---
title: "30 m line at 2 MHz, Γ SWR Zin"
subject: "Electromagnetism"
chapter: 5
key: "semester_3/electromagnetism:lossless-line-30m-2mhz-reflection-swr"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/lossless-line-30m-2mhz-reflection-swr
site: BCT Engineering
---

# 30 m line at 2 MHz, Γ SWR Zin

Electromagnetism · Chapter 5: Transmission lines

- Asked in: 2069 Chaitra, 2+2+3 marks
- Video: [Microwave Transmission Line Problems Part-2 || using Transmission line equation](https://www.youtube.com/watch?v=e4s29yZ6y4c) — M Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/lossless-line-30m-2mhz-reflection-swr
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-5

## Question

A lossless transmission line with Z0 = 50 Ω is 30 m long and operates at 2 MHz. The line is terminated with a load Z_L = (60 + j40) Ω. If velocity (v) = 3×10⁸ m/s on the line, find (a) the reflection coefficient, (b) the standing wave ratio and the input impedance.

## Short answer

Γ = (Z_L − Z₀)/(Z_L + Z₀) = 0.352∠56°. SWR = 2.09. Input impedance is 39.8 − j32.0 Ω, or 51.1∠−38.8° Ω.

## Answer

Given: Z₀ = 50 Ω, l = 30 m, f = 2 MHz, Z_L = 60 + j40 Ω, v = 3×10⁸ m/s.

```text
λ = v/f = 3×10⁸/2×10⁶ = 150 m
β = 2π/150 = 0.04189 rad/m
βl = 0.04189 × 30 = 1.2566 rad = 72°   (l = 0.2λ)
```

#### (a) Reflection coefficient

```text
Γ = (Z_L − Z₀)/(Z_L + Z₀) = (10 + j40)/(110 + j40)
  = 41.23∠75.96° / 117.05∠19.98°
  = 0.352 ∠55.98°
```

#### (b) Standing wave ratio

```text
S = (1 + |Γ|)/(1 − |Γ|) = 1.352/0.648 = 2.09
```

#### Input impedance

tan 72° = 3.0777.

```text
Z_in = Z₀ (Z_L + jZ₀ tanβl)/(Z₀ + jZ_L tanβl)

Numerator:   60 + j40 + j50(3.0777) = 60 + j193.9
Denominator: 50 + j(60 + j40)(3.0777) = 50 + j184.7 − 123.1 = −73.1 + j184.7

Z_in = 50 (60 + j193.9)/(−73.1 + j184.7)
     = 50 × 202.9∠72.8° / 198.6∠111.6°
     = 51.1 ∠−38.8° Ω
     = 39.8 − j32.0 Ω
```

**Answer: Γ = 0.352∠56°; S = 2.09; Z_in = 39.8 − j32.0 Ω (51.1∠−38.8° Ω).**

## Check yourself

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

1. With v = 3x10^8 m/s at 2 MHz, what is the electrical length of the 30 m line?
   A. lambda = 150 m, so l = 0.2 lambda (72 degrees)
   B. lambda = 15 m, so l = 2 lambda
   C. lambda = 150 m, so l = 5 lambda
   D. lambda = 1.5 km
   Answer: A. lambda = v/f = 150 m; l/lambda = 0.2. (Key point: Wavelength and length)

2. What is |Gamma| for Z_L = 60 + j40 on a 50 ohm line?
   A. 2.84
   B. 0.648
   C. 0.60
   D. 0.352 (with Gamma = (Z_L - Z0)/(Z_L + Z0))
   Answer: D. Gamma = (10 + j40)/(110 + j40) = 0.352 angle 56 degrees. (Key point: Reflection coefficient)

3. What is the standing wave ratio?
   A. (1 - 0.352)/(1 + 0.352) = 0.48
   B. 1/0.352 = 2.84
   C. (1 + 0.352)/(1 - 0.352) = 2.09
   D. 0.352
   Answer: C. S = (1 + |Gamma|)/(1 - |Gamma|). (Key point: SWR)

4. Which result is correct for Z_in using tan 72 degrees = 3.08?
   A. 60 + j40 ohm (unchanged)
   B. 39.8 - j32.0 ohm
   C. 50 ohm
   D. 16.6 + j33.8 ohm
   Answer: B. Z_in = Z0 (Z_L + jZ0 tan bl)/(Z0 + jZ_L tan bl) = 51.1 angle -38.8 degrees. (Key point: Input impedance)

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