CN104215928A - Analytical model method for current conversion ratio and phase difference of open-type current transformer - Google Patents

Abstract

The invention relates to an analytical model method for the current conversion ratio and the phase difference of an open-type current transformer. The analytical model method comprises the following steps of: (1) establishing the principle model of the open-type current transformer; (2) deriving the magnetic path time-domain equation of the open-type current transformer according to an ohm law for magnetic path, calculating a circuit time-domain equation considering a secondary side load according to a Faraday law of electromagnetic induction and the ohm law, obtaining the actual current conversion ratio and phase difference calculation formulas of the primary side and the secondary side of the open-type current transformer by the magnetic path and circuit time-domain equations, so as to complete an analytical model for the current conversion ratio and the phase difference of the open-type current transformer. The analytical model established by the method disclosed by the invention has an important theoretical direction value on realization for compatibility between the current transformer for measurement and a current transformer for protection, and is capable of providing a theoretical basis for the error compensation of a split core type current transformer used for online calibration for an electric energy meter.

Description

The analytic model method of open type current transformer electric current no-load voltage ratio and phase differential

Technical field

The invention belongs to Techniques in Electromagnetic Measurement field, the analytic model method of especially a kind of open type current transformer electric current no-load voltage ratio and phase differential.

Background technology

Current transformer is widely used in generating plant, transformer station and power distribution network; being one of key equipment of electric system electric energy metrical, state-detection, Fault diagnosing and protecting, is also under intelligent grid, to realize one of basic sensor device for electricity consumption real-time information management.Traditional current transformer generally adopts closed magnetic circuit design, and the iron core of current transformer is designed to a closed magnet ring, and winding and Secondary Winding are all wound on this closed annulus iron core.

Closed magnetic circuit design, can guarantee that magnetic circuit has less magnetoimpedance, makes the energy of primary side be delivered to more efficiently secondary side.But there is an obvious shortcoming in closed magnetic circuit design, while there is DC component in primary side current, due to the magnetic resistance very little (approaching short-circuit condition) of magnetic circuit, it can produce very high DC magnetic flux in closed magnetic circuit, makes BH working point unshakable in one's determination enter saturation region.For Verification of Measuring Current Transformer, saturated meeting unshakable in one's determination causes current transformer to occur very large measuring error; And for protective current transformer; saturated meeting unshakable in one's determination makes the ratio of primary side and secondary side current higher than the ratio of rated current; the electric current by the perception of current transformer institute will be significantly less than actual current; thereby easily cause protective relaying device misoperation, cause electric system to occur accident and bring unnecessary economic loss.

For addressing this problem, 20 century 70s, USSR (Union of Soviet Socialist Republics) has proposed the design of open type current transformer, and is applied in the protection of electric system., there is the current sensor of air-gap in the current transformer of so-called open type, its ultimate principle is the magnetic resistance that seals in an air-gap in magnetic circuit in iron circuit, thereby effectively reduce the magnetic flux that the DC component by primary side current produces in iron core.Because air permeability is much smaller than magnetic permeability unshakable in one's determination, thus even if air-gap is very little, also can produce enough large magnetic resistance, with the BH working point that guarantees current transformer iron core in linear zone.

Open type current transformer has been successfully applied to power system relay protection device, but in Verification of Measuring Current Transformer, is not widely used.This is because the variation of air-gap, can change magnetic circuit impedance more significantly, and the accuracy of current transformer is also changed, and causes measuring error thereupon.Therefore, in actual electric network, Verification of Measuring Current Transformer and protective current transformer, are two kinds of independently equipment, each other can not be compatible.Obviously, such design has increased volume and the functional redundancy of current transformer, has caused larger waste.

The analytic model of research open type current transformer; can obtain from basic electromagnetic energy conversion law the phase differential of actual current ratio, primary current and secondary current and the funtcional relationship of actual parameter (as magnetic permeability unshakable in one's determination, air gap size, secondary load etc.) of open type current transformer; and then set up the compensation model of the open type current transformer for measuring, in order finally to realize, open type current transformer is measured and the compatibility of defencive function is laid a solid foundation.

