6 002 CIRCUITS AND ELECTRONICS Damped Second Order Systems 6 002 Fall 03 1 Damped Second Order Systems 5V 5V 2K 50 2K S C A B large loop CGS Remember this Demo Our old friend the inverter driving another The parasitic inductance of the wire and the gate to source capacitance of the MOSFET are shown Review complex algebra appendix in Agarwal Lang for next class 6 002 Fall 03 2 Damped Second Order Systems 5V 5V 50 2K 2K S C A B large loop Relevant circuit 5V 6 002 Fall 03 2K CGS L B CGS 3 Observed Output 2k 5 vA 0 t vB 2k t 0 vC 0 t Now let s try to speed up our inverter by closing the switch S to lower the effective resistance 6 002 Fall 03 4 Observed Output 50 5 vA 0 t vB 50 0 t vC 0 t Huh 6 002 Fall 03 5 In the last lecture we started by analyzing the simpler LC circuit to build intuition i t L vI t 6 002 Fall 03 C v t 6 In the last lecture We solved d 2v 1 1 v vI 2 dt LC LC For input VI vI 0 t And for initial conditions v 0 0 i 0 0 ZSR 6 002 Fall 03 7 In the last lecture Total solution v t VI VI cos t o where 1 LC o v t 2VI LC VI vI 0 t i t L v I t 6 002 Fall 03 C v t 8 Today we will close the loop on our observations in the demo by analyzing the RLC circuit R L vI t i t C v t v t 2VI LC VI vI 0 add R t Damped sinusoids with R remember demo See A L Section 13 6 6 002 Fall 03 9 Let s analyze the RLC network vA vI t L i t v t R C Node method vA 1 t vA v vI v A dt L R vA v dv C R dt v Recall element rules L vL L t di dt 1 vL dt i L d 2 v R dv 1 1 v vI 2 dt L dt LC LC 6 002 Fall 03 C dvC iC C dt v i state variables 10 Let s analyze the RLC network vA vI t L i t R C v t Node method 1 t vA v v I v A dt L R vA vA v dv C R dt v 1 d 2v vI v A C 2 L dt 1 d 2v vI v A 2 dt LC dv v A RC v dt dv 1 d 2v vI RC v 2 LC dt dt d 2 v R dv 1 1 v vI 2 dt L dt LC LC 6 002 Fall 03 11 Solving Recall the method of homogeneous and particular solutions 1 Find the particular solution 2 Find the homogeneous solution L 4 steps 3 The total solution is the sum of the particular and homogeneous Use initial conditions to solve for the remaining constants v vP t vH t 6 002 Fall 03 12 Let s solve d 2 v R dv 1 1 v vI 2 dt L dt LC LC For input VI vI 0 t And for initial conditions v 0 0 i 0 0 ZSR 6 002 Fall 03 13 1 Particular solution d 2 vP R dvP 1 1 vP VI 2 dt L dt LC LC vP VI 6 002 Fall 03 is a solution 14 2 Homogeneous solution Solution to 1 d 2 vH R dvH vH 0 2 dt LC dt L Recall vH solution to homogeneous equation drive set to zero Four step method A Assume solution of the form vH Ae st A s B Form the characteristic equation f s C Find the roots of the characteristic equation s1 s2 D General solution vH A1e s1t A2 e s2t 6 002 Fall 03 15 2 Homogeneous solution 1 d 2 vH R dvH vH 0 2 dt LC dt L Solution to A Assume solution of the form vH Ae st A s so As2est R 1 Asest Aest 0 L LC characteristic equation R 1 s s 0 L LC B 2 s 2 s 2 2 o o 0 C Roots 1 LC R 2L s1 2 2 o s2 2 2 o D General solution vH A1e 6 002 Fall 03 2 2 o t A2 e 2 2 o t 16 3 Total solution v t vP t vH t v t VI A1e 2 2 o t A2 e 2 2 o t Find unknowns from initial conditions v 0 0 0 VI A1 A2 i 0 0 dv i t C dt CA e CA1 2 2 o e 2 so 2 2 o 2 2 o t 2 2 o t 0 A1 2 2 o A2 2 2 o Mathematically solve for unknowns done 6 002 Fall 03 17 Let s stare at this a while longer 2 2 o t v t VI A1e e t 2 2 o t A2 e e t 3 cases o Overdamped v t VI A1e o 1t v t VI A1e e t VI A1e e o 6 002 A2 e 2 t v t Underdamped j 2 o 2 t t VI K1e VI vI t j d t j 2 2 t o t A2 e e t j d t A2e e cos d t K 2e t sin d t d 2 o 2 e j d t cos d t j sin d t Critically damped Later Fall 03 18 Let s stare at underdamped a while longer o Underdamped contd v t VI K1e t cos d t K 2e t sin d t v 0 0 K1 VI dv i 0 0 i t C dt CK1 e t cos d t CK 2 d e t sin d t CK1 e t sin d t CK 2 d e t cos d t 0 K1 K 2 d V K2 1 d v t VI VI e t t cos d t VI e sin d t d Note For R 0 0 v t VI VI cos ot Same as LC as expected 6 002 Fall 03 19 Let s stare at underdamped a while longer o Underdamped contd v t VI VI e t t cos d t VI e sin d t d Remember scaled sum of sines of the same frequency are also sines Appendix B 7 o t v t VI VI e cos d t tan 1 d d v t 2VI LC VI vI 0 6 002 add R t Fall 03 20 o Underdamped contd v t VI VI e t t cos d t VI e sin d t d Remember scaled sum of sines of the same frequency are also sines Appendix B 7 o t 1 v t VI VI e cos d t tan d d …
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