Parallel RLC Natural Response Voltage Derivation

Описание: 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.

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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