Power calculation method of magnetic circuit

Abstract

Disclosed is a power calculation method of a magnetic circuit. In view of the power problem of a magnetic circuit and the phase problem of a magnetomotive force (MMF) and a magnetic flux in the magnetic circuit, the present disclosure draws a magnetic circuit vector diagram based on an equivalent magnetic circuit vector model, and provides a method for calculating virtual magnetic active power, virtual magnetic reactive power, and virtual magnetic complex power of the magnetic circuit by analyzing the MMF, the magnetic flux, the reluctance, and the magnetic reactance in the magnetic circuit by using the magnetic circuit vector diagram. A mathematical relationship between the virtual magnetic power of the magnetic circuit and the electric power of the corresponding equivalent electric circuit is derived according to a conversion factor between the virtual magnetic power and the electric power, so that the electric power can be directly calculated according to magnetic parameters such as the MMF and the magnetic flux in the magnetic circuit. The power calculation method of the magnetic circuit provided in the present disclosure can calculate and analyze the virtual magnetic power of the magnetic circuit according to the magnetic circuit vectors, so as to calculate the electric power from the magnetic circuit through conversion. The electric power can be solved according to the magnetic circuit vectors directly when the electric vectors are not available to calculate electric power in electromagnetic components.

Claims

1. A power calculation method of a magnetic circuit, comprising: S1. calculating, in a case that a to-be-measured magnetic circuit is in a stable operation, a magnetomotive force (MMF) {dot over (F)} and a magnetic flux {dot over (Φ)} in the magnetic circuit; S2. solving a magnetic impedance value Z.sub.mc and a magnetic impedance angle φ.sub.mc of the magnetic circuit according to a formula $\frac{\stackrel{.}{F}}{\stackrel{.}{\Phi }}=Z_{mc}{\mathrm{\angle \phi }}_{mc};$ S3. solving a reluctance value, a magnetic reactance value, and a magnetic-inductance value of the magnetic circuit respectively according to calculation formulas:
R.sub.mc=Z.sub.mc cos φ.sub.m,X.sub.mc=Z.sub.mc sin φ.sub.mc, and X.sub.mc=ωL.sub.mc; wherein R.sub.mc represents the reluctance value of the magnetic circuit, X.sub.mc represents the magnetic reactance value of the magnetic circuit, L.sub.mc represents the magnetic-inductance value, and ω represents an angular frequency of the magnetic flux varied in the magnetic circuit; S4. selecting a reference coordinate system, and drawing a magnetic circuit vector diagram; S5. performing an orthogonal decomposition on the MMF {dot over (F)} in the magnetic circuit with respect to the magnetic flux {dot over (Φ)} according to the magnetic circuit vector diagram, to obtain a magnetic voltage drop {dot over (F)}.sub.1 along a direction of the magnetic flux, and a magnetic voltage drop {dot over (F)}.sub.2 along a direction perpendicular to the direction of the magnetic flux; S6. calculating virtual magnetic active power of the magnetic circuit according to a formula P.sub.mc=∥{dot over (F)}.sub.2∥.Math.∥{dot over (Φ)}∥, and calculating virtual magnetic reactive power of the magnetic circuit according to a formula Q.sub.mc=∥{dot over (F)}.sub.1∥.Math.∥{dot over (Φ)}∥; S7. solving virtual magnetic complex power {dot over (S)}.sub.mc of the magnetic circuit according to formulas: ${\stackrel{.}{S}}_{mc}=.Math.{\stackrel{.}{S}}_{mc}.Math.{\mathrm{\angle \phi }}_{mc},.Math.{\stackrel{.}{S}}_{mc}.Math.=\sqrt{P_{mc}^2+Q_{mc}^2},\mathrm{and}{\phi }_{mc}=\mathrm{arc}\tan \left(\frac{P_{mc}}{Q_{mc}}\right);$ and S8. solving corresponding electric power according to a conversion factor between the virtual magnetic power of the magnetic circuit and corresponding electric power of an electric circuit, the conversion factor between the virtual magnetic power of the magnetic circuit and the corresponding electric power of the electric circuit being d{dot over (Φ)}/({dot over (Φ)}dt) and the conversion factor being jω in a case that the MMF and the magnetic flux in the magnetic circuit are sinusoidal, namely: electric active power P.sub.e=ωP.sub.mc, electric reactive power Q.sub.e=ω.sub.mc, and electric complex power {dot over (S)}.sub.e=jω{dot over (S)}.sub.mc.

