---
title: "Air line 400 Ω, SWR and ZL"
subject: "Electromagnetism"
chapter: 5
key: "semester_3/electromagnetism:air-line-400-ohm-swr-zl"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/air-line-400-ohm-swr-zl
site: BCT Engineering
---

# Air line 400 Ω, SWR and ZL

Electromagnetism · Chapter 5: Transmission lines

- Asked in: 2074 Asoj, 2+4+2 marks
- Video: [How to calculate distance of Vmax and Vmin (voltage maxima and minima) from the load impedance(ZL)](https://www.youtube.com/watch?v=0g6cnHbYZJg) — RF Design Basics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/air-line-400-ohm-swr-zl
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-5

## Question

A lossless line having an air dielectric has a characteristic impedance of 400 Ω. The line is operating at 200 MHz and Z_in = 200 − j200 Ω. Find (a) SWR (b) Z_L, if the line is 1 m long; (c) the distance from the load to the nearest voltage maximum.

## Short answer

SWR from the input reflection coefficient is 2.62. Using the line length of 1 m, Z_L ≈ 1042 + j70 Ω. Distance from load to nearest voltage maximum is about 0.7 cm.

## Answer

Given: air line, Z₀ = 400 Ω, f = 200 MHz, Z_in = 200 − j200 Ω.

```text
λ = c/f = 3×10⁸ / 2×10⁸ = 1.5 m
β = 2π/λ = 4.189 rad/m
```

#### (a) SWR

Reflection coefficient at the input end:

```text
Γ_in = (Z_in − Z₀)/(Z_in + Z₀) = (−200 − j200)/(600 − j200)
     = 282.8∠−135° / 632.5∠−18.43°
     = 0.447 ∠−116.6°
```

|Γ| is the same everywhere on a lossless line, so

```text
S = (1 + 0.447)/(1 − 0.447) = 1.447/0.553 = 2.62
```

**SWR = 2.62**

#### (b) Load impedance for l = 1 m

βl = 4.189 rad = 240°, tan 240° = tan 60° = 1.732.

Moving back to the load (inverse of the input-impedance formula):

```text
Z_L = Z₀ (Z_in − jZ₀ tanβl)/(Z₀ − jZ_in tanβl)

Numerator:   200 − j200 − j400(1.732) = 200 − j892.8
Denominator: 400 − j(200 − j200)(1.732) = 400 − (346.4 + j346.4)
           = 53.6 − j346.4

Z_L = 400 (200 − j892.8)/(53.6 − j346.4)
```

Equivalent check by Γ: Γ_L = Γ_in e^{+j2βl} = 0.447∠(−116.6° + 480°) = 0.447∠3.4°, and Z_L = Z₀(1 + Γ_L)/(1 − Γ_L).

```text
Z_L ≈ 1042 + j70 Ω
```

**Z_L ≈ 1.04 kΩ + j70 Ω**

#### (c) Distance from load to the nearest voltage maximum

A voltage maximum occurs where the phase of Γ(d) = Γ_L e^{−j2βd} is zero:

```text
2βd = θ_L = 3.43° = 0.0599 rad
d = 0.0599 / (2 × 4.189) = 0.00716 m ≈ 0.72 cm  (0.0048 λ)
```

**d ≈ 0.7 cm from the load** (the load is nearly at a voltage maximum because Z_L is almost real and larger than Z₀).

## Check yourself

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

1. What is lambda for an air line at 200 MHz?
   A. 3 m
   B. 0.67 m
   C. 1.5 cm
   D. c/f = 1.5 m
   Answer: D. lambda = 3x10^8/2x10^8 = 1.5 m. (Key point: Wavelength)

2. Why can |Gamma| computed at the input end be used for SWR?
   A. |Gamma| decreases along the line
   B. SWR depends on the line length
   C. |Gamma| is the same everywhere on a lossless line; S = 2.62
   D. Only the load Gamma can be used
   Answer: C. Only the phase of Gamma changes along a lossless line. 0.447 gives S = 1.447/0.553 = 2.62. (Key point: SWR from the input impedance)

3. How is Gamma_L obtained from Gamma_in for l = 1 m?
   A. Gamma_L = Gamma_in e^(-j2 beta l)
   B. Gamma_L = Gamma_in e^(+j2 beta l), with 2 beta l = 480 degrees
   C. Gamma_L = Gamma_in
   D. Gamma_L = 1/Gamma_in
   Answer: B. Moving back to the load reverses the rotation. Gamma_L = 0.447 angle 3.4 degrees and Z_L is about 1042 + j70 ohm. (Key point: Finding the load)

4. Why is the nearest voltage maximum so close to the load (about 0.7 cm)?
   A. Gamma_L is at only 3.4 degrees, so 2 beta d = 3.4 degrees
   B. Because Z_L is below Z0
   C. Because SWR is 1
   D. Because the line is a quarter wavelength
   Answer: A. A maximum occurs where the phase of Gamma is zero; here Z_L is almost real and above Z0. (Key point: Voltage maximum)

## Related in Electromagnetism

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- [Line at 120 MHz, γ v Z0](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aline-120mhz-primary-constants) — Electromagnetism, chapter 5

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