CN113033059B - Method for calculating radiation induced current of bent cable - Google Patents

Method for calculating radiation induced current of bent cable Download PDF

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CN113033059B
CN113033059B CN202110429232.9A CN202110429232A CN113033059B CN 113033059 B CN113033059 B CN 113033059B CN 202110429232 A CN202110429232 A CN 202110429232A CN 113033059 B CN113033059 B CN 113033059B
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cable
segment
representing
current
load
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CN113033059A (en
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杜平安
杨金升
魏思琦
潘泽宇
聂宝林
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a method for calculating radiation induced current of a bent cable. According to the method, the space arbitrary bent cable is uniformly segmented according to an Agrawal model, induced currents generated by an on-line distributed voltage source and lumped voltage sources at two ends at a load are solved by utilizing a chain parameter matrix respectively, and finally the induced currents are superposed to obtain the total induced current at the load. The method of the invention fits the actual space bending cable by segmenting the cable, and the fitting precision is very high because the length of each segment of the cable segment is far less than the wavelength of the incident plane wave; meanwhile, the method is based on the Agrawal transmission line theory, incident plane waves are simplified into voltage sources in the circuit, analysis of a field line coupling physical model is simplified, a grid division process of a numerical calculation method in space is avoided, the calculation process is simplified, less calculation resources are occupied, and the calculation efficiency is higher.

Description

Method for calculating radiation induced current of bent cable
Technical Field
The invention belongs to the technical field of electromagnetic interference prediction, and particularly relates to a method for calculating field line coupling induction current of a bent cable.
Background
With the continuous development of modern science and technology, the electromagnetic environment in which electronic equipment is located is increasingly severe, and the attention of people is more and more drawn to the cable as a weak link in the anti-electromagnetic interference design. Interconnection cables are important components of connecting electronic devices and systems, and have a significant impact on electromagnetic interference and electromagnetic compatibility issues of electromechanical devices. Under the irradiation effect of external electromagnetic field, the cable can become the electromagnetic wave receiving antenna that efficiency is very high, transmits external electromagnetic energy to sensitive equipment through the mode of field line coupling, arouses unnecessary electromagnetic interference phenomenon.
At present, the field line coupling system is analyzed by mainly utilizing simulation software based on numerical methods such as a finite element method, a time domain finite difference method, a moment method and the like to perform modeling calculation, although the method is high in calculation precision, the method is long in consumed time and large in consumed calculation resource, and for actual space bent cable calculation in engineering, the method usually needs to occupy a large amount of calculation time and memory resource.
Disclosure of Invention
Aiming at the technical problems in the background art, the invention provides a method for calculating the irradiation induced current of a bent cable.
The specific technical scheme of the invention is as follows: a method for calculating irradiation induced current of a bent cable comprises the following steps:
s1, acquiring a parameter equation of a space bent cable;
the parametric equation along the x-direction is:
Figure BDA0003030742410000011
wherein hy (x) represents a function of horizontal offset distance as a function of x-coordinate, hz (x) represents a function of vertical offset distance as a function of x-coordinate;
s2, segmenting the space bent cable, and representing a voltage-current relation by using a chain parameter matrix, wherein the method specifically comprises the following steps:
the spatially arbitrary bent cable is scattered into a plurality of very short uniform small segments, the length of each segment along the x direction is delta L, the non-uniform transmission line is regarded as the series connection of m segments of uniform units, the length delta L of each segment is less than lambda, lambda is the incident wavelength, and simultaneously, the requirement that m delta L is equal to L is met, wherein L represents the total length of the cable in the x direction;
the current-voltage relationship between the initial end and the load end of the spatial arbitrary bending cable can be represented by a chain parameter matrix:
Figure BDA0003030742410000021
