Chapter 12

# Linear filtering

### Fondamental :

Harmonic analysis (or frequential) of a system is its study through its harmonic response , that is to say, its response in sinusoidal permanent regime when subjected to a sinusoidal input which is varied frequency .

Linear filtering
• Filter order :

Example of a low-pass filter :

Example of a high-pass filter :

, with the cut-off frequency to .

• Filter order :

Example of a low-pass filter :

where is the natural angular frequency of the filter and the damping coefficient.

is defined as the quality factor.

Example of a high-pass filter :

Example of a bandpass filter :

here is also the band-pass filter resonance pulse.

Example of a notch filter (or band-stop) :

### Définition : General notations and definitions

We note : (in complex notations)

is the phase shift of the output voltage with respect to the filter input voltage.

is the voltage transfer function of the filter :

And :

• is the real gain :

• is the gain in decibels (  :

Bode diagram of a low pass second order filter

The previous figure gives a Bode diagram of a band-pass filter, that is to say, the graphical representation of the gain in ( ) and the phase shift as a function of the pulse and on a logarithmic scale.

### Fondamental : Fundamental filtering 1st order (eg RC series circuit)

Studying an RC series circuit and measuring the output voltage in the open output.

Qualitative study of the nature of flitre :

• At low frequencies, the capacitor is equivalent to an open switch (its impedance is infinite).

The current in the circuit is zero. Thus, the voltage at the terminal is also zero and therefore .

• At high frequency, the capacitor is equivalent to a zero resistance wire. So, .

The filter studied here is thus a low-pass filter.

RC series circuit

Open output, the voltage divider rule gives :

With :

The natural angular frequency of the filter, which is also the cut-off frequency to , that is to say the pulse for which the real gain is ,or, what is equivalent, .

It is the transfer function of a low-pass filter of the first order.

Bode amplitude for a low-pass filter 1st order

The figure above shows the Bode amplitude of this low-pass filter of the first order.

The following figure shows the Bode phase.

Bode diagram for the phase of a low-pass filter 1st order

Other filter order :

• RC series, output voltage across  :

It is a high-pass filter.

With .

• Circuit RL series, output voltage across  :

It is a high-pass filter.

With :

• Circuit RL series, output voltage across  :

It is a low-pass filter.

With :

Bode amplitude of a high-pass filter 1st order
Bode phase of a high-pass filter 1st order

The preceding figures show the Bode diagram in amplitude and phase of a high-pass filter of the first order.

### Fondamental : Filter 2nd order (eg the series RLC circuit)

It is a series RLC circuit as a quadrupole (Figure below).

Filtering with a RLC series circuit

The following table shows the nature of the filter obtained by the output dipole :

 Dipole output Filter obtained Résistance Capacitor Coil Capacitor + Coil Bandpass Low pass High pass Band cut (rejection)

### Fondamental : The terminals of C in open output : low - pass filter

The voltage divider rule gives :

We want to write this transfer function in the standard form :

Low pass filter across the capacitor

For identification :

Whence :

So :

Referring to the study of mechanical oscillators (response in elongation), one obtains :

There are "resonance voltage" across terminals ( "load resonance "), ie corresponding to a maximum voltage across , for the pulse of GBF such that :

And the maximum voltage across at "load resonance" is :

The above formulas become, using the quality factor in place of the damping coefficient ( ) :

And :

Note :

For low amortization ( «small» and «large»), then :

Thus, if , the amplitude at the resonance is times that of the excitation : the resonance is termed "sharp" and may cause destruction of the oscillating system.

The following figures give the Bode diagram in gain and phase of a low-pass filter order. Different numerical values of are used.

Bode diagram of a low-pass filter 2nd order
Bode diagram in phase of a low-pass filter 2nd order

### Fondamental : The terminals of R open output : band - pass filter

The voltage divider rule gives :

We want to write this transfer function in the standard form :

Voltage across the resistor

For identification:

Hence (Member to Member ratio) :

So :

Resonant voltage across ("resonance intensity") :

A go "resonance voltage" across (« resonance current »), corresponding to a maximum voltage terminals , for a pulse GBF equal to the oscillation frequency of the circuit (RLC) series :

And the maximum voltage across for the "resonance current" is :

The pass-band is :

The following figures give the Bode diagram in gain and phase of a band pass filter order. Different numerical values of are used.

Bode in gain of a band pass filter 2nd order
Bode in phase of a band pass filter 2nd order

### Fondamental : The terminals L open output : high - pass filter

The voltage divider rule gives :

We want to write this transfer function in the standard form :

Voltage coil terminals

For identification, we find the same characteristics as for the low-pass and band-pass filters :

So :

The following figures give the Bode diagram in gain and phase of a high-pass filter order. Different numerical values of are used.

Amplitude diagram of a high pass filter 2nd order
Phase diagram of a high pass filter 2nd order

### Fondamental : The terminals of (C + L) output open: notch or band - stop filter

The voltage divider rule gives :

We want to write this transfer function in the standard form :

Voltage across the ensemble (L + C)

For identification, we find the same characteristics as for the low-pass, band-pass and high-pass filters :

So :

The following figures give the Bode diagram in gain and phase of a notch filter order. Different numerical values of are used.

Bode in gain a 2nd order band rejection
Bode in phase of a 2nd order band rejection