2. The power calculation method of the magnetic circuit according to claim 1, further comprising: before calculating the virtual magnetic active power and the virtual magnetic reactive power of the magnetic circuit in S6, verifying whether a magnetic circuit topology comprising the magnetic-inductance component satisfies magnetic circuit Ohm's law, namely:
{dot over (F)}=(R.sub.mc+jωL.sub.mc){dot over (Φ)}; wherein j represents an imaginary unit, R.sub.mc represents the reluctance value of the magnetic circuit, ω represents the angular frequency of the magnetic flux varied in the magnetic circuit, L.sub.mc represents the magnetic-inductance value of the magnetic-inductance component, {dot over (Φ)} represents the magnetic flux in the magnetic circuit, and {dot over (F)} represents the MMF in the magnetic circuit.

3. The power calculation method of the magnetic circuit according to claim 1, wherein a magnitude of the magnetic-inductance value L.sub.mc of the magnetic-inductance component is related with a winding number N.sub.r of a short-circuited coil and a resistance value R.sub.r of the short-circuited coil, namely $L_{mc}=\frac{N_r^2}{R_r},$ and magnetic-inductance is measured in Ω.sup.−1; and the magnetic-inductance component has an obstructive effect on an alternating magnetic flux, but has no obstructive effect on a constant magnetic flux, and an expression for the magnetic reactance value is defined as X.sub.mc=ωL.sub.mc, to describe the degree of the obstructive effect of the magnetic-inductance component on the alternating magnetic flux, ω being the angular frequency of the magnetic flux varied in the magnetic circuit.

4. The power calculation method of the magnetic circuit according to claim 1, wherein an equivalent magnetic circuit vector model comprises such four magnetic circuit lumped variables as the MMF {dot over (F)}, the magnetic flux {dot over (Φ)}, the reluctance value R.sub.mc, and the magnetic-inductance value L.sub.mc; and according to the equivalent magnetic circuit vector model, the constructed electromagnetic vector diagram is capable of simultaneously showing phase relationships between electric circuit vectors and magnetic circuit vectors, in a case of linking the equivalent magnetic circuit vector model and an equivalent electric circuit model by using vectors.

5. The power calculation method of the magnetic circuit according to claim 1, wherein with reference to the magnetic circuit vector diagram, an expression for the virtual magnetic complex power of the magnetic circuit is {dot over (S)}.sub.mc={dot over (Φ)}.Math.{dot over (F)}*=∥{dot over (F)}∥.Math.∥{dot over (Φ)}∥e−.sup.jφ.sup.mc=R.sub.mc∥{dot over (Φ)}∥.sup.2−jωL.sub.mc∥{dot over (Φ)}∥.sup.2=Q.sub.mc−jP.sub.mc.

6. The power calculation method of the magnetic circuit according to claim 1, wherein the virtual magnetic active power of the magnetic circuit is defined as an imaginary part of the virtual magnetic complex power, and with reference to the magnetic circuit vector diagram, an expression for the virtual magnetic active power of the magnetic circuit is P.sub.mc=∥{dot over (F)}∥.Math.∥{dot over (Φ)}∥ sin φ.sub.mc∥{dot over (F)}.sub.2∥.Math.∥{dot over (Φ)}∥=X.sub.mc∥{dot over (Φ)}∥.sup.2=ωL.sub.mc∥{dot over (Φ)}∥.sup.2.

7. The power calculation method of the magnetic circuit according to claim 1, wherein the virtual magnetic reactive power of the magnetic circuit is defined as a real part of the virtual magnetic complex power, and with reference to the magnetic circuit vector diagram, an expression for the virtual magnetic reactive power of the magnetic circuit is Q.sub.mc=∥{dot over (F)}∥.Math.∥{dot over (Φ)}∥ cos φ.sub.mc=∥{dot over (F)}.sub.1∥.Math.∥{dot over (Φ)}∥=R.sub.mc∥{dot over (Φ)}∥.sup.2.