In addition, along with progress and the development of intelligent electric energy meter on-line testing technology, the accuracy of hook type current transformer (a kind of open type current transformer) is also had higher requirement.The analytic model of research open type current transformer, also has important references value to improving the measurement accuracy of hook type current transformer.

Summary of the invention

The object of the invention is, in order to overcome the deficiencies in the prior art, provides the analytic model method of open type current transformer electric current no-load voltage ratio and phase differential.

The present invention solves its technical matters and takes following technical scheme to realize:

An analytic model method for open type current transformer electric current no-load voltage ratio and phase differential, comprises that step is as follows:

(1) set up the principle model of open type current transformer, it is the width of air-gap that δ is set, and l is the average length of yoke unshakable in one's determination, I ₁primary side current, I ₂be secondary side current, Z is the resulting impedance of secondary coil loop and load, supposes that iron core is constant along the cross-sectional area S of the length of magnetic path, and the number of turn of primary winding is 1 circle, and the number of turn of secondary coil is N circle;

(2) domain equation while deriving the magnetic circuit of open type current transformer according to ohm's law for magnetic circuit, domain equation while calculating according to Faraday's electromagnetic induction law and Ohm law the circuit of considering secondary side load, by time domain magnetic circuit and circuit equation, by obtain magnetic circuit time when domain equation and circuit domain equation merge into One-terminal one time, when solving, directly differential term is changed into the form that j ω and electric current phasor multiply each other, by solving amplitude versus frequency characte and the phase-frequency characteristic of the transport function of the open type current transformer of considering secondary side load, obtain open type current transformer primary side and secondary side actual current no-load voltage ratio and phase difference calculating formula, complete the analytic model of open type current transformer electric current no-load voltage ratio and phase differential.

And the concrete steps that described step (2) obtains open type current transformer primary side and secondary side actual current no-load voltage ratio and phase difference calculating formula are as follows:

The principle model that A, combination are set up, sets up equation according to the Ohm law of magnetic circuit, and in impedance sum Z ≠ 0 of considering secondary coil and reometer itself, the equation of listing is:

$(\frac{l}{\mathrm{\μS}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_{0}S})\mathrm{\Φ}\left(t\right)=I_1\left(t\right)-{\mathrm{NI}}_2\left(t\right)---\left(2\right);$

Wherein, Φ (t) is the magnetic flux in current transformer iron core; μ represents the magnetic permeability of current transformer iron core material; μ ₀be permeability of vacuum, think that it equals air permeability here;

B, to secondary side circuit, make R and L be respectively resistive component and the inductive component in secondary side resulting impedance Z, by Faraday's electromagnetic induction law and Ohm law, set up equation, equation is:

$N\frac{\mathrm{d\Φ}\left(t\right)}{\mathrm{dt}}=L\frac{{\mathrm{dI}}_2\left(t\right)}{\mathrm{dt}}+{\mathrm{RI}}_2\left(t\right)---\left(3\right);$

C, simultaneous formula (2) and formula (3), obtain equation and be:

$\frac{1}{\mathrm{NS}}(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})[L\frac{{\mathrm{dI}}_2\left(t\right)}{\mathrm{dt}}+{\mathrm{RI}}_2\left(t\right)]=\frac{{\mathrm{dI}}_1\left(t\right)}{\mathrm{dt}}-N\frac{{\mathrm{dI}}_2\left(t\right)}{\mathrm{dt}}---\left(4\right);$

D, use formula (4) instead phasor method and be expressed as:

$\frac{1}{\mathrm{NS}}(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})({\mathrm{j\ωL}\stackrel{\·}{I}}_2+R{\stackrel{\·}{I}}_2)=\mathrm{j\ω}{\stackrel{\·}{I}}_1-\mathrm{j\ωNL}{\stackrel{\·}{I}}_2---\left(5\right);$

