ECE 201 Lecture 21 Borja Peleato Second order circuits RLC source free 1 Review In the previous lecture we saw the method to solve a second order differential equation and applied it to the LC oscillator In the LC circuit the b term in the characteristic equation was 0 so the discriminant b2 4c was always negative and we obtained complex roots with no real part The complete response was a perfect oscillation that did not attenuate Today we will study RLC circuits where a resistor introduces damping Two particular cases RLC series and RLC parallel Both described by a second order differential equation but with different coefficients 2 RLC series 3 RLC parallel 4 Solving the equations Characteristic equation s2 bs c 0 Discriminant b2 4c b2 4c 0 overdamped b2 4c 0 critically damped b2 4c 0 underdamped For series RLC b R L c 1 LC For parallel RLC b 1 RC c 1 LC 5 Practical considerations If the circuit is not a series or parallel RLC circuit from the start you can generally use Thevenin Norton equivalences to make it into one There are several ways to express the response in the underdamped case using trigonometric identities In all cases we have the same frequency w and exponential envelope Only the constants are different 6 Example 7 8 9
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