CN103776873A - Method for constructing current-voltage mapping by virtue of voltage-current mapping - Google Patents

Abstract

The invention relates to a method for constructing current-voltage mapping by virtue of voltage-current mapping. The method is characterized in that for a sensor with N electrodes in electric tomography, a current-voltage matrix is exclusively calculated through a derived formula through the parameters such as characteristic value and characteristic vector of a voltage-current matrix so as to construct the current-voltage mapping. The invention provides a method for directly constructing the current-voltage mapping, the method can be applied to the direct reconstruction algorithm in the field of the electric tomography, the physical significance of the method is clear, and the method is simple and feasible.

Description

A kind of method by the mapping of voltage-to-current map construction current-voltage

Technical field

The present invention relates to electricity chromatography imaging field, relate in particular to a kind of method by the mapping of voltage-to-current map construction current-voltage.

Background technology

Electricity tomography (Electrical Tomography is called for short ET) technology is the one of chromatography imaging technique.By testee is applied to excitation, and detect the variation of its boundary value, utilize the distribution of specific reconstruction algorithm reconstruct measurand internal electrical characterisitic parameter, thereby obtain the distribution situation of interior of articles.Compared with other chromatography imaging techniques, that electricity tomography has is radiationless, Noninvasive, portability, fast response time, the advantage such as cheap.

Electricity chromatography imaging reconstruction algorithm generally can be divided into two classes: the reconstruction algorithm based on sensitivity matrix and directly reconstruction algorithm.Use the former conventionally need to separate ill linear equation, this just means all pixel values that must simultaneously rebuild measured zone.The latter is shone upon to realize by calculating current-voltage mapping or voltage-to-current, and the gray-scale value of each pixel can be through directly, independently calculating acquisition.

The building method of current-voltage mapping is the important component part of the direct reconstruction algorithm of electricity tomography.For the sensor that has N electrode, can apply Law of Inner Product and construct current-voltage mapping.But also do not shine upon to construct at present the direct building method of current-voltage mapping according to voltage-to-current.

Summary of the invention

The object of the invention is to propose a kind of direct building method that shines upon to construct current-voltage mapping according to voltage-to-current.

Technical scheme of the present invention is:

Step 1, the current density having on the electrode of sensor of N electrode have general expression

being write as matrix form has:

$J=\frac{1}{A}C\·V---\left(1\right)$

Wherein J is current density vector, the internal surface area that A is each electrode, and C is capacitance matrix, and V is voltage vector, and expression formula is:

$C=\left[\begin{array}{ccccc}{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{1,1}}& -{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{1,2}}& -{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">\; <figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>\; </figure-callout>}}_{\mathrm{1,3}}& ...& -{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">C</figure-callout>}}_{1,N}\\ -{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{2,1}}& {\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{2,2}}& -{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">\; <figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>\; </figure-callout>}}_{\mathrm{2,3}}& ...& -{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{2,N}\\ -{\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{3,1}}& -{\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{3,2}& {\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">\; <figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>\; </figure-callout>}}_{\mathrm{3,3}}& ...& -{\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{3,N}\\ .& .& .& .& .\\ .& .& .& .& .\\ .& .& .& .& .\\ -C_{N,1}& -C_{N,2}& -C_{N,3}& ...& C_{N,N}\end{array}\right],$ $\left[\begin{array}{c}{j}_1\\ j_2\\ {\mathrm{<figure-callout\; id="3"\; label="j"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">j</figure-callout>}}_3\\ .\\ .\\ .\\ j_N\end{array}\right],$ $V=\left[\begin{array}{c}{V}_1\\ V_2\\ {\mathrm{<figure-callout\; id="3"\; label="V"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">V</figure-callout>}}_3\\ .\\ .\\ .\\ V_N\end{array}\right]---\left(2\right)$

J _sthe current density on the individual electrode inside surface of s (1≤s≤N), V _tt the magnitude of voltage on electrode, C _s,tit is the capacitance between electrode s and electrode t.Self-capacitance C _s,sbe defined as the electric capacity summation between s electrode and other N-1 electrode,

$C_{s,s}=\underset{t\&NotEqual;s}{\overset{N}{\underset{t=1}{\mathrm{\&Sigma;}}}}{C}_{s,t}---\left(3\right)$

So have

${\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}=\frac{1}{A}C---\left(4\right)$

being number of poles is N, voltage-to-current mapping matrix when specific inductive capacity is distributed as ε (z).

