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Filter notes.

Contents


Introduction.

This is some notes on different filters.
They are in no particular order.
Some of the calculations can be downloaded as a spread-sheet here.


Transforming cascaded 1.order filters to 2.order filter and vice versa.

f0 and Q are for the 2.order filter. Q ≤ 0.5.
f1 and f2 are the 2 1.order filters.

f0=SQRT(f1*f2).

Q=f0/(f1+f2).

f1=0.5*(f0/Q-SQRT((f0/Q)^2-4*f0^2)).

f2=0.5*(f0/Q+SQRT((f0/Q)^2-4*f0^2)).


Unity gain Sallen-Key bandpass filter with identical capacitor values.

Sallen Key bandpass filter schematic.
Fig.1: Sallen Key bandpass filter [1].

Select f, Q, C and R4.

K=V(OUT)/V(IN)=1.

C1=C2=C.

T=1/(2*Q).

A=1/(1+1/(2*Q^2)).

R2=(T*(1-A)/A+T)*1/(2*pi*f*C).

R3=Q/(pi*f*C).

R5=R4/(1/A-1).

R6=(T*A/(1-A)+T)*1/(2*pi*f*C).


Amplitude response of RIAA network.

RIAA filter frequency response
Fig.2: RIAA filter frequency response.

t1 is a low frequency zero.
t2 is an optional IEC time constant [ 7950 µs ]
t3 is a RIAA time constant [ 3180 µs ]
t4 is a RIAA time constant [ 318 µs ]
t5 is a RIAA time constant [ 75 µs ]
t6 is a high frequency zero
K is the low frequency gain of the filter. The 1 kHz gain is 20 dB lower.
f is the frequency [ Hz ]

Av=K*((SQRT(1+(2*pi*f*t1)^2)/(2*pi*f*t1))*((2*pi*f*t2)/SQRT(1+(2*pi*f*t2)^2)))*(SQRT(1+(2*pi*f*t4)^2)/SQRT(1+(2*pi*f*t3)^2))*(SQRT(1+(2*pi*f*t6)^2)/SQRT(1+(2*pi*f*t5)^2)).

If you do not want t1, set it to a very high value ( or remove it from the equation ).
If you do not want t2, set t1 and t2 to a very high value ( or remove them from the equation ).
If you do not want t6, set it to zero.


Amplitude response of 1.order high-pass filter.

f is the frequency [ Hz ]
f0 is the filter cut-off frequency [ Hz ]
K is a gain factor [ V/V ]

Av=K*((f/f0)/SQRT(1+(f/f0)^2)).


Amplitude response of 1.order high-pass filter with zero.

f is the frequency [ Hz ]
f0 is the filter cut-off frequency [ Hz ]
fz is the filter zero frequency { Hz ]
K is a gain factor [ V/V ]

Av=K*((f/f0)/SQRT(1+(f/f0)^2))*(SQRT(1+(f/fz)^2)/(f/fz)).


Amplitude response of 1.order low-pass filter.

f is the frequency [ Hz ]
f0 is the filter cut-off frequency [ Hz ]
K is a gain factor [ V/V ]

Av=K*(1/SQRT(1+(f/f0)^2)).


Amplitude response of 1.order low-pass filter with zero.

f is the frequency [ Hz ]
f0 is the filter cut-off frequency [ Hz ]
fz is the filter zero frequency { Hz ]
K is a gain factor [ V/V ]

Av=K*(SQRT(1+(f/fz)^2)/SQRT(1+(f/f0)^2)).


Amplitude response of pink noise filter.

f is the frequency [ Hz ]
f0 is the filter cut-off frequency [ Hz ]
f0 must be much lower than any frequency of interest.
K is a gain factor [ V/V ]

Av=K/SQRT(SQRT(1+(f/f0)^2)).


IEC 60268-1 filter.

This is a IEC-60268-1 noise measurement filter [2].
It is normally used in series with a pink-noise filter for shaping a white-noise signal.

IEC 60268-1 filter schematic.
Fig.3: IEC 60268-1 filter.

R31 and C31 is a 700 kHz roll-off for the input. They are not part of the IEC 60268 filter.
The input buffer is required as the filter must be driven from a low impedance.
C32, C33, R32, R33 is a 2.order Sallen-Key high-pass filter.
C34, C35, C36, R34, R35, R36 is a 3.order Sallen-Key low-pass filter.
As the roll-off frequencies for the HP and LP filter are far apart, they can share the same OP-AMP ( with some slight component value adjustment ).

IEC 60268-1 filter Monte Carlo analysis
Fig.4: IEC 60268-1 filter 500 runs Monte Carlo analysis.

The red traces are the limits from the specification and the lime traces is the simulation result.
The upper plots use the left axis and the lower plots the right.
This simulation was done with OPA134 OP-AMPs.
The upper roll-off is very close to the limits, so for some applications I might replace C34..C36 with 1% types.


Peaking frequency and amplitude for 2.order LP and HP filters.

f0 is the filter cut-off frequency [ Hz ]
Avpeak is the peak gain [ V/V ]
flp is the peak frequency for a low-pass filter [ Hz ]
fhp is the peak frequency for a high-pass filter [ Hz ]

Q>=SQRT(0.5).

Avpeak=Q/SQRT(1-1/(4*Q^2)).

flp=f0*SQRT(1-1/(2*Q^2)).

fhp=f0/SQRT(1-1/(2*Q^2)).


References.

[1] R. P. Sallen and E. L. Key, “A practical method of designing RC active filters"
You may be lucky to find a copy of this in your local technical library.
[2] UL Standard for Safety for Audio, Video and Similar Electronic Apparatus - Safety Requirements, UL 60065.

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