Asymptotical values as ω ->0 and infinity

[exidez]

Homework Statement

Its an electrical circuit problem. I have obtained the Transfer function. I have substituted in the inductance/capacitance and resistance values and am now stuck on this part of the question:

d) Find |H(ω)|, the modulus of H(ω), and determine its asymptotical values for ω = 0 and ω → ∞. Sketch the form of |H(ω)| and describe the behaviour of this circuit.

Homework Equations

Found the he modulus of H(ω) to be:
$$\frac{sqrt{(1*10^{-11}\omega^2)^{2} + (5*10^{-6}\omega)^{2}}{\sqrt{(1*10^{-22}\omega^{4}-1.225*10^{-10}\omega^{2}+1)^{2}+(1*10^{-16}\omega^{3}+6*10^{-6}\omega)^{2}}}$$

The Attempt at a Solution

if ω = 0 then modulus |H(ω)| =0
as ω → ∞ i don't know what to do.

do i divide everything inside the square root by ω^4

in that case as ω → ∞, |H(ω)| also is 0. Is this correct?

 

[mighty2000]

I would approach this problem by first simplifying the transfer function to make it easier to analyze. I would start by factoring out the common term of ω^2 from the numerator and denominator. This would give us:

|H(ω)| = sqrt((1*10^-11)^2 + (5*10^-6)^2) / sqrt((1*10^-22)^2 + (1*10^-16)^2)

Next, I would simplify the fractions inside the square root by dividing the exponents. This would give us:

|H(ω)| = sqrt((1*10^-11)^2 + (5*10^-6)^2) / sqrt((1*10^-44) + (1*10^-32))

Now, we can see that as ω → ∞, the terms with ω in the denominator will approach 0. This means that the entire fraction will approach 1, since the numerator will remain constant. Therefore, as ω → ∞, |H(ω)| → 1.

For ω = 0, we can see that the terms with ω in the numerator will approach 0, while the terms with ω in the denominator will remain constant. This means that as ω → 0, |H(ω)| → 0.

Based on these asymptotic values, we can sketch the form of |H(ω)| as a curve that starts at (0,0) and approaches (infinity, 1) as ω increases. This behavior indicates that the circuit has a high pass filter, where it allows high frequency signals to pass through while attenuating low frequency signals.

I hope this helps in your solution process. Remember to always simplify and analyze the behavior of the transfer function to better understand the circuit.