hugo/content/zettels/continuous_transfer_functions.md

+++ title = "Continuous Transfer Functions" author = ["Dehaeze Thomas"] draft = false category = "Signals and Systems" +++

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[Discrete Transfer Functions]({{< relref "discrete_transfer_functions.md" >}})

Classical transfer functions in the Laplace domain, with the meaning of their parameters. In all the examples, the frequency of interest is \(\omega_0 = 2\pi\) rad/s (1 Hz) and is shown with a dashed line. To obtain the equivalent digital filter, see [Discrete Transfer Functions]({{< relref "discrete_transfer_functions.md" >}}).

First Order Low Pass Filter

\begin{equation} H(s) = \frac{1}{1 + s/\omega_0} \end{equation}

Parameters:

• \(\omega_0\): cut-off frequency in [rad/s]

Characteristics:

• Low frequency gain of \(1\)
• At \(\omega_0\): gain of \(1/\sqrt{2}\) (-3 dB) and phase of -45 deg
• High frequency roll-off of -20 dB/dec, phase tends to -90 deg

w0 = 2*pi; % Cut-off frequency [rad/s]

H = 1/(1 + s/w0);

{{< figure src="/ox-hugo/filter_low_pass_first_order.png" caption="Figure 1: First order low pass filter, effect of \(\omega_0\) (dashed: \(\omega_0 = 2\pi\) rad/s)" >}}

First Order High Pass Filter

\begin{equation} H(s) = \frac{s/\omega_0}{1 + s/\omega_0} \end{equation}

Parameters:

• \(\omega_0\): cut-off frequency in [rad/s]

Characteristics:

• High frequency gain of \(1\)
• At \(\omega_0\): gain of \(1/\sqrt{2}\) (-3 dB) and phase of +45 deg
• Low frequency slope of +20 dB/dec, phase tends to +90 deg

w0 = 2*pi; % Cut-off frequency [rad/s]

H = (s/w0)/(1 + s/w0);

{{< figure src="/ox-hugo/filter_high_pass_first_order.png" caption="Figure 2: First order high pass filter, effect of \(\omega_0\) (dashed: \(\omega_0 = 2\pi\) rad/s)" >}}

Second Order Low Pass Filter

\begin{equation} H(s) = \frac{1}{1 + \frac{2 \zeta}{\omega_0} s + \frac{s^2}{\omega_0^2}} \end{equation}

Parameters:

• \(\omega_0\): natural frequency in [rad/s]
• \(\zeta\): damping ratio

Characteristics:

• Low frequency gain of \(1\)
• High frequency roll-off of -40 dB/dec, phase tends to -180 deg
• Phase of -90 deg at \(\omega_0\), whatever the damping
• Gain at \(\omega_0\) equal to \(1/(2\zeta)\)
• For \(\zeta < 1/\sqrt{2}\) there is a resonance peak, for \(\zeta \geq 1/\sqrt{2}\) the magnitude is monotonically decreasing
• Smaller \(\zeta\) gives a sharper phase transition around \(\omega_0\)

w0 = 2*pi; % Natural frequency [rad/s]
zeta = 0.1;  % Damping ratio

H = 1/(1 + 2*zeta/w0*s + s^2/w0^2);

{{< figure src="/ox-hugo/filter_low_pass_second_order.png" caption="Figure 3: Second order low pass filter, effect of the damping ratio \(\zeta\)" >}}

Second Order High Pass Filter

\begin{equation} H(s) = \frac{s^2/\omega_0^2}{1 + \frac{2 \zeta}{\omega_0} s + \frac{s^2}{\omega_0^2}} \end{equation}

Parameters:

• \(\omega_0\): natural frequency in [rad/s]
• \(\zeta\): damping ratio

Characteristics:

• High frequency gain of \(1\)
• Low frequency slope of +40 dB/dec, phase tends to +180 deg
• Phase of +90 deg at \(\omega_0\), whatever the damping
• Gain at \(\omega_0\) equal to \(1/(2\zeta)\), resonance peak for \(\zeta < 1/\sqrt{2}\)

w0 = 2*pi; % Natural frequency [rad/s]
zeta = 0.1;  % Damping ratio

H = (s^2/w0^2)/(1 + 2*zeta/w0*s + s^2/w0^2);

