32.9:
Power in an AC Circuit
In an AC circuit, the current and voltage are time-dependent, making the instantaneous power also time-dependent.
The average power of an alternating current circuit is defined as the time average of the instantaneous power over one cycle, where T is the period of the oscillations.
By substituting instantaneous voltage and current equations and using trigonometric relations, the average power associated with a circuit element can be determined, wherein the term cos ϕ is the power factor.
In the case of a resistor, the voltage and current are in phase; by substituting the root mean square value of current and voltage, the average power dissipated by a resistor can be determined.
In the case of a capacitor and an inductor, the power factor turns zero. Thus, the average power dissipated by either of these elements is zero.
In an RLC circuit, an AC source produces and absorbs power depending on its phase angle.
By recalling the phase angle from the phasor diagram and substituting the root mean square values, the average power of an AC source can be determined.
32.9:
Power in an AC Circuit
In a DC circuit, the power consumed is simply the product of the DC voltage times the DC current, given in watts. However, the power consumed for AC circuits with reactive components is calculated differently. Since electrical power is the "rate" at which energy is used in a circuit, all electrical and electronic components and devices have a safe operating range for electrical power.
In a DC circuit, there is no sinusoidal waveform associated with the supply; the voltages and currents are typically constant and do not change over time. However, for power in AC circuits, the supply affects the instantaneous values of the voltage, current, and consequently power, which are constantly changing. The AC circuit comprises reactance, which generates the magnetic or electric fields, resulting in a power component. As a result, unlike purely resistive components, this power is stored and then returned to the supply as the sinusoidal waveform completes one full periodic cycle. Consequently, the total power stored and returned over a single full cycle makes up the average power absorbed by a circuit. A circuit's average power consumption will therefore be the average of the instantaneous power over a complete cycle, with the instantaneous power being defined as the product of the instantaneous voltage and the instantaneous current.
It is challenging to measure the instantaneous power in AC circuits because it fluctuates constantly with the sinusoid's profile over time. Therefore, using the average or mean value of the power is more practical and simpler mathematically. Thus, given a fixed number of cycles, the following is the simple expression for the average value of the instantaneous power of the sinusoid:
where cos ϕ is the power factor, which is the amount by which the power delivered in the circuit is less than the theoretical maximum of the circuit due to voltage and current being out of phase.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version], Pg 394 – 396. Retrieved from https://openstax.org/books/university-physics-volume-2@8ede2ba/pages/15-4-power-in-an-ac-circuit