hugo/content/zettels/continuous_transfer_functions.md

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

Tags
: [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 {#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

<!--listend-->

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

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

<a id="figure--fig:filter-low-pass-first-order"></a>

{{< figure src="/ox-hugo/filter_low_pass_first_order.png" caption="<span class='figure-number'>Figure 1: </span>First order low pass filter, effect of \\(\omega\_0\\) (dashed: \\(\omega\_0 = 2\pi\\) rad/s)" >}}


## First Order High Pass Filter {#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

<!--listend-->

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

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

<a id="figure--fig:filter-high-pass-first-order"></a>

{{< figure src="/ox-hugo/filter_high_pass_first_order.png" caption="<span class='figure-number'>Figure 2: </span>First order high pass filter, effect of \\(\omega\_0\\) (dashed: \\(\omega\_0 = 2\pi\\) rad/s)" >}}


## Second Order Low Pass Filter {#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\\)

<!--listend-->

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

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

<a id="figure--fig:filter-low-pass-second-order"></a>

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


## Second Order High Pass Filter {#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}\\)

<!--listend-->

```matlab
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);
```

<a id="figure--fig:filter-high-pass-second-order"></a>

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


## Notch Filter {#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

<!--listend-->

```matlab
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);
```

<a id="figure--fig:filter-notch-gc"></a>

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

<a id="figure--fig:filter-notch-zeta"></a>

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


## Lead Filter {#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 [Figure 8](#figure--fig:filter-lead-effect-a-phase)):
    \\[ \angle H(j\omega\_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \\]

<!--listend-->

```matlab
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)));
```

<a id="figure--fig:filter-lead"></a>

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

<a id="figure--fig:filter-lead-effect-a-phase"></a>

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


## Lag Filter {#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

<!--listend-->

```matlab
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)));
```

<a id="figure--fig:filter-lag"></a>

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