The differential equation-s (their nature) describing circuit make this current direction changes you mention by itself, since they are able to differentiate positive and negative currents also (automatically).

One variable is iL and one vC.


So you need two diff. equations for two states (4 equations overall ) only.

Im not trying to be rude and i appreciate your posts but from where im sitting your posts constantly contradict each other.

You said the equation was the same for both states above then when i see in the wave form there are actually 4 equations you then agree with me.

I am not able to follow what your saying.

Even with the 4 equations the proof still does not work. With the following 4 equations the proof still fails

The 4 equations essentially just cancel each other out.

The proof says that integral should be equal to what i posted above and with these 4 equations the integral comes out to zero.

State 1 (Switch on)
Equation 1 = IC = IL - VC/R
Equation 2 = IC = VC/R - IL

Then they would be the same for the second switch state.

State 2 (Switch off)
Equation 3 = IC = IL - VC/R
Equation 4 = IC = VC/R - IL

Taking the integral of all that is zero where it should be equal to the answer in the book.









Where it should be equal to IL - VC/R

 

The four equations I posted (drawn on paper) are two differential and two algebraic.
The algebraic can be eliminated by passing them to differential.
So all you are getting is just two diff. equations for Ton.

For Toff you also need a two diff. equations. They are little different comparing to Ton.

 

The four equations I posted (drawn on paper) are two differential and two algebraic.
The algebraic can be eliminated by passing them to differential.
So all you are getting is just two diff. equations for Ton.

For Toff you also need a two diff. equations. They are little different comparing to Ton.

You say "For Toff you also need a two diff. equations"

Please post the equations. Please post all 4 equations for capacitor current both for TON and TOFF listed out as equation 1,2,3,4
PLEASE post all 4 equations that make the following proof correct.