CN102810389A - Magnetism-shunting voltage regulation type transformer and magnetism-shunting voltage regulation method thereof - Google Patents

Magnetism-shunting voltage regulation type transformer and magnetism-shunting voltage regulation method thereof Download PDF

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CN102810389A
CN102810389A CN201210321729XA CN201210321729A CN102810389A CN 102810389 A CN102810389 A CN 102810389A CN 201210321729X A CN201210321729X A CN 201210321729XA CN 201210321729 A CN201210321729 A CN 201210321729A CN 102810389 A CN102810389 A CN 102810389A
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沈广贤
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Abstract

The invention discloses a magnetism-shunting voltage regulation type transformer and a magnetism-shunting voltage regulation method thereof. The magnetism-shunting voltage regulation type transformer mainly comprises a main magnetic flux closed iron core frame (1), a primary coil (2) and a secondary coil (3). The magnetism-shunting voltage regulation type transformer is in parallel connection with one or more magnetic valves; magnetic columns positioned on two opposite edges of magnetic valve closed iron core frames (4) of the magnetic valves are respectively provided with one pair or more pairs of coils with a same number of turns; one end heads of each pair of coils are directly connected, and the other end heads are in series connection with a power supply; and a connecting iron core (5) is arranged between the main magnetic flux closed iron core frame (1) and the magnetic valve closed iron core frames (4). The magnetism-shunting voltage regulation method comprises the following steps: the main magnetic flux closed iron core frame (1) divides an integral magnetic circuit into a primary magnetic circuit and a secondary magnetic circuit according to the primary coil (2) and the secondary coil (3) and then is in parallel connection with one or more magnetic valves to form one or more magnetism-shunting magnetic circuits; and the magnetism-shunting magnetic circuits change the magnetic flux of a loop passing through a secondary winding by controlling the saturated degree of the magnetic valves to regulate the magnitude of secondary output voltages and output constant voltages. The invention can meet the requirement of a user on stabilized voltage.

Description

Magnetic-dividing voltage-regulating transformer and method for regulating voltage by dividing magnetism of transformer
Technical Field
The invention relates to a magnetic-dividing voltage-regulating transformer and a method for regulating voltage by dividing magnetism of the transformer
Background
The conventional transformer is an iron core frame around which primary and secondary coils are wound. The relationship between the primary and secondary voltages and the number of turns of the primary and secondary coils is as follows:
Figure BSA00000772249800011
one of the practical transformers has fixed turns of the secondary coil, i.e.
Figure BSA00000772249800012
This equation illustrates that the secondary voltage of the distribution transformer varies with the primary voltage of the grid and is almost linear. Once the load of the power grid changes, the voltage of the power grid fluctuates, the voltage value supplied to the user side deviates from the range of the national standard limit value, the electrical equipment of the user side cannot work near the rated voltage, the electrical equipment has low electrical energy conversion efficiency, and the electrical energy waste is large. When the voltage deviation is serious, the electric equipment can not work normally or be damaged.
The voltage regulation mode of the power grid mainly depends on voltage regulation of equipment such as a main transformer, a generator, a reactive compensation device and the like. However, the distribution transformer used in large quantities at present cannot adjust voltage in real time, and only can adjust the tap switch by manual operation seasonally and in a power failure state, so as to achieve the purpose of adjusting voltage, that is, the existing distribution transformer cannot adjust the voltage according to the voltage change of the power grid in real time, and cannot meet the requirement of stable voltage of a user terminal.
Disclosure of Invention
In order to solve the above problems, the present invention provides a magnetic-dividing voltage-regulating transformer and a method for magnetic-dividing voltage-regulating of the transformer.
The invention relates to a magnetic dividing voltage regulating transformer, which mainly comprises a main magnetic flux closed iron core frame, a primary coil and a secondary coil, wherein two sides of the main magnetic flux closed iron core frame are provided with magnetic columns, one side is provided with the primary coil, and the other side is provided with the secondary coil; the main magnetic flux closed iron core frame and the magnetic valve closed iron core frame are of an integral structure.
The magnetic valve is in parallel connection with the middle, two sides or three-dimensional space of the closed iron core frame of the main magnetic flux. The series power supply is preferably a dc power supply. The direct current power supply is preferably rectified and directly supplied by the output power supply of the secondary coil (3) of the transformer. A method for regulating voltage by dividing magnetism of transformer, its main magnetic flux closes the iron core frame, divide the whole magnetic circuit into primary magnetic circuit and secondary magnetic circuit according to the primary coil, secondary coil, connect one or more magnetic valves in parallel, its magnetic valve has magnetic valve to close the iron core frame and a secondary to link the partial main magnetic flux of the iron core frame, form one or more magnetic dividing magnetic circuit; the two opposite sides of a closed iron core frame of the magnetic valve are magnetic columns, the magnetic columns on the two sides are respectively provided with one or more pairs of coils with the same number of turns, each pair of coils is respectively arranged on the magnetic columns on the two sides, one end of each pair of coils is directly combined, and the other end of each pair of coils is connected with a power supply in series; the magnetic dividing magnetic circuit changes the magnetic flux of a loop passing through the secondary winding by controlling the saturation degree of the magnetic valve, and adjusts the magnitude of secondary output voltage to form set constant voltage output.