8. The power calculation method of the magnetic circuit according to claim 1, an expression for the electric active power being:
P.sub.e=ωP.sub.mc=ω∥{dot over (F)}∥.Math.{dot over (Φ)}∥ sin φ.sub.mc=ω∥{dot over (F)}.sub.2∥|∥{dot over (Φ)}∥=ωX.sub.mc∥{dot over (Φ)}∥.sup.2=ω.sup.2L.sub.mc∥{dot over (Φ)}∥.sup.2, an expression for the electric reactive power being:
Q.sub.e=ωQ.sub.mc=ω∥{dot over (F)}∥.Math.∥{dot over (Φ)}∥ cos φ.sub.mc=ω∥{dot over (F)}.sub.1∥.Math.∥{dot over (Φ)}∥=ωR.sub.mc∥{dot over (Φ)}∥.sup.2, and an expression for the electric complex power being: {dot over (S)}.sub.e=jω{dot over (S)}.sub.mc=ωP.sub.mc+jωQ.sub.mc=P.sub.e+jQ.sub.e.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows an equivalent magnetic circuit vector model according to the present disclosure.

(2) FIG. 2 shows a magnetic circuit vector diagram according to the present disclosure.

(3) FIG. 3 shows a flowchart of calculating power of a magnetic circuit according to the present disclosure.

(4) FIG. 4 shows waveforms of the exciting current and the magnetic flux of a to-be-measured magnetic circuit to which a magnetic-inductance component is added.

(5) FIG. 5 shows a comparison diagram of actual measured electric power and electric power calculated through conversion by applying the present disclosure.

DETAILED DESCRIPTION

(6) The technical solutions in the present disclosure are described in detail below with reference to the accompanying drawings.

(7) The present disclosure provides a power calculation method of a magnetic circuit, whose core content is as follows: Based on an equivalent magnetic circuit vector model, an MMF, a magnetic flux, a reluctance, and a magnetic reactance of a magnetic circuit are analyzed by using a provided magnetic circuit vector diagram, so as to provide a method for calculating virtual magnetic active power, virtual magnetic reactive power, and virtual magnetic complex power in the magnetic circuit, which correspond to active power, reactive power, and complex power in an electric circuit. The electric power of the magnetic circuit is calculated through conversion according to the virtual magnetic power of the magnetic circuit with reference to a provided conversion factor.

(8) The equivalent magnetic circuit vector model includes such four magnetic circuit lumped variables as an MMF {dot over (F)}, a magnetic flux {dot over (Φ)}, a reluctance R.sub.mc, and a magnetic-inductance L.sub.mc, which correspond to such four electric circuit lumped variables as a voltage {dot over (U)}, a current İ, a resistance R, and an inductance L in an equivalent electric circuit vector model, namely {dot over (F)}=(R.sub.mc+jωL.sub.mc){dot over (Φ)}. The equivalent magnetic circuit vector model is shown as in FIG. 1.

(9) In the equivalent magnetic circuit vector model, a calculation formula of the MMF {dot over (F)} is {dot over (F)}=N.sub.mİ.sub.m, ampere-turn being the unit of the MMF, N.sub.m being a winding number of an exciting coil, and İ.sub.m being a current in the exciting coil. The magnetic flux in the magnetic circuit satisfies

(10) $\stackrel{.}{\Phi }=\frac{\stackrel{.}{F}}{\left(R_{mc}+j\omega L_{mc}\right)}.$

(11) The reluctance in the magnetic circuit indicates a constant resistance effect of the magnetic circuit on the magnetic flux, which resists both an alternating magnetic flux and a constant magnetic flux. In a case of a constant MMF, a reluctance in the magnetic circuit including no magnetic-inductance component may change the magnitude of the magnetic flux, but does not change the phase of the magnetic flux. As an exciting frequency of the magnetic circuit increases, the reluctance value of the magnetic circuit increases accordingly due to a skin effect of the magnetic flux. As the magnetic flux of the magnetic circuit increases, the reluctance value of the magnetic circuit also increases due to saturation of the magnetic circuit.

(12) Further, the magnetic-inductance L.sub.mc inhibits the variation of the magnetic flux in the magnetic circuit. The magnetic-inductance has an obstructive effect on the alternating magnetic flux but no obstructive effect on the constant magnetic flux. A calculation formula of the magnetic-inductance component is

(13) $L_{mc}=\frac{N_r^2}{R_r},$
N.sub.t being a winding number of the magnetic-inductance component, and R.sub.t being a resistance value of the magnetic-inductance component. In order to describe the obstructive effect of the magnetic-inductance on the alternating magnetic flux, an expression for the magnetic reactance is defined as X.sub.mc=ωL.sub.mc, ω being an angular frequency of the magnetic flux varied in the magnetic circuit.

