Page 163 - Analog and Digital Filter Design

1 60 Analog and Digital Filter Design

problem with loading of op-amp stage outputs. Remember, as stated in the
previous chapter, loading can cause distortion and increase the power supply
current. If resistor values are much higher than lOOkR there may be problems
with noise pickup. High impedance circuits capacitively couple with the electric
field from other circuits. This coupling could cause the pickup of noise and other
unwanted signals, which may interfere with the wanted signal. Also, thermal
noise voltage increases in proportion to the resistance.

The normalized highpass active filter model uses 1F capacitors between the
filter input and the op-amp input. The normalized design is based on a cutoff
frequency of 1 rads. Denormalization is quite simple: (1) scale the impedance;
(2) scale for frequency by denormalizing the capacitance value.

Impedance scaling is simply dividing the input capacitor(s) value to give suit-
able input impedance. The input impedance of an active filter will tend towards
1 R as the frequency approaches the normalized cutoff frequency of 1 rad/s,
since the series capacitor C = 1F and its reactance is Xc = l/wC. The input
impedance will therefore change with frequency. To reduce this effect, capaci-
tors with a reactance of about 100 times the desired filter input impedance could
be used. A separate terminating resistor could then be used to provide the
correct load impedance at all frequencies.

Scaling the resistor values can now be carried out using the following equation:

Where R’is the normalized value calculated earlier and C is the denormalized
value chosen to give a suitable input impedance.

For example, suppose you want a second-order Butterworth filter using a high-
pass Sallen and Key design with an input impedance of 600R and a cutoff
frequency F, = 4kHz. The normalized lowpass poles are located at 0.7071
k j0.707 1.

1
Scaling the capacitor for a 60 kR reactance at 4 kHz, gives X, = 60,000 = -
2RF,C’

1
Thus C = = 663pF.
120, OOOR F,
This is a nonstandard value, so let C1 (and C2) = 680pF. A smaller value (higher
reactance) could have been used to increase the filter’s input impedance.