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step response of an RC circuit

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PG1995

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Hi

Please have a look on the attachments. The attached pages are: 258, 259, 260. Please help me with the queries. I need to understand this derivation soon. Thank you for your help.

If you want to have a look on the derivation of "Source Free RC Circuit" and description of "natural response", you can have look on the attached pages in first post of this thread: https://www.electro-tech-online.com/threads/source-free-rc-circuit.122570/

Regards
PG
 

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Hi,

UNIT STEP

u(t) is the step function, sometimes called the Heaviside step function.
Its value is 1 for t>0 so sometimes it is dropped because 1 times anything is just that thing.
Sometimes it is implied by the context so it would not be shown at all.

For t<0 it's value is zero and for t>0 it's value is 1, so you can see why we call it the 'step' function.
The step function isnt really defined for t=0 strictly speaking, but we do often think of it that way because for t very close to the right of zero it is 1.
't' to the right of zero is usually shown as t=0+, and to the left of zero it is shown as t=0-.
The space between 0- and 0+ isnt really defined except as an area and then it is referred to as an impulse.

So we have three different areas to think about...
The area to the left of zero 0-, the area to the right of zero 0+, and the area between 0- and 0+. These are distinct areas of interest and they are often used in different ways.


FORCED AND NATURAL RESPONSES

Take a look at a capacitor charging, charging through a resistor fed with a 1v DC source voltage:
1-e^(-t/RC)

and note what happens when t gets bigger and bigger. Pick some convenient value for RC like '1', and go from there. Start with maybe t=0.0 and then let t get larger and larger by say 0.5 second intervals while you keep repeating the calculation. Watch what the result does as t gets larger and watch what happens to the -e term by itself too. So that would mean start with t=0, then t=0.5, then t=1.0, then t=1.5, etc.
There are two terms in that expression, 1 and -e^(-t/RC). As t gets higher and higher, the term with 'e' gets smaller and smaller. Eventually we are left with only the '1' because the 'e' part dies out. To know the full response though you need both terms, not just one.
 
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