## 3.1 EM Modeling and System Performance

### Array Design Process

The antenna elements and array geometry are a major factor in determining the overall receiver system performance. The element and array design determines the operating bandwidth, aperture illumination efficiency of synthesized beams, and impedance matching to the front end LNAs. In the case of an array, the LNAs must be noise matched to the active impedances presented by the array elements to the LNAs to minimize the system noise and achieve optimal SNR. Active impedances depend on beamformer coefficients, so each formed beam or pixel corresponds to a different active impedance presented to the LNAs. This was accounted for in the design process by embedding a full electromagnetic model of the dipoles and array in an optical model for the reflector, including ground noise and spillover, sky noise, electronics noise from front end LNAs, and array signal processing to compute and apply beamformer coefficients to generate simulated formed beams. The array was designed and modeled using CST Microwave Studio by German Cortes. The reflector optics, system noise, receiver electronics, and signal processing models were in-house codes developed by Karl Warnick and research assistants.

Over the first year of the project, many candidate array designs were modeled and simulated to determine the system noise, field of view, aperture efficiency, sensitivity, and survey efficiency of the design. The design was optimized for best overall survey efficiency over the target bandwidth. Early in the process, rectangular and hexagonal array geometries were compared, and hexagonal design was selected as it achieved the best field of view for a given number of array elements.

After extensive modeling of both square and hexagonal arrays an array geometry of 69 dual polarization elements arranged in a hexagonal pattern and spaced 135 mm apart has been selected. The advantages of this array size and geometry are the smaller number of elements (69) compared with the 80 of the square array. This has reduced the number of cold heads required from 4 to 3 and the number of compressors from 2 to 1 resulting in considerable savings in complexity, cost and weight.

### Antenna Elements

The 19-element prototype dipoles had straight, cylindrical radiating arms. The new design is similar to the Flag design and has dipole arms with pie-shaped ends (see Fig.1). Design variables are the opening angle of the pie shape and the tilt angle with respect to the horizontal. Computer modeling using combinations of these variables shows that the optimum values are close to 20° (tilt angle, α) and 40°(pie angle, β). Fig. 2 shows the resultant telescope survey efficiency as a function of observing frequency and a number of opening angle and tilt angle values.

An example study from the array and antenna element design process is shown below. The team selected the design from the candidate cases that yielded the best overall survey efficiency (which in turn depends on dipole radiation pattern, beam synthesized dish illumination efficiency, and active impedance matching to the front end LNAs) over frequency.

Figure: Dipole pie and tilt angle optimization study. Values shown in the legend are half of the full dipole arm angle.

### ALPACA on GBT Performance

Using the electromagnetic model of the ALPACA front end array and the BYU reflector, receiver electronics, and array signal processing model, the estimated noise budget for the ALPACA HEX60_FIN array, with U. Calgary measured LNA noise parameters for the 1.5 GHz boresight formed beam on the GBT is:

Noise Component |
Contribution |
---|---|

Sky | 5K |

Spillover | 1K |

Loss, Scattered Ground, Unmodeled | 10K |

LNA | 10K |

Signal Transport | 1K |

Total, T_{sys} |
27 K |

The BYU reflector noise model does not include spillover noise scattered from support structures and trees and hills that extend above a flat ground at Green Bank and radiate ~280 K thermal noise towards the feed. The model does not include the effects of loss in the thermal window, supporting foam, and dipole elements, because the loss in these structures is smaller than the numerical accuracy of the CST modeling software. Dating back to before the PAF era, there is also roughly a 5K historical difference between the predicted and measured system noise temperatures for cryogenic receivers that is, in the words of Rick Fisher, "not directly attributable to any one physical mechanism." Based on our experience with previous PAF systems and prototypes, the unmodeled noise contribution to the system temperature is in the range of 5-10 K. This 10 K is included in the estimated ALPACA on GBT 1.5 GHz noise budget as the "Loss noise, scattered ground noise, and unmodeled noise contributions" line item. The estimated noise budget is conservative, and we anticipate that the measured system noise temperature will be lower than the conservatively estimated value.

For the boresight formed beam, the estimated system noise temperature over frequency, aperture efficiency, effective system temperature T_{sys}/η, and the survey efficiency are shown below.

Figure: Estimated system noise temperature over frequency. The usable system bandwidth is 1.3 to 1.72 GHz.

The modeled sensitivity map indicating the useable field of view of ALPACA on the GBT is shown below. The figure shows the peak sensitivity of steered beams at the steering angle of the beam. The 1 dB sensitivity contour at 160 m^{2}/K is about 20 arc-min in radius.

Figure: Survey efficiency over frequency. Survey efficiency is squared sensitivity (A/T_{sys})^{2} integrated over the field of view, normalized to a peak value of unity. This is a measure of the speed at which a given region of sky can be imaged to a given minimum detectable signal level and allows different instruments to be compared on an equal footing.