wherein, VLRepresenting the terminal voltage of the load, ILRepresenting the current at the load side, V0Denotes the initial terminal voltage, I0Represents the initial end current, phinA chain parameter matrix representing the nth wire;
s3, calculating distribution parameters and the characteristic impedance of each section of line element,
the voltage and current across any nth segment on the line can be expressed as:
Figure BDA0003030742410000022
wherein, VnIs the voltage of the nth segment, InIs the current of the nth segment, Vn-1Is the voltage of the n-1 th segment, In-1Is the current of the (n-1) th segment;
chain parameter matrix phi of nth section of cablenComprises the following steps:
Figure BDA0003030742410000023
wherein the content of the first and second substances,
Figure BDA0003030742410000024
representing the propagation constant of the transmission line;
Figure BDA0003030742410000025
μ0denotes the permeability of air,. epsilon.denotes the dielectric constant of air,. omega.denotes the angular frequency, Zc(n) represents the characteristic impedance of the nth segment of the cable, hz (l)n) The ground clearance of the center of the nth section of cable is shown, and a represents the radius of the conductor of the cable;
step S4, according to the Agrawal model, it can be known that the incident plane wave will generate a distributed excitation voltage source V on the cablesAnd an initial terminal lumped voltage source V1And a load side lumped voltage source V2；
Distributed voltage source VsThe excitation source is formed by superposing an incident wave electric field and a ground reflected wave electric field along the tangential direction of a cable, and the normalized direction vector of the cable can be expressed as follows:
Figure BDA0003030742410000026
wherein k isyAnd kzIs the slope of the conductor in the plane xoy and xoz;
the incident electric fields in the x, y, z directions are:
Figure BDA0003030742410000031
wherein alpha represents the included angle between the polarization direction of the incident electric field and the vertical direction, psi represents the included angle between the incident direction of the electric field and the horizontal plane, phi represents the included angle between the incident direction of the electric field and the plane where the cable and the load are located, k represents the wave number,
Figure BDA0003030742410000032
the magnitude of the incident electric field at the transmission line is:
Figure BDA0003030742410000033
wherein the content of the first and second substances,
Figure BDA0003030742410000034
electric field vector representing incident plane wave, E0Representing the magnitude of the electric field of the incident plane wave,
Figure BDA0003030742410000035
representing the direction vector of the cable, ExDenotes the x-direction component of the incident electric field, EyDenotes the y-direction component of the incident electric field, EzRepresenting the z-direction component of the incident electric field, elxRepresenting the x-direction component of the cable direction vector, elyRepresenting the y-direction component of the cable direction vector, elzA z-direction component representing a cable direction vector;
electric field E of the reflected wave at the transmission linerefComprises the following steps:
Figure BDA0003030742410000036
then the voltage source V is distributedsComprises the following steps:
Figure BDA0003030742410000037
initial lumped voltage source V1Comprises the following steps:
Figure BDA0003030742410000038
where h represents the height of the cable end from the ground.
Load end lumped voltage source V2Comprises the following steps:
V2＝V1e-jkLcosψcosφ
step 5, calculating and superposing the induced currents generated by the three excitation sources at the load end,
(1) distributed voltage source VsUnder the action of the current I generated at the load end1(L) the calculation procedure is as follows:
distributed voltage source VsPosition x axis coordinate atn-1、lnThe visual impedances from the excitation source to both sides are respectively:
Figure BDA0003030742410000041
Figure BDA0003030742410000042
wherein x issRepresenting a distributed voltage source VsX-axis coordinate of (1), ZLeq(n) is represented by the abscissa x ═ l of the cablen-1Equivalent impedance seen from the initial end, ZReq(n) is selected from x ═ lnThe equivalent impedance seen from the load end has the following recursive relationship:
Figure BDA0003030742410000043
Figure BDA0003030742410000044
wherein, Zc(n-1) represents the characteristic impedance of the n-1 th cable segment, Zc(n +1) represents the characteristic impedance of the (n +1) th cable segment, ZLeq(n-1) represents the equivalent impedance of the (n-1) th cable section as seen to the initial end, ZReq(n +1) represents the equivalent impedance of the (n +1) th cable section as seen from the load side, γn-1Denotes the propagation constant, γ, of the n-1 th cablen+1Representing the propagation constant of the n +1 th segment of the cable.
