Lecture 15
RLC Circuits
Source Free &
Transient Response
Oct. 31 & Nov. 7, 2011
Material from Textbook by Alexander & Sadiku and Electrical Engineering:
Principles & Applications, A. R. Hambley is used in lecture slides.
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Lecture 15
RLC Circuits
Source Free &
Transient Response
Oct. 31 & Nov. 7, 2011
Material from Textbook by Alexander & Sadiku and Electrical Engineering: Principles & Applications, A. R. Hambley is used in lecture slides.
2
Second-Order RLC Circuits
Chapter 8 in Your Textbook
- Examples of 2nd order RCL circuit
- The source-free series RLC circuit
- The source-free parallel RLC circuit
- Step response of a series RLC circuit
- Step response of a parallel RLC circuit
4
Source-Free Series
RLC Circuits
- The solution of the source-free series RLC circuit is called as the natural response of the circuit.
- The circuit is excited by the energy initially stored in the capacitor and inductor. The 2nd order DE turns out to be How to arrive at this DE and solve it – now read on?
Finding Initial & Final Values
- Need to find initial & final values - v across caps - i through inductors - dv/dt - di/dt - Continuity Eq’s - Across cap
v(0+) = v(0-)
- Through inductor
i(0+) = i(0-)
- Take care with signs: - Define currents - Currents flow from +
to – through C & L
Second –Order Circuits
Differentiating with respect to time:
KVL:
Second –Order Circuits
Dampening coefficient Undamped resonant frequency Define: Forcing function
Solution of the Complementary
Equation
Try i C ( t ) = Ke st : s 2 Ke st
- 2 α sKe st
- ω 0 2 Ke st = 0 Factoring : ( s 2
- 2 α s + ω 0 2 ) Ke st = 0 Characteristic equation : s 2
- 2 α s + ω 0 2 = 0 Exponential is the indestructible function by differentiation or integration
Solution of the Complementary
Equation
13
Source-Free Series
RLC Circuits
There are three possible solutions for the following 2nd order differential equation: The types of solutions for i(t) depend on the relative values of α and ω ο or ζ = α/ω ο
General 2nd order form where Resonant frequency
14 There are three possible solutions for the following 2nd order differential equation (the complementary equation):
- If α > ωo, over-damped case (ζ > 1) where
- If α = ωo, critical damped case (ζ = 1) where
- If α < ωo, under-damped case (ζ < 1) where
16
Source-Free Series
RLC Circuits
Example The circuit shown below has reached steady state at t = 0-. If the make-before-break switch moves to position b at t = 0, calculate i(t) for t > 0. Answer: i(t) = e –2.5t [5cos1.6583t – 7.538sin1.6583t] A
17
Source-Free Parallel
RLC Circuits
The 2nd order of expression Let v(0) = V 0 Apply KCL to the top node: Taking the derivative with respect to t and dividing by C
19 Source-Free Parallel RLC Circuits Example Refer to the circuit shown below. Find v(t) for t > 0. Answer: v(t) = 66.67(e –10t
- e –2.5t ) V
20
Step-Response Series
RLC Circuits
- The step response is obtained by the sudden application of a DC source. The 2nd order of expression The above equation has the same form as the equation for source-free series RLC circuit.
- The same coefficients (important in determining the frequency parameters).
- Different circuit variable in the equation.