Exercise 1.1Z: Low-Pass Filter of 1st and 2nd Order
The simplest form of a low-pass filter – for example, realisable as an RC low-pass filter according to exercise 1.1 – has the following frequency response:
- $$H_{\rm 1}(f) = \frac{1}{1+{\rm j}\cdot f/f_0}.$$
This is then referred to as a first-order low-pass filter. The diagram shows the following for this filter
- above the damping curve $a_1(f)$,
- below the phase response $b_1(f)$.
Correspondingly, for a low-pass filter of $n$–th order, the following defining equation applies:
- $$H_n(f) = H_{\rm 1}(f)^n.$$
In this task,
- based on the functions $a_1(f)$ and $b_1(f)$ of the low-pass filter of first order
- the damping curve and phase response of a low-pass filter of higher order is to be analysed.
In general, the following holds:
- $$H(f) = {\rm e}^{-a(f) - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}b(f)}.$$
Please note:
- The task belongs to the chapter System Description in Frequency Domain.
- There is the following relationship between the Np- and dB-values of an amplitude value $|H| = 1/x$ :
- $$a_{\rm Np} = \ln (x) = \ln (10) \cdot \lg (x) = \frac{\ln (10)}{20} \cdot a_{\rm dB} \approx 0.11513 \cdot a_{\rm dB}.$$
- Further consider that for two complex quantities $z_1$ and $z_2$ the following equations hold:
- $$|z_1 \cdot z_2| = |z_1| \cdot |z_2|, \hspace{0.5 cm}{\rm arc}\hspace{0.05 cm}(z_1 \cdot z_2) = {\rm arc}\hspace{0.05 cm}(z_1) + {\rm arc}\hspace{0.05 cm}(z_2).$$
Questions
Sample solution
- $$|H_{\rm 1}(f)| = \frac{1}{\sqrt{1+ (f/f_0)^2}}.$$
- This gives the damping curve in in Neper (Np):
- $$a_1(f) = \ln \frac{1}{|H_1(f)|} = {1}/{2} \cdot \ln \left[1 + ({f}/{f_0})^2 \right] \Rightarrow a_1(f = f_0) = 0.3466 \hspace{0.1 cm}{\rm Np},\hspace{0.5 cm}a_1(f = 2 f_0) = 0.8047 \hspace{0.1 cm}{\rm Np}.$$
The corresponding dB-values are obtained by multiplying by $1/0.11513 = 8.68589$ and result in
- $ \underline{3.01 \: {\rm dB} ≈ 3 \: {\rm dB}}$ für $ f = f_0$,
- $ \underline{6.99 \: {\rm dB}}$ für $ f = 2f_0$.
Thus, for the low-pass filter of first order the 3dB–cutoff frequency is $f_{\rm G} = f_0$.
(2) The frequency response $H_1(f)$ can also be represented separately by real and imaginary parts:
- $$H_{\rm 1}(f) = \frac{1}{ {1+ (f/f_0)^2} } - {\rm j} \cdot \frac{f/f_0}{ {1+ (f/f_0)^2} }.$$
- Hence, the following holds for the phase response:
- $$b_1(f) = - \arctan \hspace{0.1cm} ( {\rm Im} /{\rm Re} ) = \arctan \hspace{0.1cm} ({f}/{f_0}).$$
- $\arctan(1) = π/4 \rm \underline{\: = 0.786 \: rad}$ is obtained for $f = f_0$ and for $f = 2f_0$ the value $\arctan(2) \rm \underline{\: = 1.108 \: rad}$.
(3) For the amplitude response of a low-pass filter of $n$–th order the following is valid:
- $$|H_n(f)| = |H_{\rm 1}(f)|^n.$$
Bezüglich der (logarithmischen) Dämpfungsfunktion wird aus der $n$–fachen Multiplikation die $n$–fache Summe:
- $$a_n(f) = n \cdot a_1(f)= {n}/{2} \cdot \ln \left[ 1 + ({f}/{f_0})^2 \right].$$
Für den Tiefpass zweiter Ordnung ergibt sich daraus als Sonderfall:
- $$a_2(f) = \ln \left[ 1 + ({f}/{f_0})^2 \right]= 2 \cdot a_1(f).$$
Die dB–Werte lauten nun:
- $ \underline{6.02 \: {\rm dB} ≈ 6 \: {\rm dB}}$ für $f = ±f_0$,
- $\rm \underline{13.98 \: {\rm dB} ≈ 14 \: {\rm dB}}$ für $f = ±2f_0$.
Damit ist offensichtlich, dass für $n > 1$ der Parameter $f_0$ nicht mehr die 3 dB–Grenzfrequenz $f_{\rm G}$ angibt.
Für $n = 2$ ⇒ "Tiefpass zweiter Ordnung" gilt vielmehr der Zusammenhang:
- $${f_{\rm G} } = {f_0}/\sqrt{2}.$$
(4) Auch bezüglich der Phasenfunktion gilt:
- $$b_n(f) = n \cdot b_1(f), \hspace{0.3 cm} b_2(f) = 2 \cdot b_1(f).$$
Beim Tiefpass zweiter Ordnung sind somit alle Phasenwerte zwischen $±π$ möglich. Insbesondere ist
- $b_2(f = f_0) = π/2 \rm \underline{\: = 1.571 \: rad}$,
- $b_2(f = 2f_0) = \rm 2.216 \: rad$.
Da die Phase eine ungerade Funktion ist, gilt hier: $b_2(f = \: –2f_0) = \rm \underline{–2.216 \: rad}$.