Question

The LRC situation shown is driven by a power supply whose EMF = VoCos(wt). In steady state, the current through the ideal self-inductor is I(L) the carrent through the ideal capacitor is I(C) and the current through the resistor is I(R). Steady state means that you wait a long time so that all transient phenomena have died out Don't even THINK of writing down a differential equation. This problem is designed to see whether you have an appreciation for how a capacitor and a self inductor behave in Extreme situations. No fancy math is needed Express all your answers in terms of L, R, C and Vo

a) What are the maximum value of I(L), I(C) and I(R) in case w = 0 (zero frequency means that the power supply is now a simple battery with zero Internal resistance). We are asking you for steady state solutions, NOT transient solutions.

b) Answer the same question as under (a), for the other extreme when w approaches a value which infinitely high.

c) Do you expect the maximum value of the current I(R) to be higher or lower than the value you found under (a) in the case that the frequency is somewhere in between the above two extremes, Give your reasons

d) There is one frequency (in steady state) for which I(R) is zero. This is not so intuitive, but given the fact that this is so, what do you think that frequency is? Please, do not try to calculate this frequency.

Note: w is omega (angular frequency) ​

Answer 1

Answer:

$\sf{\green{\underline{\underline{\orange{Answer :-}}}}}$

It is well-known that there is a 90 deg phase shift between the current and voltage in the capacitor (when supplied by a sinusoidal signal) and it varies from 0 to 90 degrees in the RC integrating circuit when the frequency changes from zero to infinity. But there are not good "physical" explanations why and how the phase shift appears and why it varies with the frequency.

I have tried to explain intuitively the phenomenon on the Wikipedia talk page about the RC circuit (see the attachment below). Let's use it as an initial point for this discussion.

Answer 2

$\large{\underline{\underline{\tt \purple{ Answer }}}}$

$\bf{ a) }$V is in Volts

• R is in Ohms
• L is in Henries
• t is in Seconds
• e is the base of the Natural Logarithm = 2.71828

This value of 63.2% or 0.632x$\tiny\tt{IMAX}$ also corresponds with the transient curves shown above.

$\bf{ b) }$An electrical insulator is a material whose internal electric charges do not flow freely, and therefore make it very hard to conduct an electric current under the influence of an electric field. And those substances have infinite high electrical resistance.

Ideally, the insulation resistance would be infinite, but as no insulators are perfect, leakage currents through the dielectric will ensure that a finite (though high) resistance value is measured.

$\bf{ c) }$$\sf{Rf // RL > Vo,max / Io,lim}$

$\sf{Rf << Vo/Iin}$

$\sf{Rf/R1 << A}$

$\sf{P^{1}= VIN_2/R_1, P^{2}= VOUT_2/R_2 \: and \: the \: total}$

$\sf{power \: is \: P = (VIN + VOUT)2/(R_1 + R_2).}$

$\bf{ d) }$there is "90° phase shift between current and voltage in the capacitor": it is true only if one is studying a specific dynamical behavior i.e. the one that it is observed when the transient solution becomes negligible and one can observe the steady state solution only in the specific input configuration of a periodic single tone input.

$\tiny\tt{ hope \: this \: will \: help \: you \: don't \: report }$