Ways of producing zeros of transmission

Zeros of transmission of a two-port network are the values of complex frequency $s$ at which the transfer function ($z_{21}$, $y_{21}$, $V_2/V_1$ or $I_2/I_1$) is zero, i.e. a finite input produces no output at that frequency. For an LC ladder they lie on the $j\omega$ axis (including $s=0$ and $s=\infty$).

Ways of producing zeros of transmission

1. Series arm open — a series element whose impedance becomes infinite: series $L$ (zero at $\infty$), series $C$ (zero at $0$), series parallel-LC tank (zero at $1/\sqrt{LC}$).
2. Shunt arm short — a shunt element whose impedance becomes zero: shunt $C$ (zero at $\infty$), shunt $L$ (zero at $0$), shunt series-LC branch (zero at $1/\sqrt{LC}$).
3. Balanced bridge / lattice — in bridge or lattice networks, transmission can be zero when the bridge is balanced (cancellation of two paths), giving zeros even off the $j\omega$ axis.
4. Parallel ladders (twin-T) — two paths whose outputs cancel, e.g. the twin-T notch network.

In ladder synthesis, zeros at $0$ and $\infty$ come from complete removal of poles; zeros at finite frequencies come from zero shifting (partial pole removal) followed by complete removal.

In a ladder, a signal is blocked when a series arm is an open circuit (its impedance has a pole) or a shunt arm is a short circuit (its impedance has a zero, i.e. its admittance has a pole). So the transmission zeros of a ladder are the poles of the series-arm impedances and the poles of the shunt-arm admittances.

 zero at w1=1/sqrt(L1C1)   zero at w2=1/sqrt(L2C2)
o--[L1||C1]--+---------o
             |
            [L2]
            [C2]
             |
o------------+---------o

Poles and transmission poles

• Poles of a network function are the values of $s$ at which the function becomes infinite. For a driving-point impedance they are the natural frequencies of the network with the port open-circuited.
• Transmission poles are the poles of the transfer function ($z_{21}$, $y_{21}$ or $V_2/V_1$), i.e. the frequencies where the transmission becomes infinite. They are the natural frequencies of the network that are seen at the output. In a lossless two-port every transmission pole of $z_{21}$ is also a pole of $z_{11}$ and $z_{22}$; poles of $z_{11}$ that do not appear in $z_{21}$ are called private poles.