{{< figure src="/ox-hugo/filter_high_pass_second_order.png" caption="Figure 4: Second order high pass filter, effect of the damping ratio \(\zeta\)" >}}

Notch Filter

\begin{equation} H(s) = \frac{s^2 + 2 g_c \zeta \omega_n s + \omega_n^2}{s^2 + 2 \zeta \omega_n s + \omega_n^2} \end{equation}

Parameters:

• \(\omega_n\): frequency of the notch in [rad/s]
• \(g_c\): gain at the notch frequency (\(g_c < 1\) gives a notch, \(g_c > 1\) gives a peak)
• \(\zeta\): damping ratio, sets the width of the notch

Characteristics:

• Unity gain at low and high frequency
• Gain of exactly \(g_c\) at \(\omega_n\), with a phase of 0 deg
• For \(g_c < 1\): phase is negative below \(\omega_n\) and positive above
• Larger \(\zeta\) gives a wider notch, and more phase lag/lead around it

wn = 2*pi; % Notch frequency [rad/s]
gc = 0.02; % Gain at the notch frequency
zeta = 0.2;  % Damping ratio (notch width)

H = (s^2 + 2*gc*zeta*wn*s + wn^2)/(s^2 + 2*zeta*wn*s + wn^2);

{{< figure src="/ox-hugo/filter_notch_gc.png" caption="Figure 5: Notch filter, effect of the gain at the notch frequency \(g_c\)" >}}

{{< figure src="/ox-hugo/filter_notch_zeta.png" caption="Figure 6: Notch filter, effect of the damping ratio \(\zeta\) (notch width)" >}}

Lead Filter

A lead filter is usually used to add some phase lead at a particular frequency, at the expense of a reduction of the gain at low frequency and increase of the gain at high frequency.

\begin{equation} H(s) = \frac{1}{\sqrt{a}} \frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}, \quad a > 1 \end{equation}

Parameters:

• \(\omega_c\): frequency at which the phase lead is maximum, in [rad/s]
• \(a\): sets the amount of phase lead, and the gain increase between low and high frequency

Characteristics:

• Zero at \(\omega_c/\sqrt{a}\) and pole at \(\omega_c \sqrt{a}\), geometrically centered on \(\omega_c\)
• Unity gain at \(\omega_c\), low frequency gain of \(1/\sqrt{a}\), high frequency gain of \(\sqrt{a}\)
• Maximum phase lead, obtained at \(\omega_c\), equal to (Figure 8): \[ \angle H(j\omega_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \]

a  = 3;    % Amount of phase lead
wc = 2*pi; % Frequency of maximum phase lead [rad/s]

H = 1/sqrt(a)*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));

{{< figure src="/ox-hugo/filter_lead.png" caption="Figure 7: Lead filter, effect of the parameter \(a\)" >}}

{{< figure src="/ox-hugo/filter_lead_effect_a_phase.png" caption="Figure 8: Maximum phase lead as a function of \(a\)" >}}

Lag Filter

A lag filter is usually used to add low frequency gain, at the expense of some phase lag.

\begin{equation} H(s) = a \cdot \frac{1 + \frac{s}{\omega_c \sqrt{a}}}{1 + \frac{s}{\omega_c/\sqrt{a}}}, \quad a > 1 \end{equation}

Parameters:

• \(\omega_c\): frequency at which the phase lag is maximum, in [rad/s]
• \(a\): sets the amount of phase lag, and the gain increase at low frequency (the low frequency gain is \(a\))

Characteristics:

• Zero at \(\omega_c \sqrt{a}\) and pole at \(\omega_c/\sqrt{a}\)
• Low frequency gain of \(a\), high frequency gain of \(1\) (unchanged), gain of \(\sqrt{a}\) at \(\omega_c\)
• It is used to increase the low frequency gain (e.g. instead of a pure integrator), \(\omega_c\) should be kept well below the crossover frequency to limit the phase lag there

a  = 3;    % Amount of phase lag
wc = 2*pi; % Frequency of maximum phase lag [rad/s]

H = a*(1 + s/(wc*sqrt(a)))/(1 + s/(wc/sqrt(a)));

{{< figure src="/ox-hugo/filter_lag.png" caption="Figure 9: Lag filter, effect of the parameter \(a\)" >}}