9.5 Energy flow in transmission lines

Transmission line basics

Transmission lines carry electromagnetic energy between two points using two or more parallel conductors separated by a dielectric material. Rather than treating these structures as simple wires, you model them as a distributed network of infinitesimal inductors, capacitors, resistors, and conductors. This distributed approach is necessary whenever the physical length of the line is comparable to the wavelength of the signal, because voltage and current vary along the line's length.

Distributed parameters

Each short segment $\Delta z$ of a transmission line is characterized by four parameters, all defined per unit length:

• Inductance per unit length ($L'$) captures the magnetic field energy stored around the conductors. It depends on conductor geometry and the permeability of the surrounding medium.
• Capacitance per unit length ($C'$) captures the electric field energy stored between the conductors. It depends on conductor geometry and the permittivity of the dielectric.
• Resistance per unit length ($R'$) accounts for ohmic losses in the conductors due to their finite conductivity.
• Conductance per unit length ($G'$) accounts for leakage current through the imperfect dielectric.

These four parameters feed directly into the telegrapher's equations, which govern how voltage and current evolve along the line.

Lossless vs. lossy lines

A lossless line sets $R' \approx 0$ and $G' \approx 0$. This is the idealized case you'll use most often to build intuition: waves propagate without attenuation, and the math simplifies considerably.

A lossy line keeps $R'$ and $G'$ finite. Waves now decay exponentially as they travel, described by the attenuation constant $\alpha$. Dispersion (frequency-dependent phase velocity) can also appear, meaning different frequency components travel at different speeds and pulses spread out over distance.

Characteristic impedance

The characteristic impedance $Z_0$ is the ratio of voltage to current for a single traveling wave on an infinitely long line. It's a property of the line itself, not the load.

• Lossless line: $Z_0 = \sqrt{\frac{L'}{C'}}$
• Lossy line: $Z_0 = \sqrt{\frac{R' + j\omega L'}{G' + j\omega C'}}$

When the load impedance equals $Z_0$, there are no reflections and maximum power transfers to the load. Any mismatch between $Z_L$ and $Z_0$ sends energy back toward the source.

Voltage and current waves

Electromagnetic energy propagates along a transmission line as coupled voltage and current waves. On a lossless line, these waves travel at the phase velocity $v_p = 1/\sqrt{L'C'}$, which equals the speed of light in the dielectric medium.

Forward and reflected waves

Forward (incident) waves travel from source to load:

$V_+(z) = V_0^+ e^{-\gamma z}, \quad I_+(z) = \frac{V_0^+}{Z_0} e^{-\gamma z}$

Reflected waves travel from load back to source, arising whenever the load impedance doesn't match $Z_0$:

$V_-(z) = V_0^- e^{+\gamma z}, \quad I_-(z) = -\frac{V_0^-}{Z_0} e^{+\gamma z}$

Here $\gamma = \alpha + j\beta$ is the propagation constant ($\alpha$ = attenuation, $\beta$ = phase constant). For a lossless line, $\gamma = j\beta$.

The total voltage and current at any point $z$ are the superpositions: $V(z) = V_+(z) + V_-(z)$ and $I(z) = I_+(z) + I_-(z)$.

Note the minus sign in $I_-(z)$. The reflected current wave has the opposite sign relative to its voltage wave compared to the forward wave. This follows from the fact that the reflected wave carries power in the $-z$ direction.

Reflection coefficient

The reflection coefficient $\Gamma$ quantifies how much of the incident wave bounces back at the load:

$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$

Key values to remember:

• $\Gamma = 0$: perfect match ($Z_L = Z_0$), no reflection
• $\Gamma = -1$: short circuit ($Z_L = 0$), full reflection with phase inversion
• $\Gamma = +1$: open circuit ($Z_L \to \infty$), full reflection with no phase inversion

In general $\Gamma$ is complex, carrying both magnitude and phase information.

