- Second order filter often referred to as Biquads.
- Poles are complex conjugate in left half of the “s” plane and zeros are complex conjugate that may lie any where in ‘s’ plane.
- It is to b remembered that:

T(s)=k(s+z1)/(s+p1)

Where K=b1/a0 , z1=b0/b1 (zero)

And p1=a0/a1 (pole)

In s plane these two quantities are located at s=-z1,s=-p1

- Pole,p1, will always lie on negative side of real axis
- Zero may be either positive or negative part of real axis

* Design Parmeter “Q” and “Wo”:*

- We begin with PLC circyit shown in figure which has new familiar form of a voltage divider. The transfer function of given circuit will be:

T(S) =VL(s)/V1(s)

- The behaviourof the circuit at low frequency andhigh frequency is :

At low frequency, capacitoer behaves as open,then there is current in R, L and “c” circuit then

Vl=V1----approx equal

- At higher frequency, capacitor will act as short, so that Vl approaches the value Vl=0

- From voltage divider :

T(s)= Z2/(Z1 + Z2)

This is the equation for second order circuit given above

T(s) = 1/cs/(R+Ls+1/cs) = 1/Ls/S(^2 + (R/L)s + 1/Lc)

- The result may be put into tandard form by defining new quantites, we observe that when defining new quantity then circuit is lossless with R=0 then

S^2 =1/Lc=0

- S1,S2=+-j1/Lc =+-jW

Mean poles are on imalnary axis and are conjugate axis.

- For lossy circuit or coils, for which a quality factor “Q” defined as Q/RLoss, which is the ratio of reactance at frequency “w” to resistance

At Frequency

w=w0

Q=WoL/R =1/R1/Lc

R/L=wo/Q

Put values of this equation in previous one, we get;

T(s) = wo^2/(s^2+ (wo/Q)s +Wo^2)

- The transfer function for low pass filter in this equation can be normalized form such that

T(j)=1

- Amore generalized form for T(s), in active circuit will recognize possibility of gain and also that, the associated circuit, may be inverting or no inverting

T(s)=+-Hwo^2/(s^2 + (wo/Q)s + wo^2

This can b represented as this diagram:

- Now we scale frequency by dividing “s” by wo i.e., we use the normalized frequency, Sn=S/wo
- T(s)=+-H/(Swo)^2 + (1/Q)(S/wo+1)

=+-H/Sm^2+ (1/Q)(Sw0 +1)