---
title: "Ways of producing zeros of transmission"
subject: "Filter Design"
chapter: 4
key: "semester_5/filter_design:transmission-zeros-ways-poles"
source: https://www.bctengineer.com/q/semester_5/filter_design/transmission-zeros-ways-poles
site: BCT Engineering
---

# Ways of producing zeros of transmission

Filter Design · Chapter 4: Properties and Synthesis of Passive Networks

- Asked in: 2080 Bhadra, 5 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/transmission-zeros-ways-poles
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-4

## Question

What is zeros of transmissions? What are the different ways of producing zeros of transmission in a network realization? What is called poles and transmission poles?

## Short answer

Transmission zeros are s where transfer function is zero (no output). In ladders, produced by series arms becoming open (series L at ∞, series C at 0, parallel LC tank) or shunt arms becoming short (shunt C at ∞, shunt L at 0, series LC branch). Poles are natural frequencies; transmission poles are poles of transfer function, common to z11 and z22 except private poles.

## Answer

**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.

| Element / arm | Transmission zero at |
|---|---|
| Series $L$ | $s=\infty$ (open at high frequency) |
| Shunt $C$ | $s=\infty$ (short at high frequency) |
| Series $C$ | $s=0$ (open at dc) |
| Shunt $L$ | $s=0$ (short at dc) |
| Series arm: $L \parallel C$ tank | $\omega = 1/\sqrt{LC}$ (tank impedance → ∞) |
| Shunt arm: $L$–$C$ in series | $\omega = 1/\sqrt{LC}$ (branch impedance → 0) |

```text
 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**.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A zero of transmission is a value of s at which:
   A. The driving-point impedance becomes infinite
   B. The transfer function is zero, so no output appears for finite input
   C. The network has a natural frequency with port open
   D. The input impedance equals the load resistance
   Answer: B. Zeros of z21, y21 or V2/V1 are transmission zeros; the signal is blocked at that frequency. (Key point: Zero of transmission)

2. A parallel L-C tank in a series arm produces a transmission zero at:
   A. w = 1/sqrt(LC), where the tank is an open circuit
   B. w = 1/sqrt(LC), where the tank is a short circuit
   C. w = 0
   D. w = infinity
   Answer: A. The tank impedance goes to infinity at resonance, blocking the signal. (Key point: Series arm tank)

3. Which shunt element produces a transmission zero at s = 0?
   A. Shunt C
   B. Shunt L
   C. Series L
   D. Series parallel-LC tank
   Answer: B. A shunt inductor shorts dc to ground. A shunt C gives a zero at infinity; a series C also gives a zero at 0, but it is in the series arm. (Key point: Shunt arm element)

4. What are private poles?
   A. Poles of z11 that do not appear in z21
   B. Poles of the transfer function z21
   C. Zeros of transmission on the jw axis
   D. Poles lying in the right half plane
   Answer: A. Transmission poles are poles of z21 (shared with z11 and z22); poles of z11 absent from z21 are private poles. (Key point: Transmission poles)

## Related in Filter Design

- [Zeros of transmission and their realization](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Azeros-of-transmission-realization) — Filter Design, chapter 4, asked 10×
- [Inverse Chebyshev characteristics, order, poles and zeros](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ainverse-chebyshev-poles-zeros) — Filter Design, chapter 2, asked 2×
- [Transmission zeros and zero shifting by partial pole removal](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atransmission-zeros-zero-shifting) — Filter Design, chapter 4, asked 3×
- [Steps in realizing transmission zeros](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atransmission-zeros-realization-steps) — Filter Design, chapter 4, asked 2×
- [Terminated ladder with finite transmission zeros](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Afinite-transmission-zeros-ladder) — Filter Design, chapter 5, asked 2×

## Related in other subjects

- [Poles and zeros of a network function](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Apoles-and-zeros-significance) — Electrical Circuit Theory, chapter 5
- [Transfer function, poles and zeros from 3×3 state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-poles-zeros-3x3) — Control Systems, chapter 8, asked 3×
- [Difference equation example; poles/zeros response](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Apole-zero-define-difference-eq) — Digital Signal Analysis and Processing, chapter 3
- [Poles and zeros from state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-zeros-poles-5-6-1) — Control Systems, chapter 8