The initial conditions of the above recursion formula can be found by analyzing the boundary conditions of the first segment and the last segment as follows:
Figure BDA0003030742410000045
Figure BDA0003030742410000046
wherein Z isLeq(1) Represents the equivalent impedance, Z, of the 1 st cable section as seen from the initial endc(1) Representing the characteristic impedance, Z, of the 1 st cable segment1Representing the initial end load impedance, gamma1Denotes the propagation constant, Z, of the 1 st segment of the cableReq(m) represents the equivalent impedance of the last cable section as seen towards the load side, Zc(m) represents the characteristic impedance of the last cable segment, Z2Representing the load impedance, gamma, at the load sidemRepresenting the propagation constant of the last segment of the cable.
Distributed voltage source VsWhere x is xsThe current generated at this point is:
Figure BDA0003030742410000051
wherein the content of the first and second substances,
Figure BDA0003030742410000052
representing the equivalent impedance seen at the excitation source towards the initial end,
Figure BDA0003030742410000053
representing the equivalent impedance seen from the excitation source to the load end;
according to the chain parameter matrix:
Figure BDA0003030742410000054
wherein the content of the first and second substances,
Figure BDA0003030742410000055
V(xs) Is an equivalent impedance
Figure BDA0003030742410000056
Voltage of (V) (l)n) Is an equivalent impedance ZReq(n) voltage, I (l)n) Is an equivalent impedance ZReq(n) a current;
then the equivalent impedance ZReqCurrent I (l) at (n)n) Comprises the following steps:
Figure BDA0003030742410000057
the chain parameter matrix from segment n +1 to segment m is:
Figure BDA0003030742410000058
wherein, V (l)n)＝I(ln)ZReq(n)；
The current at the load is then: i isn(L)＝I(ln)(CZReq(n)+D)
Where C, D is the first and second elements in the second row of the chain parameter matrix from segment n +1 to segment m, then the total current generated by the distributed voltage source at the load is:
Figure BDA0003030742410000059
(2) lumped voltage source V1At x-0, load side current I2(L) the calculation procedure is as follows:
according to the chain parameter matrix:
Figure BDA00030307424100000510
wherein phi represents a cable global chain parameter matrix;
available total source V1The current at the load is:
Figure BDA0003030742410000061
wherein V (0) represents the initial terminal voltage, I (0) represents the initial terminal current, and phi21Being the first element of the second row of Φ, Φ22Is the second element of the second row in Φ;
(3) lumped voltage source V2At x ═ L, load side current I3(L) the calculation procedure is as follows:
the impedance seen from the left side of the excitation source is:
Figure BDA0003030742410000062
wherein Z isLeq(m) represents the equivalent impedance of the last cable section as seen towards the initial end;
collective source V2The current at the load is:
Figure BDA0003030742410000063
the final current at the load end is: i (L) ═ I1(L)+I2(L)+I3(L)。
The invention has the beneficial effects that: according to the method, the space arbitrary bent cable is uniformly segmented according to an Agrawal model, induced currents generated by an on-line distributed voltage source and lumped voltage sources at two ends at a load are solved by utilizing a chain parameter matrix respectively, and finally the total induced currents at the load are obtained by superposition. The method of the invention fits the actual space bending cable by segmenting the cable, and the fitting precision is very high because the length of each segment of the cable segment is far less than the wavelength of the incident plane wave; meanwhile, the method is based on the Agrawal transmission line theory, the incident plane waves are simplified into the voltage source in the circuit, the analysis of the field line coupling physical model is simplified, the grid division process of a numerical calculation method in space is avoided, the calculation process is simplified, less calculation resources are occupied, and the calculation efficiency is higher.
Drawings
Fig. 1 is a schematic flow chart of a method for calculating an irradiation induced current of a bent cable according to an embodiment of the present invention.
FIG. 2 is a schematic view of a spatially arbitrary curved cable segment according to an embodiment of the present invention.
Fig. 3 is a schematic diagram of an equivalent circuit of the Agrawal model according to the embodiment of the present invention.
FIG. 4 is a schematic plane wave incident diagram according to an embodiment of the present invention.
FIG. 5 is a schematic diagram of incident wave polarization angles according to an embodiment of the present invention.
FIG. 6 is a schematic diagram of a spatially curved cable distributed voltage source according to an embodiment of the invention.
FIG. 7 is a diagram of a lumped voltage source at the initial end according to an embodiment of the invention.
Fig. 8 is a schematic diagram of a lumped voltage source at the load end according to the embodiment of the invention.