A method for adding a shunt magnetic circuit in a transformer. When the primary winding is switched on by sine alternating current, a magnetic flux phi is generated in the iron core1And external connection voltage U1With the generated magnetic flux phi1The relationship is as follows: u shape1=4.44fNΦ1m=2πfNΦ1=ωNΦ1(Note: U)1、Φ1Is an effective value of phi1mIn amplitude). Magnetic flux phi1After passing through the "O" point, it is divided into magnetic flux phi2And magnetic flux phizMagnetic flux phi2Induced voltage U by secondary winding2,U2=4.44fNΦ2m=2πfNΦ2=ωNΦ2(Note: U)2、Φ2Is an effective value of phi2mAmplitude), magnetic flux ΦzFlows to the magnetic valve. Phi2=Φ1Z,Φ1The magnitude is controlled by an external voltage U1Size determination, i.e. U1A constant phi1And also must be. If reduced by phiZThen phi2Increase, U2Then the size is increased; conversely increase phiZThen phi2Decrease of U2And is reduced accordingly. By varying phiZThe output voltage U of the secondary winding can be realized2Namely, the function of regulating voltage is realized.
The main magnetic flux closed iron core frame, the connecting iron core and the magnetic valve closed iron core frame are of an integral structure, and are favorable for adapting to the traditional process of transformer production because the iron core of the transformer is made of ferromagnetic substances (such as silicon steel sheets, amorphous alloys and the like). For convenience of explanation, the present invention is explained with silicon steel sheets. Each silicon steel sheet is provided with a main magnetic flux closed iron core frame area, a connecting iron core area and a magnetic valve closed iron core frame area. A plurality of silicon steel sheets are superposed to form an integral structure of the main flux closed iron core frame, the connecting iron core and the magnetic valve closed iron core frame. The power supply is rectified into direct current power supply by the output power supply of the secondary coil of the transformer. And the influence on normal work caused by feeding of other direct current power supplies is prevented.
Adjusting the magnetic flux of the magnetic valve: the magnetic valve is formed by connecting two magnetic columns in parallel and then connecting the two magnetic columns with a main magnetic circuit into a whole; the two magnetic columns are respectively wound with coils with the same number of turns, one ends of the two magnetic columns are directly combined, and the other ends of the two magnetic columns are connected with a direct current power supply in series; when direct current flows in the pair of coils, the generated magnetic flux forms a loop in the two magnetic columns, and the loop magnetic flux is larger when the direct current is larger; for a fixed magnetic circuit, the larger the magnetic flux is, the higher the magnetic saturation degree is, and the larger the magnetic resistance is, namely the magnetic resistance of the magnetic valve can be controlled by controlling the direct current; the magnetic resistance of the magnetic valve is changed, namely the magnetic flux flowing through the magnetic valve can be changed. Each magnetic valve can have a plurality of pairs of direct current coils, and the effect can be achieved.
Drawings
FIG. 1 is a schematic diagram of a conventional single-phase transformer
FIG. 2 is a schematic diagram of a conventional three-phase transformer
FIG. 3 is a magnetic circuit topology
FIG. 4 is a diagram of the component magnetic circuit
FIG. 5 is a structural diagram of a single-phase magnetic division transformer with a magnetic valve in the middle according to the present invention
FIG. 6 is a structural view of a single-phase shunt magnetic transformer having a magnetic valve at one side according to the present invention
FIG. 7 is a structure diagram of a single-phase transformer shunt magnetic pressure device with magnetic valve in II shape
FIG. 8 is a diagram of the magnetic circuit equivalence of the voltage-regulating transformer
FIG. 9 is an equivalent circuit diagram of a voltage-regulating transformer
FIG. 10 is an equivalent circuit diagram of a shunt voltage-regulating transformer with leakage reactance omitted
FIG. 11 is an equivalent circuit diagram of a shunt voltage-regulating transformer with leakage reactance taken into account
FIG. 12 is the equivalent circuit diagram of the magnetic valve fully saturated magnetic-dividing voltage-regulating transformer
FIG. 13 is a secondary side off equivalent circuit diagram
FIG. 14 is a primary side open equivalent circuit diagram
FIG. 15 is a control diagram of magnetic valve excitation
FIG. 16 is a control diagram of magnetic valve excitation DC and output voltage
FIG. 17 is a parameter diagram of an equivalent magnetic circuit
FIG. 18 is a simplified diagram of an equivalent magnetic circuit
FIG. 19 is an equivalent mutual inductance circuit diagram
