## 1 Introduction  
Active noise control (ANC) is an effective method to reduce transformer noise, which is dominated by harmonic components at low frequencies. Global control of transformer noise can be achieved by placing a sufficient number of secondary sources and error sensors surrounding it [1]. However, hundreds of control channels are required due to the large size of transformers, which makes the control system complicated and expensive [2]. Therefore, local control systems, which aim to create quiet zones in a target area, are investigated [3], which provides an alternative solution. The error-sensing strategy is an important issue in the design of compact ANC systems [4]. The noise was reduced by 6 dB or more within an angle of approximately 23\({}^{\circ}\) in the far field when an error microphone was placed 30 m away from the transformer [5]. Directivity control from near to far field was achieved by adopting the sum of the squared sound pressures at several points within an angle as the cost function in the active noise control barriers [6]. However, both of the above-mentioned strategies use error points that are far from the primary source. For the stability and compactness of ANC systems, error sensors should be placed in the near field of the primary and secondary sources. Different near-field error-sensing strategies have been compared to reduce the sound radiation from a monopole and dipole primary source and it was found that the most appropriate strategy is minimizing the sum of the weighted mean active intensity in a direction normal to the surface surrounding all the primary and control sources [7]. A strategy of minimizing the near-field sound intensity perpendicular to the barrier at specific points [8] has been found to be more effective in reducing far-field noise than minimizing the squared sound pressure [9]. However, sound intensity probes are required, and the reduction performance is not always better than that achieved by minimizing the near-field sound pressure [4]. A novel error-sensing strategy was proposed recently, where a microphone array is placed near the sources and the weighted sum of sound pressures at these microphonesis minimized to achieve directivity control [10]. The weighting factor relates to the phase shift between the error microphones and the plane perpendicular to the direction where noise reduction is needed. It was found that the minimum size of the array should be twice the size of the primary source, which is consistent with the requirements in the sound power measurement of a sound source [11]. The maximum microphone spacing is half the wavelength of the frequency of interest, which is similar to the discussions in Beamforming [12]. However, there is no systematic analysis of this error-sensing strategy. The weighted sum of sound pressure has already been defined as an \"angular spectrum\" in the study of Fourier acoustics [13]. This technique decomposes the wave field into a series of plane waves that propagate independently, which are superimposed to obtain the field at the desired axial depth. The angular spectrum method can also be employed to characterize pressure fields for various sources [14; 15] and to model sound transmission in both layered homogeneous [16] and inhomogeneous media [17]. The computation time to calculate wave propagation in parallel planes can be significantly reduced by employing the angular spectrum method rather than the point-by-point direct integral methods [18]. The surface velocity of sources through acoustic holography can also be effectively predicted by the angular spectrum method [19]. In this paper, an analytical model of the angular spectrum is employed to explain the near-field error-sensing strategy proposed in Ref. [10] for active directivity control. First, the analytical model for angular spectrum estimation of a point source in free field is derived and the directivity control strategy is analyzed.