Standing wave ratio (SWR)

When forward and reflected waves coexist on a line, they interfere to create a standing wave pattern. The voltage amplitude oscillates between maxima and minima along the line. The SWR measures this variation:

$SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}$

• $SWR = 1$: perfect match, uniform voltage amplitude everywhere
• $SWR \to \infty$: total reflection, deep nulls in the standing wave pattern

High SWR means significant power is bouncing back and forth rather than being delivered to the load.

Power flow and the Poynting vector

This is where the unit's core theme connects to transmission lines. The power flowing along a transmission line can be understood through the Poynting vector $\vec{S} = \vec{E} \times \vec{H}$, integrated over the cross-sectional area between the conductors. The result is equivalent to the circuit-level expression $P = VI^*$.

Average power

The time-averaged power at any point on a lossless line is:

$P_{\text{avg}} = \frac{1}{2} \text{Re}[V(z)\, I^*(z)] = \frac{|V_0^+|^2}{2Z_0}(1 - |\Gamma|^2)$

This expression has a clean physical interpretation. The first term $|V_0^+|^2 / 2Z_0$ is the incident power. The second term $|\Gamma|^2 |V_0^+|^2 / 2Z_0$ is the reflected power. The net power delivered to the load is the difference. On a lossless line, $P_{\text{avg}}$ is the same at every point along the line, since no energy is dissipated.

Instantaneous power

The instantaneous power at position $z$ and time $t$ is:

$p(z, t) = v(z, t) \cdot i(z, t)$

This contains a DC (constant) component equal to the average power, plus an oscillating component at twice the signal frequency. The oscillating part represents reactive power sloshing back and forth between the electric and magnetic field energy stored on the line. It doesn't contribute to net energy transfer.

Power factor

The power factor is the ratio of average (real) power to apparent power:

$PF = \frac{P_{\text{avg}}}{|V_{\text{rms}}| \cdot |I_{\text{rms}}|}$

• $PF = 1$: purely resistive load, all power is delivered as real power
• $PF = 0$: purely reactive load, energy oscillates but no net transfer occurs

On a matched lossless line, the power factor equals 1 because the voltage and current are in phase everywhere.

Impedance matching

When $Z_L \neq Z_0$, reflections reduce the power delivered to the load and can cause problems like voltage standing waves, signal distortion, and even damage to sources. Impedance matching eliminates (or reduces) these reflections.

Matching techniques

Two broad categories:

• Lumped element networks (L-networks, T-networks, Pi-networks): Use discrete inductors and capacitors to transform the load impedance to $Z_0$. Practical at lower frequencies where component sizes are small relative to the wavelength.
• Distributed element networks (quarter-wave transformers, stub matching): Use transmission line sections themselves to achieve the impedance transformation. Preferred at microwave frequencies where lumped components become impractical.

Quarter-wave transformer

A transmission line section exactly $\lambda/4$ long with characteristic impedance:

$Z_T = \sqrt{Z_0 \cdot Z_L}$

This transforms a real load impedance $Z_L$ into $Z_0$ at the input. The mechanism relies on the quarter-wave section inverting the impedance: $Z_{\text{in}} = Z_T^2 / Z_L$.

The main limitation is bandwidth. The section is exactly $\lambda/4$ only at the design frequency, so the match degrades as you move away from that frequency. For broader bandwidth, you can cascade multiple quarter-wave sections with gradually tapered impedances.

Stub matching

A stub is a short section of transmission line, either open-circuited or short-circuited at one end, connected in parallel (or series) with the main line.

• Single-stub matching: Place one stub at a calculated distance from the load. You choose two parameters (stub length and position) to cancel the reactive part of the impedance and match the real part.
• Double-stub matching: Two stubs at fixed positions give more design flexibility and can handle a wider range of load impedances, though there are still some loads that can't be matched with a given stub spacing.

Transmission line losses

Real transmission lines dissipate energy, causing the signal to attenuate as it propagates.