Wherein, with respectively primary side and secondary side current phasor;

E, the transport function that obtains open type current transformer by formula (5) characterize;

$\frac{{\stackrel{\·}{I}}_2}{{\stackrel{\·}{I}}_1}=\frac{\mathrm{j\ω}}{(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{R}{\mathrm{NS}}+j[\mathrm{\ωN}+(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{\mathrm{\ωL}}{\mathrm{NS}}]}---\left(6\right);$

F, by the transport function delivery amount shown in formula (6), obtain the ratio of the actual current value of open type current transformer, that is:

$\mathrm{\α}=\frac{1}{\sqrt{{\left[(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{R}{\mathrm{\ωNS}}\right]}^2+{[N+(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{L}{\mathrm{NS}}]}^2}}---\left(7\right);$

The phase differential that can solve secondary side and primary side current by formula (6) is:

$\mathrm{\β}=\arctan \frac{(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{R}{\mathrm{\ωNS}}}{N+(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{L}{\mathrm{NS}}}---\left(8\right)$

Complete thus the analytic model of open type current transformer electric current no-load voltage ratio and phase differential.

Advantage of the present invention and good effect are:

The present invention proposes the analytic model of open type current transformer when considering secondary side load, by simultaneous time domain dynamic magnetic circuit equation and circuit equation, obtained the computing formula that open type current transformer primary side and secondary side actual current no-load voltage ratio and phase place change.The analytic model of setting up, has more important theoretical direction and is worth realizing the compatibility of Verification of Measuring Current Transformer and protective current transformer,, and meanwhile, the error compensation also can be for the split core type current transformer of electric energy meter on-line calibration provides theoretical foundation.

Accompanying drawing explanation

Fig. 1 is the principle model schematic diagram of open type current transformer, and wherein, δ is the width of air-gap, and l is the average length of yoke unshakable in one's determination, I ₁primary side current, I ₂be secondary side current, Z is the resulting impedance of secondary coil loop and load, supposes that iron core is constant along the cross-sectional area S of the length of magnetic path, and the number of turn of primary winding is 1 circle, and the number of turn of secondary coil is N circle;

Fig. 2 is the result of calculation of ε (N, f);

Fig. 3 is the result of calculation of β (N, f);

Fig. 4 is the result of calculation of ε (R, L);

Fig. 5 is the result of calculation of β (R, L);

Fig. 6 is the result of calculation of ε (δ, μ r);

Fig. 7 is the result of calculation of β (δ, μ r).

Embodiment

Below the invention process is further described, following examples are descriptive, are not determinate, can not limit protection scope of the present invention with this.

An analytic model method for open type current transformer electric current no-load voltage ratio and phase differential, comprises that step is as follows:

(1) set up the principle model of open type current transformer, as shown in Figure 1, wherein, δ is the width of air-gap, and l is the average length of yoke unshakable in one's determination, I ₁primary side current, I ₂be secondary side current, Z is the resulting impedance of secondary coil loop and load, supposes that iron core is constant along the cross-sectional area S of the length of magnetic path, and the number of turn of primary winding is 1 circle, and the number of turn of secondary coil is N circle;

(2) domain equation while deriving the magnetic circuit of open type current transformer according to ohm's law for magnetic circuit, domain equation while calculating according to Faraday's electromagnetic induction law and Ohm law the circuit of considering secondary side load, time domain magnetic circuit and the circuit equation by row, write, by obtain magnetic circuit time when domain equation and circuit domain equation merge into One-terminal one time, when solving, directly differential term is changed into the form that j ω and electric current phasor multiply each other, by solving amplitude versus frequency characte and the phase-frequency characteristic of the transport function of the open type current transformer of considering secondary side load, obtain its primary side and secondary side actual current no-load voltage ratio and phase difference calculating formula: concrete steps are as follows:

The principle model that A, integrating step (1) are set up, under ideal conditions, think the resultant load impedance Z=0 of Current Transformer Secondary side, the primary side of mutual inductor main magnetic circuit equates with the electric current ampere turns of secondary side, therefore primary side and secondary side current meet following relation:

I ₁＝NI ₂ (1)

And under actual current mutual inductor ruuning situation, due to impedance sum Z ≠ 0 of secondary coil and reometer itself, historical facts or anecdotes border primary side does not meet formula (1) with the ratio of the electric current of secondary side, and also can there is phase differential between the two.

