A 230 kV power line and a 50 Ω coaxial cable look nothing alike, yet both are transmission lines described by the same distributed equivalent circuit of R, L, G and C per unit length. From it come the characteristic impedance, propagation constant and velocity of propagation — standard questions in electromagnetic, microwave and power-system papers for BCS (EEE/Telecom), PGCB, BPDB, BTCL and university exams. This post builds the model step by step with an animation and solved problems.
- A line is modelled as a cascade of tiny sections Δz, each with series R and L and shunt G and C (per unit length).
- Characteristic impedance: Z0 = √[(R + jωL)/(G + jωC)]; lossless: Z0 = √(L/C).
- Propagation constant: γ = α + jβ = √[(R + jωL)(G + jωC)]; lossless: α = 0, β = ω√(LC).
- Phase velocity v = 1/√(LC); distortionless line when R/L = G/C (Heaviside condition).
- Use the distributed model when line length is comparable to the wavelength (rule of thumb > λ/10).
1. What Is a Transmission Line?
A transmission line is a structure of two (or more) conductors that guides electrical energy from a source to a load. Examples: overhead power lines, underground cables, coaxial cables, twisted pairs, parallel-wire (twin-lead) lines, and planar lines such as microstrip and stripline.
2. Why a Distributed Model?
In ordinary circuit theory, we treat components as lumped: voltage and current are the same at every point of a wire. This fails when the line is long compared with the wavelength (λ = v/f), because the voltage at the far end lags the sending end by a significant phase. Rule of thumb: use the distributed model when length ℓ > λ/10.
- A 50 Hz power line: λ ≈ 6000 km, so lines over a few hundred km need wave analysis (long-line model).
- A 1 GHz signal: λ = 30 cm in air, so even a few centimetres of PCB track is a transmission line.
3. The RLGC Equivalent Circuit (Animation)
Every short length Δz of line is represented by four primary constants per unit length:
| Constant | Unit | Physical origin | Connected |
|---|---|---|---|
| R – resistance | Ω/m | Finite conductivity of both conductors (rises with skin effect) | Series |
| L – inductance | H/m | Magnetic flux linking the loop formed by the conductors | Series |
| G – conductance | S/m | Leakage through the dielectric/insulators between conductors | Shunt |
| C – capacitance | F/m | Two conductors separated by a dielectric | Shunt |
4. Telegrapher’s Equations
Applying KVL and KCL to one section and letting Δz → 0 gives the telegrapher’s (transmission-line) equations. In phasor form:
dV/dz = −(R + jωL) I, dI/dz = −(G + jωC) VCombining them: d²V/dz² = γ² V, whose solution is a sum of forward and backward waves: V(z) = V+e−γz + V−e+γz.
5. Characteristic Impedance and Propagation Constant
Z0 = √[(R + jωL) / (G + jωC)] γ = α + jβ = √[(R + jωL)(G + jωC)]- Z0 = ratio of voltage to current of a single travelling wave; a line terminated in Z0 has no reflection and looks infinitely long.
- α = attenuation constant (Np/m; 1 Np = 8.686 dB). β = phase constant (rad/m) = 2π/λ.
- Phase velocity vp = ω/β.
- Typical Z0: coaxial cable 50 Ω (RF) or 75 Ω (TV/CCTV), twin-lead 300 Ω, twisted pair ≈ 100 Ω, overhead power line ≈ 300–400 Ω, power cable ≈ 30–60 Ω.
6. Lossless and Distortionless Lines
| Line type | Condition | Z0 | α | β, v |
|---|---|---|---|---|
| Lossless | R = 0, G = 0 (good approximation at high frequency) | √(L/C) (real) | 0 | β = ω√(LC), v = 1/√(LC) |
| Distortionless | R/L = G/C (Heaviside) | √(L/C) (real) | √(RG) | β = ω√(LC), v independent of f |
| General (lossy) | — | Complex | > 0, frequency-dependent | v depends on f → dispersion |
Telephone engineers once added loading coils (extra series L) to cables so that R/L ≈ G/C, reducing distortion of speech.
7. Power-Line Models: Short, Medium, Long
At 50 Hz, power engineers simplify the distributed model according to line length:
| Line | Length (approx.) | Model | What is neglected |
|---|---|---|---|
| Short | up to ~80 km (≤ 20 kV-class lines) | Series R + jX only | Shunt capacitance |
| Medium | ~80–250 km | Nominal-π or nominal-T (capacitance lumped) | Distributed nature |
| Long | above ~250 km | Exact distributed (hyperbolic ABCD) or equivalent-π | Nothing |
For long lines, the power at which reactive power generated by C equals that absorbed by L is the surge impedance loading: SIL = VL² / Z0. Light loads below SIL cause voltage rise at the receiving end (Ferranti effect).
