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Electric CircuitsDiscussion 5
Keyi YuanTeaching assistant
Apr.24 2018
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Keyi Yuan, Electric Circuit (2018 Spring)
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Contents
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β’ Discussion Review Part : Second-Order circuit
β’ Homework 6
Keyi Yuan, Electric Circuit (2018 Spring)
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1. Second-Order Circuit
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Solving second order equation
ππ2π£π£ππ2π‘π‘
+π π πΏπΏπππ£π£πππ‘π‘
+1
LCπ£π£ =
π£π£π π πΏπΏπΏπΏ
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And there are THREE cases you should know
First solving the Eigen-function of and Eigenvalues of a second order formula
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Case 1: Overdamped (Ξ±>Ο0)
2 2 2 21 2o os sΞ± Ξ± Ο Ξ± Ξ± Ο= β + β = β β β0
12RL LC
Ξ± Ο= =
π π = βπ π 2πΏπΏ
Β±π π 2πΏπΏ
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β1πΏπΏπΏπΏ
π£π£ π‘π‘ = π΄π΄1πππ π 1π‘π‘ + π΄π΄2πππ π 2π‘π‘
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Case 2: Critically Damped (Ξ±=Ο0)
π£π£(π‘π‘) = π΄π΄1π‘π‘ + π΄π΄2 ππβπΌπΌπ‘π‘
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Case 3: Underdamped (Ξ±<Ο0)
π π 1 = βπΌπΌ + πΌπΌ2 β ππ02 = βπΌπΌ + β ππ02 β πΌπΌ2 = βπΌπΌ + ππππππ
π π 2 = βπΌπΌ β πΌπΌ2 β ππ02 = βπΌπΌ β β ππ02 β πΌπΌ2 = βπΌπΌ β ππππππ
where ππ = β1 and ππππ = ππ02 β πΌπΌ2.
β’ Ο0 is often called the undamped natural frequency.β’ Οd is called the damped natural frequency.
The natural response
π£π£ π‘π‘ = π΄π΄1πππ π 1π‘π‘ + π΄π΄2πππ π 2π‘π‘becomes
π£π£ π‘π‘ = ππβπΌπΌπ‘π‘ π΅π΅1cosπππππ‘π‘ + π΅π΅2sinπππππ‘π‘7
π π = βπΌπΌ Β± πΌπΌ2 β ππ02
Recall Eulerβs formula
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Case 3: Underdamped (Ξ±<Ο0)
β’ Exponential ππβπΌπΌπ‘π‘ * Sine/Cosine termβ’ Exponentially damped, time constant =
1/πΌπΌβ’ Oscillatory, period ππ = 2ππ
ππππ
v(π‘π‘) = ππβπΌπΌπ‘π‘ π΅π΅1cosπππππ‘π‘ + π΅π΅2sinπππππ‘π‘
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Properties of Series RLC Network
β’ Behavior captured by dampingβ’ Gradual loss of the initial stored
energyβ’ πΌπΌ determines the rate of damping
β’ πΌπΌ > ππ0 (i.e., π π > 2 πΏπΏπΆπΆ
), overdamped
β’ πΌπΌ = ππ0 (i.e., π π = 2 πΏπΏπΆπΆ
), critically dampedπ£π£(π‘π‘) = π΄π΄1π‘π‘ + π΄π΄2 ππβπΌπΌπ‘π‘
β’ πΌπΌ < ππ0 (i.e., π π < 2 πΏπΏπΆπΆ
), underdamped
π£π£ π‘π‘ = ππβπΌπΌπ‘π‘ π΅π΅1cosπππππ‘π‘ + π΅π΅2sinπππππ‘π‘
π£π£ π‘π‘ = π΄π΄1πππ π 1π‘π‘ + π΄π΄2πππ π 2π‘π‘
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Series vs. Parallel (Source-Free RLC Network)
β’ Series
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β’ Parallel
π£π£ π‘π‘ = π΄π΄1πππ π 1π‘π‘ + π΄π΄2πππ π 2π‘π‘
π£π£(π‘π‘) = π΄π΄1π‘π‘ + π΄π΄2 ππβπΌπΌπ‘π‘
v(π‘π‘) = ππβπΌπΌπ‘π‘ π΅π΅1cosπππππ‘π‘ + π΅π΅2sinπππππ‘π‘
( ) 1 21 2
s t s tv t A e A e= +
( ) ( )2 1tv t A At e Ξ±β= +
( ) ( )1 2cos sintd dv t e A t A tΞ± Ο Οβ= +
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Finding Initial and Final Values
β’ Working on second order system is harder than first order in terms of finding initial and final conditions.
β’ You need to know the derivatives, dv/dt and di/dtas well.
β’ Capacitor voltage and inductor current are always continuous.
β’ For capacitor, π£π£ 0+ = π£π£ 0β ;β’ For inductor, ππ 0+ = ππ 0β .
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General Second-Order Circuits
β’ The principles of the approach to solving the series and parallel forms of RLC circuits can be applied to general second order circuits, by taking the following four steps:1. First determine the initial conditions, x(0) and dx(0)/dt.2. Turn off the independent sources and find the form of the
transient response by applying KVL and KCL.β’ Depending on the damping found, the unknown constants will be found.
3. We obtain the steady-state response as:
where x(β) is the final value of x obtained in step 1.4. The total response = transient response + steady-state response.
( ) ( )ssx t x= β
( ) ( ) ( )t ssx t x t x t= +12
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Typo:[(π·π·1cos πππππ‘π‘ +π·π·2 sin πππππ‘π‘ ) ππβπΌπΌπ‘π‘ +π₯π₯(β)] π’π’(t)
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2. Homework 6
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Question 1
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Question 2
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Question 2
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Question 3
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Question 3
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Question 4
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Question 4
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Question 5
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Question 5
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Question 6
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Question 6
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Question 7
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Question 7
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Question 8
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Question 8
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Question 8
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Question 8
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Question 9
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Question 9
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Question 9
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Question 9
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Question 10
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Question 10
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Question 10
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Question 10
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Question 10
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Question 10
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Question 10
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