26.1
理想的变压器
在单相双绕组变压器中，两个绕组绕在磁芯上，磁芯的横截面积为 A，磁导率为 μ。相量电流 i_1 进入左绕组，而 i_2 流出右绕组，通过电磁原理建立变压器的基本工作原理。
安培定律是理解变压器内磁场的基础。它指出，磁场强度沿闭合路径的切向分量的积分等于净封闭电流。对于变压器，磁路的平均长度用作此闭合…
在单相双绕组变压器中，两个绕组缠绕在一个具有横截面积 A 和磁导率 µ 的磁芯上。相量电流 i1 流入左侧绕组，而 i2 从右侧绕组流出。
安培定律指出，沿闭合路径积分的磁场强度切向分量等于该路径所包围的净电流。磁路的平均长度被选为该闭合路径，其中磁场强度相量为常数。
代入磁通密度和磁芯磁通后，可得到磁路的欧姆定律，其中净磁动势等于磁芯磁阻与磁芯磁通的乘积。
对于理想变压器，磁芯磁导率是无限的，意味着磁芯磁阻为零。因此，相电流同相。
法拉第定律定义了由随时间变化的磁通量所感应出的电压。假设磁通量为正弦稳态且频率恒定，并完全限制在铁芯内，则可确定绕组上感应的电压及其相互关系。
将匝数比代入基本关系式，有助于计算流入和流出绕组的复功率，二者相等表明没有功率损耗。
View the full transcript and gain access to JoVE Core videos
Q1: How does Ampere's Law apply to transformer magnetic circuits?
Ampere's Law states that the magnetic field intensity's tangential component integrated along a closed path equals the net enclosed current. In transformers, the mean length of the magnetic circuit serves as this closed path, where the magnetic field intensity remains constant. This principle relates the magnetomotive force from primary and secondary windings to the core flux through the magnetic circuit's reluctance.
Q2: What does infinite core permeability mean for an ideal transformer?
Infinite core permeability implies zero core reluctance, meaning the magnetic core offers no resistance to magnetic flux. This ideal condition ensures that phase currents in both windings are in phase and that the magnetomotive force relationship simplifies significantly. Real transformers approach but never achieve this ideal condition, making equivalent circuits for practical transformers necessary for accurate analysis.
Q3: How does Faraday's Law determine voltage in transformer windings?
Faraday's Law defines the voltage induced by time-varying magnetic flux. Under sinusoidal steady-state conditions at constant frequency, the induced voltage in each winding is proportional to the number of turns and the rate of flux change. The turn ratio directly relates primary and secondary voltages, enabling voltage transformation between windings.
Q4: What is the relationship between turn ratio and transformer voltage transformation?
The turn ratio, defined as the primary turns divided by secondary turns, determines the voltage relationship between windings. By substituting this ratio into Faraday's Law equations, the secondary voltage can be calculated from the primary voltage. This relationship enables transformers with off nominal turns ratios to step voltage up or down as needed in power systems.
Q5: Why is power conserved in an ideal transformer?
Complex power entering the primary winding equals complex power leaving the secondary winding in an ideal transformer. This equality, expressed as S1=S2, indicates zero power loss within the transformer core and windings. Power conservation is fundamental to ideal transformer operation, ensuring that all input power transfers to the output without dissipation.
Q6: How do magnetic flux density and core flux relate in transformer analysis?
Magnetic flux density (B) and core flux (Φ) are related through the core's cross-sectional area. Substituting this relationship into Ampere's Law yields Ohm's law for magnetic circuits, where magnetomotive force equals reluctance times flux. This magnetic circuit analogy parallels electrical circuits, simplifying transformer analysis and design calculations.
Q7: What role does the magnetic core's cross-sectional area play in transformer operation?
The magnetic core's cross-sectional area A directly influences the relationship between magnetic flux density and core flux. A larger cross-sectional area accommodates greater flux for the same flux density, affecting the transformer's capacity to transfer power. Core geometry, including area and mean length, determines the magnetic reluctance and overall transformer performance characteristics.