Page 333 - Analog and Digital Filter Design

330 Analog and Digital Filter Design

0.5 = !!d [,2. c + (KmK0)l
N
'
4

But w,"' = K@'Ko so, expanding the equation for cfurther this becomes:
(R1+ R2). N.C

K@ .KO 2
0.5 =
4C.N.(R1+ R2)
Given RI and C, the value of R2 can be found such that c= 0.7071; this can be
found by expanding and transposing the above equation.

K@ . KO [R2'.C2+( N2 )+ 2. N .C. R2
2=
C.N.(Rl+ 112) K@'. KO' , K@Ko

Multiply both sides of the equation by N.(Rl + R2), then expand the right-
hand side by multiplying through by (K@. KolC).

K@.Ko.RZ'.C+( N' )+2.N.C.R2]
K@. KO .C

Subtract 2. N. R2 from both sides leaving an equation in terms of R2' only:

N'
2.N.R1= K@.Ko.R2'.C+
(K9.Ko.C)

Rearranging this to move R2? to the left-hand side, this becomes:

K@. KO. R2' .C = 2. N. R1-

So now dividing both sides by K@. KO. to obtain:
C

R2' =

R2 =
C.K@.Ko K@'.Ko2.C2

So, using the known or selected values it is possible to find a value for R2 that
gives a damping factor of 0.7071. If a different value of [is chosen, the 2. N. R2