Hello,

I am currently going through RC circuits from https://en.wikipedia.org/wiki/RC_circuit

There are several conceptual things I am not able to appreciate ... I hope someone comments ...

The complex impedance, ZC (in ohms) of a capacitor with capacitance C (in farads) is

The complex frequency s is, in general, a complex number,

where

• 
  represents the imaginary unit:

• 
  is the exponential decay constant (in radians per second), and
• 
  is the sinusoidal angular frequency (also in radians per second).

Sinusoidal steady state
Sinusoidal steady state is a special case in which the input voltage consists of a pure sinusoid (with no exponential decay). As a result,

and the evaluation of s becomes

= 0

s = j

If I substitute Zc = 1/sC with s = jw then

Zc = 1/jCw = -j/Cw

I am sorry but I don't understand conceptually what it means when impedance is lying on the imaginary axis.

The other question I have is, conceptually what is meant by complex impedance?


Current through the RC circuit is,

Cdv/dt + V/R = 0

this is mentioned as a linear differential equation. How to check if a given differential equation is linear or not?

I assume that the voltage falls exponentially with time. Is this correct?

Also, how would current change across the resister vary?
I assume that would also fall exponentially. Is this correct?

 

Read this:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html

this is mentioned as a linear differential equation. How to check if a given differential equation is linear or not?

I assume that the voltage falls exponentially with time. Is this correct?

Also, how would current change across the resister vary?
I assume that would also fall exponentially. Is this correct?

That Wikipedia page has a link to the page for linear differential equations. I suggest you read there.

The voltage will fall exponentially if the capacitor is charged and then the voltage source is removed. If, instead, the source is a varying voltage that remains connected, the equation still describes the current through the circuit, but it will not be exponentially decaying.

I'm not sure what you mean by the last question. The current through (not across) the resistor varies if the current through the capacitor varies, because they're in series.

 

An imaginary impedance with a positive sign looks an inductor. An imaginary impedance with a negative sign looks like a capacitor. Here is the thing that is too weird for words. A given circuit as a function of frequency will sometimes look like a capacitor and sometimes look like an inductor. How this happens is described in Feynman Volume II.

http://www.feynmanlectures.caltech.edu/II_toc.html

Sections 23-1 and 23-2

 

@sharanbr

Here's a conceptual aid to perspective many find helpful:

Analogy drawn to kinetics:
Reactance may be regarded as equivalent to 'elasticity' (which being subsumptive of inertia/momentum and other 'mass-force' effects {e.g. gravitation}) and resistance to 'friction'...

Or, alternatively:

Analogy drawn to economics:
Reactance may be regarded as equivalent to 'principal' and resistance to 'interest'...

Examples of such parallels are 'endless' but I think you 'get it'

Note: the apparent 'frequency dependance' of the 'sign' on a reactance/susceptance may readily be comprehended at an intuitive level via consideration of resonance in conjunction with apprehension of 'real world proclivities' (To wit: nothing, but nothing is pure!)

Hope my (admittedly less than rigorous) response is of help!

Best regards
HP