Name: ac power and phase shift
Uploaded: 2024-07-08T13:19:51.000Z
Duration: 1 min 22 s
Channel: JoVE

对于具有正弦输入的电路，传递给无源元件的功率等于其两端电压与电流的乘积。
将电压和电流表示为正弦形式，并应用三角恒等式，可得到包含两项的瞬时功率表达式。
第一项是取决于电压与电流之间相位差的常数，而第二项是频率为电压或电流频率两倍的正弦函数。
对于包含电阻和电抗的混合电路，瞬时功率在正值和负值之间振荡。
在纯电阻电路中，电压与电流同相，导致瞬时功率耗散始终为正值。
对于纯电感和纯电容电路，电压与电流之间的相位差导致瞬时功率在正值和负值之间波动。
当通过电感器的电流或电容器两端的电压发生变化时，电感器和电容器会周期性地存储和释放能量。
在能量释放周期中，这些活性元件充当电源，将能量回馈到电路中，从而产生负的瞬时功率。

瞬时功率

瞬时功率在电路中是非常重要的，尤其是在处理正弦输入时。瞬时功率表示为 p(t)，是由元件两端的瞬时电压（v(t)）与流过元件的瞬时电流（i(t)）相乘所得出的。这种关系遵循无源符号约定，同时还代表了电气工程的基本原理。
<img alt="Equation 1" src="/files/ftp_upload/15117/15117_Equation_1.svg" style="text-align: center; height: 13px;">
在计算瞬时功率时，可以将其表示为两个不同项的组合。第一项是保持不变的，这通常会取决于电压和电流之间的相位差。第二项则是以正弦波振荡为特征，其中的频率是电压或电流的两倍。
<img alt="Equation 2" src="/files/ftp_upload/15117/15117_Equation_2.svg" style="text-align: center; height: 30px;">
在包含电阻和电抗元件的电路中，瞬时功率是动态的，会在正值和负值之间发生振荡。在电压和电流完全同相的纯电阻电路中，瞬时功率始终是正向耗散的。
相反，在以电感和电容元件为主的电路中，电压和电流之间所产生的相移会导致瞬时功率在正值和负值之间发生间歇性波动。这种现象可以归因于电感器和电容器在电流通过前者和电压出现在后者时能够循环存储和释放能量。在能量释放阶段，这些无功元件能够充当电源，可以有效地将能量回馈到电路中，从而产生负瞬时功率。电压、电流和功率动态之间的这种复杂相互作用强调了控制电路的基本原理。了解这些原理对于高效的能源管理和电气系统的设计是至关重要的。

对于一个具有正弦输入的电路，传递到无源元件的功率等于该元件两端电压与电流的乘积。  
表达式……

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

AC Steady State Power

交流稳态功率

KEY TERMS AND CONCEPTS

关键术语与概念

Instantaneous Power

ac power and phase shift

Instantaneous power in an electrical circuit comes from the voltage across an element and the current through it. It is written as p(t) and follows the passive sign convention. For sinusoidal input, this product helps show how power changes at each moment.

When instantaneous power is expanded, it has two parts. One part is constant and depends on the phase difference between voltage and current. The other part is a sinusoidal term that varies at twice the frequency of the voltage or current.

In a purely resistive circuit, voltage and current are in phase. In that case, instantaneous power stays positive and is always dissipated. In circuits with resistance and reactance, the power can swing between positive and negative values.

This change happens because inductors and capacitors store and release energy over time. As current passes through an inductor, or as voltage appears across a capacitor, these elements can give energy back to the circuit during release. That is why negative instantaneous power can appear when reactive elements act like power sources. Understanding these voltage, current, and power relationships is important for efficient energy use and electrical system design.

AC Power and Phase Shift

instantaneous power

Instantaneous power in an electrical circuit comes from the voltage across an element and the current through it. It is written as p(t) and follows the passive sign convention. For sinusoidal input, this product helps show how power changes at each moment. When instantaneous power is expanded, it has two parts. One part is constant and depends on the phase difference between voltage and current. The other part is a sinusoidal term that varies at twice the frequency of the voltage or current. <br...

average power

Power Consumption in AC Circuits

Power consumption in AC circuits is often described with average power. In practical electrical work, the changing value of instantaneous power is less useful than the average over a time period. Average power is found by integrating instantaneous power over that time and then dividing by the duration. The time-domain expression for average power comes from simplifying the equation for instantaneous power. In that form, one cosine term averages to zero over a complete cycle. The result does...

effective value of a periodic waveform

RMS Value for AC Power Measurement

The RMS value, or effective value, helps measure how much power a periodic voltage or current can deliver. It is the direct current, or DC, value that gives the same average power to a resistor as the periodic signal does. This idea is especially important in AC circuits. To find the effective value of current, calculate the root mean square, or RMS, current. First square the instantaneous current values over one period. Then find the mean of those squared values and take the square root. The...

complex power

Power Triangle in AC Circuits

Power triangle and complex power help describe how an AC load uses energy. Engineers use complex power to capture the combined effect of parallel loads. It gives a full picture of the power consumed by a specific load. Complex power is found by multiplying voltage by the complex conjugate of current. Using rms values, it is written with the rms voltage phasor and the complex conjugate of the rms current phasor. It can also be written in terms of load impedance. The magnitude of complex power...

conservation of ac power

AC Power in Series and Parallel Circuits

AC power in series and parallel circuits follows a power preservation rule. In AC circuits, the complex power, real power, and reactive power produced by the source are equal to the total complex, real, and reactive powers absorbed by the loads. When two load impedances are connected in parallel to an AC source V, the source power can be written as the sum of the complex powers delivered to loads Z1 and Z2. If those same loads are connected in series with the voltage source, the complex power...

power factor

Power Factor and AC Load Behavior

Power factor describes how voltage and current behave in an AC load. It is the ratio of average, or active, power to apparent power. The power factor angle is the phase difference between voltage and current, and the cosine of that angle gives the power factor. It also matches the angle of the load impedance. Power factor shows how much apparent power becomes real, or average, power. Its value ranges from zero to one. When a load is purely resistive, voltage and current are in phase. In that...

power factor correction

Reducing Reactive Power in AC Loads

Power factor correction helps AC systems use electrical power more efficiently. In a factory circuit, transmitted power includes apparent power, which is made of active power and reactive power. Power factor is the ratio of real power, measured in kW, to apparent power, measured in kVA. It shows how much of the supplied power becomes useful work. A low power factor often happens when an inductive load draws too much reactive power. This creates a lagging power factor. It can also cause large...

the power superposition principle

Power from Multiple AC Sources

Power from multiple AC sources is found by adding the effects of each source separately. In a circuit with two sinusoidal voltage sources, each source acts independently. The superposition principle says the total current is the sum of the currents produced by each source alone. If those currents are labeled i1 and i2, they combine to give the total current. Instantaneous power is the power at one moment in time. In this circuit, it equals the square of the instantaneous current times the...

the maximum power transfer theorem

Matching Load Impedance for Maximum Power

Maximum power transfer in an AC circuit depends on matching the load impedance to the Thevenin equivalent of the source network. The load draws current, and the circuit delivers power to that load. To analyze the current, the voltages, currents, network impedance, and load impedance are written in rectangular form. The average power delivered to the load comes from the square of the current multiplied by the load resistance. To find the condition for maximum power, the power is differentiated...

key terms and concepts

Learn how instantaneous power depends on voltage, current, and phase shift in AC circuits, including resistive, inductive, and capacitive behavior.

AC Power and Phase Shift | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

AC Power and Phase Shift

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