Lossless Transmission Line

A lossless transmission line is one in which the conductor resistance R and the shunt conductance G are both zero. While no practical line is truly lossless, the lossless model captures the essential wave behavior of real transmission lines at high frequencies where the reactive parameters L and C dominate. The lossless model is the foundation for understanding standing waves, impedance transformation, and resonance in all transmission line theory taught at the university and GATE level.

Core Concept Explanation

In the general transmission line model, the propagation constant is gamma = sqrt((R+jwL)(G+jwC)). When R=0 and G=0, this becomes gamma = sqrt(-w^2*L*C) = j*w*sqrt(L*C). The propagation constant is purely imaginary, meaning gamma = j*beta where beta = w*sqrt(L*C). This is the key result of the lossless assumption. A purely imaginary gamma means the wave propagates without any amplitude decay, only with a phase shift per unit length.

The attenuation constant alpha is the real part of gamma. For a lossless line, alpha = 0 exactly. The phase constant beta is the imaginary part of gamma and equals w*sqrt(L*C). The phase velocity is vp = w/beta = 1/sqrt(L*C). For a coaxial line filled with a dielectric of permittivity epsilon and permeability mu, vp = 1/sqrt(mu*epsilon) = c/sqrt(er), which is the speed of light in the medium.

Because alpha = 0, voltage and current waves on a lossless line maintain their amplitude as they travel. Energy is not dissipated; it is stored alternately in the magnetic field (associated with L) and the electric field (associated with C) of the line. This is analogous to a lossless LC ladder network, and the lossless transmission line can be treated as a distributed LC ladder.

The wavelength on the lossless line is lambda = 2*pi/beta = 2*pi/(w*sqrt(L*C)) = vp/f. The electrical length of a physical section of length l is beta*l radians or equivalently l/lambda wavelengths. These relations connect the physical geometry of the line to its wave behavior at a given frequency.

Mathematical Expression

The general solution for voltage on a lossless transmission line is V(z) = V+ * exp(-j*beta*z) + V- * exp(+j*beta*z). The corresponding current is I(z) = (V+/Z0)*exp(-j*beta*z) - (V-/Z0)*exp(+j*beta*z). The first terms represent the forward wave and the second terms represent the backward (reflected) wave.

The total time-average power flowing in the +z direction on a lossless line is P = (1/2)*Re[V*I*] = (1/(2*Z0)) * (|V+|^2 - |V-|^2). The net power delivered to the load is the difference between incident power |V+|^2/(2*Z0) and reflected power |V-|^2/(2*Z0). For a lossless line with a passive load, net power is always positive (more power delivered than reflected), and equals zero only for a purely reactive load where |Gamma|=1.

The standing wave pattern on a lossless line with reflection is |V(z)| = |V+| * sqrt(1 + |Gamma|^2 + 2*|Gamma|*cos(2*beta*z + angle_Gamma)). This oscillates between Vmax = |V+|(1+|Gamma|) and Vmin = |V+|(1-|Gamma|), giving VSWR = (1+|Gamma|)/(1-|Gamma|). The standing wave pattern has a spatial period of lambda/2.

Practical Understanding

The lossless model is an excellent approximation for practical transmission lines operating at microwave frequencies where wL is much greater than R, and wC is much greater than G. For example, a typical RG-58 coaxial cable has a loss of about 0.3 dB per meter at 1 GHz. Over a 30 cm connection, this is only 0.09 dB, which is negligible for most calculations. The lossless model introduces a negligible error in such cases.

Standing waves on a lossless line create localized regions of high voltage and high current. At voltage maxima, the electric field stress is highest. At current maxima, the ohmic heating in a real (slightly lossy) line is highest. In high-power RF applications such as broadcast transmitters, these hot spots must be managed carefully to prevent insulation breakdown and conductor overheating.

Numerical Example

A lossless transmission line has distributed parameters L = 0.5 uH/m and C = 200 pF/m. The line operates at f = 100 MHz. Find the characteristic impedance Z0, the phase constant beta, the wavelength lambda, and the phase velocity vp.

Given:
L = 0.5 uH/m = 0.5e-6 H/m
C = 200 pF/m = 200e-12 F/m
f = 100 MHz = 100e6 Hz
R = 0, G = 0  (lossless)

Why this formula applies:
Lossless conditions allow simplified formulas for all parameters

Formula:
Z0 = sqrt(L/C)
beta = w*sqrt(L*C) = 2*pi*f*sqrt(L*C)
vp = 1/sqrt(L*C)
lambda = vp / f

Substitution and Calculation:
Z0 = sqrt(0.5e-6 / 200e-12)
   = sqrt(2500)
   = 50 ohm

beta = 2*pi*(100e6)*sqrt(0.5e-6 * 200e-12)
     = 2*pi*(100e6)*sqrt(1e-16)
     = 2*pi*(100e6)*(1e-8)
     = 2*pi*(1)
     = 6.283 rad/m

vp = 1/sqrt(0.5e-6 * 200e-12)
   = 1/sqrt(1e-16)
   = 1/(1e-8)
   = 1e8 m/s  (one-third speed of light)

lambda = vp/f = 1e8 / 100e6 = 1 m

Final Answer:
Z0 = 50 ohm
beta = 6.283 rad/m  (= 2*pi rad/m)
vp = 1e8 m/s  (c/3)
lambda = 1 m

Exam Tip: For a lossless line in a medium with relative permittivity er, the phase velocity vp = c/sqrt(er) and beta = w*sqrt(er)/c. These two expressions let you avoid computing L and C separately. GATE often gives er and frequency and asks for beta or lambda. Also note: a line of length lambda/2 repeats its impedance, and a line of length lambda/4 inverts it. These two facts solve many GATE problems quickly.

Mechanism: Standing Wave Formation on Lossless Line

• On a lossless line, forward and reflected waves both travel without attenuation. Their coherent superposition creates a fixed spatial pattern of voltage and current amplitudes called the standing wave.
• At a short-circuit load, the voltage must be zero at the load terminal. The reflected wave must cancel the incident wave at that point, so Gamma = -1 and V- = -V+. The current adds constructively at the short, creating a current maximum.
• At an open-circuit load, the current must be zero at the load terminal. The reflected wave adds constructively for voltage (Gamma = +1), creating a voltage maximum at the open end.
• Voltage maxima and current maxima are always separated by lambda/4 along the line. This is because the electric and magnetic energy distributions are spatially offset on the lossless line, just as in a resonant LC circuit.
• For a lossless line, the time-average power flow is constant along the line. Poynting's theorem confirms that power does not accumulate anywhere in the standing wave; it simply flows past each cross-section at the same rate.
• A lossless line section of length beta*l = pi/2 (quarter wavelength) acts as a resonator when terminated in a short or open. The input impedance alternates between zero and infinity at resonant frequencies, analogous to a parallel or series LC circuit.

Quick Revision

• Lossless condition: R=0, G=0. Result: alpha=0, gamma = j*beta, Z0 = sqrt(L/C) is real.
• Key formulas: beta = w*sqrt(LC), vp = 1/sqrt(LC) = c/sqrt(er), lambda = vp/f.
• Standing wave exists when |Gamma| is nonzero. Vmax = |V+|(1+|Gamma|), Vmin = |V+|(1-|Gamma|).
• Short circuit at load: voltage node at load, current anti-node at load. Open circuit: opposite.
• Voltage and current standing wave nodes are spatially separated by lambda/4.
• Standing wave pattern repeats every lambda/2 along the line.
• GATE trap: On a lossless line, |Gamma| does not change with position. Only the phase of Gamma changes. VSWR is therefore constant along a lossless line.