Introduction To Modern Planar Transmission Lines. Anand K. Verma

components. The tip of the electric vector () moves along line A‐O‐B with respect to time. However, the slant angle θ does not change with time. If both the E‐field components are either in‐phase (A − O − B) or out of phase (A^/ − O − B^/) and have the same magnitude, i.e. E_0y = E_0z = E₀, the corresponding inclination angle of the linear polarization trace, with respect to the y‐axis, is θ = 45° and 135°, respectively. For the linear polarization, the total E‐field given by equation (4.6.1a) could also be written as follows:

      (4.6.3)

      

      4.6.2 Circular Polarization

      The circular polarization, shown in Fig. (4.10b), is obtained for two orthogonal field components of equal magnitude, and phase in quadrature. So, to get the circular polarization, two electric field components oscillate at the same frequency and meet the following conditions:

       Equal amplitude: The magnitudes of Ey and Ez are equal, i.e. |E0y| = |E0z| = E0.

       Space quadrature: The Ey and Ez field components are normal to each other.

       Time (phase) quadrature: The phase difference between the Ey and Ez field components are (φ = ± 90°), i.e. E0y = E0, and E0z = E0e±π/2 = ± j E0.

      The phasor form of the E‐field vector of the circularly polarized waves, meeting the above conditions, could be written from equation (4.6.1a) as follows:

(4.6.4)

      The e^jωt time‐harmonic factor is suppressed in the above equations. The time‐domain forms of the circularly polarized waves, using the instantaneous ‐field components, at any location in the positive x‐direction and also at the x = 0, i.e. in the (y‐z)‐plane, are expressed as follows:

(4.6.5)

The handedness, i.e. the sense of rotation of the ‐vector of the circularly polarized wave is determined by the direction of rotation of the ‐vector in the transverse (y‐z)‐plane with respect to time, whereas the EM‐wave propagates in the x‐direction. Therefore, using equation (4.6.5f), Fig. (4.10b) shows the anti‐clock rotation of the E‐field vector tracing a circle of radius E₀ in the (y‐z)‐plane. The sense of rotation is considered with respect to cos(ωt), taking t = 0. It shows the wave propagating in the positive x‐direction, i.e. the wave coming toward an observer standing on the positive x‐axis is the right‐hand circularly polarized wave (RHCP). Its field vector is given by equation (4.6.4b). Similarly, equation (4.6.5e) shows the clockwise rotation, i.e. the left‐hand circularly polarized wave (LHCP) propagating in the positive x‐direction. The magnetic fields of the RHCP and LHCP wave are obtained using equation (4.6.4) with equation (4.5.34a):

      (4.6.6)

In the case of the wave propagation in the negative x‐direction, i.e. the wave moves away from the observer standing on the positive x‐axis, the role of RHCP and LHCP gets interchanged. Further, the handedness of circular polarization can be reversed by applying 180° phase‐shift to either the y or z component of the ‐field vector. The circular polarization could be further considered as a linear combination of two linearly polarized waves. Alternatively, the linear polarization can also be considered as a linear combination of the left‐hand and right‐hand circularly polarized waves [B.9].

      4.6.3 Elliptical Polarization

      Two orthogonal field components, in the space quadrature, of unequal magnitude and arbitrary phase angle (φ) between them, generate the elliptical polarization, i.e. the resultant E‐field vector rotates in the plane of polarization such that its tip traces an elliptical path as shown in Fig. (4.10c). The instantaneous orthogonal E‐field components in the x = 0 plane are given below:

      (4.6.7)

      Using the above equation and identity the following equation of the ellipse is obtained:

      (4.6.8)

      The semi‐major axis OA, the semi‐minor axis OB of the ellipse shown in Fig. (4.10c), and the axial ratio (AR) of the polarization ellipse are given below [B.9, B.29]:

(4.6.9)

      The tilt angle θ of the polarization ellipse, i.e. inclination of the major axis OA with y‐axis is [B.9, B.29]:

      (4.6.10)

For φ ≠ π/2, the polarization ellipse is inclined with respect to the y‐axis. The linear and circular polarizations are obtained as special cases from the elliptical polarization. For instance, for E_z(t) = 0 the wave is horizontally polarized in the y‐direction. For E_0y = E_0z = E₀ and φ = ± π/2, the LHCP/RHCP wave is obtained as equation (4.6.9) is reduced to an equation of a circle with OA = OB. For the linear polarization, AR is infinity. However, for the circular polarization, AR is unity. In the case E_0y = E_0z = E₀ and φ ≠ π/2, the wave is not circularly polarized and its AR is cotφ/2. For a practical circularly polarized antenna, the axial ratio is frequency‐dependent and its axial ratio bandwidth is defined as the frequency band over which AR ≤ 3dB.

      

      4.6.4 Jones Matrix Description of Polarization States

      The polarizing devices change the state of polarization. For instance, the polarizing devices could change the rotation of the linear