Fig. 9 is a schematic diagram of a specific model according to an embodiment of the present invention.
Fig. 10 is a schematic diagram of load-side current obtained by specific calculation according to an embodiment of the present invention.
Detailed Description
The embodiments of the present invention will be further described with reference to the accompanying drawings.
As shown in fig. 1, the method for calculating the irradiation induced current of the bent cable provided by the invention comprises the following steps:
s1, acquiring a parameter equation of a space bent cable;
the parametric equation along the x-direction is:
Figure BDA0003030742410000071
where hy (x) represents the horizontal offset distance as a function of x-coordinate, and hz (x) represents the vertical offset distance as a function of x-coordinate.
Step s2, segmenting the space bending cable, specifically as shown in fig. 2, representing a voltage-current relationship by using a chain parameter matrix, specifically as follows:
the spatially arbitrary curved cable is discretized into a plurality of very short uniform small segments, each segment is delta L, the non-uniform transmission line is regarded as a series connection of m uniform units, the length delta L of each segment is less than lambda, and m delta L is satisfied, wherein L represents the total length of the cable in the x direction.
The current-voltage relationship between the initial end and the load end of the spatial arbitrary bending cable can be represented by a chain parameter matrix:
Figure BDA0003030742410000072
wherein, VLRepresenting the terminal voltage of the load, ILRepresenting the current at the load side, V0Denotes the initial terminal voltage, I0Represents the initial end current, phinA chain parameter matrix representing the nth segment of the cable.
S3, calculating distribution parameters and the characteristic impedance of each section of line element,
the voltage and current across any nth segment on the line can be expressed as:
Figure BDA0003030742410000073
wherein, VnIs the voltage of the nth segment, InIs the current of the nth segment, Vn-1Is the voltage of the n-1 th segment, In-1Is the current of the (n-1) th segment.
The chain parameter matrix of the nth segment is:
Figure BDA0003030742410000081
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003030742410000082
representing the propagation constant of the transmission line;
Figure BDA0003030742410000083
μ0denotes the permeability of air,. epsilon.denotes the dielectric constant of air,. omega.denotes the angular frequency, Zc(n) represents the characteristic impedance of the nth segment of the cable, hz (l)n) The ground clearance of the center of the nth section of cable is shown, and a represents the radius of the conductor of the cable;
step S4, according to the Agrawal model, the plane wave can generate a distributed excitation voltage source V on the cablesAnd an initial terminal lumped voltage source V1And a load side lumped voltage source V2As shown in fig. 3.
Respectively calculating expressions of a distributed voltage source and a lumped voltage source according to the spatial propagation and reflection characteristics of the electromagnetic waves: the distributed voltage source obtains a lumped voltage source as the integral of the incident plane wave and the reflected wave in the vertical direction through the projection of the incident plane wave and the reflected wave in the tangential direction of the cable in a superposition mode.
Distributed voltage source VsThe excitation source is formed by superposing an incident wave electric field and a ground reflected wave electric field along the tangential direction of a cable, and the normalized direction vector of the cable can be expressed as follows:
Figure BDA0003030742410000084
wherein k isyAnd kzIs the slope of the conductor in the plane xoy and xoz.
For the plane waves of the incident parameters shown in fig. 4 and 5, the incident electric fields in the x, y, and z directions are:
Figure BDA0003030742410000085
wherein alpha represents the included angle between the polarization direction of the incident electric field and the vertical direction, psi represents the included angle between the incident direction of the electric field and the horizontal plane, phi represents the included angle between the incident direction of the electric field and the plane where the cable and the load are located, k represents the wave number,
Figure BDA0003030742410000086
the magnitude of the incident electric field at the transmission line is:
Figure BDA0003030742410000091
wherein the content of the first and second substances,
Figure BDA0003030742410000092
electric field vector representing incident plane wave, E0Representing the magnitude of the electric field of the incident plane wave,
Figure BDA0003030742410000093
representing the direction vector of the cable, ExDenotes the x-direction component of the incident electric field, EyDenotes the y-direction component of the incident electric field, EzRepresenting the z-direction component of the incident electric field, elxRepresenting the x-direction component of the cable direction vector, elyRepresenting the y-direction component of the cable direction vector, elzRepresenting the z-direction component of the cable direction vector.