(14) A magnetic impedance value in the magnetic circuit may be calculated according to Z.sub.mc=√{square root over (R.sub.mc.sup.2+X.sub.mc.sup.2)}. A magnetic impedance angle in the magnetic circuit may be calculated according to ω.sub.mc=arctan(ωL.sub.mc/R.sub.mc). The reluctance value satisfies a formula R.sub.mc=Z.sub.mc cos φ.sub.mc. The magnetic reactance value satisfies a formula X.sub.mc=Z.sub.mc sin φ.sub.mc.

(15) Further, according to the equivalent magnetic circuit vector model, a magnetic circuit vector diagram may be drawn as shown in FIG. 2, Ė.sub.m representing an opposing electromotive force on the exciting coil. An orthogonal decomposition is performed on the MMF along a direction of the magnetic flux {dot over (Φ)} and a direction perpendicular to the direction of the magnetic flux {dot over (Φ)}, so as to obtain a magnetic voltage drop (corresponding to a virtual magnetic reactive component) on the reluctance as {dot over (F)}.sub.1={dot over (Φ)}R.sub.mc, and a magnetic voltage drop (corresponding to a virtual magnetic active component) on the magnetic-inductance as {dot over (F)}.sub.2=j{dot over (Φ)}X.sub.mc. In addition, {dot over (F)}={dot over (F)}.sub.1+{dot over (F)}.sub.2 is satisfied.

(16) According to the provided equivalent magnetic circuit vector model, a constructed electromagnetic vector diagram can simultaneously show phase relationships between electric circuit vectors and magnetic circuit vectors, in a case of linking the equivalent magnetic circuit model and an equivalent electric circuit model by using vectors (such as a magnetic flux vector).

(17) With reference to the magnetic circuit vector diagram, an expression for the virtual magnetic complex power of the magnetic circuit is:
{dot over (S)}.sub.mc={dot over (Φ)}.Math.{dot over (F)}*=∥{dot over (F)}∥.Math.∥{dot over (Φ)}∥e.sup.−jφ.sup.mc=R.sub.mc∥{dot over (Φ)}∥.sup.2−jωL.sub.mc∥{dot over (Φ)}∥.sup.2=Q.sub.mc−jP.sub.mc.

(18) The virtual magnetic active power of the magnetic circuit is defined as an imaginary part of the virtual magnetic complex power and an expression for the virtual magnetic active power of the magnetic circuit is P.sub.mc=∥{dot over (F)}∥.Math.∥{dot over (Φ)}∥ sin φ.sub.mc=∥{dot over (F)}.sub.2∥.Math.∥{dot over (Φ)}∥=X.sub.mc∥{dot over (Φ)}∥.sup.2=ωL.sub.mc∥{dot over (Φ)}∥.sup.2. The virtual magnetic reactive power of the magnetic circuit is defined as a real part of the virtual magnetic complex power, and an expression for the virtual magnetic reactive power of the magnetic circuit is Q.sub.mc=∥{dot over (F)}∥.Math.∥{dot over (Φ)}∥ cos φ.sub.mc=∥{dot over (F)}.sub.1∥.Math.∥{dot over (Φ)}∥=R.sub.mc∥{dot over (Φ)}∥.sup.2.

(19) The virtual magnetic power of the magnetic circuit satisfies the following relationships, namely, ∥{dot over (S)}.sub.mc∥=√{square root over (P.sub.mc.sup.2+Q.sub.mc.sup.2)}, and

(20) ${\phi }_{\mathrm{mc}}=\arctan \left(\frac{P_{\mathrm{mc}}}{Q_{\mathrm{mc}}}\right).$

(21) A conversion factor between the virtual magnetic power and electric power of the magnetic circuit is d{dot over (Φ)}({dot over (Φ)}dt). In particular, the conversion factor is jω in a case that the MMF and the magnetic flux in the magnetic circuit are sinusoidal.