Here from time domain equation, solve respectively when both phase differential of actual primary side and secondary side current, the main magnetic circuit to current transformer, sets up equation according to the Ohm law of magnetic circuit, the equation of listing is:

$(\frac{l}{\mathrm{\μS}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_{0}S})\mathrm{\Φ}\left(t\right)=I_1\left(t\right)-{\mathrm{NI}}_2\left(t\right)---\left(2\right)$

Wherein, Φ (t) is the magnetic flux in current transformer iron core; μ represents the magnetic permeability of current transformer iron core material; μ ₀be permeability of vacuum, think that it equals air permeability here;

B, to secondary side circuit, make R and L be respectively resistive component and the inductive component in secondary side resulting impedance Z, by Faraday's electromagnetic induction law and Ohm law, set up equation, equation is:

$N\frac{\mathrm{d\Φ}\left(t\right)}{\mathrm{dt}}=L\frac{{\mathrm{dI}}_2\left(t\right)}{\mathrm{dt}}+{\mathrm{RI}}_2\left(t\right)---\left(3\right)$

C, simultaneous formula (2) and formula (3), obtain equation and be:

$\frac{1}{\mathrm{NS}}(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})[L\frac{{\mathrm{dI}}_2\left(t\right)}{\mathrm{dt}}+{\mathrm{RI}}_2\left(t\right)]=\frac{{\mathrm{dI}}_1\left(t\right)}{\mathrm{dt}}-N\frac{{\mathrm{dI}}_2\left(t\right)}{\mathrm{dt}}---\left(4\right)$

Formula (4) demonstrates, if do not consider secondary side load, thinks Z=0 (R=0, L=0), can obviously obtain the conclusion of formula (1).Now consider secondary side load, think Z ≠ 0, solve respectively the scale-up factor α=I of primary side and secondary side current ₂/ I ₁and phase difference beta.

D, use formula (4) instead phasor method and be expressed as:

$\frac{1}{\mathrm{NS}}(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})({\mathrm{j\ωL}\stackrel{\·}{I}}_2+R{\stackrel{\·}{I}}_2)=\mathrm{j\ω}{\stackrel{\·}{I}}_1-\mathrm{j\ωNL}{\stackrel{\·}{I}}_2---\left(5\right)$

Wherein, with respectively primary side and secondary side current phasor.

E, the transport function that obtains open type current transformer by formula (5) characterize;

$\frac{{\stackrel{\·}{I}}_2}{{\stackrel{\·}{I}}_1}=\frac{\mathrm{j\ω}}{(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{R}{\mathrm{NS}}+j[\mathrm{\ωN}+(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{\mathrm{\ωL}}{\mathrm{NS}}]}---\left(6\right)$

F, by the transport function delivery amount shown in formula (6), obtain the ratio of the actual current value of open type current transformer, that is:

$\mathrm{\α}=\frac{1}{\sqrt{{\left[(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{R}{\mathrm{\ωNS}}\right]}^2+{[N+(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{L}{\mathrm{NS}}]}^2}}---\left(7\right)$

The phase differential that similarly, can solve secondary side and primary side current by formula (6) is:

$\mathrm{\β}=\arctan \frac{(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{R}{\mathrm{\ωNS}}}{N+(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\frac{L}{\mathrm{NS}}}---\left(8\right).$