The electric field of the reflected wave at the transmission line is:
Figure BDA0003030742410000094
then the voltage source V is distributedsComprises the following steps:
Figure BDA0003030742410000095
where h represents the height of the cable end from the ground.
Initial lumped voltage source V1And (3) calculating:
Figure BDA0003030742410000096
load end lumped voltage source V2And (3) calculating:
V2＝V1e-jkLcosψcosφ
step S5, calculating induced currents generated by the three excitation sources at the load end respectively and superposing the induced currents
(1) Current I at load terminal1(L) calculating:
spatially curved cable distributed voltage source as shown in FIG. 6, excitation source VsIs located atn-1、lnThe visual impedances from the excitation source to both sides are respectively:
Figure BDA0003030742410000101
Figure BDA0003030742410000102
wherein Z isLeq(n) is fromn-1Equivalent impedance seen to the left, ZReq(n) is fromnThe equivalent impedance seen to the right has the following recursive relationship:
Figure BDA0003030742410000103
Figure BDA0003030742410000104
the initial conditions of the above recursion formula can be found by analyzing the boundary conditions of the first segment and the last segment as follows:
Figure BDA0003030742410000105
Figure BDA0003030742410000106
distributed voltage source VsWhere x is xsThe current generated at this point is:
Figure BDA0003030742410000107
wherein the content of the first and second substances,
Figure BDA0003030742410000108
representing the equivalent impedance seen at the excitation source towards the initial end,
Figure BDA0003030742410000109
representing the equivalent impedance seen at the excitation source to the load side.
According to the chain parameter matrix:
Figure BDA00030307424100001010
wherein the content of the first and second substances,
Figure BDA00030307424100001011
is an equivalent impedance
Figure BDA00030307424100001012
The voltage at (c); v (l)n) Is an equivalent impedance ZReq(n) voltage, I (l)n) Is an equivalent impedance ZReq(n) the current.
Then the equivalent impedance ZReqCurrent I (l) at (n)n) Comprises the following steps:
Figure BDA0003030742410000111
the chain parameter matrix from segment n +1 to segment m is:
Figure BDA0003030742410000112
wherein, V (l)n)＝I(ln)ZReq(n)。
The current at the load is then:
In(L)＝I(ln)(CZReq+D)
where C, D is the first and second elements in the second row of the chain parameter matrix from segment n +1 to segment m, then the total current at the load from the distributed voltage source is:
Figure BDA0003030742410000113
(2) lumped voltage source V1At x-0, load side current I2(L) calculating the content of the (C),
lumped voltage sources are located at the initial end as shown in fig. 7, according to the chain parameter matrix:
Figure BDA0003030742410000114
available total source V1The current at the load is:
Figure BDA0003030742410000115
(3) lumped voltage source V2At x ═ L, load side current I3(L) calculation of
The lumped voltage source is located at the load end as shown in fig. 8, and the impedance seen from the left side of the excitation source is:
Figure BDA0003030742410000116
general source V2The current at the load is:
Figure BDA0003030742410000117
the final current at the load end is:
I(L)＝I1(L)+I2(L)+I3(L)
take a field line coupling system as an example, where the model parameters are: the radius a of the lead is 1.0mm,length L1 m, cable height hz 0.4x2-0.4x +0.2(m), hy ═ 0.1sin (2 π x) (m), line-end load Z1＝Z250 Ω, incident plane wave electric field amplitude E010V/m, frequency range 150MHz-800MHz, angle of incidence parameter α 45 °, phi 0, psi 45 °. The model is shown in fig. 9, the calculated load end current is shown in fig. 10, the solid line in fig. 10 is the variation relation of the load end induced current calculated by the method of the present invention along with the incident electromagnetic wave frequency, the dotted line is the numerical simulation result, the variation trends of the two are consistent, and the error is less than 3 dBmA. Table 1 compares the computational efficiencies of the two.
TABLE 1
Modeling time/s Calculating time/s CPU occupancy/%)
The method of the invention 30 246 20
Numerical simulation method 300 10197 100
It can be seen that the computational efficiency and the occupied resources of the method of the invention are far less than those of the traditional numerical simulation method.