Known according to Kirchhoff's law, loop current sum is zero, and the order of current density vector J is N-1, and the order of Matrix C is also N-1, and the order that can draw thus voltage vector is also N-1, therefore order

So

$V=R_{\mathrm{\&epsiv;}}^{N\&times;N}J=R_{\mathrm{\&epsiv;}}^{N\&times;N}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}V---\left(6\right)$

Wherein

being number of poles is N, current-voltage mapping matrix when specific inductive capacity is distributed as ε (z).

Front N-1 the eigenwert that can prove V is-1, and last eigenwert is 0.By calculating eigenwert and the proper vector of V, can obtain:

V=P{diag([-1-1…-10]) _N×N}P ^T (7)

Wherein diag () _{n × N}represent the diagonal matrix on N rank, P ^tp=I, P=[p ₁p ₂p _n-1p _n].Defined feature vector p _ii (1≤i≤N-1) row of P, i.e. Vp _i=(1) p _i, p _neigenwert 0 characteristic of correspondence vector, i.e. Vp _n=(0) p _n.Because the mean value of every row in equation (5) is all 0, therefore

$p_N=\frac{1}{\sqrt{\mathrm{<figure-callout\; id="1"\; label="N"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">N</figure-callout>}}}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\&times;N}^T,$ So V can be written as:

$V=-I_{N\&times;N}+p_N{p}_N^T---\left(8\right)$

Wherein I _{n × N}it is the unit matrix on N rank.

Similarly, by computation of characteristic values and proper vector, order be N-1, can be written as following form:

${\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}={\mathrm{Q\&Sigma;}}_{\mathrm{\&Lambda;}}{Q}^T=Q\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}{\mathrm{\&lambda;}}_1& {\mathrm{\&lambda;}}_2& ...& {\mathrm{\&lambda;}}_{N-1}& 0\end{array}\right]\right)}_{N\&times;N}\right\}{Q}^T---\left(9\right)$

Wherein Σ _Λ=diag ([λ ₁λ ₂λ _n-10] _{n × N}) by N the diagonal matrix that eigenwert forms of R, the matrix that Q is made up of corresponding proper vector.Q ^tq=I, Q=[q ₁q ₂q _n-1q _n], proper vector q _ii (1≤i≤N-1) row of Q, i.e. Rq _i=λ _iq _i, q _n proper vector 0 characteristic of correspondence vector, i.e. Rq _n=(0) q _n, $p_N=\frac{1}{\sqrt{\mathrm{<figure-callout\; id="1"\; label="N"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">N</figure-callout>}}}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\&times;N}^T,$ Therefore have

$Q^T\mathrm{VQ}=Q^T(-I_{N\&times;N}+p_N{p_N}^T)Q$ $=-I_{N\&times;N}+Q^T\left(q_N{q_N}^T\right)Q---\left(10\right)$

$=\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\&times;N}$

Step 2, formula (7) substitution formula (6) is had:

$P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\&times;N}\right\}{P}^T=R_{\mathrm{\&epsiv;}}^{N\&times;N}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\&times;N}\right\}{P}^T$

(11)

Both members is premultiplication P simultaneously ^t, the right side takes advantage of P to obtain:

$\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\&times;N}=P^T{R}_{\mathrm{\&epsiv;}}^{N\&times;N}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\&times;N}\right\}$

(12)

The matrix that a Matrix Multiplication obtains take diagonal matrix if easily know is still diagonal matrix, and this matrix is also diagonal matrix.So known:

$P^T{R}_{\mathrm{\&epsiv;}}^{N\&times;N}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}1& 1& ...& 1& 0\end{array}\right]\right)}_{N\&times;N}\right\}---\left(13\right)$

At formula (13) premultiplication P, P is taken advantage of on the right side again ^t:

$P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}1& 1& ...& 1& 0\end{array}\right]\right)}_{N\&times;N}\right\}{P}^T=R_{\mathrm{\&epsiv;}}^{N\&times;N}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}---\left(14\right)$

Formula (7) and formula (9) substitution formula (14) are obtained

$-V=R_{\mathrm{\&epsiv;}}^{N\&times;N}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}=R_{\mathrm{\&epsiv;}}^{N\&times;N}Q{\mathrm{\&Sigma;}}_{\mathrm{\&Lambda;}}{Q}^T---\left(15\right)$

At formula (15) premultiplication Q ^t, the right side takes advantage of Q to obtain:

