022042501/55a93f941a28ab734e8b4582/html5/thumbnails/22.jpg)
Example
Find v(t) for t>0 in the RLC circuit shown
below
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Step Response of a Series RLC Circuit
• Applying KVL around the
loop for t>0
𝐿𝑑𝑖
𝑑𝑡+ 𝑅𝑖 + 𝑣 = 𝑉𝑠
but 𝑖 = 𝐶𝑑𝑣
𝑑𝑡
substitute i in equation above
𝑑2𝑣
𝑑𝑡2+𝑅
𝐿
𝑑𝑣
𝑑𝑡+𝑣
𝐿𝐶=𝑉𝑠𝐿𝐶
![Page 24: Chapter 5 Transient and steady state response(Second-Order Circuit)](https://reader034.vdocument.in/reader034/viewer/2022042501/55a93f941a28ab734e8b4582/html5/thumbnails/24.jpg)
Step Response of a Series RLC Circuit
• There is two components in the equation (i) transient
response 𝑣𝑡 𝑡 (ii) steady-state response 𝑣𝑠𝑠 𝑡
𝑣 𝑡 = 𝑣𝑡 𝑡 + 𝑣𝑠𝑠 𝑡
• The transient response 𝑣𝑡 𝑡 is similar as discussed in
source-free circuit.
• The final value of the capacitor voltage is the same as
the source voltage Vs
𝑣𝑠𝑠 𝑡 = 𝑣 ∞ = 𝑉𝑠
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Step Response of a Series RLC Circuit
• The complete response solution are:-
𝑣 𝑡 = 𝑉𝑠 + 𝐴1𝑒𝑠1𝑡 + 𝐴2𝑒
𝑠2𝑡 (Overdamped)
𝑣 𝑡 = 𝑉𝑠 + (𝐴1+𝐴2𝑡)𝑒−𝛼𝑡 (Critically damped)
𝑣 𝑡 = 𝑉𝑠 + (𝐴1𝑐𝑜𝑠𝜔𝑑𝑡 + 𝐴2𝑠𝑖𝑛𝜔𝑑𝑡)𝑒−𝛼𝑡 (Underdamped)
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Example
For the circuit shown in figure below, find
v(t) and i(t) for t>0.
Given R = 5 Ω, C = 0.25 F
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Step Response of a Parallel RLC Circuit
• Applying KCL at the top
node for t > 0,𝑣
𝑅+ 𝑖 + 𝐶
𝑑𝑣
𝑑𝑡= 𝐼𝑠
but 𝑣 = 𝐿𝑑𝑖
𝑑𝑡
substitute vin equation above
and dividing by LC:
𝑑2𝑖
𝑑𝑡2+1
𝑅𝐶
𝑑𝑖
𝑑𝑡+𝑖
𝐿𝐶=𝐼𝑠𝐿𝐶
![Page 28: Chapter 5 Transient and steady state response(Second-Order Circuit)](https://reader034.vdocument.in/reader034/viewer/2022042501/55a93f941a28ab734e8b4582/html5/thumbnails/28.jpg)
Step Response of a Parallel RLC Circuit
• There is two components in the equation (i) transient
response 𝑖𝑡 𝑡 (ii) steady-state response 𝑖𝑠𝑠 𝑡
𝑖 𝑡 = 𝑖𝑡 𝑡 + 𝑖𝑠𝑠 𝑡
• The transient response 𝑖𝑡 𝑡 is similar as discussed in
source-free circuit.
• The final value of the current through the inductor is the
same as the source current Is
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Step Response of a Parallel RLC Circuit
• The complete response solution are:-
𝑖 𝑡 = 𝐼𝑠 + 𝐴1𝑒𝑠1𝑡 + 𝐴2𝑒
𝑠2𝑡 (Overdamped)
𝑖 𝑡 = 𝐼𝑠 + (𝐴1+𝐴2𝑡)𝑒−𝛼𝑡 (Critically damped)
𝑖 𝑡 = 𝐼𝑠 + (𝐴1𝑐𝑜𝑠𝜔𝑑𝑡 + 𝐴2𝑠𝑖𝑛𝜔𝑑𝑡)𝑒−𝛼𝑡 (Underdamped)
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Example
Find i(t) and v(t) for t > 0
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END