Euterpe Synthesizer Laboratories Vertice, Руководство пользователя

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[ 22 ] the Filters section
Why in all filters there is always a -3dB level attenuation on the Cutoff frequency:
a brief mathematical analisys
-6 and -3dB are two very well known values in music, almost iconic. At certain point studying this subject
seems that everything in the universe is -6 or -3dB related.
Well, “start to run” means just that: on the Cutoff frequency there is a "totally unsuspected" -3dB level attenuation .
This is not a characteristic of our filterbank only, in fact it is a junction point across all the Filters literature.
Understand why this “strange” coincidence appears everywhere is all about physics and mathematics. What
follow is a brief simplification of this totally not obvious argument so take it for what it is: an attempt to intrigue
and illustrate the operation baseline of one of the most complex arguments of Electronics.
Do you really want to talk about math on a musical instrument users manual?
Why not. Filters basically are made by some capacitors and some resistors connected in various ways through the
signal path. Resistors and caps are two typical passive components, like cables, diodes or inductances, so something
that work with electricity just because they are naturally involved, not forced by some sort of parallel power supply
or logic computation. They are stupid, in one word, and repetitive. So they are predictable and mathematically
understandable even if we aren't Chuck Norris (because Chuck doesn't have an intellectual quotient, he has two of
them...).
Of course there can be complex filters based on interactions of integrated circuits and other component's but
capacitor and resistor remain the basic building toys almost always.
Capacitor, the most important frequency-related component, has on the current flow an important consequence:
it blocks the direct current, or DC (frequency of a DC signal = zero Hz). Audio is, for its oscillating nature, an
alternated signal, AC (frequency of an AC signal = not zero Hz), and this is enough for the capacitor: the signal pass
through. But, like so much in physics, it is all a transient matter. This means that capacitors don't show a “square”
behavior, ON or OFF, but a gradual blocking activity on the becoming continuous signal. In other words capacitors
don't stop an alternated flow only if it is alternating enough. This is the explanation for the smooth response of a
simple passive RC filter, like the one shown below (simple filters like this one are actually used into some areas of
Vertice).
LP 1 pole RC passive filter (1 pole = 6dB/oct)
If in this simple circuit there is current flow there also must be a potential difference across R and vice versa.
We can mathematically preview the system voltage behavior with the simple following considerations.
In an alternated signal situation
ω
is called angular velocity and is equal to
ω
=
2 ∙ π ∙ f
so it is frequency directly-related.
If
V
out
=
Z
c
Z
r
+
Z
c
∙V
input
and
Z
c
=
1
ω∙c
then:
at low frequencies we have angular velocity ( ω ) very low, zero if we feed a real continuous signal (zero in this
case mean no oscillations, and consequently no calculable frequency, so a very low, or null, angular velocity).
A low ω means an high Zc ('cause 1/0=∞) and consequently a negligible
1
Zr , so the first equation can become
V
out
=
Z
c
Z
c
∙V
input
= 1
∙V
input
1 In electronics the ratio 10:1 between two elements is enough to made the lower value ignored
Signal
INPUT
Signal
OUTPUT
R
Ic
Ir
Vc
Vr
C
GND
R = resistor (passive component)
C = capacitor (passive component)
V = voltage (or potential difference)
I = current (Ampere)
ƒ = frequency (numbers of occurrences of a repeating event per unit time)
F = frequency response (which in filters theory is quite equal to Gain factor at given frequency )
ω
= angular velocity (directly related to frequency, inversely related to
τ
)
τ
= RC time constant (basics of capacitors theory)
c = capacitance (the ability to store electric charges)
Z = capacitive o resistive impedance (impedance is the measure of the opposition that a circuit present
to a flowing current)