Alternating current
Alternating current (AC) is an electric current in which the flow of electric charge periodically reverses direction, unlike direct current (DC), which maintains a constant unidirectional flow.[1] The magnitude of AC typically varies sinusoidally over time, producing a waveform that oscillates between positive and negative values. This periodic reversal enables AC to be efficiently transformed to higher or lower voltages using simple electromagnetic devices called transformers, making it ideal for long-distance power transmission with reduced energy losses compared to DC.[2] In the late 19th century, the practical implementation of AC systems revolutionized electricity distribution, culminating in the "War of the Currents" between proponents of DC, led by Thomas Edison, and AC advocates Nikola Tesla and George Westinghouse.[3] Tesla's invention of the AC induction motor in 1887-1888 provided a reliable means to convert electrical energy into mechanical work without mechanical commutation, addressing key limitations of DC motors.[4] AC ultimately prevailed due to its superior scalability for powering cities and industries, as demonstrated by Westinghouse's successful electrification of the 1893 Chicago World's Fair and Niagara Falls in 1895 using AC transmission lines.[5] Today, AC dominates global electrical grids, operating at standard frequencies of 50 Hz in most countries or 60 Hz in North America, to supply residential appliances, industrial machinery, and large-scale infrastructure like railways and renewable energy systems.[6] While DC sees niche applications in electronics and batteries, AC's versatility supports nearly all modern power generation and grid transmission, with sources such as solar and wind often involving initial DC production converted to AC.[7][8]Fundamentals
Definition and basic principles
Alternating current (AC) is an electric current in which the flow of electric charge periodically reverses direction, in contrast to direct current (DC), where the charge carriers flow unidirectionally.[9] This reversal occurs due to the periodic variation in the voltage applied across the circuit, causing the polarity of the source to switch repeatedly.[10] In practical applications, AC is most commonly generated in a sinusoidal waveform, where the current magnitude also varies smoothly between positive and negative peaks.[11] The key distinction from DC lies in the behavior of charge carriers, primarily electrons in conductors. In DC circuits, electrons experience a net drift in a single direction, resulting in steady energy transfer along the conductor.[2] With AC, the reversal of polarity causes electrons to oscillate or vibrate back and forth around a fixed position, with no net displacement over a complete cycle; however, this oscillatory motion still enables the propagation of energy through the circuit via electromagnetic fields.[12] This polarity inversion typically happens at a regular rate, defining the periodic nature of AC. A complete AC cycle consists of one full reversal from positive to negative and back, characterized by several fundamental parameters. The amplitude represents the maximum magnitude of the current or voltage during the cycle.[13] The period (T) is the duration required to complete one full cycle, measured in seconds.[14] The frequency (f) is the number of cycles per second, expressed in hertz (Hz), and is the reciprocal of the period: f = 1/T.[14] Additionally, the angular frequency (\omega), in radians per second, relates to the frequency by \omega = 2\pi f, providing a measure of the rate of oscillation in angular terms.[13] The concept of alternating current originated with the invention of the first practical AC generator by French instrument maker Hippolyte Pixii in 1832, based on Michael Faraday's principle of electromagnetic induction.[15] This device produced current that alternated direction as a permanent magnet rotated beneath a coil of wire.[16]Waveforms and cycles
