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### Transcript of Electric Circuits Discussion 1

Electric Circuits Discussion 1Contents
2
• Homework 6
1. Second-Order Circuit
And there are THREE cases you should know
First solving the Eigen-function of and Eigenvalues of a second order formula
Case 1: Overdamped (α>ω0)
2 2 2 2 1 2o os sα α ω α α ω= − + − = − − −0
1 2 R L LC
α ω= =
() = 1 + 2 −
6
1 = − + 2 − 02 = − + − 02 − 2 = − +
2 = − − 2 − 02 = − − − 02 − 2 = − −
where = −1 and = 02 − 2.
• ω0 is often called the undamped natural frequency. • ωd is called the damped natural frequency.
The natural response
= − 1cos + 2sin 7
= − ± 2 − 02
Recall Euler’s formula
• Exponential − * Sine/Cosine term • Exponentially damped, time constant =
1/ • Oscillatory, period = 2

• Behavior captured by damping • Gradual loss of the initial stored
energy • determines the rate of damping
• > 0 (i.e., > 2
), overdamped
), underdamped
= 11 + 22
• Series
10
• Parallel
( ) 1 2 1 2
( ) ( )2 1 tv t A At e α−= +
( ) ( )1 2cos sint d dv t e A t A tα ω ω−= +
Finding Initial and Final Values
• Working on second order system is harder than first order in terms of finding initial and final conditions.
• You need to know the derivatives, dv/dt and di/dt as well.
• Capacitor voltage and inductor current are always continuous.
• For capacitor, 0+ = 0− ; • For inductor, 0+ = 0− .
General Second-Order Circuits
• 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. • Depending on the damping found, the unknown constants will be found.
3. We obtain the steady-state response as:
where x(∞) is the final value of x obtained in step 1. 4. The total response = transient response + steady-state response.
( ) ( )ssx t x= ∞
13
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2. Homework 6
Case 2: Critically Damped (α=ω0)
Case 3: Underdamped (α<ω0)
Case 3: Underdamped (α<ω0)
Properties of Series RLC Network
Series vs. Parallel (Source-Free RLC Network)
Finding Initial and Final Values
General Second-Order Circuits