Question

A turnstile antenna consists of two crossed dipole antennas, positioned in this case in the \(x y\) plane. The dipoles are identical, lie along the \(x\) and \(y\) axes, and are both fed at the origin. Assume that equal currents are supplied to each antenna and that a zero phase reference is applied to the \(x\) -directed antenna. Determine the relative phase, \(\xi\), of the \(y\) -directed antenna so that the net radiated electric field as measured on the positive \(z\) axis is \((a)\) left circularly polarized; \((b)\) linearly polarized along the \(45^{\circ}\) axis between \(x\) and \(y\).

Short answer

-90° (b) What is the relative phase, ξ, of the y-directed antenna for the linearly polarized field along the 45° axis between x and y? 0°

Step-by-step solution

1. Left Circularly Polarized Case

The left circularly polarized field is produced when the electric fields in both \(x\) and \(y\) directions are equal in magnitude but have a relative phase difference of \(90^{\circ}\). Since the \(x\)-directed antenna has a zero phase reference, the phase of the \(y\)-directed antenna will be \(\xi = -90^{\circ}\).

2. Linearly Polarized along the \(45^{\circ}\) axis

In this case, we want the net electric field to be linearly polarized along the \(45^{\circ}\) axis between the \(x\) and \(y\) directions. This means that the magnitudes of the electric fields in both \(x\) and \(y\) directions should be equal. The phase difference must be zero for linear polarization, so the phase of the \(y\)-directed antenna must be equal to the phase of the \(x\)-directed antenna. Since the \(x\)-directed antenna has a zero phase reference, the phase of the \(y\)-directed antenna will be \(\xi = 0^{\circ}\). #Summary#The relative phase, \(\xi\), of the \(y\)-directed antenna is: (a) \(\xi = -90^{\circ}\) for left circularly polarized field, (b) \(\xi = 0^{\circ}\) for linearly polarized field along the \(45^{\circ}\) axis between \(x\) and \(y\).

Key concepts

Phase Difference in Antennas

Understanding the concept of phase difference in antennas is crucial when designing systems for various communication applications. The phase difference refers to the relative time offset between the peaks of two or more signal waves. In terms of antennas, particularly elements like dipoles, each antenna element can emit radio waves that are out of phase with one another. This means they reach their peak amplitude at different times.

For a turnstile antenna, which consists of two perpendicular dipole antennas operating in unison, a phase difference between the two can significantly alter its radiation pattern. By carefully manipulating the phase difference, you can create different kinds of polarization such as circular or linear. This is achieved by introducing a phase shift to one of the dipoles relative to the other. In the given exercise, creating a left circularly polarized wave requires a -90° shift from the y-axis antenna in relation to the x-axis antenna with zero phase reference. Here, the concept of a phase difference is applied to produce a specific polarization effect that is vital for the compatibility of the antenna with circularly polarized signals.

Circular Polarization

The phenomenon of circular polarization is intriguing and serves particular purposes in wireless communication. Unlike linear polarization, where the electric field oscillates in a fixed plane, circular polarization occurs when two linearly polarized waves of equal amplitude are combined with a 90° phase difference. This creates a rotating electric field that can be either left-hand or right-hand circularly polarized, depending on the direction of rotation.

In our exercise scenario, to achieve a left circularly polarized output, the electric field components from the two crossed dipoles (positioned on the x and y axes) must be identical in magnitude with precisely a 90° phase shift between them. As suggested in the solution, with the x-directed antenna set as our zero phase reference, we need to adjust the y-directed antenna by \(\xi = -90^\circ\) to get the desired left circular polarization. This form of polarization is advantageous due to its resistance to the orientation of the receiving antenna and its ability to mitigate signal fading caused by multipath interference.

Linear Polarization

Linear polarization is a more straightforward concept compared to its circular counterpart but is no less important in the realm of antenna theory. In linear polarization, the electric field oscillates in a single plane along a straight line. Common linear polarizations are horizontal, vertical, and slanted—where the tilt is often at a 45° angle.

When the exercise specifies achieving linear polarization along the 45° axis, it means we're trying to orient the polarization diagonally between the x and y axes. To accomplish this with the turnstile antenna, the radiated electric fields must have equal magnitude and no phase difference, which aligns with the definition of linear polarization. Therefore, to achieve linear polarization as required in part (b) of the problem, the y-directed antenna must be set to have the same phase as the x-directed antenna, resulting in \(\xi = 0^\circ\). This simple adjustment allows the superposition of the two dipole radiations to produce a net electric field that's linearly polarized along the desired axis, which can be particularly beneficial for communication with linearly polarized signals.