The primary waveform used in alternating current (AC) systems is the sinusoidal form, which describes the smooth, periodic variation of current or voltage over time. This waveform can be mathematically expressed for current as i(t) = I_m \sin(\omega t + \phi), where I_m represents the peak (or maximum) value of the current, \omega = 2\pi f is the angular frequency with f being the frequency in hertz, t is time, and \phi is the phase angle that indicates the waveform's shift relative to a reference.[17] The time-dependent nature of this equation arises from the rotational motion in AC generators, producing a cosine or sine function that repeats indefinitely. A cycle in an AC waveform refers to one complete sequence of variation, starting from zero, rising to the positive peak, returning to zero, descending to the negative peak, and returning to zero again, representing a full reversal of direction.[18] This cyclical behavior is fundamental to AC, with each cycle corresponding to one full rotation of the armature in a generator, linking the electrical period directly to the mechanical rotation speed.[18] The period T, or duration of one cycle, is given by T = 1/f, where f is the frequency; for instance, at 60 Hz, T is approximately 16.67 milliseconds.[19] While sinusoidal waveforms are ideal for efficient power transmission due to their purity, real-world AC systems often encounter non-sinusoidal waveforms, such as square or triangular shapes, generated by nonlinear loads like rectifiers or switching devices.[20] These distorted waveforms contain harmonics—integer multiples of the fundamental frequency—that introduce higher-frequency components, leading to potential inefficiencies or interference in circuits.[20] A brief overview of Fourier series decomposition reveals that any periodic non-sinusoidal waveform can be represented as a sum of sinusoidal terms at the fundamental frequency and its harmonics, with coefficients determining the amplitude of each component, enabling analysis and mitigation of distortions. Visual representations of AC waveforms typically plot voltage or current against time on a graph, showing the sinusoidal curve oscillating symmetrically around the zero axis, with labeled axes for amplitude (peak value), time (period), and frequency markers for cycles; for non-sinusoidal cases, the plot would display sharper transitions, overlaid with harmonic sine waves to illustrate decomposition.Mathematical formulation
Instantaneous values and RMS
In alternating current (AC) circuits, the instantaneous voltage v(t) across a component is given by v(t) = V_m \sin(\omega t), where V_m is the peak (maximum) voltage amplitude, \omega is the angular frequency, and t is time.[17] Similarly, the instantaneous current i(t) is expressed as i(t) = I_m \sin(\omega t), with I_m as the peak current amplitude, assuming no phase difference for simplicity in a purely resistive circuit.[17] These expressions describe the time-varying nature of AC quantities, which oscillate sinusoidally between positive and negative peaks over each cycle. The root mean square (RMS) value provides an effective measure of AC voltage or current, equivalent to a direct current (DC) value that produces the same average power dissipation in a resistor.[17] For a sinusoidal waveform, the RMS voltage is derived as follows: the mean square value over one period T is \frac{1}{T} \int_0^T [V_m \sin(\omega t)]^2 \, dt = \frac{V_m^2}{2}, so the RMS voltage is V_{rms} = \sqrt{\frac{V_m^2}{2}} = \frac{V_m}{\sqrt{2}} \approx 0.707 V_m.[17] The same relation holds for current: I_{rms} = \frac{I_m}{\sqrt{2}}. RMS is used because it equates the heating effect or average power delivery of AC to that of DC; for instance, a DC voltage of V_{rms} dissipates the same power P = \frac{V_{rms}^2}{R} in a resistor R as the AC waveform.[17] In standard household electrical systems in the United States, the nominal supply is 120 V RMS, corresponding to a peak voltage of approximately 170 V ($120 \times \sqrt{2}).[21] For non-sinusoidal AC waveforms, the RMS value generalizes to V_{rms} = \sqrt{\frac{1}{T} \int_0^T [v(t)]^2 \, [dt](/page/DT)}, where the integral accounts for the squared instantaneous values over the period, enabling consistent power calculations regardless of waveform shape.[22]AC power calculations
