1• Second order circuits: RLC source-freeECE 201: Lecture 21Borja PeleatoReview• In the previous lecture we saw the method to solve asecond order differential equation, and applied it to the LCoscillator– In the LC circuit, the “b” term in the characteristic equation was0, so the discriminant (b2-4c) was always negative and weobtained complex roots with no real part– The complete response was a perfect oscillation that did notattenuate• Today we will study RLC circuits, where a resistor introducesdamping– Two particular cases: RLC series and RLC parallel– Both described by a second order differential equation, but withdifferent coefficients2RLC series3RLC parallel4Solving 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)5Practical considerations• If the circuit is not a series or parallel RLC circuit fromthe start, you can generally use Thevenin/Nortonequivalences to make it into one• There are several ways to express the response in theunderdamped case, using trigonometric identities:– In all cases we have the same frequency w and exponentialenvelope. Only the constants are
View Full Document