8. Solved Problems
Problem 1. A lossless coaxial cable has L = 250 nH/m and C = 100 pF/m. Find Z0 and the phase velocity.
- Z0 = √(250 × 10⁻⁹ / 100 × 10⁻¹²) = √2500 = 50 Ω
- v = 1/√(LC) = 1/√(2.5 × 10⁻¹⁷) = 2 × 10⁸ m/s (≈ 0.67c, as for polyethylene dielectric)
Problem 2. A 132 kV overhead line has L = 1.2 mH/km and C = 9.5 nF/km. Find its surge impedance and SIL.
- Z0 = √(1.2 × 10⁻³ / 9.5 × 10⁻⁹) ≈ 355 Ω
- SIL = (132 × 10³)² / 355 ≈ 49 MW
- v = 1/√(LC per metre) ≈ 2.96 × 10⁸ m/s — close to the speed of light, as expected for air-insulated lines.
Problem 3. At 1 GHz a line has β = 30 rad/m. Find the wavelength on the line and phase velocity. λ = 2π/β = 0.209 m; v = ω/β = 2π × 10⁹/30 ≈ 2.09 × 10⁸ m/s.
9. Practice MCQs (click to see answer)
1. In the equivalent circuit of a transmission line, R and L are connected in: (a) Shunt (b) Series (c) R in series, L in shunt (d) Both in shunt
2. Characteristic impedance of a lossless line is: (a) √(LC) (b) √(L/C) (c) √(C/L) (d) L/C
3. The condition for a distortionless line is: (a) RL = GC (b) R/L = G/C (c) R = G (d) L = C
4. Velocity of propagation on a lossless line is: (a) √(LC) (b) 1/√(LC) (c) LC (d) √(L/C)
5. A line terminated in its characteristic impedance has: (a) Maximum reflection (b) No reflection (c) Standing waves only (d) Infinite VSWR
6. Standard coaxial cable for TV/CCTV has Z0 of: (a) 50 Ω (b) 75 Ω (c) 300 Ω (d) 600 Ω
7. The attenuation constant of a lossless line is: (a) 0 (b) 1 (c) ∞ (d) √(RG)
8. A medium-length power line is usually represented by: (a) Series impedance only (b) Nominal-π or T model (c) Distributed hyperbolic model only (d) Pure capacitance
9. Surge impedance loading is given by: (a) V/Z₀ (b) V²/Z₀ (c) V²Z₀ (d) Z₀/V²
10. Shunt conductance G in the model represents: (a) Conductor resistance (b) Leakage through insulation (c) Inductance (d) Skin effect
10. বাংলায় দ্রুত রিভিশন
- ট্রান্সমিশন লাইনকে প্রতি একক দৈর্ঘ্যে সিরিজ R ও L এবং শান্ট G ও C দিয়ে গঠিত অসংখ্য ক্ষুদ্র অংশের সমষ্টি হিসেবে মডেল করা হয় (ডিস্ট্রিবিউটেড মডেল)।
- বৈশিষ্ট্যসূচক ইম্পিড্যান্স Z₀ = √[(R+jωL)/(G+jωC)]; লসলেস লাইনে Z₀ = √(L/C), বেগ v = 1/√(LC)।
- প্রোপাগেশন ধ্রুবক γ = α + jβ; α = ক্ষয় ধ্রুবক, β = দশা ধ্রুবক = 2π/λ।
- বিকৃতিহীন লাইনের শর্ত: R/L = G/C (হেভিসাইড শর্ত)।
- পাওয়ার লাইন: ছোট (~৮০ কিমি পর্যন্ত), মাঝারি (৮০–২৫০ কিমি, π/T মডেল), দীর্ঘ (২৫০ কিমির বেশি)। SIL = V²/Z₀।
Related Posts
- Planar Transmission Lines: Types & Advantages
- Microstrip Line: Structure & Advantages
- Stripline Transmission Line
- Maxwell’s First Equation: Displacement Current
References: David M. Pozar, Microwave Engineering; V. K. Mehta, Principles of Power System; Sadiku, Elements of Electromagnetics. Images and animations: original work by Totalecer.