FIG. 20 is an equivalent self-inductance circuit diagram
FIG. 21 is an equivalent self-inductance mutual inductance graph
FIG. 22 is a diagram of equivalent self-inductance mutual inductance of the leakage flux
FIG. 23 is a graph of the mutual inductance conversion equivalent magnetic circuit parameters
FIG. 24 is a secondary side voltage graph
Detailed Description
According to the general single-phase transformer shown in fig. 1, a main flux closed iron core 1 is a square closed iron core, and mainly comprises a main flux closed iron core frame 1, a primary coil 2 and a secondary coil 3, wherein two sides of the main flux closed iron core frame are magnetic columns, one side of the main flux closed iron core frame is provided with the primary coil 2, and the other side of the main flux closed iron core frame is provided with the secondary coil 3. The invention connects one or more magnetic valves in parallel on the basis of the common single-phase transformer, the magnetic valve is provided with a magnetic valve closed iron core frame 4, two opposite sides of the magnetic valve closed iron core frame 4 are magnetic columns, the magnetic columns on the two sides are respectively provided with one or more pairs of coils with the same number of turns, each pair of coils is respectively arranged on the magnetic columns on the two sides, one end head of each pair of coils is directly combined, and the other end head is connected with a power supply in series. A connecting iron core 5 is arranged between the main flux closed iron core frame 1 and the magnetic valve closed iron core frame 4, and the main flux closed iron core frame 1, the connecting iron core 5 and the magnetic valve closed iron core frame 4 are of an integral structure. The invention can divide the whole magnetic circuit into a primary magnetic circuit and a secondary magnetic circuit according to the primary winding and the secondary winding. On the basis of the primary magnetic circuit and the secondary magnetic circuit, a shunt magnetic circuit is connected in parallel. The magnetic dividing magnetic circuit consists of a magnetic valve iron core and iron cores connected with the primary and secondary magnetic circuits.
The magnetic valve iron core is connected with the single-phase transformer iron core into a whole after being connected in parallel by two magnetic column iron cores 4; the two magnetic columns are respectively wound with coils with the same number of turns, and the two coils are called as a pair of coils; the connection mode is shown in fig. 15, one end head is directly combined, and the other end head is connected with direct current in series.
The magnetic valve iron core 4 has a plurality of modes in the form of two magnetic column iron cores which are connected in parallel. Fig. 5 shows a single-phase magnetic dividing transformer structure form with a magnetic valve in the middle, fig. 6 shows a single-phase magnetic dividing transformer structure form with a magnetic valve on one side, and fig. 7 shows a single-phase variable magnetic dividing device structure form with a magnetic valve in pi shape.
The iron core is formed by laminating traditional silicon steel sheets, and the simplest method is that each silicon steel sheet comprises a square frame closed iron core area, a connection iron core area and a magnetic valve iron core area, and the iron core is of an integral structure and is an integral stamping sheet. The sheets are then stacked into a unitary core.
When the primary winding is connected with sine alternating current, magnetic flux phi is generated in the iron core of the primary winding1The magnetic flux phi1Part of the current flows into the secondary winding iron core phi2A part of the magnetic flux flows into the iron core phi of the magnetic valveZSatisfy phi1=Φ2Z
According to fig. 7, the following voltage regulation principle is shown:
when the primary winding is switched on by sine alternating current, a magnetic flux phi is generated in the iron core1And external connection voltage U1With the generated magnetic flux phi1The relationship is as follows: u shape1=4.44fNΦ1m=2πfNΦ1=ωNΦ1(Note: U)1、Φ1Is an effective value of phi1mIn amplitude). Magnetic flux phi1After passing through the "O" point, it is divided into magnetic flux phi2And magnetic flux phizMagnetic flux phi2Induced voltage U by secondary winding2,U2=4.44fNΦ2m=2πfNΦ2=ωNΦ2(Note: U)2、Φ2Is an effective value of phi2mAmplitude), magnetic flux ΦzFlows to the magnetic valve. Phi2=Φ1Z,Φ1The magnitude is controlled by an external voltage U1Size determination, i.e. U1A constant phi1And also must be. If reduced by phiZThen phi2Increase, U2Then the size is increased; conversely increase phiZThen phi2Decrease of U2And is reduced accordingly. By varying phiZIs not less than twiceWinding output voltage U2Namely, the function of regulating voltage is realized.
