Transmission zeroes

A transmission zero is a frequency at which the transfer function of a linear two-port network has zero transmission. Transmission zeroes at zero frequency and infinite frequency may be found in high-pass filters and low-pass filters respectively. Transmission zeroes at finite, non-zero frequency may be found in Band-stop filters, elliptic filters, and Type II Chebyshev filters. Transfer functions with both zero and infinite frequency can be found in band-pass filters. A transfer function may have multiple zeroes at the same frequency. A transfer function may have any number of transmission zeroes at zero frequency and infinite frequency, but transmission zeroes at finite non-zero frequency always come in conjugate pairs.

Circuits with transmission zeroes

Generalized impedance converter

A GIC (generalized impedance converter) based circuit that has finite non-zero transmission zeroes.

The circuit depicted to the left, based on a GIC (generalized impedance converter), has finite non-zero transmission zeroes.

State variable derived

A state variable filter derivation that has transmission zeroes under the condition that R1/R4 = R7/R6.

The filter circuit to the right has the following transfer function:

$H(s)=-\left({\frac {R_{2}R_{8}}{R_{5}R_{6}}}\right){\frac {s^{2}R_{3}C_{1}R_{5}C_{2}+s[({\frac {R_{3}}{R_{1}}})(1-{\frac {R_{1}R_{6}}{R_{4}R_{7}}})R_{5}C_{2}]+{\frac {R_{6}}{R_{7}}}}{s^{2}R_{3}C_{1}R_{2}C_{2}+s({\frac {R_{3}}{R_{1}}})R_{2}C_{2}+{\frac {R_{8}}{R_{7}}}}}$

This circuit produces transmission zeroes at

$\omega _{0Z}=\left[{\frac {R_{6}/R_{7}}{R_{3}C_{1}R_{5}C_{2}}}\right]^{\frac {1}{2}}$

when R₁/R₄ = R₇/R₆.