Parallel RLC Natural Response Voltage Derivation

This video “Par RLC Nat Resp v(t)” derives the underdamped natural response voltage of a parallel RLC circuit due to an initial condition inductor current. The KCL differential equation, the characteristic equation, and the roots are all expressed in terms of R, L, and C. Alpha and omega-d are introduced for the complex conjugate roots related to the underdamped case. RLC Natural response derivations, for both series and parallel RLC, including the damping cases overdamped, critically damped, and underdamped, covered in this book: “First and Second Order Circuits and Equations – Technical Background and Insights, which is available from John Wiley & Sons ISBN: 978-1-119-91353-5 https://www.wiley.com/en-us/First+and... This video derives Equation 14.165 on page 279 of the “1st & 2nd” book In addition to expressing the equations and damping conditions in terms of R, L, and C, the “1st & 2nd” book also describes characteristic roots and damping cases and root locus in terms of damping coefficient alpha and resonant frequency omega-0, and also in terms of damping ratio zeta. Please subscribe and mention this channel    / @gufo377   to your friends. Thank you! 00:00 Introduction 00:02 Intro Overview 01:11 1st2nd Book Recommendation 01:29 Write KCL at Bottom Node 02:02 RLC Branch Relationships 02:25 I-sub-L-0 in Inductor Equation 02:30 I-sub-L-0 in Inductor Equation 02:50 RLC Integro Differential Equation 03:37 RLC Differential Equation to Solve 04:42 Propose Exponential Function Solution 06:59 RLC Characteristic Equation 07:44 Apply Quadratic Formula 08:03 Characteristic Roots 08:43 Underdamped Response Criteria 11:27 Underdamped Complex Roots 12:19 Solve for Constants K1 and K2 12:30 Initial Condition of Voltage 12:58 First Simultaneous Equation for Ks 13:21 Voltage Derivative Initial Conditions 15:17 Initial Inductor Current in Capacitor 16:49 Simulation Initial Conditions 18:05 Second Simultaneous Equation for Ks 18:36 Simultaneous Solution for Ks 19:35 K2 Equation 20:00 K1 Equation 20:30 Voltage Exponential Solution 21:43 Sine Solution for Voltage