$-Q^T\mathrm{VQ}=Q^T{R}_{\mathrm{\&epsiv;}}^{N\&times;N}Q{\mathrm{\&Sigma;}}_{\mathrm{\&Lambda;}}---\left(16\right)$

By formula (10) substitution formula (16) and get final product:

$-\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\&times;N}={Q^{T}R}_{\mathrm{\&epsiv;}}^{N\&times;N}\mathrm{Qdiag}{\left(\left[\begin{array}{ccccc}{\mathrm{\&lambda;}}_1& {\mathrm{\&lambda;}}_2& ...& {\mathrm{\&lambda;}}_{N-1}& 0\end{array}\right]\right)}_{N\&times;N}---\left(17\right)$

So have:

$\mathrm{diag}{\left(\left[\begin{array}{ccccc}\frac{1}{{\mathrm{\&lambda;}}_1}& \frac{1}{{\mathrm{\&lambda;}}_2}& ...& \frac{1}{{\mathrm{\&lambda;}}_{N-1}}& 0\end{array}\right]\right)}_{N\&times;N}{Q}^T{R}_{\mathrm{\&epsiv;}}^{N\&times;N}Q---\left(18\right)$

At formula (18) premultiplication Q, Q is taken advantage of on the right side ^t:

$R_{\mathrm{\&epsiv;}}^{N\&times;N}=Q\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}\frac{1}{{\mathrm{\&lambda;}}_1}& \frac{1}{{\mathrm{\&lambda;}}_2}& ...& \frac{1}{{\mathrm{\&lambda;}}_{N-1}}& 0\end{array}\right]\right)}_{N\&times;N}\right\}{Q}^T---\left(19\right)$

Formula is known thus, matrix can be determined by unique.

Further, with reference to this method, any orthogonal set of current excitation pattern may be used to calculate current-voltage mapping.

Accompanying drawing explanation

Fig. 1 is implementing procedure figure.

Fig. 2 is embodiment isoboles.

Embodiment

Referring to Fig. 1, a kind of building method algorithm block diagram that shines upon to construct current-voltage mapping according to voltage-to-current.As example, the embodiment of this method is described take 16 end to end ring resistance networks shown in Fig. 2.

Said method comprising the steps of:

Step 1, the current density having on the electrode of sensor of N=16 electrode have general expression

being write as matrix form has:

$J=\frac{1}{A}C\&CenterDot;V---\left(20\right)$

Wherein J is current density vector, the internal surface area that A is each electrode, and C is capacitance matrix, and V is voltage vector, and expression formula is:

$C=\left[\begin{array}{ccccc}{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{1,1}}& -{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{1,2}}& -{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">\; <figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>\; </figure-callout>}}_{\mathrm{1,3}}& ...& -{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HDA0000466964820000011.png"\; state="\{\{state\}\}">C</figure-callout>}}_{1,16}\\ -{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{2,1}}& {\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{2,2}}& -{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">\; <figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>\; </figure-callout>}}_{\mathrm{2,3}}& ...& -{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{2,16}\\ -{\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{3,1}}& -{\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{3,2}& {\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">\; <figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>\; </figure-callout>}}_{\mathrm{3,3}}& ...& -{\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">\; <figure-callout\; id="16"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>\; </figure-callout>}}_{3,16}\\ .& .& .& .& .\\ .& .& .& .& .\\ .& .& .& .& .\\ -{\mathrm{<figure-callout\; id="16"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{16,1}& -{\mathrm{<figure-callout\; id="16"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>}}_{16,2}& -{\mathrm{<figure-callout\; id="16"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">\; <figure-callout\; id="3"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>\; </figure-callout>}}_{16,3}& ...& {\mathrm{<figure-callout\; id="16"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">\; <figure-callout\; id="16"\; label="C"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">C</figure-callout>\; </figure-callout>}}_{16,16}\end{array}\right],$ $\left[\begin{array}{c}{j}_1\\ j_2\\ {\mathrm{<figure-callout\; id="3"\; label="j"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">j</figure-callout>}}_3\\ .\\ .\\ .\\ {\mathrm{<figure-callout\; id="16"\; label="j"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">j</figure-callout>}}_{16}\end{array}\right],$ $V=\left[\begin{array}{c}{V}_1\\ V_2\\ {\mathrm{<figure-callout\; id="3"\; label="V"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">V</figure-callout>}}_3\\ .\\ .\\ .\\ V_{16}\end{array}\right]---\left(21\right)$

