Active filters RC with a single amplifier

1) Comparison between the active filters and the filters passes to you:

The active filters are lighter and demand little space regarding the filters pass to you, moreover they can be realizes to you in integrated shape that of it it concurs the production in mass and the cost reduction.

 

2) Philosophies of realization of the active filters:

Cascata the net function is fattorizzata in products of terms of 2° the order everyone of which is individually realized from an active circuit RC and place in cascade to the others.

Diretta the net function comes completely realized from an only circuit

 

3) VCVS:

It is a source of tension controlled in tension, ideally has characteristic following:

to)       infinite stiffness of income

b)       stiffness of null escape

c)       escape tension linearly proporziona them through a gain to the income tension

In the VCVS it realizes with operational amplifiers the income stiffness to you is of some MW while the escape stiffness is smaller of 100W .

 

4) Configurations for a used operational amplifier like VCVS:

Invertente marks is applied them through the resistance R₁ to the inverting clip to which it reaches also marks them of escape through the resistance R₂ , the positive clip is to mass and therefore the gain turns out to be .

Invertente does not mark is applied them to the positive clip while to the clip negative it is connected towards mass the R resistance_{the 1} and resistance of feedback R₂ , the gain turns out to be .

Inseguitore is an not inverting amplifier in which R₂ = 0 that ₁ implies K=1ossia V ₂=V.

 

5) general amplifier Filter single:

The passive net in figure is characterized from the equations in which for the presence of the VCVS can be replaced and to eliminate 2ª the equation in how much Z_out= 0, being been moreover₃ = 0 obtains that for the realization

implica , , .

 

6) Sallen & pass-low Key:

From the equality and remembering the values it finds to you for N₃₁ , N₃₂ , N₃₃ is observed that Y₁ and Y₃ Y conductances must be constant that is₁ = G₁ e ₃Y = ₃G moreover N₃₂ must have one zero simple one in the origin therefore Y₂= sC₂ and Y₆= 0 while N₃₃ must be of the 2° order with real zeroes denied and therefore Y ₄ =sC ₄ is obtained_{to you} and Y₅= 0, of it follows che

For such outline some criteria of plan are following:

a) is R₁ = R₃= R and also ₂C = C₄ = C obtains , ,

b) is K=1, , , R₁ = R e C₂ = C , obtains and

c) is K=2, C₂ = C₄ = C , obtains and

All and the 3 plans give place to sensibility directly proporziona them to Q that it is deleterious when better obtainable filters to high Q are wanted to be realized instead using relationships 1:10 that 1:1 between R₁ and R₃ or ₂C and C₄ .

 

7) Sallen & Key pass-band:

From the equality and remembering the values it finds to you for N₃₁ , N₃₂ , N₃₃ is observed that Y₁ and Y₃ must be one a condenser and the other a resistore, assumes Y₁ = G₁ e ₃Y ₃ =sC in order to satisfy the condition on N₃₂ must be had ₂Y= G₂ and Y₆= 0 while in order to satisfy that one on N₃₃ we place ₄Y = ₄G and Y₅= sC₅ , ottiene .

For such outline some criteria of plan are following:

a) is R₁ = R₂ = R₄ = R and also ₃C = C₅ = C obtains e

b) is K=2 e R₁= C₃ = C₅ = 1 obtains ed

 

8) Sallen & pass-high Key:

From the equality and remembering the values it finds to you for N₃₁ , N₃₂ , N₃₃ it is found that ₁ = sC₁, ₂Y = ₂G mustbe Y, ₃Y ₃ =sC , ₄Y= ₄G , ₅Y = Y₆ = 0, obtains

For such outline some criteria of plan are following:

a) is R₂ = R₄ = R e C₁ = C₃ = C obtains , , .

b) sia K=1, , , C₁= C, R₂= R obtains e

c) sia K=2 e C₁ = C₃ = C , ottiene e

 

9) general Filter to infinite gain with single amplifier:

The passive net in figure is characterized from the equations in which for the operazionalesi presence of the amplifier has V₃ = 0, can be eliminated 2ª the equation in how much Z_out= 0, being moreover₃ = 0 obtains that for the realization

in figure it is reduced a .

 

10) pass-low Infinite-gain:

From the equality one deduces that as far as the denominators ₁ = Gmust be Y₁ e Y₃ = G₃ while for the denominators Y₂ = G₂ , Y₅ = sC₅ , Y₆ = sC₆ is reached therefore the equations , , .

A plan criterion consists in placing C₅ = C e C₆ = mC con , obtains , , .

Draft of an ideal filter for the realizations to Q elevated in how much introduces one sensibility of Q regarding the variations of the parameters always inferior to 1.

11) Infinite-gain pass-band:

From the equality si it has that placing Y₁= G₁ , Y₃ = sC₃ , Y₂= sC₂ , Y₅= G₅ e Y₆= G₆

A possible criterion of plan consists in porre C₂= C₃= C , ottiene , , .

 

12) pass-high Infinite-gain:

From the equality si it has that placing Y₁= sC₁,

Y₂ = sC₂ , Y₃= sC₃ , Y₅= G₅ e Y₆= G₆

A possible criterion of plan consists in porre the C₁ = C₃ = C, is obtained:

, , .

 

 

 

 

13) Criteria of plan for filters in cascade of greater order of 2:

If a filter with uneven order must be realized the section of 1° the order constituted from a RC or CR goes mail to the beginning of the cascade and continuation from a tension pursuer, the remaining sections of 2° the order go ordered with increasing Q.