(22) An expression for the electric active power is:
P.sub.e=ωP.sub.mc=ω∥{dot over (F)}∥.Math.∥{dot over (Φ)}∥ sin φ.sub.mc=ω∥{dot over (F)}.sub.2∥.Math.∥{dot over (Φ)}∥=ωX.sub.mc∥{dot over (Φ)}∥.sup.2=ω.sup.2L.sub.mc∥{dot over (Φ)}∥.sup.2.

(23) An expression for the electric reactive power is:
Q.sub.e=ωQ.sub.mc=ω∥{dot over (F)}∥.Math.∥{dot over (Φ)}∥ cos φ.sub.mc=ω∥{dot over (F)}.sub.1∥.Math.∥{dot over (Φ)}∥=ωR.sub.mc∥{dot over (Φ)}∥.sup.2.

(24) An expression for the electric complex power is:
{dot over (S)}.sub.e=jω{dot over (S)}.sub.mc=ωP.sub.mc+jωQ.sub.mc=P.sub.e+jQ.sub.e.

(25) Based on the foregoing equivalent magnetic circuit vector model, a specific process of a power calculation method of a magnetic circuit provided in the present disclosure is as follows:

(26) S1. Calculate, in a case that a to-be-measured magnetic circuit is in a stable operation, an MMF {dot over (F)} and a magnetic flux {dot over (Φ)} in the magnetic circuit.

(27) S2. Solve a magnetic impedance value Z.sub.mc and a magnetic impedance angle φ.sub.mc of the magnetic circuit according to a formula

(28) $\frac{\stackrel{.}{F}}{\stackrel{.}{\Phi }}=Z_{mc}{\mathrm{\angle \phi }}_{mc}.$

(29) S3. Solve a reluctance value of the magnetic circuit according to a formula R.sub.mc=Z.sub.mc cos φ.sub.mc, solve a magnetic reactance value of the magnetic circuit according to a formula X.sub.mc=Z.sub.mc sin φ.sub.mc, and solve a magnetic-inductance value of the magnetic circuit according to a formula X.sub.mc=ωL.sub.mc.

(30) S4. Select a reference coordinate system, and draw a magnetic circuit vector diagram.

(31) S5. Perform an orthogonal decomposition on the MMF {dot over (F)} in the magnetic circuit with respect to the magnetic flux {dot over (Φ)} according to the magnetic circuit vector diagram, to obtain a magnetic voltage drop {dot over (F)}.sub.1 corresponding to a virtual magnetic reactive component along a direction of the magnetic flux, and a magnetic voltage drop {dot over (F)}.sub.2 corresponding to a virtual magnetic active component along a direction perpendicular to the direction of the magnetic flux.

(32) S6. Calculate virtual magnetic active power of the magnetic circuit according to a formula P.sub.mc=∥{dot over (F)}.sub.2∥.Math.∥{dot over (Φ)}∥), and calculate virtual magnetic reactive power of the magnetic circuit according to a formula Q.sub.mc=∥{dot over (F)}.sub.1∥.Math.∥{dot over (Φ)}∥.

(33) S7. Solve virtual magnetic complex power of the magnetic circuit according to formulas

(34) $.Math.{\stackrel{.}{S}}_{mc}.Math.=\sqrt{P_{mc}^2+Q_{mc}^2}\mathrm{and}{\phi }_{mc}=\mathrm{arc}\tan \left(\frac{P_{mc}}{Q_{mc}}\right).$

(35) S8. Solve corresponding electric power according to a conversion factor jω between the virtual magnetic power and the electric power of the magnetic circuit, namely, P.sub.e=ωP.sub.mc, Q.sub.e=ωQ.sub.mc, and {dot over (S)}.sub.e=jω{dot over (S)}.sub.mc.

(36) In order to calculate the active power and the reactive power of a transformer, a flowchart of a power calculation method of a magnetic circuit is drawn as in FIG. 3. First, in a case that the transformer is running with no load, a reluctance value R.sub.mc=22343.6 H.sup.−1 and an initial magnetic-inductance value L.sub.mc0=43.34 Ω.sup.−1 of the magnetic circuit are solved according to the formula {dot over (F)}.sub.0=(R.sub.mc+jωL.sub.mc0){dot over (Φ)}. The reluctance value R.sub.mc basically remains unchanged, in a case that an exciting frequency of the magnetic circuit and the magnitude of the magnetic flux in the magnetic circuit are kept unchanged.

