Vector Rotator Solves Satcom Antenna Skew

1. This block diagram shows the hybrid construction of an RF vector rotator.

Figure 1 shows a block diagram of the vector rotator circuit, based on a low-frequency operational-amplifier (op-amp) circuit described in an earlier article.¹ Whereas the low-frequency op-amp version had the advantage of being frequency-invariant over the usable range of op amps, it was limited in frequency by those same devices. That basic low-frequency version is shown in Figure 2.

2. This is a low-frequency version of the vector rotation circuit.

The proposed hybrid circuit performs the same vector rotation, but is better suited for microwave and millimeter-wave frequencies. The op amps in the original circuit perform the function of summing and subtraction of the scaled in-phase (I) and quadrature (Q) input signals. In the hybrid circuit, the inverting and adding functions are provided by 180-deg. hybrids, and the purely summation function is provided by 0-deg. power combiners. The op-amp circuit used resistors to set the scale factor of the input I and Q vectors, whereas in the hybrid circuit, attenuators set the scale factors.

The theoretical operation of the circuit is based on the vector representation of the received signals represented in eq. 1:

S_R =     [H_Tcos(ϕ) – V_T sin(ϕ)]xˆ_H + [H_Tsin(ϕ) +V_Tcos(ϕ)]ŷ_V                                    (1)

where S_R is the received signal; H_T is the transmitted horizontal signal; V_T is the transmitted vertical signal; xˆ_H is the received horizontal signal reference axis; ŷ_V is the received vertical signal reference axis (where x_H is the horizontal channel and y_V is the vertical channel); and ϕ is the angle of misalignment between the transmit and receive antennas.

3. Satellite antenna misalignment results in cross-channel coupling.

Figure 3 illustrates the relationship between the satellite's antenna reference system and the earth, or in-flight transceiver’s antenna reference system. If you assume that all hybrids have the same insertion loss (−3 dB) for each input to output, and normalize the output amplitude by the overall loss factor, then the output can be written as:

V₀ = H_T[m₁cos(ϕ) + m₂sin(ϕ)]

+V_T[m₂cos(ϕ) – m₁sin(ϕ)]

where m₁ = k₁ – G₁ and m₂ = k₂ – G₂.

If m₁ = cos(ϕ) and m₂ = sin(ϕ), then:

V₀ = H_T[cos²(ϕ) + sin²(ϕ)] + V_T[cos(ϕ)sin(ϕ) – cos(ϕ)sin(ϕ)]  → V₀ = H_T

In a similar derivation for the output of the vertical channel:

V₀ = H_T[m₃cos(ϕ) + m₄sin(ϕ)] + V_T[m₄cos(ϕ) – m₃sin(ϕ)]

where m₃ = k₃ – G₃ and m₄ = k₄ – G₄.

If m₄ = cos(ϕ) and m₃ = –sin(ϕ), then V₀ = H_T[–sin(ϕ)cos(ϕ) + sin(ϕ)cos(ϕ)] + V_T[cos²(ϕ) + sin²(ϕ)] → V₀ = V_T

Using the preceding determination for m₁, m₂, m₃, and m₄ dictates the following relationships:

k₁ – G₁ = cos(ϕ);

k₂ – G₂ = sin(ϕ);

k₃ – G₃ = -sin(ϕ);

k₄ – G₄ = -cos(ϕ);

These relationships can be realized with the following equations:

k₁ = cos²(ϕ); G₁ = k₁ – cos(ϕ);

k₂ = 0.5 + 0.5sin(ϕ); G₂ = k₂ – sin(ϕ);

G₃ = 0.5 + 0.5sin(ϕ); k₃ = G₃ – sin(ϕ);

k₄ = cos²(ϕ); G₁ = k₁ – cos(ϕ);

Plots of the k and G values are shown in Figures 4, 5, and 6.

4. Attenuation factors K₁ and K₄ and G₁ and G₄ are plotted here.  

5. Attenuation factors K₂ and G₂ are plotted as a function of rotation angle.

6. Attenuation factors K₃ and G₃ are plotted as a function of rotation angle.

Detailed derivation along with a spreadsheet model (Fig. 7) that allows for variation of all hybrid losses, phase differences, and interconnecting parameters is available upon request. The spreadsheet model does not lend itself to complicated variations of the components and interconnecting lines, but does allow investigations of real circuit tolerances to the extent of manual effort tolerated by the investigator. An Octave-based program is being developed to make it easy to manipulate the myriad of variables in an actual circuit fabrication.

7. This spreadsheet model of the solid-state vector rotator includes the effects of variations due to hybrid losses, phase differences, and interconnecting parameters.

Reference:

1. R.C. Monzello, "Circuit Provides Frequency-Independent Quadrature Rotation," Microwaves & RF, September 1995.