Example

For further quantizing and showing more intuitively the analytic model of the open type current transformer that present patent application is set up, existing in conjunction with a specific embodiment, the groundwork feature of the open type current transformer of considering secondary load is discussed.The definition actual current ratio of open type current transformer and the relative difference of ideal current transformer current ratio are:

$\begin{array}{c}\mathrm{\ϵ}=\frac{1}{\mathrm{\αN}}-1\\ =\sqrt{{[1+\frac{L}{N^{2}S}(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})]}^2+{\left[\frac{R}{\mathrm{\ω}{N}^{2}S}(\frac{l}{\mathrm{\μ}}+\frac{\mathrm{\δ}}{{\mathrm{\μ}}_0})\right]}^2}-1\end{array}---\left(9\right)$

During calculating, the typical set value of each parameter is: N=200 circle; R=5Ohm; L=1mH; L=100mm; δ=0.5mm; μ=5000*4 π * 1e-7H/m; F=50Hz; S=0.01*0.01mm ².For the purpose of convenient, by affecting the relative difference ε of open type current transformer current ratio and the parameter of phase difference beta, be divided into three groups and discuss, be specifically respectively (N, f), (R, L) and (μ, δ).

First calculate the relative variation of actual current no-load voltage ratio and phase differential and be related to ε (N with the two-dimensional function between the frequency of operation f of secondary coil number of turn N (being the turn ratio of secondary, primary winding) and this open type current transformer, f), β (N, f).During calculating, N and f are made as to variable, other parameters are all got standard value.The result of calculation of ε (N, f) and β (N, f), respectively as shown in Figures 2 and 3.

From the result of calculation of Fig. 2 and Fig. 3, can obtain as drawn a conclusion:

1) number of turn of open type current transformer secondary coil is more, and actual current no-load voltage ratio approaches desirable no-load voltage ratio, and the phase differential of primary side and secondary side current is also less.This be because, along with the increase of the secondary coil number of turn, secondary side current is corresponding to diminish, the active loss being caused by secondary side load shared ratio in total load will reduce, the change of actual current no-load voltage ratio and phase deviation also will correspondingly reduce.

2) frequency of operation of open type current transformer is higher, and actual current no-load voltage ratio and phase place also more approach the situation of ideal current mutual inductor.Therefore, if open type current transformer is done to measure, use, its accuracy of measuring high-frequency current is better than measuring the accuracy of low-frequency current.

Secondly, the relative variation of calculating actual current no-load voltage ratio and phase differential are with function ε (R, L), the β (R, L) of secondary side load variations.During calculating, adopt similar Variable Control method, obtained result of calculation as shown in Figure 4 and Figure 5.

Fig. 4 and Fig. 5 result of calculation demonstrate, and the relative variation ε of the actual no-load voltage ratio of open type current transformer and desirable no-load voltage ratio can increase along with the increase of secondary resistance or inductance; Phase difference beta between first and second side electric current, under the definite condition of inductance, can increase along with the increase of resistance, but at resistance fixedly in the situation that, with the increase of inductance, can reduce on the contrary.

Finally, the relative variation of calculating actual current no-load voltage ratio and phase differential are with the width of air-gap and function ε (δ, the μ of core material relative permeability variation _r), β (δ, μ _r).During concrete calculating, by the relative permeability μ of air-gap width δ and core material _rbe made as variable, other parameters are got standard value, and result of calculation as shown in Figure 6 and Figure 7.