J _sthe current density on the individual electrode inside surface of s (1≤s≤16), V _tt the magnitude of voltage on electrode, C _s,tit is the capacitance between electrode s and electrode t.Self-capacitance C _s,sbe defined as the electric capacity summation between s electrode and other 15 electrodes,

$C_{s,s}=\underset{t\&NotEqual;s}{\overset{N}{\underset{t=1}{\mathrm{\&Sigma;}}}}{C}_{s,t}---\left(22\right)$

So have

${\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{16\&times;16}=\frac{1}{A}C---\left(23\right)$

Known according to Kirchhoff's law, loop current sum is zero, and the order of current density vector J is 15, and the order of Matrix C is also 15, and the order that can draw thus voltage vector is also 15, therefore order

So

$V=R_{\mathrm{\&epsiv;}}^{16\&times;16}J=R_{\mathrm{\&epsiv;}}^{16\&times;16}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{16\&times;16}V---\left(25\right)$

Front 15 eigenwerts that can prove V are-1, and last eigenwert is 0.By calculating eigenwert and the proper vector of V, can obtain:

V=P{diag([-1-1…-10]) _16×16}P ^T (26)

Wherein diag () _{16 × 16}represent the diagonal matrix on 16 rank, P ^tp=I, P=[p ₁p ₂p ₁₅p ₁₆].Defined feature vector p _ii (1≤i≤15) row of P, i.e. Vp _i=(1) p _i, p ₁₆eigenwert 0 characteristic of correspondence vector, i.e. Vp ₁₆=(0) p ₁₆.Because the mean value of every a line in equation (24) is all 0, therefore ${\mathrm{<figure-callout\; id="16"\; label="p"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_{16}=\frac{1}{4}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\&times;16}^T,$

So V can be written as:

$V=-I_{16\&times;16}+p_{16}{p}_{16}^T---\left(27\right)$

Wherein I _{16 × 16}it is the unit matrix on 16 rank.

Similarly, by computation of characteristic values and proper vector,

order be 15, can be written as following form:

${\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{16\&times;16}={\mathrm{Q\&Sigma;}}_{\mathrm{\&Lambda;}}{Q}^T=Q\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}{\mathrm{\&lambda;}}_1& {\mathrm{\&lambda;}}_2& ...& {\mathrm{\&lambda;}}_{15}& 0\end{array}\right]\right)}_{16\&times;16}\right\}{Q}^T---\left(28\right)$

Wherein Σ _Λ=diag ([λ ₁λ ₂λ ₁₅0] _{16 × 16}) by 16 diagonal matrix that eigenwert forms of R, the matrix that Q is made up of corresponding proper vector.Q ^tq=I, Q=[q ₁q ₂q ₁₅q ₁₆], proper vector q _ii (1≤i≤15) row of Q, i.e. Rq _i=λ _iq _i, q ₁₆ proper vector 0 characteristic of correspondence vector, i.e. Rq ₁₆=(0) q ₁₆,

${\mathrm{<figure-callout\; id="16"\; label="q"\; filenames="HDA0000466964820000012.png"\; state="\{\{state\}\}">q</figure-callout>}}_{16}=\frac{1}{4}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\&times;16}^T.$

From formula (24), in ring resistance network, can be calculated current matrix:

Hence one can see that, and voltage-to-current mapping matrix is

Calculate according to formula (28)

(31)

Step 2, formula (26) substitution formula (25) is had:

$P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{16\&times;16}\right\}{P}^T=R_{\mathrm{\&epsiv;}}^{16\&times;16}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{16\&times;16}P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{16\&times;16}\right\}{P}^T$

(33)

Obtain

$R_{\mathrm{\&epsiv;}}^{16\&times;16}=\mathrm{Qdiag}{\left(\left[\begin{array}{ccccc}\frac{1}{{\mathrm{\&lambda;}}_1}& \frac{1}{{\mathrm{\&lambda;}}_2}& ...& \frac{1}{{\mathrm{\&lambda;}}_{15}}& 0\end{array}\right]\right)}_{16\&times;16}{Q}^T$

$=\mathrm{Qdiag}{\left(\left[\begin{array}{ccccc}\frac{1}{0.5+0.2071\mathrm{i\&lambda;}}& \frac{1}{0.5-0.2071i}& ...& \frac{1}{0.5-2.5137i}& 0\end{array}\right]\right)}_{16\&times;16}{Q}^T---\left(34\right)$

Formula is known thus, matrix

can be determined by unique.

