I’m doing a class AC Theory and it’s asking me what is the frequency at infinity times the resonant frequency of the circuit. Then the next questions are similar asking what’s the XL and XC of the circuit at infinite frequency. I DO NOT UNDERSTAND WHY WE ARE MULTIPLYING TIMES INFINITY AND HAVE NO IDEA HOW TO
They are asking you,
in the limit as the frequency
approaches infinity, what is the reactance?
The impedance of a capacitor is 1/(j*ω*C)
The impedance of an inductor is j*ω*L
L and C are component properties, ω is the frequency, in units of radians/second (that is, 2*π*f, if you prefer f in Hertz), and j is the imaginary unit. The imaginary component of impedance is reactance, so this means, once you clear all j's from the bottom, the number multiplied by j on the top, is the reactance.
For a capacitor, infinite ω means the impedance might as well be zero. So at high frequencies, a capacitor behaves as a short.
For an inductor, infinite ω means impedance might as well be infinite. So at high frequencies, an inductor behaves as an open gap.
The opposite behavior happens in the limit as ω approaches zero for steady state DC.
At resonance between an inductor and capacitor, there's a similar behavior of interest, when the reactances cancel each other. Ideal versions of these components in parallel, behave as an open gap. Ideal versions of these components in series, behave as a short.
Example:
A capacitor (C) is in series, with a parallel combination of a resistor (R) & inductor (L). The total impedance of this combination would be:
Znet = 1/(j*ω*C) + parallel(R, j*ω*L) =
Znet = 1/(j*ω*C) + (j*ω*L*R)/(R + (j*ω*L))
Simplify:
Znet = (L*ω + j*R*L*C*ω^2 - j*R)/(R*C*ω + j*L*C*ω^2)
As frequency approaches zero, only the lowest power of ω governs. Then set ω=0:
Znet0 = (-j*R)/(R*C*ω) = -j/(C*ω)
Take the limit as ω approaches zero from positive, and this approaches infinity.
At high frequencies, only the highest power of ω governs, so ignore all other terms. This gives us:
ZnetHF = (+j*R*L*C*ω^2)/(+j*L*C*ω^2) = R
This is consistent with what we expect. Low frequencies cause the capacitor acts as an open gap, blocking steady state current. At high frequencies, the inductor blocks its parallel path, the capacitor shorts the resistor to the source, so only the resistor governs the impedance.
