step response of an RC circuit

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.

Regards
PG

Looks as though yet another textbook writer has generated verbiage about concepts that are easier for us to grasp by building a circuit and watching a meter, than by our having to memorize idiosyncratic terminology.

It is true that a capacitor shows behavior A in one instance, and behavior B in the other instance. After seeing these demonstrated a few times you'll think of these behaviors with a more intuitive part of your brain.

Nevertheless for the time being you may as well let the author have his day. The math has a beauty to it, and the concepts he uses will turn up as applied to other topics, most likely in vital connections.

So take in whatever concepts jibe with what you learn as you work with real components.

He says something about transient response. As for me the only real criterion I know of is that capacitor response is transient when enough time goes by that the cap attains whatever voltage is applied to it. Five time constants is the convention. The curve has the well-known look to it: drops quickly at first, then tapers to horizontal.

On the other hand if voltage changes before 5 time constants go by, the capacitor has a new situation to adapt to, and it manifests the other type of response. (Whatever the author calls it. Didn't he use the term 'forced response' somewhere? I cannot fathom what that might mean as applied to a capacitor.)

There are some important distinctions the author brings out. There are cases where you do want a capacitor to fully charge (or discharge). And other cases where you do not.

How do we talk in everyday life? We might say 'it's discharging too fast, use more microfarads', or 'substitute a larger value resistor', etc. A textbook on the other hand can't talk like we do in everyday life.

A textbook needs to be hard to read, or else the purchaser will get the idea it's only for beginners, and won't buy it.

In any case you'll think of your own terminology for these concepts as time goes by. Or maybe you won't think in words at all. It will become second nature to you.

Certainly the book uses non conventional terminology.
When the switch of 7.40a is switched on what all happens in the initial stages (may be 3-5 time constants) is called as transient response and what happens later is called as forced.
Regarding Q1 related to fig 7.40b what I understand is that the switching is replaced by assigning ma mathematical feature to the input voltage source itself. This simplifies appearance of the circuit but included additional feature to the representation of the voltage source (including the property of switch).
This is accomplished using the function u(t). It seems a unit step function whose value is zero for t < 0 and is 1 for t > 0, implying that the voltage is applied to the circuit at t = 0 (as shown in Fig 7.40a).
Doing this assignment (use of u(t)) provides many advantages in circuit analysis like solving the differential equation using Laplace transform.