Claims (1)

1. A method for calculating irradiation induced current of a bent cable comprises the following steps:
s1, acquiring a parameter equation of a space bent cable;
the parametric equation along the x-direction is:
Figure FDA0003030742400000011
wherein hy (x) represents a function of horizontal offset distance as a function of x-coordinate, hz (x) represents a function of vertical offset distance as a function of x-coordinate;
s2, segmenting the space bent cable, and representing a voltage-current relation by using a chain parameter matrix, wherein the method specifically comprises the following steps:
the spatially arbitrary bent cable is scattered into a plurality of very short uniform small segments, the length of each segment along the x direction is delta L, the non-uniform transmission line is regarded as the series connection of m segments of uniform units, the length delta L of each segment is less than lambda, lambda is the incident wavelength, and simultaneously, the requirement that m delta L is equal to L is met, wherein L represents the total length of the cable in the x direction;
the current-voltage relationship between the initial end and the load end of the spatial arbitrary bending cable can be represented by a chain parameter matrix:
Figure FDA0003030742400000012
wherein, VLRepresenting the terminal voltage of the load, ILRepresenting the current at the load side, V0Denotes the initial terminal voltage, I0Represents the initial end current, phinA chain parameter matrix representing the nth wire;
s3, calculating distribution parameters and the characteristic impedance of each section of line element,
the voltage and current across any nth segment on the line can be expressed as:
Figure FDA0003030742400000013
wherein, VnIs the voltage of the nth segment, InIs the current of the nth segment, Vn-1Is the voltage of the n-1 th segment, In-1Is the current of the (n-1) th segment;
chain parameter matrix phi of nth section of cablenComprises the following steps:
Figure FDA0003030742400000014
wherein the content of the first and second substances,
Figure FDA0003030742400000015
representing the propagation constant of the transmission line;
Figure FDA0003030742400000016
μ0denotes the permeability of air,. epsilon.denotes the dielectric constant of air,. omega.denotes the angular frequency, Zc(n) represents the characteristic impedance of the nth segment of the cable, hz (l)n) The ground clearance of the center of the nth section of cable is shown, and a represents the radius of the conductor of the cable;
step S4, according to the Agrawal model, it can be known that the incident plane wave will generate a distributed excitation voltage source V on the cablesAnd an initial terminal lumped voltage source V1And a load side lumped voltage source V2；
Distributed voltage source VsThe excitation source is formed by superposing an incident wave electric field and a ground reflected wave electric field along the tangential direction of a cable, and the normalized direction vector of the cable can be expressed as follows:
Figure FDA0003030742400000021
wherein k isyAnd kzIs the slope of the conductor in the plane xoy and xoz;
the incident electric fields in the x, y, z directions are:
Figure FDA0003030742400000022
wherein alpha represents the included angle between the polarization direction of the incident electric field and the vertical direction, psi represents the included angle between the incident direction of the electric field and the horizontal plane, phi represents the included angle between the incident direction of the electric field and the plane where the cable and the load are located, k represents the wave number,
Figure FDA0003030742400000023
the magnitude of the incident electric field at the transmission line is:
Figure FDA0003030742400000024
wherein the content of the first and second substances,
Figure FDA0003030742400000025
electric field vector representing incident plane wave, E0Representing the magnitude of the electric field of the incident plane wave,
Figure FDA0003030742400000026
representing the direction vector of the cable, ExDenotes the x-direction component of the incident electric field, EyDenotes the y-direction component of the incident electric field, EzRepresenting the z-direction component of the incident electric field, elxRepresenting the x-direction component of the cable direction vector, elyRepresenting the y-direction component of the cable direction vector, elzA z-direction component representing a cable direction vector;
electric field E of the reflected wave at the transmission linerefComprises the following steps:
Figure FDA0003030742400000027
then the voltage source V is distributedsComprises the following steps:
