Name: power consumption in ac circuits
Uploaded: 2024-07-08T13:19:51.000Z
Duration: 1 min 13 s
Channel: JoVE

For most practical applications, the time-varying instantaneous power is not a commonly used quantity.
Instead, the average power is used as the measurable quantity. It is calculated by integrating the instantaneous power over the period and dividing it by the period.
The expression for instantaneous power is substituted and further simplified to obtain a time domain expression for the average power.
The second term is the average value of the cosine function over a period and is zero.
The final expression of the average power is time-independent and is proportional to the phase difference between the voltage and current.
The term resulting from half the product of voltage and current in phasor form comprises both real and imaginary parts.
Comparing this phasor expression with the average power equation indicates that the real part corresponds to the average power.
In a purely resistive circuit, the in-phase voltage and current lead to a positive average power.
However, in purely reactive circuits, a ninety-degree phase shift between voltage and current results in zero average power.

average power

In practical electrical applications, the concept of time-varying instantaneous power is not frequently utilized. Instead, focus shifts to the more practical quantity known as average power. Average power is determined by integrating the instantaneous power over a specified time period and subsequently dividing it by that duration.
<img alt="Power average calculation, formula P=(1/T)&#8747;p(t)dt, integral symbol." src="/files/ftp_upload/15118/15118_Equation_1.svg" style="text-align: center; height: 42px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
The equation for instantaneous power is simplified to arrive at a time-domain expression for average power. Notably, the second term, involving a cosine function, averages out to zero over a complete cycle.
<img alt="Electrical power formula, integral equation, symbol analysis in physics concepts." src="/files/ftp_upload/15118/15118_Equation_2.svg" style="text-align: center; height: 42px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
As a result, the final expression for average power is independent of time and contingent upon the phase difference between voltage and current.
<img alt="Power equation: P=(1/2)V_mi_mcos(&#952;_v&#8722;&#952;_i); electrical engineering analysis." src="/files/ftp_upload/15118/15118_Equation_3.svg" style="text-align: center; height: 28px;" data-altupdatedat="1762173717000" data-alt="Equation 3" />
Upon comparing this expression with the phasor representation of the product of voltage and current, it becomes evident that the real part of the phasor corresponds to the average power.
<img alt="Power system equation, phasor analysis, in mathematical formula for electrical engineering concepts." src="/files/ftp_upload/15118/15118_Equation_4.svg" style="text-align: center; height: 26px;" data-altupdatedat="1762173717000" data-alt="Equation 4" />
In purely resistive circuits, where voltage and current are perfectly in phase, the average power is positive, signifying continuous power consumption. Conversely, the average power dwindles to zero in purely reactive circuits characterized by a ninety-degree phase shift between voltage and current. This phenomenon is attributed to the cyclic storage and release of energy, where, on average, no net power is consumed.
It's important to recognize that while instantaneous power varies with respect to time, average power remains constant. Calculating instantaneous power requires voltage and current in the time domain, but average power can also be determined when voltage and current are represented in the frequency domain using phasors.
Understanding average power is vital in practical applications, as it facilitates the assessment of power consumption and efficiency, especially in circuits featuring diverse combinations of resistive and reactive components.

Average Power

For most practical applications, the time-varying instantaneous power is not a commonly used quantity.
Instead, the average power is used as the ...

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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 not change with time and depends on the phase difference between voltage and current.

The same result appears in the phasor form of voltage and current. The real part of the phasor product matches the average power. In a purely resistive circuit, voltage and current are in phase, so the average power is positive and the circuit keeps consuming power.

In a purely reactive circuit, voltage and current are ninety degrees apart. In that case, the average power drops to zero because energy is stored and then released in a cycle. Instantaneous power still changes with time, but average power stays constant. It can be calculated from time-domain signals or from phasors in the frequency domain. This makes average power useful for judging power use and efficiency in circuits with resistive and reactive parts.

Power Consumption in AC Circuits

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...

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 average power describes power consumption in AC circuits, why phase difference matters, and why reactive circuits can have zero net power.

Power Consumption in AC Circuits | 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 Consumption in AC Circuits

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