Design Article
Understand ripples in RF performance measurements: Theory and Experiments
Habeeb Ur Rahman Mohammed, Ph.D., Texas Instruments
2/4/2013 10:46 AM EST
Understand ripple--Page 2.
Theoretical analysis
Reflection and transmission coefficients are functions of the constitutive parameters (permittivity, permeability, and conductivity) of the material or medium across the boundary, directions of wave travel (angle of incidence) and the direction of the electric and magnetic fields (wave polarization). Electromagnetic waves propagate in transverse electromagnetic mode (TEM) which is characterized by the absence of longitudinal field in transmissions lines that consist of two or more conductors (co-axial or microstrip lines). Wave propagation with no electric E and magnetic H field components in the direction of propagation is considered at the boundary of two mediums shown in Figure 1d at a certain angle of incidence θi (oblique incidence angle).
Oblique incident wave Reflection and transmission coefficients at oblique angle of incidence θi are obtained considering parallel ║ or perpendicular ┴ polarization of electromagnetic waves. For most of the cables and transmission lines dielectric material relative permeability µr is unity. Fresnel equations of reflection coefficient in parallel polarization Γ║, transmission coefficient in perpendicular polarization τ║, reflection coefficient in perpendicular polarization Γ┴ and transmission coefficient in perpendicular polarization τ┴ with µr = 1 are shown in Equations 3 – 6, respectively. A detailed explanation of these equations is provided in reference [1]. Subscripts ‘i’, ‘r’ and ‘t’ represent incident, reflected and transmitted fields.
Wave propagation within two conductor transmission lines is through the length of transmission line that is the normal angle of incidence θi = 0o. The Fresnel field reflection and transmission coefficients become polarization independent in the limit as θi goes to zero. Reflection coefficients in Equations 3 and 5 and transmission coefficient in Equations 4 and 6 yield the same results as in Equations 7 and 8. Subscript ‘12’ represents the wave is incident from medium 1 and transmitted into medium 2 of Figure 1d.
The reflection and transmission coefficients with multiple reflections of dielectric block shown in Figure 1e will be calculated in this section. Figure 2 shows the normal incident plane wave multiple interactions within this dielectric block.
Normal parallel or perpendicularly polarized plane wave incident on the dielectric block can be considered as:
Theoretical analysis
Reflection and transmission coefficients are functions of the constitutive parameters (permittivity, permeability, and conductivity) of the material or medium across the boundary, directions of wave travel (angle of incidence) and the direction of the electric and magnetic fields (wave polarization). Electromagnetic waves propagate in transverse electromagnetic mode (TEM) which is characterized by the absence of longitudinal field in transmissions lines that consist of two or more conductors (co-axial or microstrip lines). Wave propagation with no electric E and magnetic H field components in the direction of propagation is considered at the boundary of two mediums shown in Figure 1d at a certain angle of incidence θi (oblique incidence angle).
Oblique incident wave Reflection and transmission coefficients at oblique angle of incidence θi are obtained considering parallel ║ or perpendicular ┴ polarization of electromagnetic waves. For most of the cables and transmission lines dielectric material relative permeability µr is unity. Fresnel equations of reflection coefficient in parallel polarization Γ║, transmission coefficient in perpendicular polarization τ║, reflection coefficient in perpendicular polarization Γ┴ and transmission coefficient in perpendicular polarization τ┴ with µr = 1 are shown in Equations 3 – 6, respectively. A detailed explanation of these equations is provided in reference [1]. Subscripts ‘i’, ‘r’ and ‘t’ represent incident, reflected and transmitted fields.
Normal incident wave
Wave propagation within two conductor transmission lines is through the length of transmission line that is the normal angle of incidence θi = 0o. The Fresnel field reflection and transmission coefficients become polarization independent in the limit as θi goes to zero. Reflection coefficients in Equations 3 and 5 and transmission coefficient in Equations 4 and 6 yield the same results as in Equations 7 and 8. Subscript ‘12’ represents the wave is incident from medium 1 and transmitted into medium 2 of Figure 1d.
Reflection and transmission coefficients with multiple reflections
The reflection and transmission coefficients with multiple reflections of dielectric block shown in Figure 1e will be calculated in this section. Figure 2 shows the normal incident plane wave multiple interactions within this dielectric block.
Where w is angular frequency, d is the distance travelled by wave within the dielectric block, and γ is propagation constant of dielectric block and is given in Equation 10. The real part of propagation constant is attenuation constant α (Np/m) and imaginary is phase constant β (rad/m). εr and μr in Equation 10 is relative permittivity and permeability of dielectric block (or wave travelling medium).
Considering Equation 7 the reflection coefficient of the wave incident from medium 1 into medium 2 and from medium 2 into medium 1 is given in Equations 11 and 12, respectively.
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