Figure FDA0003030742400000031
initial lumped voltage source V1Comprises the following steps:
Figure FDA0003030742400000032
wherein h represents the height of the cable tip from the ground;
load end lumped voltage source V2Comprises the following steps:
V2＝V1e-jkLcosψcosφ
step 5, calculating and superposing the induced currents generated by the three excitation sources at the load end,
(1) distributed voltage source VsUnder the action of the current I generated at the load end1(L) the calculation procedure is as follows:
distributed voltage source VsPosition x-axis coordinate atn-1、lnThe visual impedances from the excitation source to both sides are respectively:
Figure FDA0003030742400000033
Figure FDA0003030742400000034
wherein x issRepresenting a distributed voltage source VsX-axis coordinate of (1), ZLeq(n) is the abscissa x ═ l of the cablen-1Equivalent impedance seen from the initial end, ZReq(n) is selected from x ═ lnThe equivalent impedance seen from the load end has the following recursive relationship:
Figure FDA0003030742400000035
Figure FDA0003030742400000036
wherein Z isc(n-1) represents the characteristic impedance of the n-1 th cable segment, Zc(n +1) represents the characteristic impedance of the (n +1) th cable segment, ZLeq(n-1) represents the equivalent impedance of the (n-1) th cable section as seen to the initial end, ZReq(n +1) represents the equivalent impedance of the (n +1) th cable section as seen from the load side, γn-1Denotes the propagation constant, γ, of the n-1 th cablen+1Representing the propagation constant of the n +1 th segment of the cable.
The initial conditions of the above recursion formula can be found by analyzing the boundary conditions of the first segment and the last segment as follows:
Figure FDA0003030742400000041
Figure FDA0003030742400000042
wherein Z isLeq(1) Represents the equivalent impedance, Z, of the 1 st cable section as seen from the initial endc(1) Representing the characteristic impedance, Z, of the 1 st cable segment1Representing the initial end load impedance, gamma1Denotes the propagation constant, Z, of the 1 st segment of the cableReq(m) represents the equivalent impedance of the last cable section as seen towards the load side, Zc(m) represents the characteristic impedance of the last cable segment, Z2Representing the load impedance, gamma, at the load sidemRepresenting the propagation constant of the last segment of the cable.
Distributed voltage source VsWhere x is xsThe current generated at this point is:
Figure FDA0003030742400000043
wherein the content of the first and second substances,
Figure FDA0003030742400000044
representing the equivalent impedance seen at the excitation source towards the initial end,
Figure FDA0003030742400000045
representing the equivalent impedance seen from the excitation source to the load end;
according to the chain parameter matrix:
Figure FDA0003030742400000046
wherein the content of the first and second substances,
Figure FDA0003030742400000047
V(xs) Is an equivalent impedance
Figure FDA0003030742400000048
Voltage of (V) (l)n) Is an equivalent impedance ZReq(n) voltage, I (l)n) Is an equivalent impedance ZReq(n) the current.
Then the equivalent impedance ZReqCurrent I (l) at (n)n) Comprises the following steps:
Figure FDA0003030742400000049
the chain parameter matrix from segment n +1 to segment m is:
Figure FDA00030307424000000410
wherein, V (l)n)＝I(ln)ZReq(n)。
The current at the load is then: i isn(L)＝I(ln)(CZReq(n)+D)
Where C, D is the first and second elements in the second row of the chain parameter matrix from segment n +1 to segment m, then the total current generated by the distributed voltage source at the load is:
Figure FDA0003030742400000051
(2) lumped voltage source V1At x-0, load side current I2(L) the calculation procedure is as follows:
according to the chain parameter matrix:
Figure FDA0003030742400000052
wherein phi represents a cable global chain parameter matrix;
available total source V1The current at the load is:
Figure FDA0003030742400000053
wherein V (0) represents the initial terminal voltage, I (0) represents the initial terminal current, phi21Being the first element of the second row in Φ, Φ22Is the second element of the second row in Φ;
(3) lumped voltage source V2At x ═ L, load side current I3(L) the calculation procedure is as follows:
the impedance seen from the left side of the excitation source is:
Figure FDA0003030742400000054
wherein Z isLeq(m) represents the equivalent impedance of the last cable section as seen towards the initial end;
general source V2The current at the load is:
Figure FDA0003030742400000055
the final current at the load end is: i (L) ═ I1(L)+I2(L)+I3(L)。
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