Adjusting the magnetic flux of the magnetic valve: the magnetic valve is formed by connecting two magnetic columns in parallel and then connecting the two magnetic columns with a main magnetic circuit into a whole; the two magnetic columns are respectively wound with coils with the same number of turns, the two coils are called as a pair of coils, the connection mode is shown in figure 7, one ends of the two coils are directly combined, and the other ends of the two coils are connected with a direct current power supply in series; when direct current flows in the pair of coils, the generated magnetic flux forms a loop in the two magnetic columns, and the loop magnetic flux is larger when the direct current is larger; for a fixed magnetic circuit, the larger the magnetic flux is, the higher the magnetic saturation degree is, and the larger the magnetic resistance is, namely the magnetic resistance of the magnetic valve can be controlled by controlling the direct current; the magnetic resistance of the magnetic valve is changed, namely the magnetic flux flowing through the magnetic valve can be changed. There may be several pairs of dc coils per magnetic valve.
2. Voltage regulating method for magnetic dividing and voltage regulating transformer
As shown in fig. 7 (for convenience of explanation, the leakage magnetic flux is ignored).
(1) The voltage regulating method is explained by the magnetic circuit induction principle of the magnetic dividing voltage regulating transformer:
when the primary winding N1After being connected with sine alternating current, magnetic flux phi is generated in the iron core of the primary winding1The magnetic flux phi1A part of the current flows into the secondary winding N2Iron core phi2A part of the magnetic flux flows into the iron core phi of the magnetic valveZSatisfy phi1=Φ2Z
When the primary winding is switched on by sine alternating current, a magnetic flux phi is generated in the iron core1And external connection voltage U1With the generated magnetic flux phi1The relationship is as follows: u shape1=4.44fNΦ1m=2πfNΦ1=ωNΦ1. Magnetic flux phi1After passing through the "O" point, it is divided into magnetic flux phi2And magnetic flux phizMagnetic flux phi2Induced voltage U by secondary winding2(U2=4.44fNΦ2m=2πfNΦ2=ωNΦ2) Magnetic flux phizFlows to the magnetic valve. Phi2=Φ1Z,Φ1The magnitude is controlled by an external voltage U1Size determination, i.e. U1A constant phi1And also must be. If reduced by phiZThen phi2Increase, U2Then the size is increased; conversely increase phiZThen phi2Decrease of U2And is reduced accordingly. By varying phiZThe output voltage U of the secondary winding can be realized2Namely, the function of regulating voltage is realized.
Magnetic flux phi of magnetic valveZSize adjustment: the magnetic valve is formed by connecting two magnetic columns in parallel and then connecting the two magnetic columns with a main magnetic circuit into a whole; the two magnetic columns are respectively wound with coils with the same number of turns, the two coils are called as a pair of coils, the connection mode is shown in figure 7, one ends of the two coils are directly combined, and the other ends of the two coils are connected with a direct current power supply in series; when direct current flows in the pair of coils, the generated magnetic flux forms a loop in the two magnetic columns, and the loop magnetic flux is larger when the direct current is larger; for a fixed magnetic circuit, the larger the magnetic flux is, the higher the magnetic saturation degree is, and the larger the magnetic resistance is, namely the magnetic resistance of the magnetic valve can be controlled by controlling the direct current; the magnetic resistance of the magnetic valve is changed, namely the magnetic flux flowing through the magnetic valve can be changed. There may be several pairs of dc coils per magnetic valve.
(2) The voltage regulating method is explained by an equivalent circuit of a magnetic dividing voltage regulating transformer:
fig. 8 shows an equivalent diagram of a magnetic circuit of a voltage-regulating transformer. Rm1、Rm2、RmzThe magnetic resistances of the primary winding, the secondary winding and the magnetic valve iron core are respectively.
Fig. 9 shows an equivalent circuit diagram of the magnetic-dividing voltage-regulating transformer.
<math> <mrow> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math> <math> <mrow> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math>
<math> <mrow> <msub> <mi>X</mi> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>.</mo> </mrow> </math>
Figure BSA00000772249800064
lc、Sc、μcRespectively the equivalent length, the sectional area, the magnetic conductivity and the mu of the magnetic circuit of the magnetic valvecB/H. With saturation of the magnetic shunt circuit, mucConstantly decreasing, RmzThen it is increased continuously, at this time, Xc1、Xc2Constantly decreasing, XmAre increasing continuously.
When R ismz→∞,Xc1→0、Xc2→0、
Figure BSA00000772249800065
The magnetic-dividing voltage-regulating transformer at this time has become a conventional transformer.
It is clear that with Xc1→0、Xc2→0,U′2→U1Namely, the transmission voltage drop of the transformer is constantly changed, and the voltage regulation process is realized.
(3) The voltage regulation method is explained by the relation between the primary side voltage and the secondary side voltage:
<math> <mrow> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mfrac> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mi>j&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>/</mo> <msub> <mi>Z</mi> <mi>L</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> </mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>j&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>/</mo> <msub> <mi>Z</mi> <mi>L</mi> </msub> <mo>)</mo> </mrow> </mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> </mrow> </math> ZL=RL+jXL
with saturation of the magnetic shunt circuit, mucConstantly decreasing, RmzThen increase continuously, at this time U2Continuously increasing; when R ismz→∞,
Figure BSA00000772249800072
The magnetic voltage regulating transformer at this time has become a conventional transformer.
