Question

1. What is the cutoff frequency?
   

2. What is the Butterworth response of a filter?
   
   

3. What are the differences between first-order and second-order filters?

Answer 1

i) CUTOFF FREQUENCY

The cutoff frequency is given by the equation:

Cutoff Frequency

Cutoff frequency is the frequency at which magnitude if filter is is 3db lower than maximum magnitude.

thus, all high frequencies above the cut-off point band rolls down to zero at 20dB per decade or 6dB per octave in the stop band for first order filter.

ii) Frequency response of Butterworth filter

Butterworth Filter Ideal Frequency Response

The frequency response of the nth order Butterworth filter is given as

Where ‘n’ indicates the filter order, ‘ω’ = 2πƒ, Epsilon ε is maximum pass band gain, (Amax). If we define Amax at cut-off frequency -3dB corner point (ƒc), then ε will be equal to one and thus ε2 will also be equal to one. But, if we want to define Amax at another voltage gain value, consider 1dB, or 1.1220 (1dB = 20logAmax) then the value of ε can be found by:

Where, H0 represents the maximum pass band gain and H1 represents the minimum pass band gain. Now, if we transpose the above equation, then we will get

By using the standard voltage transfer function, we can define the frequency response of Butterworth filter as

Where, Vout indicates voltage of output signal, Vin indicates input voltage signal, j is square root of -1, and ‘ω’ = 2πƒ is the radian frequency. The above equation can be represented in S-domain as given below

iii) Difference between First order and second order filter

a.)

Order basically defines the rolloff rate how does it attenuate the signals a first order filter has -20db/decade roll off past of cutoff frequency.

Second order has -40 db/ decade.

b.)

Basically 2nd order is better in filtering than first order.

c.)

A first order filter would have one capacitor or one inductor, that affects the filters frequency response. A second order filter would have two capacitors or two inductors, or one capacitor and one inductor, that affects the filter's frequency response.