--- title: "Transmission and reflection coefficients; 2nd order LPF" subject: "Filter Design" chapter: 5 key: "semester_5/filter_design:transmission-reflection-second-order" source: https://www.bctengineer.com/q/semester_5/filter_design/transmission-reflection-second-order site: BCT Engineering --- # Transmission and reflection coefficients; 2nd order LPF Filter Design · Chapter 5: Design of Resistively-Terminated Lossless Filters - Asked in: 2075 Chaitra, 2+5 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/transmission-reflection-second-order - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-5 ## Question What information do you get from transmission coefficient and reflection coefficient? Design a second order Butterworth low pass filter using resistively terminated lossless ladder with equal termination of 1Ω. T(s) = 1/(s² + √2 s + 1) ## Short answer The transmission coefficient t(s)=(2√(R₁/R₂))V₂/V₁ indicates the fraction of maximum available source power delivered to the load, defining the magnitude response. The reflection coefficient ρ(s) gives the reflected power fraction and input impedance Z₁ for synthesis, with |ρ|²+|t|²=1. For the 2nd-order Butterworth LPF (1 Ω terminations, ωc=1): series L₁=1.414 H, shunt C₂=1.414 F. ## Answer #### Information from transmission and reflection coefficients The **transmission coefficient** $t(s)$ tells how much of the source's maximum available power $P_{\max} = |V_1|^2/4R_1$ is actually delivered to the load $R_2$: $$ |t(j\omega)|^2 = \frac{P_2}{P_{\max}} = \frac{4R_1}{R_2}\left|\frac{V_2(j\omega)}{V_1(j\omega)}\right|^2 , \qquad t(s) = 2\sqrt{\frac{R_1}{R_2}}\,\frac{V_2(s)}{V_1(s)} $$ The **reflection coefficient** $\rho(s)$ of a resistively terminated lossless ladder compares the power reflected back towards the source with the maximum power the source can give. Looking into the ladder (input impedance $Z_1$) from a source of resistance $R_1$: $$ \rho(s) = \frac{Z_1(s) - R_1}{Z_1(s) + R_1}, \qquad |\rho(j\omega)|^2 = \frac{P_{\text{reflected}}}{P_{\max}}, \qquad Z_1(s) = R_1\frac{1+\rho(s)}{1-\rho(s)} $$ | Transmission coefficient $t$ | Reflection coefficient $\rho$ | |---|---| | Fraction of available power reaching the load | Fraction of available power reflected back | | Gives the magnitude response directly | Gives the input impedance $Z_1$ to synthesize | | $\lvert t\rvert = 1$: maximum power transfer | $\rho = 0$: perfect match | | $t = 0$: transmission zero (stopband) | $\lvert\rho\rvert = 1$: total reflection | For a lossless ladder $|t|^2 + |\rho|^2 = 1$. #### Design of 2nd order Butterworth LPF ($R_1 = R_2 = 1\ \Omega$) **Step 1: Transmission coefficient.** For $n = 2$ (Table 1), $B(s) = s^2 + \sqrt2 s + 1$. With equal terminations $|t(0)|$ can be 1: $$ t(s) = \frac{1}{s^2+\sqrt2 s+1}, \qquad |t(j\omega)|^2 = \frac{1}{1+\omega^4} $$ (The actual voltage gain is $V_2/V_1 = \tfrac12 t(s)$, since $t = 2\sqrt{R_1/R_2}\,V_2/V_1$.) **Step 2: Reflection coefficient.** $$ |\rho(j\omega)|^2 = 1 - \frac{1}{1+\omega^4} = \frac{\omega^4}{1+\omega^4} \;\Rightarrow\; \rho(s)\rho(-s) = \frac{s^4}{B(s)B(-s)}, \quad \rho(s) = \frac{s^2}{s^2+\sqrt2 s+1} $$ **Step 3: Input impedance** ($R_1 = 1$): $$ Z_1(s) = \frac{1+\rho}{1-\rho} = \frac{2s^2+\sqrt2 s+1}{\sqrt2 s+1} $$ **Step 4: Continued fraction.** $$ \begin{aligned} Z_1 &= \sqrt2 s + \frac{1}{\sqrt2 s + 1} \\ &= 1.414\,s + \cfrac{1}{1.414\,s + 1} \end{aligned} $$ So series $L_1 = 1.414$ H, then a shunt $C_2 = 1.414$ F, and the last "1" is $R_2 = 1\ \Omega$. **Step 5: Circuit.** ```text R1=1 L1=1.414 H o-/\/\---UUU----+------+ | | | (~) V1 C2=1.414 F R2=1 | | | o---------------+------+ ``` **Answer:** $R_1 = 1\ \Omega$, $L_1 = 1.414$ H, $C_2 = 1.414$ F, $R_2 = 1\ \Omega$ (normalized, $\omega_c = 1$ rad/s). The dual form $Z_1 = R_1(1-\rho)/(1+\rho)$ gives shunt $C_1 = 1.414$ F and series $L_2 = 1.414$ H. These agree with the standard table values (1.414, 1.414). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which statement about t is correct? A. t = 0 is a transmission zero where no power reaches the load B. |t| = 1 means total reflection C. |t| can exceed 1 in a passive ladder D. t gives the input impedance directly Answer: A. t = 0 means no power is delivered; |t| = 1 means maximum transfer and |t| <= 1 for passive ladders. (Key point: Information from t) 2. For T(s) = 1/(s^2+sqrt2 s+1) with equal terminations, rho(s) = A. 1/(s^2+1.414s+1) B. s/(s^2+1.414s+1) C. s^2/(s^2+1.414s+1) D. (s^2+1)/(s^2+1.414s+1) Answer: C. |rho|^2 = w^4/(1+w^4), so rho = s^2/B(s). (Key point: Reflection coefficient) 3. The continued fraction Z1 = 1.414 s + 1/(1.414 s + 1) gives: A. L1 = 1 H, C2 = 1 F B. L1 = 1.414 H, C2 = 1.414 F, R2 = 1 ohm C. L1 = 0.707 H, C2 = 0.707 F D. C1 = 1.414 F, L2 = 1 H Answer: B. Series L1 = 1.414 H, shunt C2 = 1.414 F, remaining 1 is the load R2. 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