Physics and Engineering of Radiation Detection

8.4. Filtering 491

Vin Vout

(b)

(a)Low Pass Filters

High Pass Filters

(b)Band Pass Filter

Figure 8.4.1: (a) (b) Schematics of different types of low

pass and high pass filters. (c) A simple band pass filter.

8.4.A LowPassFilter..........................

The RC integrator we discussed earlier (see Fig.8.3.4(b)) is actually a low pass filter
as can be seen from the following expression for its output voltage.

Vout=Vin

1

√

1+(2πfRC)^2

(8.4.1)

HereVinis the input voltage andfis the frequency of the input signal. It is obvious
that the response of the circuit is weaker at higher frequencies. The frequency at
which the response becomes too weak is called the cutoff frequency of the filter. Its
value depends not only on the circuit components but also on the definition of the
too weak output. The most commonly agreed upon value of the cutoff frequency is
the value at which the response becomes -3dB. To translate this value in terms of
resistance and capacitance of the circuit, let us take the base-10 logarithm of both
sides of the above equation and multiply by 2. We are doing this to convert the
ratioVout/Vininto decibel (dB) units.

2log

(

Vout

Vin

)

=2log

[

1

√

1+(2πfRC)^2

]

(8.4.2)