In alternating current (AC) circuits, the instantaneous power p(t) delivered to a load is the product of the instantaneous voltage v(t) and current i(t), expressed as p(t) = v(t) i(t) = V_m I_m \sin(\omega t) \sin(\omega t + \phi), where V_m and I_m are the peak voltage and current, \omega is the angular frequency, and \phi is the phase angle between voltage and current. This formulation arises from the sinusoidal nature of AC signals, capturing the varying power flow over each cycle.[23] The average power P, which represents the usable real power over a complete cycle, is calculated using root-mean-square (RMS) values as P = V_{\rms} I_{\rms} \cos(\phi), where V_{\rms} and I_{\rms} are the RMS voltage and current, and \cos(\phi) accounts for the phase difference. This average is obtained by integrating the instantaneous power over one period and dividing by the period length, yielding the time-averaged energy transfer rate in watts (W).[24] Apparent power S, measured in volt-amperes (VA), quantifies the total power capacity of the circuit without considering phase effects and is given by S = V_{\rms} I_{\rms}.[25] Reactive power Q, in reactive volt-amperes (VAR), represents the portion of power oscillating between source and load due to inductive or capacitive elements, calculated as Q = V_{\rms} I_{\rms} \sin(\phi).[23] Together, these form the power triangle, where S = \sqrt{P^2 + Q^2}, illustrating the relationship between real, reactive, and apparent power.[26] The power factor, defined as \cos(\phi), is the ratio of average power to apparent power and indicates the efficiency of power utilization in the circuit, ranging from 0 (purely reactive) to 1 (purely resistive). A low power factor increases current draw for the same real power, leading to higher losses in conductors and reduced system efficiency, which is why utilities often impose penalties or require correction via capacitors.[27] For a single-phase resistive load, where \phi = 0 and power factor equals 1, the average power simplifies to P = V_{\rms} I_{\rms}. Consider a 120 V RMS household outlet connected to a 10 Ω resistor: the RMS current is I_{\rms} = V_{\rms} / R = 120 / 10 = 12 A, yielding P = 120 \times 12 = 1440 W of real power delivered entirely without reactive components.[25]Generation and frequencies
Common power frequencies
The two predominant frequencies for alternating current (AC) power systems worldwide are 50 Hz and 60 Hz. The 50 Hz standard is used in most of Europe, Asia, Africa, Australia, and parts of South America, while 60 Hz is standard in North America, much of South America, and parts of Asia such as Japan and South Korea.[28] These frequencies originated in the late 19th century during the early commercialization of AC power. In 1891, engineers at Westinghouse Electric in Pittsburgh selected 60 Hz for its balance of efficiency in generators and reduced flicker in lighting systems, influenced by the design of alternating current generators driven by steam engines at typical speeds. That same year, engineers at the Allgemeine Elektricitäts-Gesellschaft (AEG) in Berlin chose 50 Hz, prioritizing lower transmission losses over long distances and compatibility with European generator designs, which set the standard for much of the world outside North America.[29][30] Specialized applications employ other frequencies to meet specific operational needs. For instance, 16.7 Hz AC is used in railway electrification systems in Germany, Austria, and Switzerland at 15 kV, a legacy of early 20th-century single-phase AC traction technology that allowed efficient power delivery to locomotives without excessive transformer size. In aviation, 400 Hz AC is standard for onboard electrical systems in most civil and military aircraft, as the higher frequency enables smaller, lighter transformers and motors while maintaining power output, critical for weight-sensitive designs.[31][32] Power grid operators maintain frequency stability within tight tolerances—typically ±0.05 Hz for 60 Hz systems and similar for 50 Hz—to ensure reliable operation. Deviations occur due to imbalances between generation and load; for example, excess demand can cause frequency to drop, while overgeneration leads to an increase. Such variations directly affect synchronous appliances like electric clocks, which derive timing from line frequency; a sustained 0.1 Hz deviation over a day can cause a time error of about 144 seconds, potentially disrupting schedules or requiring manual adjustments.[33][34] The following table summarizes AC power frequencies by selected countries and regions, illustrating global patterns (note: some nations have mixed or transitional systems):| Region/Country | Frequency (Hz) | Notes |
|---|---|---|
| United States | 60 | Standard across North America |
| Canada | 60 | Aligns with U.S. grid |
| Brazil | 60 | Predominant in South America |
| United Kingdom | 50 | European standard |
| Germany | 50 | Widespread in continental Europe |
| China | 50 | Adopted across Asia |
| India | 50 | Common in South Asia |
| Japan | 50/60 | Eastern regions 50 Hz, western 60 Hz |
| Saudi Arabia | 60 | Middle East variation |