Name: power from multiple ac sources
Uploaded: 2024-07-08T13:19:54.000Z
Duration: 1 min 19 s
Channel: JoVE

考虑一个包含两个正弦电压源的电路。
根据叠加原理，两个电源共同作用产生的总电流等于每个电源单独作用时产生的电流之和。
此外，瞬时功率等于瞬时电流的平方乘以电阻。
对一个周期内进行积分可得到平均功率，该平均功率等于各独立电源产生的平均功率之和，再加上一个第三项——对于频率不同的电源，该项等于零。
简化后的方程表示功率叠加原理，即不同频率的电源所提供的平均功率等于各个电源单独作用时所提供平均功率的总和。
这一原理在电信领域（如无线电广播）中至关重要。
多个广播电台同时发射信号，每个电台都作为电磁波的独立波源。
根据功率叠加原理，无线电接收到的总功率等于各发射站发射功率之和。
将收音机调谐至特定频率，可使用户在接收多个电台信号的同时，一次仅收听一个电台。

功率叠加原理

The Power Superposition Principle

假设有一个具有两个正弦电压源的电路。每个正弦电压源都独立地影响着电路，叠加原理能够通过将每个源的响应进行相加来帮助我们理解综合效应。
<img alt="Figure 1" src="/files/ftp_upload/15124/15124_Figure_1.svg" style="text-align: center; height: 150px;">
根据叠加原理可以得知，两个源的总电流是每个电源中独立电流的总和。当两个电源一起工作时，将它们各自的电流（i_1和i_2）进行相加便能够得到总电流。此外，瞬时功率等于瞬时电流的平方乘以电阻。
<img alt="Equation 1" src="/files/ftp_upload/15124/15124_Equation_1.svg" style="text-align: center; height: 20px;">
这一概念也同样适用于功率，即能量使用的速率。在任一给定的时刻内，电路中的瞬时功率等于电流的平方乘以电阻。在一个周期内对其进行积分便能够得出平均功率，该功率等于每个电源的平均功率之和，对于具有不同频率的电源来说，第三项则会变为零。
<img alt="Equation 2" src="/files/ftp_upload/15124/15124_Equation_2.svg" style="text-align: center; height: 42px;">
<img alt="Equation 3" src="/files/ftp_upload/15124/15124_Equation_3.svg" style="text-align: center; height: 42px;">
简化后的公式能够用来表示功率叠加原理，即不同频率的电源所输出的平均功率等于每个源输出的平均功率之和。这一原理在电信领域（例如无线电广播）是至关重要的。多个广播电台能够同时广播信号，每个电台都可以作为独立的电磁波源。根据功率叠加原理可以得知，无线电接收的总功率等于每个电台发射功率之和。将收音机调到特定的频率，用户就可以一次只收听一个电台，尽管收音机会收到来自多个电台的信号。

考虑一个包含两个正弦电压源的电路。  
根据叠加原理，由这两个电源共同产生的总电流等于各个电源单独作用时所产生的电流之和……

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

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 resistance. To find the average power, this expression is integrated over one period. The result shows that the average power from both sources is the sum of the average powers from each source, and a third term becomes zero when the sources have different frequencies.

This idea is called the power superposition principle. It means that the average power delivered by sources with different frequencies is the sum of the average power delivered by each source. The principle is important in telecommunications, including radio broadcasting. Multiple radio stations can broadcast at the same time, and each one acts as an independent source of electromagnetic waves. A radio receives the total power from all of them, but tuning to one frequency lets the listener hear one station at a time.

Power from Multiple AC Sources

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. <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 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 superposition adds currents from two sinusoidal sources and why average power from different frequencies can be combined in AC circuits and radio.

Power from Multiple AC Sources | 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

Power from Multiple AC Sources

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