CN107091858A - A kind of method mapped by Current Voltage map construction voltage x current - Google Patents
A kind of method mapped by Current Voltage map construction voltage x current Download PDFInfo
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- CN107091858A CN107091858A CN201710187323.XA CN201710187323A CN107091858A CN 107091858 A CN107091858 A CN 107091858A CN 201710187323 A CN201710187323 A CN 201710187323A CN 107091858 A CN107091858 A CN 107091858A
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Abstract
The present invention relates to a kind of method mapped by Current Voltage map construction voltage x current, for having the sensor of N number of electrode in electricity tomography, the parameters such as characteristic value and characteristic vector by calculating current voltage matrix, by the formula derived, voltage x current matrix uniquely is calculated, voltage x current mapping is constructed.The present invention gives the direct building method of voltage x current mapping, it can be applied to electricity chromatography imaging field and directly reconstruct in algorithm, its explicit physical meaning, and it is simple and easy to apply.
Description
Technical field
Imaging field is chromatographed the present invention relates to electricity, more particularly to one kind is reflected by current-voltage map construction voltage-to-current
The method penetrated.
Background technology
Electricity tomography (Electrical Tomography, abbreviation ET) technology is one kind of chromatography imaging technique.Its
Encouraged by applying to testee, and detect the change of its boundary value, rebuild using specific algorithm for reconstructing in measurand
The distribution of portion's electrical characteristic parameter, so as to obtain the distribution situation of interior of articles.Compared with other chromatography imaging techniques, electricity chromatography
Imaging is with the advantage such as radiationless, Noninvasive, portability, fast response time, cheap.
Electricity tomographic reconstruction algorithm generally can be divided into two classes：Algorithm for reconstructing based on sensitivity matrix and directly reconstruct
Algorithm.Using the former it is generally necessary to solve ill linear equation, this means that must be while rebuild all pixels of measured zone
Value.The latter realizes that the gray value of each pixel can be straight by calculating current-voltage mapping or voltage-to-current mapping
Connect, independent calculating.
The building method of voltage-to-current mapping is the important component that electricity tomography directly reconstructs algorithm.For N
The sensor of individual electrode, the data that voltage-to-current mapping is generally obtained by adjacent actuators measurement pattern are rebuild, because this
Measurement pattern is easier the realization on hardware, and the voltage-to-current map construction method based on adjacent actuators measurement pattern can give
Go out the direct physical significance of voltage-to-current mapping.But there is presently no construct voltage-to-current according to current-voltage mapping
The method of mapping.
The content of the invention
It is an object of the invention to propose a kind of direct structure that voltage-to-current mapping is constructed according to current-voltage mapping
Make method.
The technical scheme is that：
Step 1: the voltage on the electrode of the sensor with N number of electrode has general expression V=RAj, write as matrix
Form has：
V=ARJ (1)
Wherein J is current density vector, and A is the internal surface area of each electrode, and R is resistor matrix, and V is voltage vector, table
It is up to formula：
jsIt is the current density on s (1≤s≤N) individual electrode inner surface, VtIt is the magnitude of voltage on t-th of electrode,Table
Show that unitary current flows into tested field domain from i-th electrode, during j-th of electrode outflow, i-th and j electrode and s and t electrode
Between mutual impedance.
Then have：
As number of poles is N, and dielectric constant is distributed as current-voltage mapping matrix during ε (z).
It can be seen from Kirchhoff's law, loop current sum is zero, i.e., current density vector J order is N-1, therefore
Order：
Then：
WhereinAs number of poles is N, and dielectric constant is distributed as voltage-to-current mapping matrix during ε (z).
Definition：
So have：
BTSubscript T represent transposition to matrix, B can be provedTJ preceding N-1 characteristic value is -1, and last
Characteristic value is 0.By calculating BTJ characteristic value and characteristic vector, can be obtained：
BTJ=P { diag ([- 1-1 ...-1 0])N×N}PT (8)
Wherein diag ()N×NRepresent the diagonal matrix of N ranks, PTP=I, P=[p1 p2 … pN-1 pN].Defined feature to
Measure piIt is P i-th (1≤i≤N-1) row, i.e. BTJ·pi=(- 1) pi, pNIt is the corresponding characteristic vector of characteristic value 0, i.e. BTJ·pN
=(0) pN.Because average value often capable in equation (7) is all 0, thereforeSo BTJ can be with
It is written as：
Wherein IN×NIt is N rank unit matrixs.
Similar, by calculating characteristic value and characteristic vector,Order be N-1, can be written as following form：
∑R=diag ([λ1 λ2 … λN-1 0])N×NBe byN number of eigenvalue cluster into diagonal matrix, Q is by phase
The matrix of corresponding characteristic vector composition.QTQ=I, Q=[q1 q2 … qN-1 qN], characteristic vector qiRepresent Q i-th (1≤i≤
N-1) arrange, i.e.,qNIt is the corresponding characteristic vector of characteristic value 0, i.e.,
Therefore have：
Step 2: BTOrder be N, formula (8) both members simultaneously premultiplication (BT)-1：
J=(BT)-1P{diag([-1 -1 … -1 0])N×N}PT (12)
Formula (12) is substituted into formula (8) to obtain：
Both members multiply P, premultiplication B in the right side simultaneouslyT, then premultiplication PT：
If being apparent from matrix that a Matrix Multiplication obtains with diagonal matrix still for diagonal matrix, the matrix is also to angular moment
Battle array.Then understand：
Both members are while premultiplication P, the right side multiplies PT：
Formula (8) and formula (10) are substituted into formula (16) to obtain：
Both members are while premultiplication (BT)-1, the right side multiplies BT：
And BTJ=JBT, then：
Both members are while premultiplication QT, the right side multiplies Q and obtained：
Formula (12) is substituted into formula (20) to obtain：
Then have：
Both members are while premultiplication Q, the right side multiplies QT：
Thus formula is understood, matrixIt can be now uniquely determined.
