--- title: "Zeros of transmission and their realization" subject: "Filter Design" chapter: 4 key: "semester_5/filter_design:zeros-of-transmission-realization" source: https://www.bctengineer.com/q/semester_5/filter_design/zeros-of-transmission-realization site: BCT Engineering --- # Zeros of transmission and their realization Filter Design · Chapter 4: Properties and Synthesis of Passive Networks - Asked in: 2083 Baisakh, 1+4 marks; 2081 Baisakh, 5 marks; 2079 Bhadra, 2+4 marks; 2079 Baisakh, 1+4 marks; 2075 Chaitra, 5 marks; 2074 Chaitra, 4 marks; 2073 Chaitra, 5 marks; 2072 Chaitra, 5 marks; 2080 Baisakh, 1+4 marks; 2070 Asar, 5 marks (asked 10 times) - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/zeros-of-transmission-realization - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-4 ## Question What are zeros of transmission (in a two port network)? How can zeros of transmission be realized in a network? Explain with suitable examples. ## Short answer Zeros of transmission are s-values where the transfer function is zero. In an LC ladder they come from a series arm becoming an open circuit or a shunt arm a short circuit. Zeros at 0 and ∞ result from complete pole removal; finite zeros need zero shifting. e.g., a series capacitor yields a zero at s=0, a shunt series-LC branch yields a finite zero at 1/√(LC). ## 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 $s=\infty$). #### Realization of transmission zeros 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 ``` Zeros at $0$ and $\infty$ are obtained by **complete removal** of poles of $Z$ or $Y$ at $0$ or $\infty$ (Cauer-type steps). A zero at a finite frequency $\omega_z$ is obtained by **zero shifting** (partial pole removal) followed by complete removal of the pole at $\pm j\omega_z$. **Example.** Realize $Z(s) = \dfrac{(s^2+1)(s^2+3)}{s(s^2+2)}$ as a ladder with a transmission zero at $\omega = 2$ rad/s. The zeros of $Z$ are at $\omega = 1$ and $1.732$; we need one at $\omega=2$, so remove part of the pole at $\infty$: $Z_1(s) = Z(s) - k s$ with $Z_1(j2) = 0$. $$ \begin{aligned} Z(j2) &= \frac{(-4+1)(-4+3)}{j2(-4+2)} = \frac{3}{-j4} = j0.75 \\ Z_1(j2) &= j0.75 - j2k = 0 \;\Rightarrow\; k = \tfrac{3}{8} \\ Z_1(s) &= Z(s) - \tfrac{3}{8}s = \frac{(s^2+4)(5s^2+6)}{8s(s^2+2)} \end{aligned} $$ So series $L_1 = 3/8$ H, and $Z_1$ now has a zero at $s=\pm j2$ (the other zero moved from 1 to 1.095 rad/s). Its reciprocal has a pole at $\pm j2$, removed completely as a shunt branch: $$ \begin{aligned} Y_1(s) &= \frac{8s(s^2+2)}{(s^2+4)(5s^2+6)} = \frac{\tfrac{8}{7}s}{s^2+4} + \frac{16s}{7(5s^2+6)} \\ \tfrac{8}{7}\tfrac{s}{s^2+4} &\Rightarrow L_2 = \tfrac{7}{8}\ \text{H},\ C_2 = \tfrac{2}{7}\ \text{F}\quad (1/\sqrt{L_2C_2} = 2) \\ Z_2(s) &= \frac{7(5s^2+6)}{16s} = \tfrac{35}{16}s + \frac{21}{8s} \Rightarrow L_3 = \tfrac{35}{16}\ \text{H},\ C_3 = \tfrac{8}{21}\ \text{F} \end{aligned} $$ ```text o--[L1=3/8H]--+--[L3=35/16H]--[C3=8/21F]--+ | | [L2=7/8H] | [C2=2/7F] | | | o-------------+---------------------------+ ``` The shunt branch $L_2$–$C_2$ shorts the signal at $\omega=2$ rad/s, which is the required transmission zero. (Check: rebuilding the ladder gives back the original $Z(s)$.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is a zero of transmission of a two-port network? A. A frequency at which the input impedance is zero B. A complex frequency at which the transfer function is zero, so a finite input gives no output C. A frequency at which the transfer function has a pole D. A frequency at which all elements are short-circuited Answer: B. Transmission zeros are values of s where the transfer function (z21, y21, V2/V1, I2/I1) is zero, i.e. no output for a finite input. (Key point: Definition of transmission zero) 2. In a ladder network, which condition blocks the signal and creates a transmission zero? A. A series arm that is a short circuit B. A shunt arm that is an open circuit C. A series arm that is an open circuit (impedance pole) or a shunt arm that is a short circuit D. Any arm containing a resistor Answer: C. An open series arm or a shorted shunt arm prevents the signal reaching the output, so transmission zeros are poles of series impedances and poles of shunt admittances. (Key point: How a ladder blocks the signal) 3. Which element placement gives a transmission zero at s = 0? A. Series inductor B. Shunt capacitor C. Series capacitor D. Series L parallel C tank Answer: C. A series capacitor is open at dc, so it blocks transmission at s = 0. A series L and a shunt C give zeros at infinity. (Key point: Zeros from simple elements) 4. In the example Z(s) = (s^2+1)(s^2+3)/(s(s^2+2)), how is the zero at 2 rad/s obtained and what is the series inductor? A. Remove the pole at infinity completely; L1 = 1 H B. Remove part of the pole at infinity, Z1 = Z - ks with Z1(j2) = 0; L1 = 3/8 H C. Remove the pole at the origin completely; L1 = 3/8 H D. Remove part of the pole at infinity; L1 = 3/4 H Answer: B. Z(j2) = j0.75, so j0.75 - j2k = 0 gives k = 3/8. This partial removal makes Z1 zero at +-j2, and its reciprocal pole is then removed as the shunt L-C branch. (Key point: Finite-frequency zero by partial removal) ## Related in Filter Design - [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× - [Ways of producing zeros of transmission](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atransmission-zeros-ways-poles) — Filter Design, chapter 4 - [Two-port impedance model and series connection](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atwo-port-impedance-model-series) — Filter Design, chapter 4 - [Series connection of two-port networks](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atwo-port-series-connection) — Filter Design, chapter 4 ## Related in other subjects - [Two-port network model and parameters](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Atwo-port-network-model) — RF and Microwave Engineering, chapter 3 - [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 - [S-parameters and three-port network S-matrix](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3As-matrix-three-port-network) — RF and Microwave Engineering, chapter 3, asked 4× - [Importance of S-parameters and two-port derivation](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3As-parameters-two-port-derivation) — RF and Microwave Engineering, chapter 3, asked 3×