Fig. 6 and Fig. 7 result of calculation shows, relative variation ε and the phase difference beta of the actual no-load voltage ratio of open type current transformer and desirable no-load voltage ratio, can be along with the change generation acute variation of air-gap width; And ε and β are very inresponsive to the variation of core material relative permeability, this is mainly that relative variation by current transformer main magnetic circuit magnetic resistance determines.The magnetic resistance of closed magnetic circuit iron core is

And after increasing opening and being air-gap, the magnetic resistance change rate of main magnetic circuit is

As δ <<l/ μ _rtime, magnetoimpedance component unshakable in one's determination is the major part of main magnetic circuit magnetic resistance, the component being caused by secondary resistance and inductance in formula (8) and formula (9) mainly determines by the relative permeability of core material, but its value is much smaller than 1, now the value of ε and β all close to 0.But along with the increase of δ, gap reluctance becomes the principal component of main magnetic circuit magnetic resistance, l/ μ _rδ becomes in a small amount relatively, so main magnetic circuit magnetic resistance is to μ _rvariation insensitive, and δ >>l/ μ _r, with the width change of opening air-gap, can there is violent variation in the value that causes ε and β.

The diamagnetic saturation coefficient ζ of definition open type current transformer is

ζ has reflected the multiple that open type current transformer changes with respect to the diamagnetic saturability of closed magnetic circuit current transformer.From formula (12), be not difficult to find out, only needing increases less air-gap, just can obviously improve the anti-saturation magnetization property of current transformer.

It should be noted that in setting up above-mentioned analytic model process, do not consider the eddy current loss of open type current transformer self.The introducing of open type current transformer air-gap, can obviously increase the magnetic resistance of main magnetic circuit, thereby reduces the magnetic flux change of main magnetic circuit.Therefore, the eddy current loss of open type current transformer self will be starkly lower than the current transformer of closing magnetic path structure.

In addition, in the process of establishing of above-mentioned analytic model, for simplicity, the magnetic permeability of core material is similar to thought it is a constant.And in fact, the magnetic permeability of ferromagnetic material should be the nonlinear function of internal magnetic field intensity unshakable in one's determination.But analyze, find, the analytic model itself of setting up changes magnetic permeability unshakable in one's determination and is not obvious, therefore, due to the error of not considering that saturation of magnetic material causes, should be an a small amount of in theory.

Claims (2)

1. an analytic model method for open type current transformer electric current no-load voltage ratio and phase differential, is characterized in that comprising that step is as follows:

(1) set up the principle model of open type current transformer, it is the width of air-gap that δ is set, and l is the average length of yoke unshakable in one's determination, I ₁primary side current, I ₂be secondary side current, Z is the resulting impedance of secondary coil loop and load, supposes that iron core is constant along the cross-sectional area S of the length of magnetic path, and the number of turn of primary winding is 1 circle, and the number of turn of secondary coil is N circle;

(2) domain equation while deriving the magnetic circuit of open type current transformer according to ohm's law for magnetic circuit, domain equation while calculating according to Faraday's electromagnetic induction law and Ohm law the circuit of considering secondary side load, by time domain magnetic circuit and circuit equation, by obtain magnetic circuit time when domain equation and circuit domain equation merge into One-terminal one time, when solving, directly differential term is changed into the form that j ω and electric current phasor multiply each other, by solving amplitude versus frequency characte and the phase-frequency characteristic of the transport function of the open type current transformer of considering secondary side load, obtain open type current transformer primary side and secondary side actual current no-load voltage ratio and phase difference calculating formula, complete the analytic model of open type current transformer electric current no-load voltage ratio and phase differential.

2. the analytic model method of open type current transformer electric current no-load voltage ratio according to claim 1 and phase differential, is characterized in that: the concrete steps that described step (2) obtains open type current transformer primary side and secondary side actual current no-load voltage ratio and phase difference calculating formula are as follows:

The principle model that A, combination are set up, sets up equation according to the Ohm law of magnetic circuit, and in impedance sum Z ≠ 0 of considering secondary coil and reometer itself, the equation of listing is:

$(\frac{l}{\mathrm{\&mu;S}}+\frac{\mathrm{\&delta;}}{{\mathrm{\&mu;}}_{0}S})\mathrm{\&Phi;}\left(t\right)=I_1\left(t\right)-{\mathrm{NI}}_2\left(t\right)---\left(2\right);$