According to aboveIn the principle, when the primary partial magnetic circuit magnetic valve is not saturated, the voltage U 'is output'2And the voltage is at the lower limit value of the voltage, and then the saturation degree of the magnetic circuit is controlled, so that the voltage can be regulated.
3. Parameter measuring method for magnetic-dividing voltage-regulating transformer
(1) The equivalent circuit of the shunt voltage-regulating transformer neglecting leakage reactance is shown in figure 10:
wherein, <math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mfrac> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> <msub> <mi>U</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> <msub> <mi>I</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow> </math>
<math> <mrow> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math> <math> <mrow> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math>
<math> <mrow> <msub> <mi>X</mi> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>.</mo> </mrow> </math>
(2) the equivalent circuit of the shunt voltage-regulating transformer considering leakage reactance is shown in fig. 11:
X1、X2is leakage reactance, Xc1、Xc2The magnetic dividing reactance generated by the magnetic dividing magnetic circuit to the primary coil and the secondary coil neglects the coil loss, and can be obtained by the following steps:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>X</mi> <mi>M</mi> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <msup> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>+</mo> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>-</mo> <msub> <mi>X</mi> <mi>M</mi> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mo>-</mo> <msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <msup> <msub> <mi>Z</mi> <mi>L</mi> </msub> <mo>&prime;</mo> </msup> </mtd> </mtr> </mtable> </mfenced> </math>
wherein <math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfrac> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> </mfrac> <mo>=</mo> <mfrac> <msup> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <msub> <mi>X</mi> <mn>2</mn> </msub> </mfrac> <mo>=</mo> <mfrac> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <msup> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <msub> <mi>U</mi> <mn>2</mn> </msub> </mfrac> <mo>=</mo> <mfrac> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <msup> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <msub> <mi>I</mi> <mn>2</mn> </msub> </mfrac> <mo>=</mo> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
(3) Leakage reactance X1、X2Is measured by
The leakage reactance actual measurement method fully saturates the magnetic valve iron core of the sub-magnetic circuit:
namely, it is <math> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>l</mi> <mi>c</mi> </msub> <mrow> <msub> <mi>&mu;</mi> <mi>c</mi> </msub> <msub> <mi>S</mi> <mi>c</mi> </msub> </mrow> </mfrac> <mo>&RightArrow;</mo> <mo>&infin;</mo> </mrow> </math>
At this time: <math> <mrow> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&RightArrow;</mo> <mn>0</mn> <mo>,</mo> </mrow> </math>
<math> <mrow> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&RightArrow;</mo> <mn>0</mn> <mo>,</mo> </mrow> </math>
<math> <mrow> <msub> <mi>X</mi> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&RightArrow;</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> </mfrac> <mo>.</mo> </mrow> </math>
the equivalent circuit of the magnetic-dividing voltage-regulating transformer is shown in fig. 12:
the above circuit is a conventional transformer, and the leakage reactance can be measured by a conventional method.
(4) Partial magnetic impedance Xc1、Xc2And excitation impedance XmMethod for actually measuring value
The magnetic circuit magnetic valve iron cores of the partial magnetic circuits have different saturation degrees and corresponding partial magnetic impedances Xc1、Xc2And excitation impedance XmThe values are also different, and the specific actual measurement method is as follows:
firstly, the secondary side is disconnected, and the following equivalent circuit diagram 13 can be obtained:
thus, it is possible to obtain:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> <mo>+</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mo>-</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </math>
let the primary side turn off, the following equivalent circuit diagram 14 can be obtained:
thus, it is possible to obtain:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msup> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <msup> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>+</mo> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> <mo>&prime;</mo> </msup> <mo>-</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> <msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> <mo>&prime;</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> <msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> <mo>&prime;</mo> </msup> </mtd> </mtr> </mtable> </mfenced> </math>
the impedance X can be calculated by no-load experimentc1、Xc2And excitation impedance XmThe value is obtained.
4. The magnetic valve structure and the control method of the magnetic valve excitation direct current magnitude are as follows:
the magnetic valve is formed by connecting two magnetic columns in parallel and then connecting the two magnetic columns with a main magnetic circuit into a whole; the two magnetic columns are respectively wound with coils with the same number of turns, the two coils are called as a pair of coils, each magnetic valve can have a plurality of pairs of coils, and the connection mode is shown in the following figure. One end is directly combined, and the other end is connected with a direct current power supply in series. When direct current flows in the pair of coils, the generated magnetic flux forms a loop in the two magnetic columns; the larger the direct current is, the larger the loop magnetic flux is; for a fixed iron core magnetic circuit, the larger the magnetic flux is, the higher the magnetic saturation degree is, namely the larger the magnetic resistance is; therefore, the magnetic resistance of the magnetic valve can be controlled by controlling the direct current, and the alternating current magnetic flux in the magnetic valve can be controlled.
