--- title: "Derive reflection coefficient for terminated LC ladder" subject: "Filter Design" chapter: 5 key: "semester_5/filter_design:reflection-coefficient-derivation" source: https://www.bctengineer.com/q/semester_5/filter_design/reflection-coefficient-derivation site: BCT Engineering --- # Derive reflection coefficient for terminated LC ladder Filter Design · Chapter 5: Design of Resistively-Terminated Lossless Filters - Asked in: 2080 Baisakh, 2+5 marks; 2071 Chaitra, 2+5 marks (asked 2 times) - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/reflection-coefficient-derivation - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-5 ## Question What is transmission coefficient? What information do we get from it? Derive the expression for reflection coefficient for a resistively terminated LC ladder circuit. ## Short answer Transmission coefficient t(s)=2√(R1/R2)·V2/V1, |t|²=P2/Pmax indicates power response. Derive ρ: from Pmax=|V1|²/(4R1), input power P1, reflected power fraction is (Pmax−P1)/Pmax = |(R1−Z1)/(R1+Z1)|², so ρ = ±(R1−Z1))/(R1+Z1) and |ρ|²+|t|²=1. ## Answer #### Transmission coefficient The **transmission coefficient** $t(s)$ is the ratio (in power terms) of the power actually delivered to the load $R_2$ to the maximum available power of the source $P_{\max} = |V_1|^2/4R_1$: $$ |t(j\omega)|^2 = \frac{P_2}{P_{\max}} = \frac{4R_1}{R_2}\left|\frac{V_2(j\omega)}{V_1(j\omega)}\right|^2 , \qquad t(s) = 2\sqrt{\frac{R_1}{R_2}}\,\frac{V_2(s)}{V_1(s)} $$ **Information obtained:** $|t|^2$ is the fraction of available power reaching the load, so it is the filter's power response. $|t| = 1$ means perfect matching and maximum power transfer; $|t| = 0$ marks a transmission zero; and it bounds the gain of a passive ladder ($|t| \le 1$). #### Derivation of reflection coefficient ```text R1 I1 --> o-/\/\----+-------------+------+ | | | | (~) V1 Z1(s) -> Lossless LC R2 V2 | | ladder | | o---------+-------------+------+ ``` 1. **Maximum available power** of the source (delivered to a matched load $R_1$): $$ P_{\max} = \frac{|V_1|^2}{4R_1} $$ 2. **Input power** to the ladder with $Z_1 = R_{in} + jX_{in}$ and $I_1 = \dfrac{V_1}{R_1+Z_1}$: $$ P_1 = |I_1|^2 R_{in} = \frac{|V_1|^2\,\text{Re}(Z_1)}{|R_1+Z_1|^2} $$ 3. **Power not delivered** ("reflected") is $P_{\max} - P_1$: $$ \begin{aligned} P_{\max} - P_1 &= \frac{|V_1|^2}{4R_1}\left[1 - \frac{4R_1\,\text{Re}(Z_1)}{|R_1+Z_1|^2}\right] \\ &= \frac{|V_1|^2}{4R_1}\cdot\frac{|R_1+Z_1|^2 - 4R_1\,\text{Re}(Z_1)}{|R_1+Z_1|^2} \end{aligned} $$ 4. Using $|R_1+Z_1|^2 - 4R_1 R_{in} = (R_1+R_{in})^2 + X_{in}^2 - 4R_1R_{in} = (R_1-R_{in})^2 + X_{in}^2 = |R_1-Z_1|^2$: $$ \frac{P_{\max}-P_1}{P_{\max}} = \left|\frac{R_1 - Z_1}{R_1+Z_1}\right|^2 $$ 5. This ratio is defined as $|\rho(j\omega)|^2$, so the reflection coefficient is $$ \rho(s) = \pm\frac{R_1 - Z_1(s)}{R_1 + Z_1(s)} $$ 6. Since the ladder is lossless, $P_1 = P_2$, so $|t|^2 = P_1/P_{\max}$ and $$ |\rho(j\omega)|^2 + |t(j\omega)|^2 = 1 \quad \text{(Feldtkeller equation)} $$ 7. Solving for $Z_1$ gives the synthesis formula: $$ Z_1(s) = R_1\frac{1 \mp \rho(s)}{1 \pm \rho(s)} $$ **Use:** given $t(s)$, find $|\rho|^2 = 1-|t|^2$, factor $\rho(s)\rho(-s)$, pick $\rho(s)$ with a Hurwitz denominator, find $Z_1$ and expand it into an LC ladder. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does |t(jw)|^2 represent for a resistively terminated ladder? A. Ratio of load voltage to source voltage B. Ratio of power delivered to the load to the maximum available source power C. Ratio of reflected power to input power D. Ratio of load resistance to source resistance Answer: B. |t|^2 = P2/Pmax, where Pmax = |V1|^2/4R1 is the maximum available power of the source. (Key point: Definition of transmission coefficient) 2. The maximum available power of a source with voltage V1 and resistance R1 is: A. |V1|^2/R1 B. |V1|^2/2R1 C. |V1|^2/4R1 D. |V1|^2/8R1 Answer: C. Delivered to a matched load R1, the power is |V1|^2/(4R1). (Key point: Maximum available power) 3. Which identity turns (Pmax - P1)/Pmax into a reflection-coefficient form? A. |R1+Z1|^2 - 4R1 Re(Z1) = |R1 - Z1|^2 B. |R1+Z1|^2 + 4R1 Re(Z1) = |R1 - Z1|^2 C. |R1+Z1|^2 - 2R1 Re(Z1) = |R1 - Z1|^2 D. |R1+Z1|^2 = |R1 - Z1|^2 Answer: A. Expanding (R1+Rin)^2 + Xin^2 - 4R1Rin gives (R1-Rin)^2 + Xin^2 = |R1-Z1|^2. (Key point: Key algebraic identity) 4. Why does |rho|^2 + |t|^2 = 1 hold? A. Because the source is ideal B. Because the ladder is lossless, so P1 = P2 and reflected + delivered = Pmax C. Because R1 = R2 always D. Because Z1 is purely real Answer: B. No power is dissipated in the lossless ladder, so power not reflected reaches the load. 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