Back to Electromagnetic Theory

Study notes

Qwestrum Engineering360 · Electrical & Electronics · Electromagnetic Theory

Transmission Lines

A transmission line supports forward and reflected waves; its characteristic impedance Z₀ = √(L/C) and the load mismatch determine the reflection coefficient, VSWR and the input impedance seen at any distance.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

$Quarter-wave transformer: Z_{in} = \frac{Z_{0}^{2}}{Z_{L}} when l = \frac{\lambda}{4}$
Smith chart for impedance matching
VSWR and reflection from mismatched load

Topic details

Introduction

For a lossless line Z₀ = √(L/C) is real. When the load Z_L differs from Z₀, a reflected wave appears with Γ_L = (Z_L − Z₀)/(Z_L + Z₀), producing standing waves along the line.

Scope in B.Tech and GATE syllabus

The input impedance transforms with distance via Z_in = Z₀(Z_L + jZ₀ tanβl)/(Z₀ + jZ_L tanβl). Two special cases are heavily examined: a quarter-wave line inverts impedance (Z_in = Z₀²/Z_L) and a half-wave line repeats it (Z_in = Z_L).

Key relations & formulas

Formulas (Indian textbook notation)

$V(x) = V^{+} e^(-\gamma x) + V^{-} e^(\gamma x)$

Formulas (Indian textbook notation)

$Z_{0} = \sqrt{\frac{Z}{Y}} = \sqrt{\frac{L}{C}} lossless$

Formulas (Indian textbook notation)

$Input impedance Z_{in} = Z_{0} \frac{(Z_{L} + j Z_{0} tan \beta l)}{(Z_{0} + j Z_{L} tan \beta l)}$

Notation and sign conventions

Relation 1 —

V

Formulas (Indian textbook notation)

$V(x) = V^{+} e^(-\gamma x) + V^{-} e^(\gamma x)$

Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

Z_{0} = √

Formulas (Indian textbook notation)

$Z_{0} = \sqrt{\frac{Z}{Y}} = \sqrt{\frac{L}{C}} lossless$

Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 3 —

Input impedance Z_{in} = Z_{0}

Formulas (Indian textbook notation)

$Input impedance Z_{in} = Z_{0} \frac{(Z_{L} + j Z_{0} tan \beta l)}{(Z_{0} + j Z_{L} tan \beta l)}$

Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

The quarter-wave transformer matches a real load R_L to a line Z₀ by inserting a section of impedance √(Z₀R_L). This is the go-to matching trick in exam problems and antenna feeds.

Governing relations in practice

VSWR = (1+|Γ|)/(1−|Γ|) is measurable and ranges from 1 (matched) to ∞ (open/short). A short-circuited stub of adjustable length provides a pure reactance used for matching on the Smith chart.

Design and analysis considerations

At βl = π/2 (quarter wavelength) tanβl → ∞, giving the impedance-inversion formula; at βl = π (half wavelength) tanβl = 0, returning Z_in = Z_L.

Assumptions and validity limits

State assumptions explicitly before using any relation for transmission lines — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Electromagnetic Theory viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Electromagnetic Theory papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to transmission lines.
4. Use equation 1:

V

.
5. Use equation 2:

Z_{0} = √

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Transmission Lines appears in RF, power apparatus, and communications. In Indian electrical curricula this topic is tested because it connects theory to fields, Maxwell equations, and waves.

GATE and semester exams often combine transmission lines with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use transmission lines?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using electrical length in metres instead of βl in radians
• Forgetting a quarter-wave section inverts impedance (Z₀²/Z_L), not repeats it
• Sign error in Γ_L by swapping Z_L and Z₀
• Treating a lossy line’s Z₀ as purely real

Quick revision checklist

Before attempting transmission lines problems, confirm you can:

1.

Quarter-wave transformer: Z_{in} = \frac{Z_{0}^{2}}{Z_{L}} when l = \frac{\lambda}{4}

2. Smith chart for impedance matching
3. VSWR and reflection from mismatched load

Revise the solved examples in Elements of Electromagnetics — Matthew Sadiku and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Quarter-wave matching section

Problem

A 50 Ω line must feed a 200 Ω resistive load. Find the characteristic impedance of a quarter-wave matching section and the reflection coefficient at the load before matching.

Solution

Before matching: Γ = (Z_L − Z₀)/(Z_L + Z₀) = (200 − 50)/(200 + 50) = 150/250 = 0.6.
Quarter-wave transformer impedance = √(Z₀ × Z_L) = √(50 × 200) = √10000 = 100 Ω.
A 100 Ω, λ/4 section between the line and load gives Z_in = 100²/200 = 50 Ω, matching the line.

Conceptual check — Transmission Lines

Problem

In a Electromagnetic Theory semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of transmission lines." What should a complete answer include?

Exams & GATE

Sadiku — calculate Z_in for given Z_L and line length.

📖 Standard books (India)

Elements of Electromagnetics — Matthew Sadiku
Read: Syllabus unit
Fields, Maxwell equations, and waves

Explore related topics

See real electrical & electronics careers

After exams and interviews, see how engineers actually built careers — milestones and decisions from people in the field.

Browse Electrical & Electronics journeys Plan your path