Referring to FIGS. 15 and 16, a method of controlling the magnitude of magnetic valve excitation DC is shown. Real-time acquisition of secondary voltage U of transformer2According to a given desired value U of the secondary voltage of the transformer2eCalculating Δ U2=U2-U2eAccording to Δ U2Calculating magnetic valve excitation direct current IdThe magnitude of the direct current is changed by adjusting the magnitude of the potential of the direct current power supply or the magnitude of the loop resistance of the direct current power supply, and finally the secondary output voltage of the transformer is changed; secondly, collecting the secondary voltage U of the transformer in real time2… … to continuously correct the secondary side output voltage U of the transformer2To make it approach the expected value U2e
5. Applications of
The transformer can be widely applied to urban and rural public distribution and transformation, is more important for independent users, especially rural long-distance power transmission, and can stabilize the voltage quality of a user side and ensure the safe and energy-saving operation of electrical equipment.
6. The equivalent circuit reasoning method of the magnetic-dividing voltage-regulating transformer comprises the following steps:
(1) the equivalent circuit parameters are shown in FIG. 17
In the magnetic valve, the direct currents are equal in magnitude and opposite in direction, so that the magnetic valve has no influence on an external circuit, the saturation of the magnetic valve is influenced only by the magnitude of the direct currents, and R is influencedmzSince the magnitude of (c) has an influence, the magnetic circuit diagram is simplified as shown in fig. 18:
from kirchhoff's law for magnetic circuits:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>&phi;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>&phi;</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>i</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>&phi;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> <mo>=</mo> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>i</mi> <mn>1</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </math>
and by sigma phii0, gives phiz=φ12
Thus, there were obtained:
<math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>N</mi> <mi>i</mi> </msub> <msub> <mi>i</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>i</mi> <mn>1</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mo>+</mo> <mi>R</mi> </mrow> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow> </math> is provided with <math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>k</mi> <mn>11</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>i</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>12</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>k</mi> <mn>21</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>i</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>22</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </math>
Namely, it is <math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow> </math> Is provided with <math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>k</mi> <mn>11</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>12</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>k</mi> <mn>21</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>22</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </math>
The magnetic circuit is then analyzed from a phasor perspective:
it is known that
Figure BSA00000772249800111
Then it can be derived (neglecting the leakage flux):
<math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mover> <mi>E</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mi>j&omega;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <msub> <mover> <mi>E</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>&DoubleRightArrow;</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mi>j&omega;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>11</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>12</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>21</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>22</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>
from the mutual inductance analysis, the following circuit diagram can be obtained as shown in fig. 19:
by definition of mutual inductance <math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mi>j&omega;</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mi>j&omega;M</mi> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <mi>j&omega;M</mi> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </math>
Wherein, <math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>&omega;M</mi> <mo>=</mo> <mi>&omega;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>k</mi> <mn>12</mn> </msub> <mo>=</mo> <mi>&omega;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <mfrac> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mi>&omega;</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>k</mi> <mn>11</mn> </msub> <mo>=</mo> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mfrac> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mi>&omega;</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>k</mi> <mn>22</mn> </msub> <mo>=</mo> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mfrac> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
namely, it is <math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>&omega;M</mi> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mfrac> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mi>&omega;</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> <mrow> <mfrac> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mfrac> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mi>&omega;</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mfrac> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
As can be seen from the above equation,
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>&omega;</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mfrac> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mi>&omega;</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mfrac> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
therefore, the transformer can be separated from a conventional transformer (neglecting leakage flux), and the equivalent is that the following circuit diagram is shown in fig. 20:
wherein:
<math> <mrow> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math> <math> <mrow> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </math>
<math> <mrow> <mi>&omega;</mi> <msubsup> <mi>L</mi> <mn>1</mn> <mo>&prime;</mo> </msubsup> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math> <math> <mrow> <mi>&omega;</mi> <msubsup> <mi>L</mi> <mn>2</mn> <mo>&prime;</mo> </msubsup> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </math>
<math> <mrow> <mi>&omega;M</mi> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </math>
from fig. 20, the following equation can be derived:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mi>j</mi> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mfrac> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mi>j</mi> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mfrac> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mfrac> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mi>j</mi> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <mi>&omega;M</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </math>
from the above equation, the following equivalent circuit diagram can be drawn as shown in fig. 21:
wherein, <math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msup> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mfrac> <msub> <mi>X</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> <msub> <mi>U</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>&CenterDot;</mo> </msup> <mo>=</mo> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> <msub> <mi>I</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow> </math>
<math> <mrow> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math> <math> <mrow> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math>
<math> <mrow> <msub> <mi>X</mi> <mi>m</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>.</mo> </mrow> </math>
