### 3.1 Radiation Field with Facet Displacement

Facet displacement leads to the variety of induced current, and the radiation field will vary correspondingly. Since the facet displacement is very small, the induced current varies only on phase but not on amplitude [27]. If the facet offset vector is assumed as *Δ*
*r*
_{
i
}’ shown in Fig. 3, the alterant current can be obtained as

$${\varvec J}^{\Delta } \left( {{\varvec r}_{i} ' + \Delta {\varvec r}_{i} '} \right) = {\varvec J}\left( {{\varvec r}_{i} '} \right)\exp \left( { - {\text{j}}k{\hat{\varvec p}}_{i} \cdot \Delta {\varvec r}_{i} '} \right),$$

(6)

where \({\hat{\varvec p}}_{i}\)—Unit Poynting vector.

Put Eq. (6) into Eq. (5), the variation of radiation field in point *r*
_{
l
} contributed by the *i*th triangle facet because of the *n*th facet displacement can be expressed as

$$ {\varvec E}_{l,ni}^{\Delta } = {\varvec E}_{l,ni}^{\text{s}} \exp \left( {{\text{j}}\delta_{ni} } \right), $$

(7)

$$\delta_{ni} = k\left( {{\hat{\varvec r}}_{l} - {\hat{\varvec p}}_{i} } \right) \cdot \Delta {\varvec r}_{i} '.$$

(8)

If all facets displaced, as a result, the total scattered field in point *r*
_{
l
} can be expressed as the sum of every single contribution by

$${\varvec E}_{l}^{\Delta } = \sum\limits_{n = 1}^{N} {E_{l,n}^{\Delta } } = \sum\limits_{n = 1}^{N} {\sum\limits_{i = 1}^{{N_{n} }} {E_{l,ni}^{\Delta } } } .$$

(9)

From Eq. (7) and Eq. (8), the variety of radiation field only by phase error is proportional to facet offset. If the phase error in Eq. (7) is small enough, the exponent function can be approximated by Taylor expansion. By taking the first two items, the variety of radiation field with phase error can be linearized as

$${\varvec E}_{l,ni}^{\Delta } = {\varvec E}_{l,ni}^{\text{s}} \left( { 1 {\text{ + j}}\delta_{ni} } \right).$$

(10)

Based on Eq. (10), if all triangle facets displaced in the whole reflector, the influence of facet displacement on the radiation field forms a linear system.

### 3.2 Relationship Between Adjustment and Displacement

Because individual panels are generally fabricated much more precisely than fixed onto a large BUS, the panels can be assumed to be rigid plate. Thus, the panel position is only geometrically decided by three adjustors selected in all corner screws in Fig. 4.

Once the panel is adjusted by three adjustors, the facet displacement in this panel can be written as [28, 29]

$$\Delta {\varvec r}_{i} '{ = }{\varvec S}_{ni} {\varvec a}_{n} ,$$

(11)

where *a*
_{
n
}—Panel adjustment vector, *S*
_{
ni
}—Transform matrix between adjustment vector and displacement of the *i*th facet.

$${\varvec a}_{n} = \left[ {\begin{array}{*{20}c} {a_{\text{A}} } & {a_{\text{B}} } & {a_{\text{C}} } \\ \end{array} } \right]^{\text{T}} ,$$

(12)

$${\varvec S}_{ni} = \left[ {\begin{array}{*{20}c} {{\text{signA}}_{i} \frac{{d_{{{\text{A}}_{i} }} }}{{d_{\text{A}} }}{\bar{\varvec n}}_{{{\text{A}}_{\text{i}} }} } & {{\text{signB}}_{i} \frac{{d_{{{\text{B}}_{i} }} }}{{d_{\text{B}} }}{\bar{\varvec n}}_{{{\text{B}}_{\text{i}} }} } & {{\text{signC}}_{i} \frac{{d_{{{\text{C}}_{i} }} }}{{d_{\text{C}} }}{\bar{\varvec n}}_{{{\text{C}}_{\text{i}} }} } \\ \end{array} } \right]^{\text{T}}$$

(13)

where \({\bar{\varvec n}}\)—Unit normal vector on the facet.

$${\bar{\varvec n}} = \left[ {\begin{array}{*{20}c} {n_{x} } &{n_{y} }& {n_{z} } \\ \end{array} } \right]^{\text{T}}$$

(14)

Actually, reflector panels are designed for curve fitting to a parabolic surface. As a result, facet displacement includes components in three different directions when different adjustors adjusted. Therefore, the whole offset vector is the superposition of three components.

