---
title: "Reflection coefficient; 3rd order LPF with R2 = 4 Ω"
subject: "Filter Design"
chapter: 5
key: "semester_5/filter_design:reflection-coefficient-unequal-termination"
source: https://www.bctengineer.com/q/semester_5/filter_design/reflection-coefficient-unequal-termination
site: BCT Engineering
---

# Reflection coefficient; 3rd order LPF with R2 = 4 Ω

Filter Design · Chapter 5: Design of Resistively-Terminated Lossless Filters

- Asked in: 2079 Bhadra, 1+5 marks; 2078 Bhadra, 1+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-unequal-termination
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-5

## Question

What is (define) reflection coefficient? Design (realize) a third order Butterworth lowpass filter using resistively terminated lossless ladder with unequal termination of R1 = 1 Ω and R2 = 4 Ω. [Refer Table]

## Short answer

Reflection coefficient ρ=(R1−Z1)/(R1+Z1), with |ρ|²=reflected power fraction. For third-order Butterworth LPF, R1=1 Ω, R2=4 Ω, using t(s)=0.8/(s³+2s²+2s+1), ρ(s) and Z1 expansion yield L1=6.387 H, C2=0.3608 F, L3=2.170 H, R2=4 Ω (ωc=1 rad/s).

## Answer

#### Reflection coefficient

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 $R_1$:

$$
\rho(s) = \frac{R_1 - Z_1(s)}{R_1 + Z_1(s)}, \qquad |\rho(j\omega)|^2 = \frac{P_{\text{reflected}}}{P_{\max}}
$$

$|\rho|$ lies between 0 (perfect match, all available power delivered) and 1 (total reflection, no power to the load). For a lossless ladder $|\rho|^2 + |t|^2 = 1$.

#### Design ($R_1 = 1\ \Omega$, $R_2 = 4\ \Omega$)

**Step 1: Transmission coefficient.** Third order Butterworth (Table 1): $B(s) = (s+1)(s^2+s+1) = s^3+2s^2+2s+1$. At dc the inductors are shorts and capacitors open, so the ladder reduces to a divider: $V_2/V_1 = \frac{R_2}{R_1+R_2} = \frac{4}{5} = 0.8$. Hence the largest possible dc transmission is

$$
|t(0)|^2 = \frac{4R_1R_2}{(R_1+R_2)^2} = \frac{4(1)(4)}{(1+4)^2} = \frac{16}{25} = 0.64
$$

so $|t(j\omega)|^2 = \dfrac{0.64}{1+\omega^6}$, i.e. $t(s) = \dfrac{0.8}{s^3+2s^2+2s+1}$.

**Step 2: Reflection coefficient.**

$$
|\rho(j\omega)|^2 = 1 - \frac{0.64}{1+\omega^6} = \frac{0.36 + \omega^6}{1+\omega^6}
\;\Rightarrow\;
\rho(s)\rho(-s) = \frac{0.36 - s^6}{1 - s^6}
$$

The zeros satisfy $s^6 = 0.36$, i.e. they lie on a circle of radius $a = (0.36)^{1/6} = 0.8434$ at the Butterworth angles. Choosing the left-half-plane zeros (like the Butterworth poles, scaled by $a$):

$$
\rho(s) = \frac{(s+a)(s^2+as+a^2)}{B(s)} = \frac{s^3 + 1.6869 s^2 + 1.4228 s + 0.6}{s^3+2s^2+2s+1}
$$

**Step 3: Input impedance** ($R_1 = 1$):

$$
Z_1(s) = R_1\frac{1+\rho(s)}{1-\rho(s)} = \frac{2s^3+3.6869s^2+3.4228s+1.6}{0.3131s^2+0.5772s+0.4}
$$

**Step 4: Continued fraction expansion** (computed by repeated long division):

$$
Z_1(s) = 6.387\,s + \cfrac{1}{0.3608\,s + \cfrac{1}{2.170\,s + 4}}
$$

So series $L_1 = 6.387$ H, shunt $C_2 = 0.3608$ F, series $L_3 = 2.170$ H, and the remainder is the load $R_2 = 4\ \Omega$, as required (this choice of $\rho$ sign gives the correct $R_2$; the other sign ends in $1/4\ \Omega$ and is rejected).

**Step 5: Circuit.**

```text
  R1=1  L1=6.387 H     L3=2.170 H
 o-/\/\---UUU----+----UUU---+
 |               |          |
(~) V1     C2=0.3608 F   R2=4
 |               |          |
 o---------------+----------+
```

**Check:** analysing this ladder gives $V_2/V_1 = \dfrac{0.8}{s^3+2s^2+2s+1}$, the required Butterworth response.

**Answer:** $R_1 = 1\ \Omega$, $L_1 = 6.387$ H, $C_2 = 0.3608$ F, $L_3 = 2.170$ H, $R_2 = 4\ \Omega$ (normalized to $\omega_c = 1$ rad/s).

## Check yourself

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

1. The reflection coefficient of a terminated ladder with input impedance Z1 and source R1 is:
   A. rho = (R1 - Z1)/(R1 + Z1)
   B. rho = (R1 + Z1)/(R1 - Z1)
   C. rho = Z1/R1
   D. rho = R1*Z1
   Answer: A. It measures the mismatch between source and ladder. (Key point: Definition of rho)

2. For a lossless ladder:
   A. |rho|^2 + |t|^2 = 1
   B. |rho| + |t| = 1
   C. |rho|^2 = |t|^2
   D. |rho|^2 - |t|^2 = 1
   Answer: A. Power not reflected is delivered to the load. (Key point: Power relation)

3. For R1 = 1 and R2 = 4, the dc voltage ratio V2/V1 is:
   A. 0.8
   B. 0.5
   C. 0.64
   D. 4
   Answer: A. At dc inductors are shorts and capacitors open, so V2/V1 = R2/(R1+R2) = 4/5. (Key point: DC transmission with unequal terminations)

## Close variants set in other exams

- [Reflection coefficient; 3rd order Butterworth HPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Areflection-coefficient-third-order-hpf) — Filter Design, chapter 5, asked 7×

## Related in Filter Design

- [3rd order Butterworth LPF, R1 = 1 Ω, R2 = 4 Ω](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abutterworth-lp-third-order-unequal-termination) — Filter Design, chapter 5, asked 4×
- [Zero reflection coefficient; 3rd order LPF equal termination](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Areflection-zero-third-order-equal) — Filter Design, chapter 5, asked 4×
- [Reflection coefficient; realize 3rd order LPF (1 Ω/1 Ω)](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Areflection-coefficient-third-order-lpf-equal) — Filter Design, chapter 5
- [Significance of reflection coefficient; R1 = 1 Ω, R2 = 4 Ω](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Areflection-significance-unequal-termination) — Filter Design, chapter 5
- [Realize 3rd order Butterworth TF with R2 = 2 Ω](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aladder-third-order-r2-2-ohm) — Filter Design, chapter 5
- [Information from reflection coefficient; 3rd order LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Areflection-info-third-order-equal) — Filter Design, chapter 5, asked 4×

## Related in other subjects

- [Ladder programming of a third-order filter](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aladder-third-order-128) — Digital Control System, chapter 4, asked 2×