# Elliptic Low Pass Filter with the State Variable Biquad -Applet.

Cuthbert Nyack
For an Elliptic filter, a circuit with both poles and zeros is required. These circuits are usually referred to as Biquadratic (or Biquad) circuits. Several single opamp Biquad circuits are available in the literature. Here the Biquad based on the State Variable circuit is used. Although expensive in opamps it is easy to control, less sensitive and the easiest to understand.

The circuit for a second order section is shown below. If the Capacitor C6 is added in parallel with R6 then the circuit can be used for a third order section.
The transfer function of the circuit has 2 poles and 2 zeros and is shown below. When R1 = R4, the zeros occur on the imaginary axis. Component values can then be chosen to make the transfer function equal to that of a second order Elliptic filter. Several circuits can be combined to produce higher order filters.
The applet below shows component values for Elliptic filters for orders n = 2 to 9. Filter order is set by scrollbar 0. For any n, Elliptic Filters are characterised by the reflection coefficient r and the modular angle q. r and q can be set by scrollbars 2 and 3. Changes in the transfer function when any one of the component values are changed by up to ± 20% can also be seen.
Values of R and C and the upper 3dB frequency in Hz can be set by scrollbars (4,5), (6,7) and (38,39,40). These values are used to calculate denormalized component values which are shown in green.

Image below shows component values for a 9th order Elliptic high pass filter implemented as 3 second order and 1 third order section.
r = 20 and q = 43°.
R is set to 100KW, C = 0.1 mF and the 3dB frequency is 2kHz.
The normalized transition BW is ~0.466rad/s, the minimum stop band attenuation is > 100dB and the pass band ripple is 0.177dB.
Image below show the magnitude(pink) and phase(cyan) error introduced when the resistor R2 in the first section is 5% larger than its correct value.