Conductor losses

These arise from the finite conductivity of the metal conductors. Two effects make them worse at high frequencies:

• Skin effect: At high frequencies, current crowds into a thin layer near the conductor surface (the skin depth $\delta_s = \sqrt{2/\omega\mu\sigma}$). This reduces the effective cross-sectional area carrying current, increasing resistance.
• Proximity effect: Current distributions in nearby conductors distort each other, further concentrating current and increasing effective resistance.

Dielectric losses

The dielectric material between conductors isn't a perfect insulator. The loss tangent $\tan \delta$ quantifies how "lossy" the dielectric is, representing the ratio of conduction current to displacement current in the material. Dielectric losses generally increase with both frequency and temperature.

Radiation losses

Energy can leak out of the transmission line as radiated electromagnetic waves. This is more significant for:

• Higher frequencies (shorter wavelengths relative to conductor spacing)
• Unshielded or poorly shielded geometries
• Bends, discontinuities, and asymmetries in the line

Coaxial cables and waveguides are inherently better shielded than open two-wire lines, which is one reason they're preferred at high frequencies.

Transient behavior

When a transmission line is suddenly excited (a switch closes, a digital pulse launches), the steady-state standing wave pattern doesn't appear instantly. Instead, waves bounce back and forth between source and load, and the voltages and currents build up over multiple transit times.

Step response

When a step voltage is applied to a transmission line:

1. A forward wave launches from the source and propagates toward the load at velocity $v_p$.

2. After one transit time $T_d = \ell / v_p$ (where $\ell$ is the line length), the wave reaches the load.

3. If $Z_L \neq Z_0$, a reflected wave with amplitude $\Gamma_L \cdot V_0^+$ heads back toward the source.

4. At the source, if the source impedance $Z_S \neq Z_0$, another reflection occurs with coefficient $\Gamma_S = (Z_S - Z_0)/(Z_S + Z_0)$.

5. This process repeats, with each successive reflection smaller (assuming $|\Gamma_L \Gamma_S| < 1$), until the line settles to its DC steady state.

Overshoots, undershoots, and ringing in the transient response are direct consequences of these multiple reflections.

Pulse response

A pulse can be thought of as two step functions separated in time. Each edge launches its own set of reflections. On lossy or dispersive lines, the pulse also broadens and distorts as it propagates. In digital systems, this distortion can cause inter-symbol interference (ISI), where one pulse bleeds into the time slot of the next, increasing bit error rates.

Bounce diagrams

A bounce diagram is a space-time plot that tracks wave reflections visually:

• The horizontal axis represents position along the line (source on the left, load on the right).
• The vertical axis represents time, increasing downward.
• Diagonal lines show waves propagating back and forth, with their amplitudes labeled at each reflection.

To read the voltage at any point and time, you sum all the wave amplitudes that have passed through that point up to that time. Bounce diagrams are extremely useful for working through transient problems step by step without getting lost in the algebra.

Transmission line applications

Coaxial cables

An inner conductor is surrounded by a dielectric insulator and an outer conducting shield. The fields are entirely contained between the two conductors, providing excellent shielding against electromagnetic interference. Coaxial cables are standard for RF signal transmission (antenna feeds, cable TV, test equipment connections). Common characteristic impedances are 50 $\Omega$ (lab/RF) and 75 $\Omega$ (video/broadcast).

Microstrip lines

A conducting strip on one side of a dielectric substrate, with a ground plane on the other side. Microstrip is the workhorse of microwave integrated circuits and printed circuit boards because it's easy to fabricate using standard lithography. The fields are partly in the dielectric and partly in air above the strip, making the effective permittivity a weighted average of the two. This "quasi-TEM" mode means microstrip is slightly dispersive, but it works well up to tens of GHz.

Waveguides

Hollow metallic tubes (rectangular or circular cross-section) that guide electromagnetic waves through internal reflections off the conducting walls. Unlike coaxial cables and microstrip, waveguides have a cutoff frequency below which waves cannot propagate. Above cutoff, they offer very low loss and can handle high power levels, making them ideal for radar systems, satellite communication feeds, and industrial microwave applications.