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

For a circuit with a sinusoidal input, the power delivered to a passive element equals the product of the voltage and current across it.
Expressing voltage and current in a sinusoidal form and applying the trigonometric identity gives the expression for instantaneous power with two terms.
The first term is a constant dependent on the phase difference between the voltage and current, while the second term is a sinusoidal function with twice the frequency of the voltage or current.
For circuits with mixed resistance and reactance, the instantaneous power oscillates between positive and negative values.
In a purely resistive circuit, the voltage and current are in phase, resulting in a consistently positive instantaneous power dissipation.
For purely inductive and capacitive circuits, the phase shift between the voltage and current causes the instantaneous power to fluctuate between positive and negative values.
Inductors and capacitors cyclically store and release energy as the current through the inductor or voltage across the capacitor changes.
During the energy release cycle, the reactive elements act as a power source, supplying energy back to the circuit, leading to negative instantaneous power.

instantaneous power

Instantaneous power is important in electrical circuits, mainly when dealing with sinusoidal input. Instantaneous power, denoted as p(t), results from the multiplication of the instantaneous voltage (v(t)) across an element and the instantaneous current (i(t)) flowing through it. This relationship adheres to the passive sign convention and represents a fundamental principle in electrical engineering.
<img alt="power calculation formula \(p(t)=v(t)i(t)\), symbol for electrical engineering" src="/files/ftp_upload/15117/15117_Equation_1.svg" style="text-align: center; height: 13px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
When examining instantaneous power, it can be expressed as a combination of two distinct terms. The first term, which remains constant, is dependent on the phase difference between voltage and current. The second term, characterized by sinusoidal oscillation, exhibits a frequency twice that of either the voltage or current.
<img alt="Power equation in trigonometric form, featuring amplitudes and phase angles, static analysis." src="/files/ftp_upload/15117/15117_Equation_2.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
In circuits encompassing resistance and reactance elements, instantaneous power is dynamic, oscillating between positive and negative values. In cases of purely resistive circuits, where voltage and current are perfectly in phase, instantaneous power consistently dissipates positively.
Conversely, in circuits dominated by inductive and capacitive components, the phase shift between voltage and current leads to intermittent fluctuations in instantaneous power between positive and negative values. This phenomenon can be attributed to the cyclic storage and release of energy by inductors and capacitors as current traverses through the former and voltage appear across the latter. During the energy release phase, these reactive elements act as power sources, effectively supplying energy back into the circuit resulting in the manifestation of negative instantaneous power. This intricate interplay of voltage, current, and power dynamics underscores the fundamental principles governing electrical circuits. An understanding of these principles is critical for efficient energy management and the design of electrical systems.

Instantaneous Power

For a circuit with a sinusoidal input, the power delivered to a passive element equals the product of the voltage and current across it.
Expressing ...

Education

JoVE Core

Electrical Engineering

Unit 1

AC Steady State Power

KEY TERMS AND CONCEPTS

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 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

Description

Transcript

Tags