taking the leakage magnetic flux into consideration, the equivalent circuit diagram is shown in fig. 22:
X1、X2is leakage reactance, Xc1、Xc2The magnetic shunt reactance generated by the magnetic valve to the primary coil and the secondary coil neglects the coil loss, and can be obtained by the following steps:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <msup> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>+</mo> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>-</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <mo>=</mo> <mo>-</mo> <msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <msup> <msub> <mi>Z</mi> <mi>L</mi> </msub> <mo>&prime;</mo> </msup> </mtd> </mtr> </mtable> </mfenced> </math>
wherein <math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfrac> <msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>&prime;</mo> </msup> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> </mfrac> <mo>=</mo> <mfrac> <msup> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <msub> <mi>X</mi> <mn>2</mn> </msub> </mfrac> <mo>=</mo> <mfrac> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <msup> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <msub> <mi>U</mi> <mn>2</mn> </msub> </mfrac> <mo>=</mo> <mfrac> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <msup> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>&prime;</mo> </msup> <msub> <mi>I</mi> <mn>2</mn> </msub> </mfrac> <mo>=</mo> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
(2) Conversion of mutual inductance
-
As shown in fig. 23;
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>N</mi> <mi>i</mi> </msub> <msub> <mi>i</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>i</mi> <mn>1</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mo>+</mo> <mi>R</mi> </mrow> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
carrying out no-load test on the transformer:
1) opening a secondary side:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>10</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>N</mi> <mi>i</mi> </msub> <msub> <mi>i</mi> <mn>10</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>20</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>i</mi> <mn>10</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mo>+</mo> <mi>R</mi> </mrow> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
this can result in:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mi>j&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
defined by mutual and self inductance:
mutual inductance: <math> <mrow> <mfrac> <mrow> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> </mrow> <mrow> <mi>j</mi> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> </mrow> </mfrac> <mo>=</mo> <mi>&omega;M</mi> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </math>
self-induction: <math> <mrow> <mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> <mrow> <mi>j</mi> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>j&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msup> <msub> <mi>N</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mfrac> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> <mi>&omega;M</mi> </mrow> </math>
2) opening the primary side:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>10</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>20</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&phi;</mi> <mn>20</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>20</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
this can result in:
<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mi>j&omega;</mi> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mrow> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <mi>j&omega;</mi> <msub> <mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> </mrow> <mn>20</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </math>
defined by mutual and self inductance:
mutual inductance: <math> <mrow> <mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>10</mn> </msub> <mrow> <mi>j</mi> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> </mrow> </mfrac> <mo>=</mo> <mi>&omega;M</mi> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </math>
self-induction: <math> <mrow> <mo>-</mo> <mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> <mrow> <mi>j</mi> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>20</mn> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <msub> <mi>X</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> <mi>&omega;M</mi> </mrow> </math>
(3) the primary and secondary voltage relationships are shown in fig. 24:
by <math> <mrow> <msub> <mi>&phi;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <msub> <mi>&phi;</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>i</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </math>
Can obtain the product <math> <mrow> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </math>
Thus, it is possible to prevent the occurrence of, <math> <mrow> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>&ap;</mo> <msub> <mover> <mi>E</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mi>j&omega;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>R</mi> <mi>mz</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mrow> <mi>j&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </math>
by <math> <mrow> <msub> <mover> <mi>&Phi;</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mrow> <mi>j&omega;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> </mrow> </mfrac> </mrow> </math> The obtained product is substituted into the raw material to obtain,
<math> <mrow> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mrow> <mi>j&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </math>
handle <math> <mrow> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <msub> <mi>Z</mi> <mi>L</mi> </msub> </mfrac> </mrow> </math> Substituting to obtain:
<math> <mrow> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mfrac> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mi>j&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>/</mo> <msub> <mi>Z</mi> <mi>L</mi> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> </mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mfrac> <msub> <mi>N</mi> <mn>2</mn> </msub> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>j&omega;</mi> <msup> <msub> <mi>N</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>/</mo> <msub> <mi>Z</mi> <mi>L</mi> </msub> <mo>)</mo> </mrow> </mfrac> <msub> <mover> <mi>U</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> </mrow> </math> ZL=RL+jXL
(4)Rmzand (3) calculating:
the direct current exciting current only has an effect on the magnetic conductivity of the magnetic valve iron core, and has no influence on the primary and secondary magnetic circuits. i.e. icIs caused by a change in ucThe following are specific changes:
by H c = N c i c l c , B c = f ( H c ) = f ( N c i c l c ) , To obtain <math> <mrow> <msub> <mi>&mu;</mi> <mi>c</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>B</mi> <mi>c</mi> </msub> <msub> <mi>H</mi> <mi>c</mi> </msub> </mfrac> </mrow> </math> Then the magnetic valve reluctance <math> <mrow> <msub> <mi>R</mi> <mi>mc</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>l</mi> <mi>c</mi> </msub> <mrow> <msub> <mi>&mu;</mi> <mi>c</mi> </msub> <msub> <mi>S</mi> <mi>c</mi> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mi>c</mi> </msub> <msub> <mi>H</mi> <mi>c</mi> </msub> </mrow> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <msub> <mi>S</mi> <mi>c</mi> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </math> Magnetic resistance of magnetic circuit <math> <mrow> <msub> <mi>R</mi> <mi>mz</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>l</mi> <mi>c</mi> </msub> <mrow> <mn>2</mn> <msub> <mi>&mu;</mi> <mi>c</mi> </msub> <msub> <mi>S</mi> <mi>c</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>l</mi> <mn>3</mn> </msub> </mrow> <mrow> <msub> <mi>&mu;</mi> <mn>3</mn> </msub> <msub> <mi>S</mi> <mn>3</mn> </msub> </mrow> </mfrac> <mo>.</mo> </mrow> </math>
(whereincLength of the magnetic valve, /)3The length of the magnetic valve connected with the primary iron core magnetic circuit and the secondary iron core magnetic circuit can be 0).