Further, with reference to this method, any orthogonal set of current excitation pattern is shared in calculating voltage-to-current mapping.
Brief description of the drawings
Fig. 1 is implementing procedure figure.
Fig. 2 is embodiment isoboles.
Embodiment
Participate in Fig. 1, a kind of building method algorithm block diagram mapped according to current-voltage map construction voltage-to-current.To scheme
Exemplified by 16 end to end ring resistance networks shown in 2, illustrate the embodiment of this method.
It the described method comprises the following steps：
Step 1: the voltage on the electrode of the sensor with N=16 electrode has general expression V=RAj, write as
Matrix form has：
V=ARJ (24)
Wherein J is current density vector, and A is the internal surface area of each electrode, and R is resistor matrix, and V is voltage vector, table
It is up to formula：
jsIt is the current density on the individual electrode inner surfaces of s (1≤s≤16),Represent unitary current from i-th of electrode
Tested field domain is flowed into, during j-th of electrode outflow, the mutual impedance between i-th and j electrode and s and t electrode.
Then have：
As number of poles is 16, and dielectric constant is distributed as current-voltage mapping matrix during ε (z).
It can be seen from Kirchhoff's law, loop current sum is zero, i.e., current density vector J order is 15, therefore order：
Then：
WhereinAs number of poles is 16, and dielectric constant is distributed as voltage-to-current mapping matrix during ε (z).
Order：
So：
B can be provedTJ preceding 15 characteristic values are -1, and last characteristic value is 0.By calculating BTJ feature
Value and characteristic vector, can be obtained：
BTJ=P { diag [- 1-1 ...-1 0]16×16}PT (31)
Wherein diag ()16×16Represent the diagonal matrix of 16 ranks, PTP=I, P=[p1 p2 … p15 p16].Defined feature
Vectorial piIt is P i-th (1≤i≤15) row, i.e. BTJ·pi=(- 1) pi, p16It is the corresponding characteristic vector of characteristic value 0, i.e. BTJ·
p16=(0) p16.Because average value often capable in equation (30) is all 0, thereforeSo BTJ can
To be written as：
Wherein I16×16It is 16 rank unit matrixs.
Similar, by calculating characteristic value and characteristic vector,Order be 15, can be written as following form：
∑R=diag ([λ1 λ2 … λ15 0])16×16Be by16 eigenvalue clusters into diagonal matrix, Q be by
The matrix of corresponding characteristic vector composition.QTQ=I, Q=[q1 q2 … q15 q16], characteristic vector qiRepresent Q i-th (1≤
I≤15) arrange, i.e.,q16It is the corresponding characteristic vector of characteristic value 0, i.e.,
In ring resistance network, voltage matrix can be calculated：
It follows that current-voltage mapping matrix is：
Calculated according to formula (33)：
Obtain：
Thus formula is understood, matrixIt can be now uniquely determined.
To prove this conclusion, following checking is done：Under neighboring modes, current density matrix is calculated：
I with setting is consistent, thus can verify that this method.
A kind of described method that voltage-to-current mapping is constructed by current-voltage, gives the one of voltage-to-current mapping
Computational methods are planted, this method explicit physical meaning is simple and easy to apply.With reference to this method, any orthogonal set of measurement pattern is encouraged
It may be used to calculate voltage-to-current mapping.
Above to the description of the present invention and embodiments thereof, it is not limited to which this, is only the reality of the present invention shown in accompanying drawing
Apply one of mode.Without departing from the spirit of the invention, it is similar with the technical scheme without designing with creating
Structure or embodiment, belong to the scope of the present invention.
Claims (1)
1. a kind of direct building method this method that voltage-to-current mapping is constructed according to current-voltage mapping includes following steps
Suddenly：
Step 1: the voltage on the electrode of the sensor with N number of electrode has general expression V=RAj, dielectric constant distribution
Current-voltage mapping matrix during for ε (z)：
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Definition：
So have：
Calculate BTJ characteristic value and characteristic vector, can be obtained：
BTJ=P { diag ([- 1-1 ...-1 0])N×N}PT (6)
Wherein diag ()N×NRepresent the diagonal matrix of N ranks, PTP=I, P=[p1 p2 … pN-1 pN], defined feature vector pi
It is P i-th (1≤i≤N-1) row, i.e. BTJ·pi=(- 1) pi, pNIt is the corresponding characteristic vector of characteristic value 0, i.e. BTJ·pN=
(0)pN,BTJ can be written as：
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I.e.qNIt is the corresponding characteristic vector of characteristic value 0, i.e.,
Therefore have：
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Formula (10) is substituted into formula (3) to obtain：
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Thus formula is understood, matrixIt can be now uniquely determined.
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