### 3.3 Modeling of Linear System

The total radiation field of point *r*
_{
l
} can be expressed as the overall contribution of the *n*th panel because of its facet displacement, say, putting Eq. (10) and Eq. (11) into Eq. (9) as

$${\varvec E}_{l,n}^{\Delta } = {\varvec E}_{l,n}^{\text{s}} + {\varvec B}_{n} {\varvec a}_{n} ,$$

(15)

$${\varvec B}_{n} = \sum\limits_{i = 1}^{{N_{n} }} {\left[ {{\varvec E}_{l,ni}^{\text{s}} {\text{j}}k\left( {{\hat{\varvec r}}_{l} - {\hat{\varvec p}}_{ni} } \right){\varvec S}_{ni} } \right]} .$$

(16)

The total radiation field of all *m* observing points in the far region can be obtained by adding the contributions of all panels together. Noting that

$${\varvec Q} = \left[ {\begin{array}{*{20}c} {\underbrace {{\begin{array}{*{20}c} {B_{11} } & {B_{12} } & {B_{13} } \\ \end{array} }}_{{{\varvec B}_{1} }}} & {\underbrace {{\begin{array}{*{20}c} {B_{21} } & {B_{22} } & {B_{23} } \\ \end{array} }}_{{{\varvec B}_{2} }}} & \cdots {\underbrace {{\begin{array}{*{20}c} {B_{n1} } & {B_{n2} } & {B_{n3} } \\ \end{array} }}_{{{\varvec B}_{n} }}} \\ \end{array} } \right],$$

(17)

$${\varvec M} = \left[ {\begin{array}{*{20}c} {{\varvec Q}_{1} } & {{\varvec Q}_{2} } & \cdots & {{\varvec Q}_{m} } \\ \end{array} } \right]^{\text{T}} ,$$

(18)

$${\varvec a} = \left[ {\begin{array}{*{20}c} {\underbrace {{\begin{array}{*{20}c} {a_{11} } & {a_{12} } & {a_{13} } \\ \end{array} }}_{{{\varvec a}_{1} }}} & {\underbrace {{\begin{array}{*{20}c} & {a_{21} } & {a_{22} } & {a_{23} } \\ \end{array} }}_{{{\varvec a}_{2} }}} & \cdots & {\underbrace {{\begin{array}{*{20}c}{a_{n1} } & {a_{n2} } & {a_{n3} } \\ \end{array} }}_{{{\varvec a}_{n} }}} \\ \end{array} } \right]^{\text{T}} ,$$

(19)

thus, the relationship between the adjustment values of all panels and the electric field in far region can be written into a linear system as

$${\varvec E}^{\Delta } = {\varvec E}^{\text{S}} + {\varvec {Ma}},$$

(20)

### 3.4 Solution of Linear Equations Using SVD

The method of Singular Value Decomposition (SVD) is used to calculate the pseudo inverse matrix *B*. In SVD the *m* × *n* matrix *A*(where *m* ≥ *n*)can be expressed as the product of three matrices:

$${\varvec A} = {\varvec {UWV}}^{\text{T}} ,$$

(21)

where *U*—An *m* × *n* column orthogonal matrix,*W*—An *n* × *n* diagonal matrix whose elements *w*
_{
i
} are positive or zero and are referred to as the singular values of the matrix *A*,*V*—An *n* × *n* orthogonal matrix,T—Transpose operator.

The matrix *B* can be obtained as

$${\varvec B} = {\varvec {VW}}^{ - 1} {\varvec U}^{\text{T}} ,$$

(22)

where 1/1*w*
_{
j
} in *W*
^{−1} can be replaced by 0 in the event that *w*
_{
j
} = 0. The matrix *V* defines a basic set of modes which are essentially unique orthonormal, so that any arbitrary configuration of the system can be written as a unique linear combination of these basic modes [30]. Using SVD to solve Eq. (23), the generalized reverse of transformation matrix can be calculated. After that, the panel adjustment values on the whole reflector can be obtained from the following equations

$${\varvec {Ma}} = {\varvec E}^{\Delta } - {\varvec E}^{\text{S}} .$$

(23)