Quarter-Wave Transformer Matching

• Two common transmission-line techniques are used to match a transmission line to a load that has an impedance not equal to Z₀: quarter wavelength transformer matching and stub matching.

Quarter Wavelength Transformer Matching

• This technique utilizes a quarter-wavelength section of transmission line to match a transmission line to a purely resistive load.
• This load's resistance is not equal to the characteristic impedance of the line.
• Z'₀ = √(Z₀ Zₗ) is the formula to calculate the characteristic impedance (Z'₀) of the quarter-wave transformer required to match a load impedance (Zₗ) to a transmission line with characteristic impedance (Z₀).
• L = λ/4 indicates the length (L) of the quarter-wavelength transformer, where λ is the wavelength of operation.
• λ = c/f is used to relate wavelength λ to the speed of light c and frequency f.
• The quarter-wave transformer acts as a transformer with a 1:1 turns ratio when Zₗ = Z₀ and α = 1.
• The quarter-wave transformer acts as a step-down transformer when Zₗ > Z₀ and α > 1.
• The quarter-wave transformer acts as a step-up transformer when Zₗ < Z₀ and α <1.

Stub Matching

• Stub matching involves using a short section of transmission line, typically short-circuited at one end.
• It is employed for impedance matching.
• It is placed across the primary line as close as possible to the load.
• This method removes the reactive component of the complex impedance of the load, facilitating matching to the transmission line.

Microstrips and Stripline Transmission Lines

• Standard transmission lines are impractical at low frequencies because they would be too long for use as reactive components or tuned circuits.
• Special transmission lines with copper patterns on a Printed Circuit Board were developed for high-frequency applications in order to interconnect components on PC boards.
• Stripline and Microstrip represent two PCB implementations of transmission lines.

Microstrip

• Microstrip is a flat conductor separated from a ground plane by an insulating dielectric material.
• The characteristic impedance depends on its physical characteristics.
• It has advantages over stripline: simpler construction and easier integration with semiconductor devices, which is suitable for printed circuit and thin film techniques.
• L = µT/L is a formula related to inductance.
• C = εT/l is a capacitance equation.

Stripline

• A stripline consists of a flat conductor sandwiched between two ground planes.
• It is more difficult to manufacture than microstrip, but radiates less, resulting in lower losses compared to microstrip.

Waveguides

• Transmission line losses increase rapidly with frequency, making conventional transmission lines impractical at gigahertz frequencies for long cable runs.
• Parallel wire transmission lines (including coaxial cables) cannot propagate signals with high powers because high voltages can cause the dielectric separating the conductors to break down.
• A waveguide is a type of transmission line used at low frequencies but is very useful in the microwave region.
• It generally consists of a hollow, air-filled tube made of conducting material.

Modes and Cutoff Frequency

• Modes are ways in which electrical energy can propagate along a waveguide.
• Modes are understood as a wave moving through the guide like a ray of light.
• For each mode, the ray strikes the waveguide walls at a different angle, and the distance the ray must travel to reach the far end of the guide becomes larger as the angle increases.
• Each mode has a cutoff frequency below which it will not propagate.
• Single-mode propagation is achieved using the mode with the lowest cutoff frequency: the dominant mode.
• Waveguides are used at frequencies between the cutoff frequency for the dominant mode and that of the mode with the next lowest cutoff frequency.
• Transverse Electric (TE) modes have all electric components transverse to the direction of propagation.
• Transverse Magnetic (TM) modes have all magnetic field components transverse to the direction of propagation.

Phase Velocity and Group Velocity

• Group velocity is the actual speed at which a signal travels along the guide.
• The velocity of propagation of a wave along a waveguide is less than its velocity through free space (speed of light) because of the zigzag path taken by the wavefront.
• Phase velocity is the rate at which the wave appears on the way the angle varies along the walls.

Wavelength Impedance

• A waveguide has a characteristic impedance.
• Waveguide impedance is a function of frequency unlike wire lines.

Guide Wavelength

• Impedance matching with waveguides is similar to other transmission lines: the phase velocity vp must be used in calculating λg, the wavelength in the guide.