(37) In a case that an exciting frequency of the magnetic circuit is kept at 50 Hz and the amplitude of the magnetic flux is kept constant, a magnetic-inductance component L.sub.md constructed by using a short-circuited coil is added to the magnetic circuit, and a magnetic-inductance value is calculated as 68.353 Ω.sup.−1 according to the formula

(38) $L_{mc1}=\frac{N_r^2}{R_r}.$
In this case, an equivalent magnetic circuit vector model of the transformer has a reluctance of R.sub.mc=22343.6 H.sup.−1 and a magnetic-inductance value of L.sub.mc2=L.sub.mc0+L.sub.mc1=111.6 Ω.sup.−1.

(39) Waveforms of an exciting current and the magnetic flux of the transformer to which the magnetic-inductance component is added are shown in FIG. 4. According to FIG. 4, the amplitudes and the phases of the MMF vector {dot over (F)}=N.sub.mİ.sub.m and the magnetic flux vector {dot over (Φ)} of the transformer are calculated. A magnetic impedance value Z.sub.mc and a magnetic impedance angle φ.sub.mc of the magnetic circuit are calculated according to a formula

(40) 0$\frac{\stackrel{.}{F}}{\stackrel{.}{\Phi }}=Z_{mc}{\mathrm{\angle \phi }}_{mc}.$
Through calculation, the magnetic impedance value is 41038.6 Ω.sup.−1, and the magnetic impedance angle is 57.7°. According to the formula R.sub.mc=Z.sub.mc cos φ.sub.mc, the reluctance value may be obtained as 21929.07 H.sup.−1. According to the formula ωL.sub.mc=Z.sub.mc sin φ.sub.mc, the magnetic-inductance value may be obtained as 110.4165 Ω.sup.−1. Therefore, the equation {dot over (F)}=(R.sub.mc+jωL.sub.mc2){dot over (Φ)} is verified to be true.

(41) According to the MMF {dot over (F)}, the magnetic flux {dot over (Φ)}, the reluctance R.sub.mc, and the magnetic reactance X.sub.mc of the magnetic circuit of the transformer, the vector diagram of the equivalent magnetic circuit model of the transformer may be drawn as shown in FIG. 2, Ė.sub.m representing an opposing electromotive force on an exciting coil. By performing the orthogonal decomposition on the MMF {dot over (F)} along the direction of the magnetic flux {dot over (Φ)} and the direction perpendicular to the direction of the magnetic flux {dot over (Φ)}, the magnetic voltage drop on the reluctance may be obtained as {dot over (F)}.sub.1={dot over (Φ)}R.sub.mc and the magnetic voltage drop on the magnetic-inductance may be obtained as {dot over (F)}.sub.2=j{dot over (Φ)}X.sub.mc. The virtual magnetic active power of the magnetic circuit is solved according to the formula P.sub.mc=∥{dot over (F)}.sub.2∥.Math.∥{dot over (Φ)}∥, and the virtual magnetic reactive power of the magnetic circuit is solved according to the formula Q.sub.mc=∥{dot over (F)}.sub.1∥.Math.∥{dot over (Φ)}∥. Then, according to the conversion factor jω, the active power of the transformer is calculated as P.sub.e=ωP.sub.mc, and the reactive power of the transformer is calculated as Q.sub.e=ωQ.sub.mc. The calculated active power P.sub.e of the magnetic circuit is compared with the active power measured by a power analyzer, the calculated reactive power Q.sub.e of the magnetic circuit is compared with the measured reactive power, and the result is shown in FIG. 5. The calculation error of the active power calculated using the magnetic circuit theory is 2.86%, and the calculation error of the reactive power calculated using the magnetic circuit theory is 4%. The result better indicates the correctness and effectiveness of the magnetic circuit calculation method provided in the present disclosure.

(42) In conclusion, the present disclosure provides a power calculation method of a magnetic circuit. The foregoing descriptions are merely exemplary implementations of the present disclosure, and the protection scope of the present disclosure is not limited by the foregoing implementations. A person of ordinary skill in the art may make equivalent modifications or changes according to the contents disclosed by the present disclosure, and such equivalent modifications or changes shall fall within the protection scope recorded in the claims.