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RC Natural Response Derivation v(t)

This same derivation, of the natural response voltage of an RC circuit, appears in chapter 8, section 2, of my circuit theory book “First and Second Order Circuits and Equations – Technical Background and Insights”. That same section 8.2 also has a dual derivation of the natural response current in an RL circuit. Published by John Wiley & Sons, Inc., the book “First and Second Order Circuits and Equations”, (called “1st & 2nd” for short) is available here: https://www.wiley.com/en-us/First+and... The derivations in section 8.2 start with switch models in the RC and RL circuits to generate the natural response. This video “RC Natural Response Derivation v(t)” derives the RC natural response voltage function using separation of variables and direct integration. Every step is shown and described in detail. Kirchhoff’s current law is used to generate the first order ordinary homogeneous differential equation for zero input response (ZIR), also called natural response. Timestamps for your convenience: 0:10 Brief Outline of the Derivation Process 1:10 Removing Sources to Isolate Natural Response 2:47 Start of Detailed Derivation of v(t) ZIR 3:10 Current and Voltage Polarity Insights 3:59 More Detailed Derivation – Branch Relationships 5:39 Solving the ZIR Differential Equation 6:56 Performing the Integration 7:51 Natural Logarithm and Exponential Math 9:07 Evaluating the Unknown Constant k There is brief mention of the often-confusing topic of current and voltage reference polarities versus the physical current and voltage, starting at 3:10. For more detail on this topic for RC natural response exponential decay, please view our video "RC Circuit Natural Response Polarity Reference"    • RC Circuit Natural Response Polarity Refer...   If you are brand new to RC natural response and first order exponential decay and the related time constants, please view our video: RC Natural Response Description    • RC Natural Response Description