9.5 Energy flow in transmission lines

Transmission lines are crucial for efficiently transmitting electromagnetic energy. They consist of parallel conductors separated by dielectric material, forming a network of inductors and capacitors. Understanding their behavior requires knowledge of distributed parameters like inductance and capacitance per unit length.

Energy flows through transmission lines as voltage and current waves. These waves can be forward or reflected, depending on impedance matching. The power flow is determined by these waves, with average power representing net flow and instantaneous power including reactive components.

Transmission line basics

• Transmission lines are structures designed to efficiently transmit electromagnetic energy from one point to another
• They consist of two or more parallel conductors separated by a dielectric material, forming a distributed network of inductors and capacitors
• The behavior of transmission lines depends on their distributed parameters, which are the inductance and capacitance per unit length

Distributed parameters

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• Inductance per unit length ($L$) represents the magnetic field energy stored around the conductors
  • Depends on the geometry of the conductors and the permeability of the surrounding medium
• Capacitance per unit length ($C$) represents the electric field energy stored between the conductors
  • Depends on the geometry of the conductors and the permittivity of the dielectric material
• Resistance per unit length ($R$) accounts for the losses due to the finite conductivity of the conductors
• Conductance per unit length ($G$) accounts for the losses due to the imperfect dielectric material

Lossless vs lossy lines

• Lossless transmission lines have negligible resistance and conductance ($R \approx 0$ and $G \approx 0$)
  • Ideal case used for simplifying calculations and understanding fundamental behavior
• Lossy transmission lines have non-negligible resistance and conductance
  • More realistic representation of practical transmission lines
  • Attenuation and dispersion effects become significant

Characteristic impedance

• Characteristic impedance ($Z_0$) is the ratio of the voltage to the current for a wave traveling in one direction on an infinite transmission line
  • For a lossless line: $Z_0 = \sqrt{\frac{L}{C}}$
  • For a lossy line: $Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}$
• Determines the maximum power transfer and minimizes reflections when the load impedance matches the characteristic impedance

Voltage and current waves

• Electromagnetic energy propagates along a transmission line as voltage and current waves
• The waves travel at the speed of light in the dielectric medium, determined by the permittivity and permeability

Forward and reflected waves

• Forward (incident) waves propagate from the source to the load
  • Voltage: $V_+(z) = V_0^+ e^{-\gamma z}$, Current: $I_+(z) = \frac{V_0^+}{Z_0} e^{-\gamma z}$
• Reflected waves propagate from the load back to the source due to impedance mismatches
  • Voltage: $V_-(z) = V_0^- e^{\gamma z}$, Current: $I_-(z) = -\frac{V_0^-}{Z_0} e^{\gamma z}$
• Total voltage and current at any point on the line are the sum of the forward and reflected waves

Reflection coefficient

• Reflection coefficient ($\Gamma$) quantifies the fraction of the incident wave that is reflected at the load
  • $\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$, where $Z_L$ is the load impedance
• Ranges from -1 (short circuit) to 1 (open circuit), with 0 indicating a perfect match

Standing wave ratio (SWR)

• Standing wave ratio (SWR) is a measure of the impedance mismatch between the transmission line and the load
  • $SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}$
• Ranges from 1 (perfect match) to infinity (total reflection)
• High SWR indicates significant reflections and reduced power transfer efficiency

Power flow

• Power flow in transmission lines is determined by the voltage and current waves

Average power

• Average power is the time-averaged product of the voltage and current at a given point on the line
  • For a lossless line: $P_{avg} = \frac{1}{2} \text{Re}[V I^*] = \frac{|V_0^+|^2}{2Z_0} (1 - |\Gamma|^2)$
• Represents the net power flow in the direction of the load

Instantaneous power

• Instantaneous power is the product of the instantaneous voltage and current at a given point and time
  • $p(z, t) = v(z, t) \cdot i(z, t)$
• Consists of a constant term (average power) and a time-varying term (reactive power)

Power factor

• Power factor is the ratio of the average power to the apparent power (product of RMS voltage and current)
  • $PF = \frac{P_{avg}}{|V_{rms}| |I_{rms}|}$
• Ranges from 0 (purely reactive) to 1 (purely resistive)
• Higher power factor indicates more efficient power transfer

Impedance matching

• Impedance matching is the process of adjusting the load impedance to match the characteristic impedance of the transmission line
• Minimizes reflections and maximizes power transfer

Matching techniques

• Lumped element matching networks (L, T, or Pi networks)
  • Use discrete inductors and capacitors to transform the load impedance
• Distributed element matching networks (quarter-wave transformer, stub matching)
  • Use transmission line sections to transform the load impedance

Quarter-wave transformer

• A quarter-wavelength section of transmission line with a characteristic impedance $Z_T = \sqrt{Z_0 Z_L}$
• Transforms the load impedance $Z_L$ to $Z_0$ at the design frequency
• Bandwidth limited by the frequency dependence of the quarter-wavelength condition

Stub matching

• Open or short-circuited transmission line sections (stubs) connected in parallel or series with the main line
• Adjusts the input impedance to match the characteristic impedance
• Single-stub matching: One stub placed at a specific distance from the load
• Double-stub matching: Two stubs placed at fixed locations, offering more flexibility and bandwidth

Transmission line losses

• Losses in transmission lines lead to attenuation and dispersion of the propagating waves

Conductor losses

• Caused by the finite conductivity of the transmission line conductors
• Skin effect: Current concentration near the conductor surface at high frequencies, increasing the effective resistance
• Proximity effect: Current distribution influenced by nearby conductors, further increasing the resistance

Dielectric losses

• Caused by the imperfect insulating properties of the dielectric material
• Dielectric loss tangent ($\tan \delta$) quantifies the ratio of the conduction current to the displacement current
• Increases with frequency and temperature

Radiation losses

• Caused by the unintended emission of electromagnetic energy from the transmission line
• More significant at high frequencies and for unshielded or poorly shielded lines
• Bends, discontinuities, and asymmetries in the line geometry can enhance radiation losses

Transient behavior

• Transient behavior refers to the transmission line's response to sudden changes in the applied voltage or current

Step response

• Response of the transmission line to a unit step input voltage or current
• Characterized by the propagation delay and the rise time
• Reflections at the load and source can cause overshoots, undershoots, and ringing

Pulse response

• Response of the transmission line to a pulse input voltage or current
• Pulse width and shape are affected by the line's dispersion and attenuation
• Reflections can cause pulse distortion and inter-symbol interference (ISI)

Bounce diagrams

• Graphical method for visualizing the propagation and reflection of voltage and current waves on a transmission line
• Helps in understanding the transient behavior and locating impedance discontinuities
• Each bounce represents a reflection at the load or source, with the amplitude determined by the reflection coefficient

Transmission line applications

• Transmission lines are used in various applications to transmit signals and power

Coaxial cables

• Consist of an inner conductor surrounded by a dielectric insulator and an outer conductor (shield)
• Commonly used for RF signal transmission (radio, television, cable internet)
• Provides good shielding against electromagnetic interference (EMI)

Microstrip lines

• Consist of a conducting strip separated from a ground plane by a dielectric substrate
• Widely used in microwave integrated circuits (MICs) and printed circuit boards (PCBs)
• Easy to fabricate and integrate with other components

Waveguides

• Hollow metallic structures that guide electromagnetic waves
• Operate above a certain cutoff frequency determined by their geometry
• Used for high-power and high-frequency applications (radar, satellite communication, microwave ovens)
• Low loss and high power handling capacity compared to other transmission lines