For proving this conclusion, do following checking: under adjacent incentive mode, calculate node potential matrix

Be consistent with the V setting, can verify thus this method.

A kind of described method by the mapping of voltage-to-current structure current-voltage, has provided a kind of computing method that current-voltage shines upon, the method explicit physical meaning, simple.With reference to this method, any orthogonal set of excitation measurement pattern may be used to calculate current-voltage mapping.

Description to the present invention and embodiment thereof, is not limited to this above, is only one of embodiments of the present invention shown in accompanying drawing.In the situation that not departing from the invention aim, design and the similar structure of this technical scheme or embodiment without creating, all belong to protection domain of the present invention.

Claims (1)

1. a direct building method that shines upon to construct current-voltage mapping according to voltage-to-current, is characterized in that, the method comprises the steps:

Step 1, the current density having on the each electrode of sensor of N electrode have general expression

so voltage-to-current mapping matrix while having specific inductive capacity to be distributed as ε (z):

${\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}=\frac{1}{A}C---\left(1\right)$

Known according to Kirchhoff's law, loop current sum is zero, and the order of current density vector J is N-1, and the order of Matrix C is also N-1, and the order that can draw thus voltage vector is also N-1, therefore order

So

$V=R_{\mathrm{\&epsiv;}}^{N\&times;N}J=R_{\mathrm{\&epsiv;}}^{N\&times;N}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}V---\left(3\right)$

Eigenwert and the proper vector of calculating V, can obtain:

V=P{diag([-1-1…-10]) _N×N}P ^T (4)

Wherein diag () _{n × N}represent the diagonal matrix on N rank, P ^tp=I, P=[p ₁p ₂p _n-1p _n], proper vector p _ii (1≤i≤n-1) row of P, i.e. Vp _i=(1) p _i, p _neigenwert 0 characteristic of correspondence vector,

Be Vp _n=(0) p _n, $p_N=\frac{1}{\sqrt{N}}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\&times;N}^T,$ Be that V can be written as:

$V=-I_{N\&times;N}+p_N{p}_N^T---\left(5\right)$

Similarly, by computation of characteristic values and proper vector,

order be N-1, can be written as following form:

${\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}={\mathrm{Q\&Sigma;}}_{\mathrm{\&Lambda;}}{Q}^T=Q\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}{\mathrm{\&lambda;}}_1& {\mathrm{\&lambda;}}_2& ...& {\mathrm{\&lambda;}}_{N-1}& 0\end{array}\right]\right)}_{N\&times;N}\right\}{Q}^T---\left(6\right)$

Wherein Σ _Λ=diag ([λ ₁λ ₂λ _n-10] _{n × N}) by N the diagonal matrix that eigenwert forms of R, the matrix that Q is made up of corresponding proper vector, Q ^tq=I, Q=[q ₁q ₂q _n-1q _n], proper vector q _ibe

I (1≤i≤N-1) row of Q, i.e. Rq _i=λ _iq _i, q _nproper vector 0 characteristic of correspondence vector,

Rq _N=(0)q _N， $p_N=\frac{1}{\sqrt{N}}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\&times;N}^T,$ So have

$Q^T\mathrm{VQ}=Q^T(-I_{N\&times;N}+p_N{p_N}^T)Q$

$=-I_{N\&times;N}+Q^T\left(q_N{q_N}^T\right)Q---\left(7\right)$

$=\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\&times;N}$

Step 2, formula (4) substitution formula (3) is had:

$P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\&times;N}\right\}{P}^T=R_{\mathrm{\&epsiv;}}^{N\&times;N}{\mathrm{\&Lambda;}}_{\mathrm{\&epsiv;}}^{N\&times;N}P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\&times;N}\right\}{P}^T$

(8)

, through deriving, current-voltage shines upon corresponding matrix

can be written as:

$R_{\mathrm{\&epsiv;}}^{N\&times;N}=\mathrm{Qdiag}{\left(\left[\begin{array}{ccccc}\frac{1}{{\mathrm{\&lambda;}}_1}& \frac{1}{{\mathrm{\&lambda;}}_2}& ...& \frac{1}{{\mathrm{\&lambda;}}_{N-1}}& 0\end{array}\right]\right)}_{N\&times;N}{Q}^T---\left(9\right)$

Formula is known thus, matrix

can be determined by unique.

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