Wherein, Φ (t) is the magnetic flux in current transformer iron core; μ represents the magnetic permeability of current transformer iron core material; μ ₀be permeability of vacuum, think that it equals air permeability here;

B, to secondary side circuit, make R and L be respectively resistive component and the inductive component in secondary side resulting impedance Z, by Faraday's electromagnetic induction law and Ohm law, set up equation, equation is:

$N\frac{\mathrm{d\&Phi;}\left(t\right)}{\mathrm{dt}}=L\frac{{\mathrm{dI}}_2\left(t\right)}{\mathrm{dt}}+{\mathrm{RI}}_2\left(t\right)---\left(3\right);$

C, simultaneous formula (2) and formula (3), obtain equation and be:

$\frac{1}{\mathrm{NS}}(\frac{l}{\mathrm{\&mu;}}+\frac{\mathrm{\&delta;}}{{\mathrm{\&mu;}}_0})[L\frac{{\mathrm{dI}}_2\left(t\right)}{\mathrm{dt}}+{\mathrm{RI}}_2\left(t\right)]=\frac{{\mathrm{dI}}_1\left(t\right)}{\mathrm{dt}}-N\frac{{\mathrm{dI}}_2\left(t\right)}{\mathrm{dt}}---\left(4\right);$

D, use formula (4) instead phasor method and be expressed as:

$\frac{1}{\mathrm{NS}}(\frac{l}{\mathrm{\&mu;}}+\frac{\mathrm{\&delta;}}{{\mathrm{\&mu;}}_0})({\mathrm{j\&omega;L}\stackrel{\&CenterDot;}{I}}_2+R{\stackrel{\&CenterDot;}{I}}_2)=\mathrm{j\&omega;}{\stackrel{\&CenterDot;}{I}}_1-\mathrm{j\&omega;NL}{\stackrel{\&CenterDot;}{I}}_2---\left(5\right);$

Wherein, with respectively primary side and secondary side current phasor;

E, the transport function that obtains open type current transformer by formula (5) characterize;

$\frac{{\stackrel{\&CenterDot;}{I}}_2}{{\stackrel{\&CenterDot;}{I}}_1}=\frac{\mathrm{j\&omega;}}{(\frac{l}{\mathrm{\&mu;}}+\frac{\mathrm{\&delta;}}{{\mathrm{\&mu;}}_0})\frac{R}{\mathrm{NS}}+j[\mathrm{\&omega;N}+(\frac{l}{\mathrm{\&mu;}}+\frac{\mathrm{\&delta;}}{{\mathrm{\&mu;}}_0})\frac{\mathrm{\&omega;L}}{\mathrm{NS}}]}---\left(6\right);$

F, by the transport function delivery amount shown in formula (6), obtain the ratio of the actual current value of open type current transformer, that is:

$\mathrm{\&alpha;}=\frac{1}{\sqrt{{\left[(\frac{l}{\mathrm{\&mu;}}+\frac{\mathrm{\&delta;}}{{\mathrm{\&mu;}}_0})\frac{R}{\mathrm{\&omega;NS}}\right]}^2+{[N+(\frac{l}{\mathrm{\&mu;}}+\frac{\mathrm{\&delta;}}{{\mathrm{\&mu;}}_0})\frac{L}{\mathrm{NS}}]}^2}}---\left(7\right);$

The phase differential that can solve secondary side and primary side current by formula (6) is:

$\mathrm{\&beta;}=\arctan \frac{(\frac{l}{\mathrm{\&mu;}}+\frac{\mathrm{\&delta;}}{{\mathrm{\&mu;}}_0})\frac{R}{\mathrm{\&omega;NS}}}{N+(\frac{l}{\mathrm{\&mu;}}+\frac{\mathrm{\&delta;}}{{\mathrm{\&mu;}}_0})\frac{L}{\mathrm{NS}}}---\left(8\right)$

Complete thus the analytic model of open type current transformer electric current no-load voltage ratio and phase differential.