Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 61

Fig. 1.41 Active first-order
Butterworth LPF

R 2

R 1

v −

_

C Y

v + +

+ +

X

R

Fig. 1.42 Active first-order
Butterworth HPF

R 2

R 1

v − _

v +

Y

R

+ + +

X

C

: V þ ¼ X

V þ ¼ 1 þ R 2

Y X c

R þ X c

1 =ð jxC Þ
R þ 1 =ð jxC Þ ¼ X

1
1 þ jxRC

¼ X

R 1

1
1 þ jxRC ¼

V þ

Y

X ¼ Y X ¼ 1 þ R 2

G m
1 þ jxRC

with G m ¼ 1 þ R 2 )

R 1 :

V þ

R 1

G m
1 þ x o 2

where x o ¼ 1

) j H ð x Þj¼ r p is the cutoff frequency :

RC

Comparing this with the known form for the B-LPF magnitude gain:

q

1 þð x = x c Þ 2n j H ð x Þj¼ G m = ;

the above is seen to be a first-order B-LPF. Exchanging R and C as in Fig. 1.42
gives the B-HPF with magnitude response:

j H ð x Þj¼ G m ð x = x o Þ

r

1 þ x o 2

This is the standard form for the B-HPF as can be found from Tables:

j¼ G m

G m
1 þ x o = s

j H LP s ðÞ

1 þ s N ; hence

j H HP ð s Þ j¼