I'm unsure about the RLC low-pass filter transfer and frequency response functions I've been trying to calculate.
I need $$ H(s)=\frac{Y(s)}{X(s)} $$ where x(t) is Vin and y(t) is Vr. I calculated $$\frac{d^2v_c(t)}{dt^2}+\frac{1}{LC}v_c(t)=\frac{1}{LC}x(t)$$or$$ H(s)=\frac{1}{s^2LC+1} $$
This transfer/frequency response has been giving me the correct magnitudes but obviously doesn't give a phase angle. Have I calculated the transfer function correctly in this instance? Any assistance would be appreciated.
Best Answer
The transfer function starts from the potential divider equation for the components: -
$$\dfrac{\dfrac{\frac{R}{sC}}{R + \frac{1}{sC}}}{\dfrac{\frac{R}{sC}}{R + \frac{1}{sC}}+sL} \Longrightarrow \dfrac{R}{R+s^2RLC+sL}$$
And when you drill down it becomes this: -
$$\dfrac{\dfrac{1}{LC}}{s^2+s\dfrac{1}{CR}+\dfrac{1}{LC}}$$
So immediately you can see that your first error is in not including "R" in the formula.
Only the formula above will give you the correct magnitudes at all frequencies. And, it does give you the phase angle. You need to substitute s with jw: -
$$H(j\omega) = \dfrac{\dfrac{1}{LC}}{\dfrac{1}{LC}-\omega^2 +j\omega\dfrac{1}{CR}}$$
So, you have a complex number that can give you amplitude and phase angles.
It might be easier if you convert it into "the standard form" of a 2nd order low pass filter namely: -
$$\dfrac{1}{1-\dfrac{\omega^2}{\omega_n^2 } +j\dfrac{\omega}{\omega_n}\cdot 2\zeta}$$
Where \$\omega_n^2 = \dfrac{1}{LC}\$
and, \$2\zeta = \dfrac{1}{CR\cdot \omega_n}\$
I recommend this route because it's intuitively easier to examine the phase angles. For instance, at DC, \$\omega\$ is zero and the formula reduces to unity hence gain is 1 and phase angle is 0 degrees. At very high frequencies, the formula reduces to a very small negative number thus impling that the phase is 180 degrees and the amplitude is very small.
At resonance (\$\omega = \omega_n\$), the formula reduces to \$\dfrac{1}{j2\zeta}\$ hence the phase angle is -90 degrees and the amplitude response equals the Q of the circuit (Q = \$\dfrac{1}{2\zeta}\$}.
See this web page for an interactive RLC filter tool/calculator: -