Claims (6)

1. A kind of voltage-regulating transformer of dividing magnetism, mainly by the closed iron core frame of main magnetic flux (1), primary coil (2), the secondary coil (3) makes up, its closed iron core frame of main magnetic flux is both sides to be the magnetic pole, there are primary coils (2) on one side, another side has secondary coils (3), its characteristic is to connect in parallel one or more magnetic valves, its magnetic valve mainly has magnetic valve closed iron core frame (4) and coil, its closed iron core frame of magnetic valve (4) is the magnetic pole relatively both sides, there are one or more pairs of coils of the same number of turns on the magnetic pole of both sides respectively, each pair of coils is divided and set up on the magnetic pole of both sides, one end of each pair of coils is combined directly, another end connects the power in series; a connecting iron core (5) is arranged between the main flux closed iron core frame (1) and the magnetic valve closed iron core frame (4), and the main flux closed iron core frame (1), the connecting iron core (5) and the magnetic valve closed iron core frame (4) are of an integral structure.
2. A transformer according to claim 1, characterized in that the magnetic valves are in parallel relation in the middle, both sides or three-dimensional space of the main flux closed core frame (1).
3. The transformer according to claim 1, the magnetic valve core frame (4) is rectangular, circular, oval, polygonal.
4. The transformer of claim 1, wherein the series power supply is a dc power supply.
5. A transformer according to claim 4, characterized in that the DC power is rectified from the output power of the secondary winding (3) of the transformer.
6. A method for regulating voltage by dividing magnetism of a transformer is characterized in that a main magnetic flux closes an iron core frame (1), the whole magnetic circuit is divided into a primary magnetic circuit and a secondary magnetic circuit according to a primary coil (2) and a secondary coil (3), and one or more magnetic valves are connected in parallel to form one or more magnetic dividing magnetic circuits; the magnetic dividing magnetic circuit changes the magnetic flux of a loop passing through the secondary winding by controlling the saturation degree of the magnetic valve, adjusts the size of secondary output voltage, and outputs constant voltage according to set requirements.
CN201210321729XA 2012-09-04 2012-09-04 Magnetism-shunting voltage regulation type transformer and magnetism-shunting voltage regulation method thereof Pending CN102810389A (en)

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Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3659191A (en) * 1971-04-23 1972-04-25 Westinghouse Electric Corp Regulating transformer with non-saturating input and output regions
US4177418A (en) * 1977-08-04 1979-12-04 International Business Machines Corporation Flux controlled shunt regulated transformer
US4213084A (en) * 1977-05-20 1980-07-15 Tdk Electronics Company Limited Variable leakage transformer
US4841428A (en) * 1988-03-18 1989-06-20 Hughes Aircraft Company Non-saturating magnetic amplifier controller
CN2066614U (en) * 1990-03-16 1990-11-28 沈阳变压器厂 T shape binding transformer of continuous adjusting voltage

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3659191A (en) * 1971-04-23 1972-04-25 Westinghouse Electric Corp Regulating transformer with non-saturating input and output regions
US4213084A (en) * 1977-05-20 1980-07-15 Tdk Electronics Company Limited Variable leakage transformer
US4177418A (en) * 1977-08-04 1979-12-04 International Business Machines Corporation Flux controlled shunt regulated transformer
US4841428A (en) * 1988-03-18 1989-06-20 Hughes Aircraft Company Non-saturating magnetic amplifier controller
CN2066614U (en) * 1990-03-16 1990-11-28 沈阳变压器厂 T shape binding transformer of continuous adjusting voltage

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Application publication date: 20121205